Local Moduli of Semisimple Frobenius Coalescent Structures
We extend the analytic theory of Frobenius manifolds to semisimple points with coalescing eigenvalues of the operator of multiplication by the Euler vector field. We clarify which freedoms, ambiguities, and mutual constraints are allowed in the definition of monodromy data, in view of their importan...
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| description | We extend the analytic theory of Frobenius manifolds to semisimple points with coalescing eigenvalues of the operator of multiplication by the Euler vector field. We clarify which freedoms, ambiguities, and mutual constraints are allowed in the definition of monodromy data, in view of their importance for conjectural relationships between Frobenius manifolds and derived categories. Detailed examples and applications are taken from singularity and quantum cohomology theories. We explicitly compute the monodromy data at points of the Maxwell Stratum of the A₃-Frobenius manifold, as well as at the small quantum cohomology of the Grassmannian ₂(ℂ⁴). In the latter case, we analyse in detail the action of the braid group on the monodromy data. This proves that these data can be expressed in terms of characteristic classes of mutations of Kapranov's exceptional 5-block collection, as conjectured by one of the authors.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 040, 105 pages
Local Moduli of Semisimple Frobenius Coalescent
Structures
Giordano COTTI †, Boris DUBROVIN ‡ and Davide GUZZETTI ‡
† Max-Planck-Institut für Mathematik, Vivatsgasse 7 - 53111 Bonn, Germany
E-mail: gcotti@sissa.it, G.Cotti.1@bham.ac.uk
URL: https://www.birmingham.ac.uk/staff/profiles/maths/cotti-giordano.aspx
‡ SISSA, Via Bonomea 265 - 34136 Trieste, Italy
E-mail: guzzetti@sissa.it
Received June 18, 2019, in final form April 13, 2020; Published online May 07, 2020
https://doi.org/10.3842/SIGMA.2020.040
Abstract. We extend the analytic theory of Frobenius manifolds to semisimple points
with coalescing eigenvalues of the operator of multiplication by the Euler vector field. We
clarify which freedoms, ambiguities and mutual constraints are allowed in the definition of
monodromy data, in view of their importance for conjectural relationships between Frobe-
nius manifolds and derived categories. Detailed examples and applications are taken from
singularity and quantum cohomology theories. We explicitly compute the monodromy data
at points of the Maxwell Stratum of the A3-Frobenius manifold, as well as at the small
quantum cohomology of the Grassmannian G2
(
C4
)
. In the latter case, we analyse in details
the action of the braid group on the monodromy data. This proves that these data can be
expressed in terms of characteristic classes of mutations of Kapranov’s exceptional 5-block
collection, as conjectured by one of the authors.
Key words: Frobenius manifolds; isomonodromic deformations; singularity theory; quantum
cohomology; derived categories
2020 Mathematics Subject Classification: 34M56; 53D45; 18E30
Contents
1 Introduction and results 2
1.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Plan of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 Moduli of semisimple Frobenius manifolds 16
2.1 Spectrum of a Frobenius manifold and its monodromy data at z = 0 . . . . . . . 18
2.2 Semisimple Frobenius manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3 Monodromy data for a semisimple Frobenius manifold . . . . . . . . . . . . . . . 27
3 Ambiguity in definition of monodromy data and braid group action 35
3.1 Action of the braid group Bn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4 Isomonodromy theorem at coalescence points 39
4.1 Reconstruction of monodromy data of the whole manifold . . . . . . . . . . . . . 43
5 First detailed example of application of Theorem 4.5: the A3 Frobenius
manifold. Stokes phenomenon for Pearcey-type oscillating integrals
from Hankel functions 43
5.1 Singularity theory and Frobenius manifolds . . . . . . . . . . . . . . . . . . . . . 44
mailto:gcotti@sissa.it
mailto:G.Cotti.1@bham.ac.uk
https://www.birmingham.ac.uk/staff/profiles/maths/cotti-giordano.aspx
mailto:guzzetti@sissa.it
https://doi.org/10.3842/SIGMA.2020.040
2 G. Cotti, B. Dubrovin and D. Guzzetti
5.2 Frobenius structure of type An . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.3 The case of A3: reduction of the system for deformed flat coordinates . . . . . . 46
5.4 Computation of Stokes and central connection matrices . . . . . . . . . . . . . . 50
5.5 A “tour” in the Maxwell stratum: reconstruction of neighboring monodromy data 55
5.6 Monodromy data as computed outside the Maxwell stratum . . . . . . . . . . . . 59
6 Second example of application of Theorem 4.5: quantum cohomology
of the Grassmannian G2
(
C4
)
and Γ-conjecture 66
6.1 Notations in Gromov–Witten theory . . . . . . . . . . . . . . . . . . . . . . . . . 67
6.2 Small quantum cohomology of G2
(
C4
)
. . . . . . . . . . . . . . . . . . . . . . . . 68
6.2.1 Generalities and proof of its semisimplicity . . . . . . . . . . . . . . . . . 68
6.2.2 Idempotents at the points
(
0, t2, 0, . . . , 0
)
. . . . . . . . . . . . . . . . . . 70
6.3 Differential system for deformed flat coordinates . . . . . . . . . . . . . . . . . . 71
6.3.1 Expected asymptotic expansions . . . . . . . . . . . . . . . . . . . . . . . 73
6.4 Solutions of the differential system . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6.5 Computation of monodromy data . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.5.1 Solution at the origin and computation of C0(η, µ,R) . . . . . . . . . . . . 76
6.5.2 Stokes rays and computation of Ξleft, Ξright . . . . . . . . . . . . . . . . . 78
6.5.3 Computation of Stokes and central connection matrices,
using constraint (2) of Theorem 2.43 . . . . . . . . . . . . . . . . . . . . . 82
6.6 Monodromy data of the small quantum cohomology and exceptional collections
in Db(G) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.7 Reconstruction of monodromy data along the small quantum locus . . . . . . . . 88
7 A note on the topological solution for Fano manifolds 89
A Proofs of Lemmata 6.2 and 6.4 95
B Computation of the central connection matrix 100
References 103
1 Introduction and results
There is a conjectural relation, formulated by the second author ([25, 28], see also [10, 45] and
references therein), between the enumerative geometry of a wide class of smooth projective va-
rieties and their derived category of coherent sheaves. In particular, there is an increasing interest
for an explicit description of certain local invariants, called monodromy data, of semisimple
quantum cohomologies in terms of characteristic classes of exceptional collections in the derived
categories [28, 33]. Being intentioned to address this problem, which, to our opinion, is still not
well understood, we have realized that some issues in the theory of Frobenius manifolds need to
be preliminarily clarified, and that an extension of the theory itself is necessary, in view of the
fact that quantum cohomologies of certain classes of homogeneous spaces may show a coalescence
phenomenon.
In this paper, after reviewing the definition of the monodromy data, such as the Stokes
matrix and the central connection matrix, we clarify their mutual constraints, the freedom and
the natural operations allowed when we associate the data to a chart of the manifold. See
Theorem 1.1 and 1.2 in the Introduction and Sections 2 and 3. This issue does not seem to be
sufficiently clear in the existing literature (some minor imprecisions are found also in [23, 25, 26],
especially concerning the central connection matrix), and it is fundamental in order to study
the above mentioned conjectures.
Local Moduli of Semisimple Frobenius Coalescent Structures 3
Then, we extend the analytic theory of Frobenius manifolds in order to take into account
a coalescence phenomenon, which occurs already for simple classes of varieties (e.g., for almost all
complex Grassmannians [17]). By this we mean that the operator of multiplication by the Euler
vector field does not have a simple spectrum at some points where nevertheless the Frobenius
algebra is semisimple. We call these points semisimple coalescence points (see Definition 1.3).
Such a phenomenon forbids an immediate application of the analytic theory of Frobenius mani-
folds to the computation of monodromy data. On the other hand, typically, the Frobenius
structure is explicitly known only at the locus of semisimple coalescence points. Thus, we need
to prove that the monodromy data associated with each region of the manifold can be computed
starting only from the knowledge of the manifold at coalescence points. From the analytic point
of view, coalescence implies that we have to deal with isomonodromic linear differential systems
which violate one of the main assumptions1 of the monodromy preserving deformations theory
of M. Jimbo, T. Miwa and K. Ueno [51]. Applying the results of [19], where the isomonodromy
deformation theory has been extended to coalescence loci, we will show that the monodromy data
computed at a semisimple coalescence point are the data associated with a whole neighbourhood
of the point. The result is in Theorem 1.4 of the Introduction and in Theorem 4.5. Moreover,
by an action of the braid group, these data suffice to compute the data of the whole manifold
(see Section 4.1).
We give two explicit examples of the above procedure. One is from singularity theory in
Section 5, where we compute the monodromy data at points of the Maxwell Stratum of the
A3-Frobenius manifold. The monodromy data of the A3-Frobenius manifold are well known,
but here we present for the first time in the literature their computation at a coalescence point,
where the isomonodromic system highly simplifies and can be solved in terms of Hankel functions.
The validity of the result of the computation for the whole A3-Frobenius manifold is justified
by Theorem 4.5.
The second example is new: in Section 6, we explicitly compute the Stokes matrix and the
central connection matrix of the quantum cohomology of the Grassmannian G2
(
C4
)
. The com-
putation can be done only at the locus of small cohomology, which is the only locus where the
structure of the manifold is known. This is a locus of coalescence points. Thus, we will have to
do the computation at a coalescence point, explicitly obtaining Stokes and central connection
matrices there. Theorem 4.5 becomes crucial in order to give geometrical meaning to our com-
putation: it guarantees that the result obtained at a coalescence point provides the monodromy
data in a whole neighbourhood of the point. Consequently the result can be extended to the
whole manifold by an action of the braid group (explained in the paper). In this way, we will
obtain the monodromy data of QH•
(
G2
(
C4
))
and explicitly relate them to characteristic classes
of objects of an exceptional collection in Db
(
G2
(
C4
))
, establishing a correspondence between
each region of the quantum cohomology and a full exceptional collection. See Theorem 1.5
in the Introduction and Theorem 6.10. To our best knowledge, it seems that such an explicit
description has not been done in the literature.
The results of the example in Section 6 are important, because they make it evident that both
formulations in [28] and [33], refining the original conjecture of [25], require more investigations.
A refinement of the conjecture of [25, 28], its proof for the case of all complex Grassmannians,
and the relation to the version given in [33], will be the content of our paper [18].
In conclusion, the example in Section 6 yields an explicit result of great theoretical significance
in the theory of Frobenius manifolds and quantum cohomology, because it clarifies the conjecture
explained above and because it shows the crucial role of Theorem 4.5. From another point of
view, it may also be considered as a non-trivial example of analysis of a linear differential system
with coalescing eigenvalues, showing a high level of complexity. As such, it may serve as a useful
example in the field of linear systems and isomonodromy deformations.
1See [51, p. 312], assumption that the eigenvalues of Aν−rν are distinct. See also condition (2) at p. 133 of [31].
4 G. Cotti, B. Dubrovin and D. Guzzetti
Before explaining the results of the paper in more detail, we briefly recall preliminary basic
facts. A Frobenius manifold M is a complex manifold, with finite dimension
n := dimC(M),
endowed with a structure of associative, commutative algebra with product ◦p and unit on each
tangent space TpM , analytically depending on the point p ∈ M ; in order to be Frobenius the
algebra must also satisfy an invariance property with respect to a symmetric non degenerate
bilinear form η on TM , called metric, invariant w.r.t. the product ◦p, i.e.,
η(a ◦p b, c) = η(a, b ◦p c) for all a, b, c ∈ TpM, p ∈M,
whose corresponding Levi-Civita connection ∇ is flat and, moreover, the unit vector field is
flat. The above structure is required to be compatible with a C∗-action on M (the so-called
quasihomogeneity assumption). A precise definition and description will be given in Section 2,
including the potentiality (namely, the fact that the structure constants of the algebra product ◦p
are third derivatives of a function at p).
The geometry of a Frobenius manifold is (almost) equivalent to the flatness condition for
an extended connection ∇̂ defined on the pull-back π∗TM of the tangent vector bundle along
the projection map π : C∗ ×M → M . Consequently, we can look for n holomorphic functions
t̃1, . . . , t̃n : C∗ ×M → C such that
(
z, t̃1, . . . , t̃n
)
are ∇̂-flat coordinates. In ∇-flat coordinates
t =
(
t1, . . . , tn
)
, the ∇̂-flatness condition ∇̂dt̃(z, t) = 0 for a single function t̃ reads
∂ζ
∂z
=
(
U(t) +
1
z
µ
)
ζ, (1.1)
∂ζ
∂tα
= zCα(t)ζ, α = 1, . . . , n, (1.2)
where the entries of the column vector ζ(z, t) are the components of the η-gradient of t̃
grad t̃ := ζα(z, t)
∂
∂tα
, ζα(z, t) := ηαν
∂t̃
∂tν
, ηαβ := η
(
∂
∂tα
,
∂
∂tβ
)
, (1.3)
and Cα(t), U(t) and
µ := diag(µ1, . . . , µn)
are n× n matrices described in Section 2, satisfying ηU = UTη and ηµ+ µTη = 0.
A fundamental matrix solution of (1.1)–(1.2) provides n independent ∇̂-flat coordinates(
t̃1, . . . , t̃n
)
. For fixed t, the equation (1.1) is an ordinary linear differential system with ra-
tional coefficients, with Fuchsian singularity at z = 0 and an irregular singularity of Poincaré
rank 1 at z =∞.
A point p ∈M is called semisimple if the Frobenius algebra TpM is semisimple, i.e., without
nilpotents. A Frobenius manifold is semisimple if it contains an open dense subset Mss of
semisimple points. If the matrix U is diagonalizable at p with pairwise distinct eigenvalues, then
p ∈ Mss, see [23, 26]; this is a simple consequence of the definition of U , given in the paper in
expression (2.1). The condition is not necessary: there exist semisimple points p ∈ Mss where
U does not have a simple spectrum. In this case, if we move in Mss along a curve terminating
at p then some eigenvalues of U(t) coalesce.
The eigenvalues u := (u1, . . . , un) of the operator U , with chosen labelling, define a local
system of coordinates p 7→ u = u(p) in a neighborhood of any semisimple point p, called
canonical. In canonical coordinates, we set
grad t̃α(u, z) ≡
∑
i
Y i
α(u, z)fi(u), fi(u) :=
1
η
(
∂
∂ui
|u, ∂
∂ui
|u
) 1
2
∂
∂ui
∣∣∣∣
u
(1.4)
Local Moduli of Semisimple Frobenius Coalescent Structures 5
for some choice of the square roots. Then the equations (1.1), (1.2), are equivalent to the
following system
∂Y
∂z
=
(
U +
V (u)
z
)
Y, (1.5)
∂Y
∂uk
= (zEk + Vk(u))Y, 1 ≤ k ≤ n, (1.6)
where (Ek)ij := δikδjk, U = diag(u1, . . . , un), V is skew-symmetric and
U := ΨUΨ−1, V := ΨµΨ−1, Vk(u) :=
∂Ψ(u)
∂uk
Ψ(u)−1.
Here, Ψ(u) is a matrix defined by the change of basis between
(
∂
∂t1
, . . . , ∂
∂tn
)
and the normalized
canonical vielbein (f1, . . . , fn)
∂
∂tα
=
n∑
i=1
Ψiαfi.
The compatibility conditions of the equations (1.5)–(1.6) are
[U, Vk] = [Ek, V ],
∂V
∂uk
= [Vk, V ]. (1.7)
When ui 6= uj for i 6= j, equations (1.7) coincide with the Jimbo–Miwa–Ueno isomonodromy de-
formation equations for system (1.5), with deformation parameters (u1, . . . , un) [49, 50, 51]. This
isomonodromic property allows to classify semisimple Frobenius manifolds by locally constant
monodromy data of (1.5). Conversely, such local invariants allow to reconstruct the Frobenius
structure by means of an inverse Riemann–Hilbert problem [23, 26, 42]. Below, we briefly recall
how they are defined in [23, 26].
In [23, 26] it was shown that system (1.5) has a fundamental solution near z = 0 in Levelt
normal form
Y0(z, u) = Ψ(u)Φ(z, u)zµzR, Φ(z, u) := 1 +
∞∑
k=1
Φk(u)zk, (1.8)
satisfying the orthogonality condition
Φ(−z, u)TηΦ(z, u) = η for all z ∈ R, u ∈M. (1.9)
Here,
R := C̃\{0}
denotes the universal cover of C \ {0} and R is a certain nilpotent matrix, which is non-zero
only if µ has some eigenvalues differing by non-zero integers. Since z = 0 is a regular singularity,
Φ(z, u) is convergent.
If u = (u1, . . . , un) are pairwise distinct, so that U has distinct eigenvalues, then the sys-
tem (1.5) admits a formal solution of the form
Yformal(z, u) = G(z, u)ezU ,
G(z, u) = 1 +
∞∑
k=1
Gk(u)
1
zk
, G(−z, u)TG(z, u) = 1. (1.10)
6 G. Cotti, B. Dubrovin and D. Guzzetti
Although Yformal in general does not converge, it always defines the asymptotic expansion of
a unique genuine solution on any sectors in the universal covering R, having central opening
angle π + ε for ε > 0 sufficiently small.
The choice of a ray `+(φ) := {z ∈ R : arg z = φ} with directional angle φ ∈ R induces
a decomposition of the Frobenius manifold into disjoint chambers.2 An `-chamber is defined
(see Definition 2.35) to be any connected component of the open dense subset of points p ∈M
such that the eigenvalues of U at p are all distinct (so, in particular, they are points of Mss),
and the ray `+(φ) does not coincide with any Stokes rays at p, namely <(z(ui(p)− uj(p))) 6= 0
for i 6= j and z ∈ `+(φ).
Let p belong to an `-chamber, and let u = (u1, . . . , un) be the canonical coordinates in
a neighbourhood of p contained in the chamber. Then, there exist unique solutions Yleft/right(z, u)
such that
Yleft/right(z, u) ∼ Yformal(z, u) for z →∞,
respectively in the sectors
Πε
right(φ) := {z ∈ R : φ− π − ε < arg z < φ+ ε},
Πε
left(φ) := {z ∈ R : φ− ε < arg z < φ+ π + ε}.
The two solutions Yleft/right(z, u) are connected by the multiplication by an invertible matrix S,
called Stokes matrix :
Yleft(z, u) = Yright(z, u)S, for all z ∈ R.
S has the “triangular structure” described in Theorem 2.42. Namely, Sij 6= 0 implies Sji = 0.
In particular, diag(S) = (1, . . . , 1) and Sij = Sji = 0 whenever ui = uj . Moreover, there exists
a central connection matrix C, whose properties will be described later, such that
Yright(z, u) = Y0(z, u)C, for all z ∈ R.
In [23] and [26] it is shown that the coefficients Φk’s and Gk’s are holomorphic at any point of
every `-chamber and that the monodromy data µ,R, S,C are constant over a `-chamber (the
isomonodromy Theorem I and II of [26], cf. Theorems 2.16 and 2.44 below). They define local
invariants of the semisimple Frobenius manifold M . In this sense, there is a local identification
of a semisimple Frobenius manifold with the space of isomonodromy deformation parameters
(u1, . . . , un) of the equation (1.5).
1.1 Results
We now describe the results of the paper at points 1, 2 and 3 below.
1. Ambiguity in associating monodromy data with a point of the manifold (cf.
Sections 2 and 3).
From the above discussion, we see that with a point p ∈ Mss such that u1(p), . . . , un(p) are
pairwise distinct, we associate the monodromy data (µ,R, S,C). These data are constant on the
whole `-chamber containing p. Nevertheless, there is not a unique choice of (µ,R, S,C) at p. The
understanding of this issue is crucial in order to undertake a meaningful and well-founded study
of the conjectured relationships of the monodromy data coming from quantum cohomology of
smooth projective varieties with derived categories of coherent sheaves on these varieties.
2This definition does not appear in [23, 26]. See also Remark 2.36.
Local Moduli of Semisimple Frobenius Coalescent Structures 7
The starting point is the observation that a normal form (1.8) is not unique because of some
freedom in the choice of Φ and R (in particular, even for a fixed R there is a freedom in Φ). The
description of this freedom was given in [26], with a minor imprecision, to be corrected below.
Let us identify3 all tangent spaces TpM , for p ∈M , using the Levi-Civita connection on M , with
a n-dimensional complex vector space V , so that µ ∈ End(V ) is a linear operator antisymmetric
w.r.t. the bilinear form η. Let G(η, µ) be the complex (η, µ)-parabolic orthogonal Lie group,
consisting of all endomorphisms G : V → V of the form G = 1V + ∆, with ∆ a µ-nilpotent
endomorphism, and such that η
(
eiπµGa,Gb
)
= η
(
eiπµa, b
)
for any a, b ∈ V (see Section 2.1 and
Definition 2.6). We denote by g(η, µ) its Lie algebra.
Theorem 1.1 (Section 2.14). Given a fundamental matrix solution of system (1.5) in Levelt
form (1.8) near z = 0 holomorphically depending on (u1, . . . , un) and satisfying the orthogonality
condition (1.9), with µ = Ψ(u)−1V (u)Ψ(u) constant and diagonal, then the holomorphic function
R = R(u) takes values in the Lie algebra g(η, µ). Moreover,
1. All other solutions in Levelt form near z = 0 are Y0(z, u)G(u), where G is a holomorphic
function with values in G(η, µ); the Levelt normal form of Y0(z, u)G(u) has again the
structure (1.8) with R(u) replaced with R̃(u) := G(u)R(u)G(u)−1 (cf. Theorem 2.14).
2. Because of the compatibility of (1.5) and (1.6), G(u) can be chosen so that R̃ is independent
of u (isomonodromy Theorem I in [26], Theorem 2.16).
3. For a fixed R ∈ g(η, µ), the isotropy subgroup G(η, µ)R of transformations G ∈ G(η, µ),
such that GRG−1 = R, can be identified with the group
C0(η, µ,R) :=
G ∈ GL(V ) :
PG(z) := zµzRGz−Rz−µ ∈ End(V )[z],
PG(0) ≡ 1V ,
η(PG(−z)v1, PG(z)v2) = η(v1, v2),
for all v1, v2 ∈ V
. (1.11)
The definition (1.11) can be re-written in coordinates as follows
C0(η, µ,R) :=
{
G ∈ GL(n,C) : PG(z) := zµzRGz−Rz−µ is a matrix-valued
polynomial such that PG(0) = 1, and PG(−z)TηPG(z) = η
}
.
If G ∈ C0(η, µ,R) and Y0(z, u) = Ψ(u)Φ(z, u)zµzR, then Y0(z, u)G = Ψ(u)Φ(z, u)PG(z)zµzR.
The refinement introduced here consists in the restriction of the group C0(µ,R) introduced
in [26] by adding the condition
PG(−z)TηPG(z) = η,
which does not appear in [26]. In [26] neither the η-orthogonality conditions appeared in the
definition of the group C0(µ,R), nor this group was identified with the isotropy subgroup of R
w.r.t. the adjoint action of G(η, µ) on its Lie algebra g(η, µ). These η-orthogonality conditions
are crucial for preserving (1.9) and the constraints (1.12) of all monodromy data (µ,R, S,C)
(see also Theorem 2.43).
Let us now summarize the freedom in assigning the monodromy data (µ,R, S,C) to a given
semi-simple point p of the Frobenius manifold. It has various origins: it can come from a re-
ordering of the canonical coordinates u1(p), . . . , un(p), from changing signs of the normalized
3See Section 2.1 for the precise definition.
4In the description of the monodromy phenomenon of solutions of the system (1.5) near z = 0 the assumption
of semisimplicity is not used. This will be crucial only for the description of solutions near z =∞. Theorem 1.1
can be formulated for system (1.1), having the fundamental solution Ξ0 = Ψ−1Y0.
8 G. Cotti, B. Dubrovin and D. Guzzetti
idempotents, from changing the Levelt fundamental solution at z = 0 and, last but not least,
from changing the slope of the oriented line `+(φ). Taking into account all these possibilities,
we have the following
Theorem 1.2 (Section 3). Let p ∈ Mss be such that (u1(p), . . . , un(p)) are pairwise distinct.
If (µ,R, S,C) is a set of monodromy data computed at p, then with a different labelling of the
eigenvalues, different signs, different choice of Y0(z, u) and different φ, another set of monodromy
data can be computed at the same p, which lies in the orbit of (µ,R, S,C) under the following
actions:
• the action of the group of permutations Sn
S 7−→ PSP−1, C 7−→ CP−1,
which corresponds to a relabelling (u1, . . . , un) 7→ (uτ(1), . . . , uτ(n)), where τ ∈ Sn and the
invertible matrix P has entries Pij = δjτ(i). For a suitable choice of the permutation,
PSP−1 is in upper-triangular form;
• the action of the group (Z/2Z)×n
S 7−→ ISI, C 7−→ C I,
where I is a diagonal matrix with entries equal to 1 or −1, which corresponds to a change
of signs of the square roots in (1.4);
• the action of the group C0(η, µ,R)
S 7−→ S, C 7−→ GC, G ∈ C0(η, µ,R),
which corresponds to a change Y0(z, u) 7→ Y0(z, u)G as in Theorem 1.1.
• the action of the braid group, as in formulae (3.4) and (3.5),
S 7→ Aβ(S) · S ·
(
Aβ(S)
)T
, C 7→ C ·
(
Aβ(S)
)−1
,
where β is a specific braid associated with a translation of φ, corresponding to a rotation
of `+(φ). More details are in Section 3.
Any representative of µ, R, S, C in the orbit of the above actions satisfies the monodromy
identity
CSTS−1C−1 = e2πiµe2πiR,
and the constraints
S = C−1e−πiRe−πiµη−1
(
CT
)−1
, ST = C−1eπiReπiµη−1
(
CT
)−1
. (1.12)
We stress again that the freedoms in Theorem 1.2 must be taken into account when we want
to investigate the relationship between monodromy data and similar objects in the theory of
derived categories
2. Isomonodromy theorem at semisimple coalescence points (cf. Section 4).
Definition 1.3. A point p ∈ Mss such that the eigenvalues of U at p are not pairwise distinct
is called a semisimple coalescence point.
Local Moduli of Semisimple Frobenius Coalescent Structures 9
The isomonodromy deformations results presented above apply if U has distinct eigenval-
ues. If two or more eigenvalues coalesce, as it happens at semisimple coalescence points of
Definition 1.3, then a priori solutions Yleft/right(z, u) are expected to have a singular behaviour
(branching at ui−uj = 0 and/or divergent limits for ui−uj → 0 along any direction), the coef-
ficients of the formal expansion of Yformal(z, u) may have poles at ui − uj = 0, and monodromy
data must be redefined.
In almost all studied cases of quantum cohomology the structure of the manifold is explic-
itly known only at the locus of small quantum cohomology defined in terms of the three-points
genus zero Gromov–Witten invariants. Along this locus the coalescence phenomenon may occur
(for example, coalescence occurs in case of the quantum cohomology of almost all Grassman-
nians [17]). Therefore, if we want to compute the monodromy data, we can only rely on the
information available at coalescence points. Thus, we need to extend the analytic theory of
Frobenius manifolds, in order to include this case, showing that the monodromy data are well
defined at a semisimple coalescence point, and locally constant. Moreover, from these data we
must be able to reconstruct the data for the whole manifold. We stress that this extension of
the theory is essential in order to study the conjectural links to derived categories.
The extension is based on the observation that the matrix Ψ(u) is holomorphic at semisimple
points including those of coalescence (see Lemma 2.29). Consequently, the matrices Vk’s and V
are holomorphic at any semisimple point, and V is holomorphically similar to µ. These are
exactly the sufficient conditions allowing the application of the general results obtained in [19],
which yield the following
Theorem 1.4 (cf. Theorem 4.5 below). Let p0 be a semisimple coalescence point with canonical
coordinates u(p0) =
(
u
(0)
1 , . . . , u
(0)
n
)
.5 Moreover,
• Let φ ∈ R be fixed so that `+(φ) does not coincide with any Stokes ray at p0, namely
<
(
z
(
u
(0)
i − u
(0)
j
))
6= 0 for u
(0)
i 6= u
(0)
j and z ∈ `+(φ).
• Consider the closed polydisc centered at u(0) =
(
u
(0)
1 , . . . , u
(0)
n
)
and of size ε0 > 0
Uε0
(
u(0)
)
:=
{
u ∈ Cn : max
1≤i≤n
∣∣ui − u(0)
i
∣∣ ≤ ε0}.
Let ε0 > 0 be sufficiently small, so that Uε0
(
u(0)
)
is homeomorphic by the coordinate map to
a neighbourhood of p0 contained in Mss. An additional upper bound for ε0 will be specified
in Section 4, see equation (4.2).
• Let ∆ ⊂ Uε0
(
u(0)
)
be the locus in the polydisc Uε0
(
u(0)
)
where some eigenvalues of U(u) =
diag(u1, . . . , un) coalesce.6
Then, the following results hold:
5Up to permutation, these coordinates can be arranged as
u
(0)
1 = · · · = u(0)
r1 ,
u
(0)
r1+1 = · · · = u
(0)
r1+r2
,
· · · · · · · · · · · · · · · · · · · · · · · ·
u
(0)
r1+···+rs−1+1 = · · · = u
(0)
r1+···+rs−1+rs
,
where r1, . . . , rs (r1 + · · ·+ rs−1 + rs = n) are the multiplicities of the eigenvalues of U
(
u0
)
= diag
(
u
(0)
1 , . . . , u
(0)
n
)
.
6Namely, ui = uj for some 1 ≤ i 6= j ≤ n whenever u ∈ ∆. The bound on ε0, to be clarified later, implies that,
with the arrangement of footnote 5 the sets
{u1, . . . , ur1}, {ur1+1, . . . , ur1+r2}, . . . , {ur1+···+rs−1+1, . . . , ur1+···+rs−1+rs}
do not intersect for any u ∈ Uε0
(
u(0)
)
. In particular, u(0) ∈ ∆ is a point of “maximal coalescence”.
10 G. Cotti, B. Dubrovin and D. Guzzetti
1. System (1.5) at the fixed value u = u(0) admits a unique formal solution, which we denote
with Y̊formal(z), having the structure (1.10), namely Y̊formal(z) =
(
1 +
∞∑
k=1
G̊kz
−k
)
ezU ;
moreover, it admits unique fundamental solutions, which we denote with Y̊left/right(z), ha-
ving asymptotic representation Y̊formal(z) in sectors Πε
left/right(φ), for suitable ε > 0.7 Let S̊
be the Stokes matrix such that
Y̊left(z) = Y̊right(z)S̊.
2. The coefficients Gk(u), k ≥ 1, in (1.10) are holomorphic over Uε0
(
u(0)
)
, and Gk
(
u(0)
)
=
G̊k; moreover Yformal
(
z, u(0)
)
= Y̊formal(z).
3. For fixed z, Yleft(z, u), Yright(z, u), computed in a neighbourhood of a point u ∈ Uε0
(
u(0)
)
\∆,
can be u-analytically continued as single-valued holomorphic functions on the whole
Uε0
(
u(0)
)
. Moreover
Yleft/right
(
z, u(0)
)
= Y̊left/right(z).
4. For any ε1 < ε0 the asymptotic relations
Yleft/right(z, u)e−zU ∼ I +
∞∑
k=1
Gk(u)z−k, for z →∞ in Πε
left/right(φ),
hold uniformly in u ∈ Uε1
(
u(0)
)
. In particular they also hold at u ∈ ∆.
5. Denote by Y̊0(z) a solution of system (1.5) with the fixed value u = u(0), in Levelt form
Y̊0(z) = Ψ
(
u(0)
)
(1+O(z))zµzR̊, having monodromy data µ and R̊. For any such Y̊0(z) there
exists a fundamental solution Y0(z, u) in Levelt form (1.8), (1.9) holomorphic in Uε0
(
u(0)
)
,
such that its monodromy data µ and R are independent of u and
Y0
(
z, u(0)
)
= Y̊0(z), R = R̊. (1.13)
Let C̊ be the central connection matrix for Y̊0 and Y̊right; namely
Y̊right(z) = Y̊0(z)C̊.
6. For any ε1 < ε0 the monodromy data µ, R, S, C of system (1.5) are well defined and
constant in the whole Uε1
(
u(0)
)
, so that the system is isomonodromic in Uε1
(
u(0)
)
. They
coincide with the data associated with the fundamental solutions Y̊left/right(z) and Y̊0(z)
above, namely
R = R̊, S = S̊, C = C̊.
The entries of S = (Sij)
n
i,j=1 with indices corresponding to coalescing canonical coordinates
vanish:
Sij = Sji = 0 for all i 6= j such that u
(0)
i = u
(0)
j . (1.14)
7A more precise characterisation of the angular amplitude of the sectors will be given later.
Local Moduli of Semisimple Frobenius Coalescent Structures 11
Theorem 1.4 implies that, in order to compute the monodromy data µ, R, S, C in the whole
Uε0
(
u(0)
)
, it suffices to compute µ, R̊, S̊, C̊ at u(0). These can be used to obtain the monodromy
data at any other point of Mss (including semisimple coalescence points, by Theorem 1.4), by
the action of the braid group Bn introduced in [23] and [26]
S 7−→ AβS
(
Aβ
)T
, C 7−→ C
(
Aβ
)−1
, (1.15)
as in formulae (3.4) and (3.5). This action is well defined whenever u1, . . . , un are pairwise
distinct. It allows to obtain the monodromy data associated with all `-chambers. Therefore, the
action can be applied to S, C as defined by the above theorem starting from a point of Uε0
(
u(0)
)
where u1, . . . , un are pairwise distinct.
We will give two detailed applications of the above theorem. The first example, in Section 5,
is the analysis of the monodromy data at the points of one of the two irreducible components
of the bifurcation diagram of the Frobenius manifold associated with the Coxeter group A3.
This is the simplest polynomial Frobenius structure in which semisimple coalescence points
appear. The whole structure is globally and explicitly known, and the system (1.5) at generic
points is solvable in terms of oscillatory integrals. At semisimple points of coalescence, however,
the system considerably simplifies, and it reduces to a Bessel equation. Thus, the asymptotic
analysis of its solutions can be easily completed using Hankel functions, and S and C can be
immediately computed. By Theorem 1.4 above, these are monodromy data of points in a whole
neighbourhood of the coalescence point. We explicitly verify that the fundamental solutions
expressed by means of oscillatory integrals converge to those expressed in terms of Hankel
functions at a coalescence point, and that the computation done away from the coalescence
point provides the same S and C, as Theorem 1.4 predicts. In particular, the Stokes matrix S
computed invoking Theorem 1.4 is in agreement with both the well-known results of [23, 26],
stating that S+ST coincides with the Coxeter matrix of the group W (A3) (group of symmetries
of the regular tetrahedron), and with the analysis of [29] for monodromy data of the algebraic
solutions of PVIµ corresponding to A3 (see also [19] for this last point).
The second example is the quantum cohomology QH•
(
G2
(
C4
))
of the Grassmannian G2
(
C4
)
,
which sheds new light on the conjecture mentioned in the beginning.
3. Quantum cohomology of the Grassmannian G2
(
C4
)
(cf. Section 6).
We consider the Frobenius structure on QH•
(
G2
(
C4
))
. The small quantum ring – or small
quantum cohomology – of Grassmannians has been one of the first cases of quantum cohomol-
ogy rings to be studied both in physics [70, 76] and mathematical literature [11, 65], so that
a quantum extension of the classical Schubert calculus has been obtained [14]. However, the
ring structure of the big quantum cohomology is not explicitly known, so that the computation
of the monodromy data can only be done at the small quantum cohomology locus. It happens
that the small quantum locus of almost all Grassmannians Gk
(
Cn
)
is made of semisimple coa-
lescence points, see [17]; the case of G2
(
C4
)
is the simplest case where this phenomenon occurs.
Therefore, in order to compute the monodromy data, we invoke Theorem 1.4 above.
In Section 6, we carry out the asymptotic analysis of the system (1.5) at the coalescence
locus, corresponding to t = 0 ∈ QH•
(
G2
(
C4
))
. We explicitly compute the monodromy data µ
and R (see (6.8) and (6.20)) and the Stokes matrix, receiving the matrix S in expression (6.31)
(with v = 6). For the computation of S, we take an admissible8 line ` :=
{
z ∈ C : z =
eiφ
}
with the slope 0 < φ < π
4 . The signs in the square roots in (1.4) and the labelling of
(u1, . . . , u6) are chosen in Section 6.2. In order to compute the central connection matrix, we
choose a specific fundamental solution (1.13) of (1.5) with fixed t = 0, namely the enumerative-
topological fundamental solution9 Y0(z) := Ψ
∣∣
t=0
Φ(z)zµzR, whose coefficients are the genus 0
8Namely, `+(φ) defined above is an admissible ray.
9This is the solution Ψ(0)Y (z, 0) = Ψ(0)H(z, 0)zµzR in Proposition 7.2, where Φ is called H.
12 G. Cotti, B. Dubrovin and D. Guzzetti
Gromov–Witten invariants with descendants
Φ(z)αβ = δαβ +
∞∑
n=0
∑
λ
∑
ν∈Eff(G)\{0}
〈τnTβ, Tλ〉G0,2,νηλαzn+1,
with
〈τnTβ, Tλ〉G0,2,ν :=
∫
[G0,2,ν ]vir
ψn1 ∪ ev1(Tβ) ∪ ev2(Tν),
and (ηµν) the inverse of Poincaré metric. This solution will be precisely described in Section 7
(cf. Proposition 7.2). The computation yields the connection matrix C reported in Appendix B
(with v = 6).
Given S and C computed as explained above, let us denote by S′ and C ′ the data obtained
by the action
S 7−→ IPS(IP )−1 =: S′,
C 7−→ GC(IP )−1 =: C ′, (1.16)
of the groups of Theorem 1.2, where P and I will be explicitly written in the proof of Theo-
rem 6.10, while G = A or G = AB ∈ C0(η, µ,R), for certain matrices A and B given in (1.17),
(1.18) below. Geometrically, P , I and G correspond respectively to
• an appropriate re-ordering of the canonical coordinates u1, . . . , u6 near 0 ∈ QH•(G), yield-
ing the Stokes matrix in upper-triangular form.
• another determination of signs in the square roots of (1.4) of the normalized idempotents
vector fields (fi)i
• another choice of the fundamental solution of the equation (1.5) in Levelt-normal form
(1.8), obtained from the enumerative-topological solution by the action Y0 7→ Y0G of
C0(η, µ,R).
Given these explicit data, we prove Theorem 1.5 below, which clarifies for G2
(
C4
)
the conjecture,
formulated by the second author in [25] (see also [10, 45] and references therein) and then refined
in [28], relating the enumerative geometry of a Fano manifold with its derived category (see also
Remark 1.7). More details and new more general results about this conjecture are the contents
of our paper [18] (see also Remark 1.7).
For brevity, let G := G2
(
C4
)
. Fix the Schubert basis
(T0, T1, T2, T3, T4, T5) = (1, σ1, σ2, σ1,1, σ2,1, σ2,2)
of H•(G;C). Let c ∈ C∗ be defined by∫
G
σ2,2 = c.
Denote by S the tautological bundle on G and by Sλ the Schur functor associated with the
Young diagram λ. Let (E1, . . . , E6) be the 5-block10 exceptional collection, obtained from the
Kapranov exceptional 5-block collection(
S0S∗, S1S∗, S2S∗
S1,1S∗, S2,1S∗, S2,2S∗
)
,
10This means that χ(E3, E4) = χ(E4, E3) = 0 and thus that both (E1, E2, E3, E4, E5, E6) and (E1, E2, E4, E3,
E5, E6) are exceptional collections: we will write(
E1, E2,
E3
E4
, E5, E6
)
if we consider the exceptional collection with an unspecified order.
Local Moduli of Semisimple Frobenius Coalescent Structures 13
by mutation11 under the inverse of any of the following braids12 in B6:
β34β12β56β23β45β34 β12β56β23β45 β12β56β23β45β34
Consider the Taylor expansion of the Γ-function
Γ(1− x) = exp
{
γx+
∞∑
n=2
ζ(n)
n
xn
}
,
where
γ := lim
n→∞
(
− log n+
n∑
k=1
1
k
)
denotes the Euler–Mascheroni constant and ζ(n) is the Riemann zeta function. Let us introduce
the following characteristic classes:
Γ̂±(G) :=
∏
j
Γ(1± δj) where δj ’s are the Chern roots of TG,
Ch(V ) :=
∑
k
e2πixk , xk’s are the Chern roots of a vector bundle V on G,
Ch(V •) :=
∑
j
(−1)j Ch
(
V j
)
for a bounded complex V • of vector bundles on G.
Theorem 1.5 (monodromy data of QH•(G) cf. Theorem 6.10). The Stokes matrix and the
central connection matrix at t = 0 ∈ QH•(G) are related to the full exceptional collection
(E1, . . . , E6) in the following way.
The central connection matrix C ′ in (1.16) equals the matrix (one for both choices of sign ±)
associated with the C-linear morphism
X±G : K0(G)⊗Z C→ H•(G;C),
[E] 7→ 1
(2π)2c
1
2
Γ̂±(G) ∪ Ch(E)
computed w.r.t. the basis ([E1], . . . , [E6]) of K0(G), and the Schubert basis (T0, T1, T2, T3, T4, T5).
In both cases (±), the Stokes matrix S′ coincides with the inverse of the Gram matrix of the
Grothendieck–Euler–Poincaré pairing (χ(Ei, Ej))
n
i,j=1.
The matrix C ′ in the above theorem is obtained by formula (1.16) for G = A and G = AB
as follows:
11The definition of the action of the braid group on the set of exceptional collections will be given in Section 6.6,
slightly modifying (by a shift) the classical definitions that the reader can find, e.g., in [40]. Our convention for
the composition of action of braids is the following: braids act on an exceptional collection/monodromy datum
on the right. The braid β34 acts on the 5-block collection (E1, . . . , E6) above just as a permutation of the third
and fourth elements of the block.
12Curiously, these braids show a mere mirror symmetry: notice that they are indeed equal to their specular
reflection. Any contingent geometrical meaning of this fact deserves further investigations.
14 G. Cotti, B. Dubrovin and D. Guzzetti
• the matrix representing X−G is equal to C ′ computed w.r.t. the solution Y0(z) · A, where
A ∈ C0(η, µ,R) is
A =
1 0 0 0 0 0
2iπ 1 0 0 0 0
−2π2 2iπ 1 0 0 0
−2π2 2iπ 0 1 0 0
−1
3
(
8iπ3
)
−4π2 2iπ 2iπ 1 0
4π4
3 −1
3
(
8iπ3
)
−2π2 −2π2 2iπ 1
; (1.17)
• the matrix representing X+
G is equal C ′ computed w.r.t. the solution Y0(z) · A · B, where
B ∈ C0(η, µ,R) is
B =
1 0 0 0 0 0
−8γ 1 0 0 0 0
32γ2 −8γ 1 0 0 0
32γ2 −8γ 0 1 0 0
8
3
(
ζ(3)− 64γ3
)
64γ2 −8γ −8γ 1 0
64
3
(
16γ4 − γζ(3)
)
8
3
(
ζ(3)− 64γ3
)
32γ2 32γ2 −8γ 1
. (1.18)
A few more words about Theorem 1.5 are in order. The theorem provides the monodromy
data at t = 0 ∈ QH•(G). It is worth stressing that the result acquires its geometrical significance
because of Theorem 1.4 (i.e., Theorem 4.5), which guarantees that the data computed at the
coalescence point t = 0 are the monodromy data of the manifold in any chamber whose closure
contains the point t = 0. Without Theorem 1.4, the results of the computations of Section 6
would have little geometrical meaning for the theory of Frobenius manifolds.
From the data computed at t = 0, the Stokes and the central connection matrices at all other
points of the small quantum cohomology and/or w.r.t. other possible admissible lines `, will be
computed in Section 6.7, by a careful and pedagogical application of the action of the braid
group. They satisfy the same properties as in Theorem 1.5 w.r.t. other full exceptional 5-block
collections, obtained from (E1, . . . , E6) by alternate mutation under the braids
ω1 := β12β56, ω2 := β23β45β34β23β45, ω̂1 := β12β34β56.
See Section 6.7 for details.
It is important to remark that the Kapranov 5-block exceptional collection itself appears
neither at t = 0 nor anywhere else along the locus of the small quantum cohomology. See
Corollary 6.13.
The monodromy data in any other chamber of QH•(G) can be obtained from the data S′, C ′
computed at 0 ∈ QH•(G) (or from PSP−1 and CP−1), by the action (1.15) of the braid group.
It is also worth noticing that the coalescence phenomenon should be taken into account in
order to understand the appearance of the two exceptional collections (E1, E2, E3, E4, E5, E6)
and (E1, E2, E4, E3, E5, E6) in Theorem 1.5. In Section 4 we will explain that, as a consequence of
coalescence, a small neighbourhood of 0 ∈ QH•(G) is divided into two `-cells (see Definition 4.3).
The passage from one exceptional collection to the other reflects the passage from one `-cell to
the other.
Remark 1.6. In Theorem 1.5, we have two morphisms X±G; the sign (−) is the one taken in [28],
whereas (+) is the one taken in [33].
Remark 1.7. Our explicit results suggest that the conjecture formulated in [28] and [33] requires
some refinements, at least as far as the central connection matrix C is concerned. Indeed, the
Local Moduli of Semisimple Frobenius Coalescent Structures 15
connection matrix C ′ in Theorem 1.5 belongs to the C0(η, µ,R)-orbit of the connection matrix
obtained from the topological-enumerative fundamental solution, but is not the connection
matrix w.r.t. the topological-enumerative solution. These refinements are fully discussed in our
paper [18].
Remark 1.8. In [9] it is shown that the class of smooth projective varieties admitting generically
semisimple quantum cohomology is closed w.r.t. the operation of blowing up at a finite number of
points. Since this holds true also for the class of varieties which admit full exceptional collections
in their derived categories, it is tempting to conjecture that the mentioned relationship between
monodromy data and exceptional collections can be extended also for non-Fano varieties. This
is already suggested in [9]. To the best of our knowledge, no explicit computations of the
monodromy data have been done in the non-Fano case. The computations of the monodromy
data for the 1
2K3-surface, the rational elliptic surface obtained by blowing up 9 points in P2,
could represent a significant step in this direction. This will represent a future research project
of the authors.
Remarkably, our results suggest the validity of a constraint on the kind of exceptional collec-
tions associated with the monodromy data in a neighborhood of a semisimple coalescing point
of the quantum cohomology QH•(X) of a smooth projective variety X. If the eigenvalues ui’s
coalesce, at some semisimple point t0, to s < n values λ1, . . . , λs with multiplicities p1, . . . , ps
(with p1 + · · · + ps = n, here n is the sum of the Betti numbers of X), then the correspon-
ding monodromy data can be expressed in terms of Gram matrices and characteristic classes of
objects of a full s-block exceptional collection, i.e., a collection of the type
E := (E1, . . . , Ep1︸ ︷︷ ︸
B1
, Ep1+1, . . . , Ep1+p2︸ ︷︷ ︸
B2
, . . . , Ep1+···+ps−1+1, . . . , Ep1+···+ps︸ ︷︷ ︸
Bs
),
Ej ∈ Obj
(
Db(X)
)
, where for each pair (Ei, Ej) in a same block Bk the orthogonality conditions
hold
Ext`(Ei, Ej) = 0, for any `.
In particular, any reordering of the objects inside a single block Bj preserves the exceptiona-
lity of E . More results about the nature of exceptional collections arising in this context and
about their dispositions in the locus of small quantum cohomology for the class of complex
Grassmannians will be explained in our paper [18].
1.2 Plan of the paper
In Section 2, we review the analytic theory of Frobenius manifolds, their monodromy data and
the isomonodromy theorems, according to [23, 25, 26]. In particular, we characterise the freedom
in the choice of the central connection matrix C, introducing the group C0(η, µ,R). We define
a chamber-decomposition of the manifold, which depends on the choice of an oriented line ` in
the complex plane: this is a natural structure related to the local invariance of the monodromy
data (isomonodromy Theorems 2.16 and 2.44), as well as of their discontinuous jumps from one
chamber to another one, encoded in the action of the braid group, as a wall-crossing phenomenon.
In Section 3 we review all freedoms and all other natural transformations on the monodromy
data.
In Section 4 we extend the isomonodromy theorems and give a complete description of mon-
odromy data in a neighborhood of semisimple coalescence points, specialising the result of [19]
in Theorem 4.5.
In Section 5, we study the A3 Frobenius manifold near the Maxwell stratum. We compute
monodromy data both invoking Theorem 4.5 above and using oscillatory integrals. We compare
16 G. Cotti, B. Dubrovin and D. Guzzetti
the two approaches, so providing an explicit example of how Theorem 4.5 works. We also show
how monodromy data mutate along a loop inside the Maxwell stratum.
In Section 6.1, we explicitly compute all monodromy data of the quantum cohomology of
the Grassmannian G2
(
C4
)
. The result allows us to explicitly verify the conjecture of [25, 28]
relating the monodromy data to characteristic classes of objects of an exceptional collection
in Db
(
G2
(
C4
))
.
In Section 7 we give an analytic characterisation of the enumerative-topological solution, in
a different way with respect to [33].
2 Moduli of semisimple Frobenius manifolds
We denote with
⊙
the symmetric tensor product of vector bundles, and with (−)[ the standard
operation of lowering the index of a (1, k)-tensor using a fixed inner product.
Definition 2.1. A Frobenius manifold structure on a complex manifold M of dimension n is
defined by giving
(FM1) a symmetric nondegenerate O(M)-bilinear tensor η ∈ Γ
(⊙2 T ∗M
)
, called metric, whose
corresponding Levi-Civita connection ∇ is flat;
(FM2) a (1, 2)-tensor c ∈ Γ
(
TM ⊗
⊙2 T ∗M
)
such that
• the induced multiplication of vector fields X ◦ Y := c(−, X, Y ), for X,Y ∈ Γ(TM),
is associative,
• c[ ∈ Γ
(⊙3 T ∗M
)
,
• ∇c[ ∈ Γ
(⊙4 T ∗M
)
;
(FM3) a vector field e ∈ Γ(TM), called the unity vector field, such that
• the bundle morphism c(−, e,−) : TM → TM is the identity morphism,
• ∇e = 0;
(FM4) a vector field E ∈ Γ(TM), called the Euler vector field, such that
• LEc = c,
• LEη = (2− d) · η, where d ∈ C is called the charge of the Frobenius manifold.
For simplicity it will be assumed that the tensor ∇E ∈ TM ⊗ T ∗M is diagonalizable.
Since the connection ∇ is flat, there exist local flat coordinates that we denote
(
t1, . . . , tn
)
,
w.r.t. which the metric η is constant and the connection ∇ coincides with the partial derivatives
∂α = ∂/∂tα, α = 1, . . . , n. Because of flatness and the conformal Killing condition, the Euler
vector field is affine, i.e.,
∇∇E = 0, so that E =
n∑
α=1
(
(1− qα)tα + rα
) ∂
∂tα
, qα, rα ∈ C.
Following [23, 25, 26], we choose flat coordinates so that ∂
∂t1
≡ e and rα 6= 0 only if qα = 1
(this can always be done, up to an affine change of coordinates). In flat coordinates, let ηαβ =
η(∂α, ∂β), and cγαβ = c
(
dtγ , ∂α, ∂β
)
, so that ∂α ◦ ∂β = cγαβ∂γ . Condition (FM2) means that
cαβγ := ηαρc
ρ
βγ and ∂αcβγδ are symmetric in all indices. This implies the local existence of
a function F such that
cαβγ = ∂α∂β∂γF.
Local Moduli of Semisimple Frobenius Coalescent Structures 17
The associativity of the algebra is equivalent to the following conditions for F , called WDVV-
equations
∂α∂β∂γFη
γδ∂δ∂ε∂νF = ∂ν∂β∂γFη
γδ∂δ∂ε∂αF,
while axiom (FM4) is equivalent to
ηαβ = ∂1∂α∂βF, LEF = (3− d)F +Q(t),
with Q(t) a quadratic expression in tα’s. Conversely, given a solution of the WDVV equations,
satisfying the quasi-homogeneity conditions above, a structure of Frobenius manifold is naturally
defined on an open subset of the space of parameters tα’s.
Let us consider the canonical projection π : P1
C ×M → M , and the pull-back of the tangent
bundle TM :
π∗TM //
��
TM
��
P1
C ×M
π //M.
We will denote by
1) TM the sheaf of sections of TM ,
2) π∗TM the pull-back sheaf, i.e., the sheaf of sections of π∗TM ,
3) π−1TM the sheaf of sections of π∗TM constant on the fibers of π.
Introduce two (1, 1)-tensors U , µ on M defined by
U(X) := E ◦X, µ(X) :=
2− d
2
X −∇XE (2.1)
for all X ∈ Γ(TM). In flat coordinates (tα)nα=1 chosen as above, the operator µ is constant and
in diagonal form
µ = diag(µ1, . . . , µn), µα = qα −
d
2
∈ C.
All the tensors η, e, c, E, U , µ can be lifted to π∗TM , and their lift will be denoted with the
same symbol. So, also the Levi-Civita connection ∇ is lifted on π∗TM , and it acts so that
∇∂zY = 0 for Y ∈
(
π−1TM
)
(M).
Let us now twist this connection by using the multiplication of vectors and the operators U , µ.
Definition 2.2. Let M̂ := C∗×M . The deformed connection ∇̂ on the vector bundle π∗TM |
M̂
→ M̂ is defined by
∇̂XY = ∇XY + z ·X ◦ Y,
∇̂∂zY = ∇∂zY + U(Y )− 1
z
µ(Y )
for X,Y ∈ (π∗TM )
(
M̂
)
.
The crucial fact is that the deformed extended connection ∇̂ is flat.
18 G. Cotti, B. Dubrovin and D. Guzzetti
Theorem 2.3 ([23, 26]). The flatness of ∇̂ is equivalent to the following conditions on M
• ∇c[ is completely symmetric,
• the product on each tangent space of M is associative,
• ∇∇E = 0,
• LEc = c.
Because of this integrability condition, we can look for deformed flat coordinates
(
t̃1, . . . , t̃n
)
,
with t̃α = t̃α(t, z). These coordinates are defined by n independent solutions of the equation
∇̂dt̃ = 0.
Let ξ denote a column vector of components of the differential dt̃. The above equation becomes
the linear system∂αξ = zCT
α (t)ξ,
∂zξ =
(
UT(t)− 1
z
µT
)
ξ,
(2.2)
where Cα is the matrix (Cα)βγ = cβαγ . We can rewrite the system in the form∂αζ = zCαζ,
∂zζ =
(
U +
1
z
µ
)
ζ,
(2.3)
where ζ := η−1ξ. In order to obtain (2.3), we have also used the invariance of the product,
encoded in the relations
η−1CT
α η = Cα,
UTη = ηU , (2.4)
and the η-skew-symmetry of µ
µTη + ηµ = 0. (2.5)
Geometrically, ζ is the η-gradient of a deformed flat coordinate as in (1.3). Monodromy data of
system (2.3) define local invariants of the Frobenius manifold, as explained below.
2.1 Spectrum of a Frobenius manifold and its monodromy data at z = 0
Let us fix a point t of the Frobenius manifold M and let us focus on the associated equation
∂zζ =
(
U(t) +
1
z
µ(t)
)
ζ. (2.6)
Remark 2.4. If ζ1, ζ2 are solutions of the equation (2.6), then the two products
〈ζ1, ζ2〉± := ζT
1
(
e±πiz
)
ηζ2(z)
are independent of z. Indeed we have
∂z
(
ζT
1
(
e±πiz
)
ηζ2(z)
)
= ∂z
(
ζT
1
(
e±πiz
))
ηζ2(z) + ζT
1
(
e±πiz
)
η∂zζ2(z)
= ζT
1
(
e±πiz
) [
ηU − UTη +
1
z
(µη + ηµ)
]
ζ2(z)
= 0 by (2.4) and (2.5).
Local Moduli of Semisimple Frobenius Coalescent Structures 19
In order to give an intrinsic description of the structure of the normal forms of solutions of
equation (2.6), as well as a geometric characterization of the ambiguity and freedom up to which
they are defined, we introduce the concept of the spectrum of a Frobenius manifold (see also
[26, 27]). Let (V, η, µ) be the datum of
• an n-dimensional complex vector space V ,
• a bilinear symmetric non-degenerate form η on V ,
• a diagonalizable endomorphism µ : V → V which is η-antisymmetric
η(µa, b) + η(a, µb) = 0 for any a, b ∈ V.
Let spec(µ) = (µ1, . . . , µn) and let Vµα be the eigenspace of a µα.
Definition 2.5. Let (V, η, µ) as above. We say that an endomorphism A ∈ End(V ) is µ-nilpotent
if
AVµα ⊆
⊕
m∈Z≥1
Vµα+m for any µα ∈ spec(µ).
In particular such an operator is nilpotent in the usual sense. We can decompose a µ-nilpotent
operator A in components Ak, k ≥ 1, such that
AkVµα ⊆ Vµα+k for any µα ∈ spec(µ),
so that the following identities hold:
zµAz−µ = A1z +A2z
2 +A3z
3 + · · · , [µ,Ak] = kAk for k = 1, 2, 3, . . . .
Definition 2.6. Let (V, η, µ) as above. Let us define on V a second non-degenerate bilinear
form {·, ·} by the equation
{a, b} := η
(
eiπµa, b
)
, for all a, b ∈ V.
The set of all {·, ·}-isometries G ∈ End(V ) of the form
G = 1V + ∆,
with ∆ a µ-nilpotent operator, is a Lie group G(η, µ), called (η, µ)-parabolic orthogonal group.
Its Lie algebra g(η, µ) coincides with the set of all µ-nilpotent operators R which are also {·, ·}-
skew-symmetric in the sense that
{Rx, y}+ {x,Ry} = 0.
In particular, any such matrix R commutes with the operator e2πiµ.
The following result gives a description, in coordinates, of both µ-nilpotents operators and
elements of g(η, µ) and also describes some of their properties.
Lemma 2.7. Let (V, η, µ) as above, and let us fix a basis (vi)
n
i=1 of eigenvectors of µ.
1. The operator A ∈ End(V ) is µ-nilpotent if and only if its associate matrix w.r.t. the basis
(vi)
n
i=1 satisfies the condition
(A)αβ = 0 unless µα − µβ ∈ N∗.
20 G. Cotti, B. Dubrovin and D. Guzzetti
2. If A ∈ End(V ) is a µ-nilpotent operator, then the matrices associated with its components
(Ak)k≥1 w.r.t. the basis (vi)
n
i=1 satisfy the condition
(Ak)
α
β = 0 unless µα − µβ = k, k ∈ N∗. (2.7)
3. A µ-nilpotent operator A ∈ End(V ) is an element of g(η, µ) if and only if the matrices of
its components (Ak)k≥1 w.r.t. (vi)
n
i=1 satisfy the further conditions
AT
k = (−1)k+1ηAkη
−1, k ≥ 1. (2.8)
4. If A ∈ g(η, µ), then the following identity holds
zA
T
ηe±iπµzA = ηe±iπµ, (2.9)
for any z ∈ C∗.
Proof. The proof for points (1), (2), (3) can be found in [26]. For the identity (2.9), notice
that (2.8) implies
zA
T
= η
(
zA1−A2+A3−A4+···)η−1.
Moreover, from (2.7) we deduce that
e∓iπµAke
±iπµ = (−1)kAk.
So, we conclude that
zA
T
ηe±iπµzA = η
(
zA1−A2+A3−A4+···)(e±iπµzAe∓iπµ
)
e±iπµ
= η
(
zA1−A2+A3−A4+···)(z−A1+A2−A3+A4−···)e±iπµ = ηe±iπµ. �
The parabolic orthogonal group G(η, µ) acts canonically on its Lie algebra g(η, µ) by the
adjoint representation Ad: G(η, µ)→ Aut(g(η, µ)):
AdG(R) := G ·R ·G−1, for all G ∈ G(η, µ), R ∈ g(η, µ).
Such an action, in general is not free.
Definition 2.8. Let R ∈ g(η, µ). We define the group C0(η, µ,R) as the isotropy group of R for
the adjoint representation Ad: G(η, µ)→ Aut(g(η, µ)).
The following Lemma can be easily directly proved from Definitions 2.5, 2.6, 2.8 and from
results of Lemma 2.7.
Lemma 2.9. Let (V, η, µ) a triple as above. If G ∈ G(η, µ), and R ∈ g(η, µ) then
zµzAdGRGz−Rz−µ
is an element of End(V )[z], i.e., it is polynomial in the indeterminate z. Furthermore, the
following is an equivalent characterization of the isotropy subgroup C0(η, µ,R):
C0(η, µ,R) =
G ∈ GL(V ) :
PG(z) := zµzRGz−Rz−µ ∈ End(V )[z],
PG(0) ≡ 1V ,
η(PG(−z)v1, PG(z)v2) = η(v1, v2),
for all v1, v2 ∈ V
.
Local Moduli of Semisimple Frobenius Coalescent Structures 21
Definition 2.10. If (V1, η1, µ1), (V2, η2, µ2) are two triples as above, a morphism of triples
f : (V1, η1, µ1) → (V2, η2, µ2), is the datum of a linear morphism f : V1 → V2, compatible with
the metrics and the operators µ1, µ2, i.e.,
η1(v, w) = η2(f(v), f(w)), v, w ∈ V1, µ2 ◦ f = f ◦ µ1.
Given a Frobenius manifold M (not necessarily semisimple), we can canonically associate to
it an isomorphism class [(V, η, µ)] of triples as above, which will be called the spectrum of M .
Attached with any point p ∈ M , indeed, we have a triple (TpM,ηp, µp). Given p1, p2 ∈ M ,
the two triples are (non-canonically) isomorphic: using the Levi-Civita connection, for any
path γ : [0, 1] → M with γ(0) = p1 and γ(1) = p2, the parallel transport along γ provides an
isomoprhism of the triples at p1 and p2.
Definition 2.11 ([26, 27]). A Frobenius manifold M is called resonant if µα − µβ ∈ Z\{0} for
some α 6= β; otherwise, M is called non-resonant.
We can now give a complete (componentwise) description of normal forms of solutions of the
system (2.6).
Theorem 2.12 ([23, 26]). Let M be a Frobenius manifold (not necessarily semisimple). The
system (2.6) admits fundamental matrix solutions of the form
Z(z, t) = Φ(z, t) · zµzR(t), Φ(z, t) =
∑
k∈N
Φk(t)z
k, Φ0(t) ≡ 1, (2.10)
Φ(−z, t)T · η · Φ(z, t) = η, (2.11)
where Φk ∈ O(M)⊗ gln(C), and R ∈ O(M)⊗ g(η, µ). A solution of such a form will be said to
be in Levelt normal form at z = 0.
Remark 2.13. In the general case, although not related to Frobenius manifolds, when µ is not
diagonalizable and has a non-trivial nilpotent part, analogous results can be proved. However,
the normal form becomes a little more complicated: e.g., it is no more defined by requiring that
some entries of matrices Rk are nonzero, but that some blocks are. For a detailed analysis of
such case, we recommend the book by F.R. Gantmacher [34].
Because of the Fuchsian character of the singularity z = 0, the power series Φ of Theorem 2.12
is convergent, and defines a genuine analytic solution. In general, solutions in Levelt normal form
are not unique. As the following result shows, the freedom in the choice of solutions in normal
form are suitably quantified by the Lie groups G(η, µ) and its isotropic subgroups C0(η, µ,R).
Theorem 2.14 ([23, 26]). Let M be a Frobenius manifold (not necessarily semisimple). Solu-
tions of (2.6) in normal form are not unique. Given two of them
Z(z, t) = Φ(z, t) · zµzR(t), Z̃(z, t) = Φ̃(z, t) · zµzR̃(t),
there exists a unique holomorphic G(η, µ)-valued function
G(t) = 1 + ∆(t)
on M such that
Z̃(z, t) = Z(z, t) ·G(t), R̃(t) = G(t)−1 ·R(t) ·G(t), Φ̃(z, t) = Φ(z, t) · PG(z, t),
where
PG(z, t) := zµ ·G(t) · z−µ = 1 + z∆1(t) + z2∆2(t) + · · · ,
(∆k)k≥1 being the components of ∆. In particular, if R̃ = R, then G is C0(η, µ,R)-valued.
22 G. Cotti, B. Dubrovin and D. Guzzetti
Remark 2.15. A first description of the freedom and ambiguities in the definition of the mon-
odromy data was given in [23, 26]. In particular, a complex Lie group C0(µ,R) was introduced
in order to describe the freedom of normal forms of solutions of (2.6). Such a group is too big,
and in particular does not preserve the orthogonality condition (2.11). It must be replaced by
C0(η, µ,R) of Definition 2.8, which is the correct one.
For non-resonant Frobenius manifolds the corresponding (η, µ)-parabolic orthogonal group
G(η, µ) together with all its subgroups C0(η, µ,R) are trivial. Since these groups are the respon-
sible of a certain freedom in the choice of a normal form for solutions of (2.6) (according to
Theorem 2.14), it follows that for non-resonant Frobenius manifolds such a choice is unique.
So far, we have focused on the system (2.6) at a fixed point of the manifold. Now let us vary
the point t in system (2.6), so that a fundamental solution Φ(z, t)zµzR(t), as in (2.10), depends
on t. If instead of considering only the equation (2.6), we focus on the whole system (2.3), then
the previous results can be further refined: namely, a t-independent choice for the exponent R
is allowed. Again, even for a fixed exponent R, solutions on normal forms are not unique, and
they are parametrized by the isotropy group C0(η, µ,R).
Theorem 2.16 (isomonodromy Theorem I, [23, 26]). Let M be a Frobenius manifold (not
necessarily semisimple).
1. The system (2.3) admits fundamental matrix solutions of the form
Z(z, t) = Φ(z, t) · zµzR,
Φ(z, t) =
∑
k∈N
Φk(t)z
k, Φ0(t) ≡ 1, Φ(−z, t)T · η · Φ(z, t) = η,
where Φk ∈ O(M) ⊗ gln(C), and R ∈ g(η, µ) is independent of t. In particular the
monodromy M0 = exp(2πiµ) exp(2πiR) at z = 0 does not depend on t.
2. Solutions of the whole system (2.3) in normal form are not unique. Given two of them
Z(z, t) = Φ(z, t) · zµzR, Z̃(z, t) = Φ̃(z, t) · zµzR̃,
there exists a unique matrix G ∈ G(η, µ), say G = 1 + ∆, such that
Z̃(z, t) = Z(z, t) ·G, R̃ = G−1 ·R ·G, Φ̃(z, t) = Φ(z, t) · PG(z, t),
where
PG(z, t) := zµ ·G · z−µ = 1 + z∆1 + z2∆2 + · · · ,
(∆k)k≥1 being the components of ∆. In particular, if R̃ = R, then G ∈ C0(η, µ,R).
Proof. Let Z(z, t) be a solution of (2.3), and let M0(t) be the monodromy of Z(·, t) at z = 0:
Z
(
e2πiz, t
)
= Z(z, t) ·M0(t).
The coefficients of the equations
∂αZ(z, t) = zCα(t) · Z(z, t), α = 1, . . . , n
being holomorphic in z, we have that
∂αZ(z, t) · Z(z, t)−1 = ∂αZ
(
e2πiz, t
)
· Z(e2πiz, t)−1
= ∂α(Z(z, t) ·M0(t)) · (Z(z, t) ·M0(t))−1
Local Moduli of Semisimple Frobenius Coalescent Structures 23
= ∂αZ(z, t) · Z(z, t)−1 + Z(z, t) · ∂αM0(t) ·M0(t)−1 · Z(z, t)−1,
for any α. Hence
∂αM0(t) = 0, α = 1, . . . , n.
By Theorem 2.14, we necessarily conclude that R is t-independent. �
Definition 2.17 ([23, 26]). Given a Frobenius manifold M , we will call monodromy data of M
at z = 0 the data (µ, [R]), where [R] denotes the G(η, µ)-class of exponents of formal solutions
in Levelt normal form of the system (2.3) as in Theorem 2.12. According to Theorem 2.16,
a representative R can be chosen independent of the point t ∈M .
We conclude this section with a result giving sufficient conditions on solutions of the sys-
tem (2.3) for resonant Frobenius manifolds in order that they satisfy the η-orthogonality condi-
tion (2.11). In its essence, this result is stated and proved in [33], in the specific case of quantum
cohomologies of Fano manifolds.
Proposition 2.18. Let M be a resonant Frobenius manifold, and t0 ∈M a fixed point.
1. Suppose that there exists a fundamental solution of (2.3) of the form
Z(z, t) = Φ(z, t)zµzR, Φ(t) = 1 +
∞∑
j=1
Φj(t)z
j ,
with R satisfying all the properties of the Theorem 2.12, such that
H(z) := z−µΦ(z, t0)zµ
is a holomorphic function at z = 0 and H(0) ≡ 1. Then Φ(z, t) satisfies the constraint
Φ(−z, t)TηΦ(z, t) = η
for all points t ∈M .
2. If a solution with the properties above exists, then it is unique.
Proof. From Remark 2.4, we already know that the following bracket must be independent
of z:
〈Z(z, t0), Z(z, t0)〉+ =
(
Φ(−z, t0)
(
eiπz
)µ(
eiπz
)R)T
η
(
Φ(z, t0)zµzR
)
=
((
eiπz
)µ
H(−z)
(
eiπz
)R)T
η
(
zµH(z)zR
)
= eiπRT
zR
T
H(−z)TeiπµzµηzµH(z)zR
= eiπRT
zR
T
H(−z)TeiπµηH(z)zR.
By taking the first term of the Taylor expansion in z of the r.h.s., and using (2.9), we get
〈Z(z, t0), Z(z, t0)〉+ = eiπRT
eiπµη.
So, using again the equation zµ
T
ηzµ = η and (2.9), we can conclude that
Φ(−z, t0)TηΦ(z, t0) =
((
eiπz
)µ(
eiπz
)R)−T〈Z(z, t0), Z(z, t0)〉+
(
zµzR
)−1
= η.
24 G. Cotti, B. Dubrovin and D. Guzzetti
Because of (2.3) and the property of η-compatibility of the Frobenius product, we have that
∂
∂tα
(
Φ(−z, t)TηΦ(z, t)
)
= z · Φ(−z, t)T ·
(
ηCα − CT
α η
)
· Φ(z, t) = 0.
This concludes the proof of (1). Let us now suppose that there are two solutions
Φ1(z, t)zµzR, Φ2(z, t)zµzR
such that
z−µΦ1(z, t0)zµ = 1 + zK1 + z2K2 + · · · , (2.12)
z−µΦ2(z, t0)zµ = 1 + zK ′1 + z2K ′2 + · · · . (2.13)
The two solutions must be related by
Φ2(z, t)zµzR = Φ1(z, t)zµzR · C
for some matrix C ∈ C0(η, µ,R). This implies that Φ2(z, t) = Φ1(z, t) · P (z), where P (z) is
a matrix valued polynomial of the form
P (z) = 1 + z∆1 + z2∆2 + · · · , with (∆k)
α
β = 0 unless µα − µβ = k, and P (1) ≡ C.
We thus have z−µΦ−1
1 Φ2z
µ = z−µP (z)zµ, and(
z−µP (z)zµ
)α
β
= δαβ +
∑
k
(∆k)
α
β z
k−µα+µβ = δαβ +
∑
k
(∆k)
α
β ≡ C.
Then, from (2.12), (2.13) it immediately follows that C = 1, which proves that Φ1 = Φ2. �
2.2 Semisimple Frobenius manifolds
Definition 2.19. A finite dimensional commutative and associative K-algebra A with unit is
called semisimple if there is no nilpotent element, i.e., an element a ∈ A \ {0} such that ak = 0
for some k ∈ N.
In what follows we will always assume that the ground field is C.
Lemma 2.20. Let A be a C-Frobenius algebra of dimension n. The following are equivalent:
1) A is semisimple;
2) A is isomorphic to C⊕n;
3) A has a basis of idempotents, i.e., elements π1, . . . , πn such that
πi ◦ πj = δijπi, η(πi, πj) = ηiiδij ,
for a suitable non-degenerate multiplication invariant pairing η on A;
4) there is a vector E ∈ A such that the multiplication operator E◦ : A → A has n pairwise
distinct eigenvalues.
Proof. All these equivalences are well known. For the equivalence of (1) and (2), see for example
[13, Chapter V, Section 6, Proposition 5, p. A.V.34]. Another elementary proof can be found in
the lectures notes [26]. The fact that (2) and (3) are equivalent is trivial. Let us prove that (3)
and (4) are equivalent. If (3) holds it is sufficient just to take
E =
∑
k
kπk.
Local Moduli of Semisimple Frobenius Coalescent Structures 25
So E◦ has spectrum {1, . . . , n}. Let us now suppose that (4) holds. The commutativity of
the algebra implies that all operators a◦ : A → A commute. Therefore they preserve the one
dimensional eigenspaces of E◦ and thus they are all diagonalizable. It follows that all oper-
ators a◦, being commuting and diagonalizable linear operators on a finite dimensional vector
space, are simultaneously diagonalizable. Thus idempotents are constructed by suitably resca-
ling the eigenvectors of E◦. �
Definition 2.21 (semisimple Frobenius manifolds). A point p of a Frobenius manifold M is
semisimple if the corresponding Frobenius algebra TpM is semisimple. If there is an open dense
subset Mss ⊂M of semisimple points, then M is called a semisimple Frobenius manifold.
It is evident from point (4) of Lemma 2.20 that semisimplicity is an open property : if p is
semisimple, then all points in a neighborhood of p are semisimple.
Definition 2.22 (caustic and bifurcation set). Let M be a semisimple Frobenius manifold. We
call caustic the set
KM := M \Mss = {p ∈M : TpM is not a semisimple Frobenius algebra}.
We call bifurcation set of the Frobenius manifold the set
BM := {p ∈M : spec (E◦p : TpM → TpM) is not simple}.
By Lemma 2.20, we have KM ⊆ BM . Semisimple points in BM \ KM are called semisimple
coalescence points.
The bifurcation set BM and the caustic KM are either empty or a hypersurface (in general
a singular one), invariant w.r.t. the unit vector field e (see [44]). For Frobenius manifolds
defined on the base space of semiuniversal unfoldings of a singularity, these sets coincide with
the bifurcation diagram and the caustic as defined in the classical setting of singularity theory
[3, 4]. In this context, the set BM \KM is called Maxwell stratum. Remarkably, all these subsets
typically admit a naturally induced Frobenius submanifold structure [66, 67]. In what follows
we will assume that the semisimple Frobenius manifold M admits nonempty bifurcation set BM ,
and set of semisimple coalescence points BM \ KM .
At each point p in the open dense semisimple subset Mss ⊆ M , there are n idempotent
vectors
π1(p), . . . , πn(p) ∈ TpM,
unique up to a permutation. By Lemma 2.20 there exists a suitable local vector field E such
that π1(p), . . . , πn(p) are eigenvectors of the multiplication E◦, with simple spectrum at p and
consequently in a whole neighborhood of p. Using the results exposed in [52] about analytic
deformation of operators with simple spectrum w.r.t. one complex parameter, in particular the
results stating analyticity of eigenvectors and eigenprojections, and extending them to the case
of more parameters using Hartogs’ theorem, we deduce the following
Lemma 2.23. The idempotent vector fields are holomorphic at a semisimple point p, in the sense
that, chosen an ordering π1(p), . . . , πn(p), there exist a neighborhood of p where the resulting local
vector fields are holomorphic.
Notice that, although the idempotents are defined (and unique up to a permutation) at each
point of Mss, it is not true that there exist n globally well-defined holomorphic idempotent
vector fields. Indeed, the caustic KM is in general a locus of algebraic branch points: if we
26 G. Cotti, B. Dubrovin and D. Guzzetti
consider a semisimple point p and a close loop γ : [0, 1]→M , with base point p, encircling KM ,
along which a coherent ordering is chosen, then(
π1(γ(0)), . . . , πn(γ(0))
)
and
(
π1(γ(1)), . . . , πn(γ(1))
)
may differ by a permutation. Thus, the idempotent vector fields are holomorphic and single-
valued on simply connected open subsets not containing points of the caustic.
Remark 2.24. More generally, under the assumption KM 6= ∅, the idempotent vector fields
define single-valued and holomorphic local sections of the tangent bundle TM on any connected
open set Ω ⊆M \ KM = Mss satisfying the following property: for any z ∈ Ω the inclusions
Ω
α
↪−−−−→Mss
β
↪−−−−→M
induce morphisms in homotopy
π1(Ω, z)
α∗ // π1(Mss, z)
β∗
// π1(M, z)
such that im(α∗)∩ ker(β∗) = {0}. Moreover, this means that the structure group of the tangent
bundle of Mss is reduced to the symmetric group Sn, and that the local isomorphism of OMss-
algebras
TMss
∼=
loc
O⊕nMss
,
existing everywhere, can be replaced by a global one by considering a Frobenius structure pro-
longed to an unramified covering of degree at most n! (see [56]).
Theorem 2.25 ([22, 23, 26]). Let p ∈ Mss be a semisimple point, and (πi(p))
n
i=1 a basis of
idempotents in TpM . Then
[πi, πj ] = 0;
as a consequence there exist local coordinates u1, . . . , un such that
πi =
∂
∂ui
.
Definition 2.26 (canonical coordinates [23, 26]). Let M a Frobenius manifold and p ∈ M
a semisimple point. The coordinates defined in a neighborhood of p of Theorem 2.25 are called
canonical coordinates.
Canonical coordinates are defined only up to permutations and shifts. They are holomorphic
local coordinates in a simply connected neighbourhood of a semisimple point not containing
points of the caustic KM , or more generally on domains with the property of Remark 2.24.
Holomorphy holds also at semisimple coalescence points.
Theorem 2.27 ([26]). If u1, . . . , un are canonical coordinates near a semisimple point of a Frobe-
nius manifold M , then (up to shifts) the following relations hold
∂
∂ui
◦ ∂
∂ui
= δij
∂
∂ui
, e =
n∑
i=1
∂
∂ui
, E =
n∑
i=1
ui
∂
∂ui
.
In this paper we will fix the shifts of canonical coordinates so that they coincide with the
eigenvalues of the (1, 1)-tensor E◦.
Local Moduli of Semisimple Frobenius Coalescent Structures 27
Definition 2.28 (matrix Ψ). Let M be a semisimple Frobenius manifold, t1, . . . , tn be local
flat coordinates such that ∂
∂t1
= e and u1, . . . , un be canonical coordinates. Introducing the
orthonormal basis
fi :=
1
η
(
∂
∂ui
, ∂
∂ui
) 1
2
∂
∂ui
(2.14)
for an arbitrary choice of signs in the square roots, we define a matrix Ψ (depending on the
point of the Frobenius manifold) whose elements Ψiα (i-th row, α-th column) are defined by the
relation
∂
∂tα
=
n∑
i=1
Ψiαfi, α = 1, . . . , n.
Lemma 2.29. The matrix Ψ is a single-valued holomorphic function on any simply connected
open subset not containing points of the caustic KM , or more generally on any open domain Ω
as in Remark 2.24. Moreover, it satisfies the following relations:
ΨTΨ = η, Ψi1 = η
(
∂
∂ui
,
∂
∂ui
) 1
2
,
∂
∂ui
= Ψi1
n∑
α,β=1
Ψiαη
αβ ∂
∂tβ
, cαβγ =
n∑
i=1
ΨiαΨiβΨiγ
Ψi1
.
If U is the operator of multiplication by the Euler vector field, then Ψ diagonalizes it:
ΨUΨ−1 = U := diag(u1, . . . , un).
Proof. The first assertion is a direct consequence of the analogous property of the idempotents
vector fields, as in Lemma 2.23. All the other relations follow by computations (see [26], notice
a misprint in formula (3.16) there, where fi must be replaced by πi). �
We stress that Ψ and the coordinates ui’s are holomorphic also at semisimple coalescence
points, due to the same property of the idempotents.
2.3 Monodromy data for a semisimple Frobenius manifold
Monodromy data at z = ∞ are defined in [23, 25, 26] at a point of a semisimple Frobenius
manifold not belonging to the bifurcation set. In the present section we review these issues, and
we enlarge the definition to all semisimple points, including the bifurcation ones, namely the
semisimple coalescence points of Definition 1.3.
In this section, we fix an open subset Ω ⊆ Mss satisfying the property of Remark 2.24, so
that we can choose and fix on Ω
• an ordering of idempotent vector fields and canonical local coordinates p 7→ u(p), p ∈ Ω,
• a choice of the square roots in the definition of normalized idempotent vector fields fi’s,
and hence a determination of the matrix Ψ.
In this way, system (2.3) and system (2.16) below, are determined. In the idempotent frame
y = Ψζ, (2.15)
28 G. Cotti, B. Dubrovin and D. Guzzetti
system (2.3) becomes∂iy = (zEi + Vi)y,
∂zy =
(
U +
1
z
V
)
y,
(2.16)
where (Ei)
α
β = δαi δ
β
i and
V := ΨµΨ−1, Vi := ∂iΨ ·Ψ−1, (2.17)
U := ΨUΨ−1 = diag(u1, . . . , un),
with not necessarily ui 6= uj when i 6= j. By Lemma 2.29, Ψ(u), V (u) and Vi(u)’s are holomor-
phic on Ω.
Lemma 2.30. The matrix V = ΨµΨ−1 is antisymmetric, i.e., V T + V = 0. Moreover,
if ui = uj, then Vij = Vji = 0.
Proof. Antisymmetry is an easy consequence of (2.5) and the η-orthogonality of Ψ (see [26]).
Moreover, compatibility conditions of the system (2.16) imply that
[Ei, V ] = [Vi, U ].
Reading this equation for entries at place (i, j), we find that
Vij = (uj − ui)(Vi)ij .
Now, (Vi)ij is holomorphic, by Lemma 2.29 and (2.17), so that if i 6= j, but ui = uj , then
Vij = 0. �
We focus on the second linear system
∂zy =
(
U +
1
z
V
)
y, (2.18)
and study it at a fixed point p ∈ Ω.
Theorem 2.31. Let Ω ⊆Mss as in Remark 2.24. At a (fixed) point p ∈ Ω, there exists a unique
formal (in general divergent) series
F (z) := 1 +
∞∑
k=1
Ak
zk
with
FT(−z)F (z) = 1,
such that the transformation ỹ = F (z)y reduces the corresponding system (2.18) at p to the one
with constant coefficients
∂z ỹ = Uỹ.
Hence, system (2.18) has a unique formal solution
Yformal(z) = G(z)ezU , G(z) := F (z)−1 = 1 +
∞∑
k=1
Gk
zk
. (2.19)
Local Moduli of Semisimple Frobenius Coalescent Structures 29
Proof. By a direct substitution, one finds the following recursive equations for the coeffi-
cients Ak:
[U,A1] = V, [U,Ak+1] = AkV − kAk, k = 1, 2, . . . .
If (i, j) is such that ui 6= uj then we can determine (Ak+1)ij by the second equation in terms of
entries of Ak; if ui = uj then we can determine (Ak+1)ij from the successive equation:
[U,Ak+2] = Ak+1V − (k + 1)Ak+1.
Indeed, the (i, j)-entry of the l.h.s. is 0 and, by Lemma 2.30 (Ak+1V )ij is a linear combination
of already determined entries (Ak+1)ih, with ui 6= uh. In such a way we can construct F (z). Let
us now prove that FT(−z)F (z) = 1. Let us take any (formal or analytic) solution Y of the
original system, and pose
A := Y
(
e−iπz
)T
Y (z).
A is a constant matrix, since it does not depend on z. Thus, for an appropriate constant
matrix C we have
F (z)Y (z) = ezUC,
from which we deduce that
F (z)−1 = Y (z)C−1e−zU , F (−z)−T = ezUC−TY (e−iπz)T.
So
F (−z)−TF (z)−1 = ezUC−TAC−1e−zU .
Comparing the constant terms of the expansion of the r.h.s and the l.h.s. we conclude that
C−TAC−1 = 1. �
Notice in the above proof that the equation [U,Ak+1] = AkV −kAk, that is (ui−uj)(Ak+1)ij =
(AkV −kAk)ij , implies that, if we let p vary in Ω then the Gk’s define holomorphic matrix valued
functions Gk(u) at points u, lying in u(Ω), such that ui 6= uj for i 6= j. Accordingly, the formal
matrix solution
Yformal(z, u) = G(z, u)ezU , G(z, u) = 1 +
∞∑
k=1
Gk(u)
zk
, (2.20)
is well defined and holomorphic w.r.t. u = u(p) away from semisimple coalescence points in Ω.
In Theorem 4.5 below, we will show that Yformal(z, u) extends holomorphically also at semisimple
coalescence points.
Remark 2.32. The proof of Theorem 2.31 is based on a simple computation, which holds both
at a coalescence and a non-coalescence semisimple point. The statement can also be deduced
from the more general results of [8] (see also [19]). A similar computation can be found also
in [68] and [33]. Notice however that this computation does not provide any information about
the analiticity of G(u) in case of coalescence ui → uj , i 6= j. The analiticity of Yformal(z, u) –
and of actual fundamental solutions – at a semisimple coalescence point follows from the results
proved in [19], and will be the content of Theorem 4.5 below.
30 G. Cotti, B. Dubrovin and D. Guzzetti
In order to study actual solutions at p ∈ Ω, we introduce Stokes rays. In what follows, we
denote by pr: R → C\{0} the covering map. For pairs (ui, uj) such that ui 6= uj , we locally
choose arguments αij of arg(ui − uj) within the interval [0; 2π[, and we let
τij :=
3π
2
− αij .
Definition 2.33 (Stokes rays). We call Stokes rays of the system (2.18) the rays in the universal
covering R defined by
Rij,k := {z ∈ R : arg z = τij + 2kπ} , k ∈ Z.
The characterisation of Stokes rays is as follows: z ∈ Rij,k if and only if
Re((ui − uj)z) = 0, Im((ui − uj)z) < 0, z ∈ R.
For given 1 ≤ i 6= j ≤ n, the projection of the rays Rij,k, k ∈ Z, on the C-plane
Rij := pr(Rij,k)
does not depend on k and is also called a Stokes ray. It coincides with the ray defined in [26],
namely
Rij = {z ∈ C : z = −iρ(ui − uj), ρ > 0}. (2.21)
Stokes rays have a natural orientation from 0 to ∞. For z ∈ C we have∣∣ezui∣∣ =
∣∣ezuj ∣∣ if z ∈ Rij ,∣∣ezui∣∣ > ∣∣ezuj ∣∣ if z is on the left of Rij ,∣∣ezui∣∣ < ∣∣ezuj ∣∣ if z is on the right of Rij .
Definition 2.34 (admissible rays and line). Let φ ∈ R and let us define the rays in R
`+(φ) := {z ∈ R : arg z = φ},
`−(φ) := {z ∈ R : arg z = φ− π}.
We will say that these rays are admissible at u, for the system (2.18), if they do not coincide with
any Stokes rays Rij,k for any i, j s.t. ui 6= uj and any k ∈ Z. Moreover, a line `(φ) :=
{
z = ρeiφ,
ρ ∈ R
}
of the complex plane, with the orientation induced by R, is called admissible at u for
the system (2.18) if
Re z(ui − uj)|z∈`\0 6= 0
for any i, j s.t. ui 6= uj . In other words, a line is admissible if it does not contain (projected)
Stokes ray Rij .
Notice that the rays pr (`±(φ)) are contained in the line `(φ) =
{
z = ρeiφ, ρ ∈ R
}
, and that
the orientation induced by R is such that the positive part of `(φ) is pr (`+(φ)).
Definition 2.35 (`-chambers). Given a semisimple Frobenius manifold M , and fixed an oriented
line `(φ) = {z = ρeiφ, ρ ∈ R} in the complex plane, consider the open dense subset of points
p ∈M such that
• the eigenvalues of U at p are pairwise distinct,
• the line ` is admissible at u(p) = (u1(p), . . . , un(p)).
We call `-chamber any connected component Ω` of this set.
Local Moduli of Semisimple Frobenius Coalescent Structures 31
The definition is well posed, since it does not depend on the ordering of the idempotents (i.e.,
the labelling of the canonical coordinates) and on the signs in the square roots defining Ψ. Any
`-chamber satisfies the property of Remark 2.24: hence, idempotent vector fields and canonical
coordinates are single-valued and holomorphic on any `-chamber. The topology of an `-chamber
in M can be highly non-trivial (it should not be confused with the simple topology in Cn of an
`-cell of Definition 4.3 below). For example, in [43] the analytic continuation of the Frobenius
structure of the Quantum Cohomology of P2 is studied: it is shown that there exist points
(u1, u2, u3) ∈ C3 with ui 6= uj , which do not correspond to any true geometric point of the
Frobenius manifold. This is due to singularities of the change of coordinates u 7→ t.
Remark 2.36. In [25, Section 3.4], the second author introduced a strictly related notion of
charts of semisimple Frobenius manifolds. Although both definitions of chambers and charts are
subordinate to the choice of an oriented line `, notice some differences between the two concepts.
Basically, `-chambers are a non-coordinatized version of charts. Given a semisimple Frobenius
manifold, its decomposition is intrinsically defined and it depends on the spectrum of U as a set,
without particular reference to any ordering of canonical coordinates.
Conversely, adopting an inverse-problem point of view, as in Section 3.4 of [25], charts are
identified with open sets of n-tuples (u1, . . . , un) ∈ Cn with pairwise distinct values of ui’s in
`-lexicographical order (see Definition 3.1), and in correspondence to which a suitable Riemann-
Hilbert problem is solvable, so that the local Frobenius structure can be reconstructed. Further-
more, it is also required a condition guaranteeing that the changes of coordinates t 7→ u, u 7→ t
are not singular. Note that in both cases (charts or chambers), semisimple coalescence points
are not considered: hence, despite of their name, charts do not really constitute an atlas of the
Frobenius manifold.
For a fixed φ ∈ R, we define the sectors
Πright(φ) := {z ∈ R : φ− π < arg z < φ} ,
Πleft(φ) := {z ∈ R : φ < arg z < φ+ π} .
Theorem 2.37. Let Ω ⊂ Mss be as in Remark 2.24 and let system (2.16) be determined as in
the beginning of this section. Let φ ∈ R be fixed. Then the following statements hold.
1. At any p ∈ Ω such that `(φ) is admissible at u(p) and, for any k ∈ Z there exist two
fundamental matrix solutions Y
(k)
left/right(z) uniquely determined by the asymptotic condition
Y
(k)
left/right(z) ∼ Yformal(z), |z| → ∞, z ∈ e2πikΠleft/right(φ).
2. The above solutions Y
(k)
left/right satisfy
Y
(k)
left/right
(
e2πikz
)
= Y
(0)
left/right(z), z ∈ R. (2.22)
3. In case Ω ≡ Ω` is an `(φ)-chamber if p varies in Ω`, then the solutions Y
(k)
left/right(z) define
holomorphic functions
Y
(k)
left/right(z, u)
w.r.t. u = u(p). Moreover, the asymptotic expansion
Y
(k)
left/right(z, u) ∼ Yformal(z, u), |z| → ∞, z ∈ e2πikΠleft/right(φ), (2.23)
holds uniformly in u corresponding to p varying in Ω`. Here Yformal(z, u) is the u-holo-
morphic formal solution (2.20).
32 G. Cotti, B. Dubrovin and D. Guzzetti
R R
Πleft
Πright
`−(φ) `−(φ)
`+(φ+ 2π)
Figure 1. The figure shows Πε
right(φ),Πε
left(φ) as dashed sectors, `±(φ) in (black) and Stokes rays (in
color).
Proof. The proof of (1) and (2) away from coalescence points is standard (see [7, 26, 27, 73]),
while at coalescence points it follows from the results of [19] and [8]. Point (3) is stated in
[26, 27], though the name “`-chamber” does not appear there. �
Remark 2.38. The holomorphic properties at point (3) of Theorem 2.37 hold in a `-chamber,
where there are no coalescence points. In our Theorem 4.5 below, we will see that point (3)
actually holds in a set Ω ⊂ Mss as in Remark 2.24, no matter whether it contains semisimple
coalescence points or not. The only requirement is that `(φ) is admissible at u = u(p) for any
p ∈ Ω.
Remark 2.39. The asymptotic relation (2.23) means that for any compact K b Ω`, for any
h ∈ N and for any proper closed subsector S ( e2πikΠright/left(φ) there exists CK,h,S > 0 such
that, if z ∈ S \ {0}, then
sup
u∈K
∥∥∥∥∥Y (k)
right/left(z, u) · exp(−zU)−
h−1∑
m=0
Gm(u)
zm
∥∥∥∥∥ < CK,h,S
|z|h
.
Actually, the solutions Y
(k)
right/left(z, u) maintain their asymptotic expansions (2.23) in sectors
wider than e2πikΠright/left(φ) after extending at least up to the nearest Stokes rays outside
e2πikΠright/left(φ). In particular, for any p ∈ K b Ω` and suitably small ε = ε(K) > 0, then the
asymptotics holds in e2πikΠε
right/left(φ), where
Πε
right(φ) := {z ∈ R : φ− π − ε < arg z < φ+ ε} ,
Πε
left(φ) := {z ∈ R : φ− ε < arg z < φ+ π + ε} .
The positive number ε is chosen small enough in such a way that, as p varies in the compact
set K, no Stokes ray is contained in the following sectors:
Πε
+(φ) := {z ∈ R : φ− ε < arg z < φ+ ε},
Πε
−(φ) := {z ∈ R : φ− π − ε < arg z < φ− π + ε}.
Lemma 2.40. In the assumptions of Theorem 2.37, for any k ∈ Z and any z ∈ R the following
orthogonality relation holds:
Y
(k)
left
(
eiπz
)T
Y
(k)
right(z) = 1.
Local Moduli of Semisimple Frobenius Coalescent Structures 33
Proof. From Remark 2.4 we already know that the product above is independent of z ∈ R.
According to Remark 2.39, if ε > 0 is a sufficiently small positive number, then
Y
(k)
left/right(z) ∼ Yformal(z), |z| → ∞, z ∈ e2πikΠε
left/right(φ).
Consequently,
Y
(k)
left
(
eiπz
)
∼ G(−z)e−zU , Y
(k)
right(z) ∼ G(z)ezU , |z| → ∞, z ∈ e2πikΠε
+(φ).
Thus, Y
(k)
left
(
eiπz
)T
Y
(k)
right(z) = 1 for all z ∈ e2πikΠε
+(φ), and by analytic continuation for all
z ∈ R. �
Let Y0(z, u) be a fundamental solution of (2.18) near z = 0 of the form (1.8), i.e.,
Y0(z, u) = Ψ(u)Φ(z, u)zµzR, Φ(z, u) = 1 +
∞∑
k=1
Φk(u)zk,
Φ(−z, u)TηΦ(z, u) = η, (2.24)
with ΨTΨ = η, obtained from (2.10) and (2.11) through the constant gauge (2.15). This solution
is not affected by coalescence phenomenon and since µ and R are independent of p ∈ Ω, it is
holomorphic w.r.t. u (see [26, 27]). Recall that Y0(z, u) is not uniquely determined by the choice
of R.
Definition 2.41 (Stokes and central connection matrices). Let Ω ⊂Mss be as in Remark 2.24
and let the system (2.16) be determined as in the beginning of this section. Let φ ∈ R be fixed.
Let p ∈ Ω be such that `(φ) is admissible at u(p). Finally, let Y
(0)
right/left(z) be the fundamental
solutions of Theorem 2.37 at p. The matrices S and S− defined at u(p) by the relations
Y
(0)
left (z) = Y
(0)
right(z)S, z ∈ R, (2.25)
Y
(0)
left
(
e2πiz
)
= Y
(0)
right(z)S−, z ∈ R
are called Stokes matrices of the system (2.18) at the point p w.r.t. the line `(φ). The matrix C
such that
Y
(0)
right(z) = Y0(z, u(p))C, z ∈ R
is called central connection matrix of the system (2.16) at p, w.r.t. the line ` and the fundamental
solution Y0.
Theorem 2.42. The Stokes matrices S, S− and the central connection matrix C of Defini-
tion 2.41 at a point p ∈ Ω satisfy the following properties, for all k ∈ Z and all z ∈ R:
1.
Y
(k)
left (z) = Y
(k)
right(z)S,
Y
(k)
left (z) = Y
(k+1)
right (z)S−,
Y
(k)
right(z) = Y0(z, u(p)) M−k0 C,
where M0 = exp(2πiµ) exp(2πiR);
2.
Y
(k)
right
(
e2πiz
)
= Y
(k)
right(z)S−S
−1,
Y
(k)
left
(
e2πiz
)
= Y
(k)
right(z)S
−1S−;
34 G. Cotti, B. Dubrovin and D. Guzzetti
3.
S− = ST,
Sii = 1, i = 1, . . . , n,
Sij 6= 0 with i 6= j only if ui 6= uj and Rij ⊂ pr(Πleft(φ)).
Proof. The first and second identities of (1) follow from equation (2.22). For the third note
that
Y
(k)
right(z) = Y
(0)
right
(
e−2ikπz
)
= Y0
(
e−2ikπz
)
C = Y0(z)M−k0 C.
Point (2) follows easily from the vanishing of the exponent of formal monodromy (diag V = 0).
By definition of Stokes matrices we have that
Y
(0)
left
(
eiπz
)
= Y
(0)
right
(
e−iπz
)
S−, Y
(0)
right(z) = Y
(0)
left (z)S−1,
and by Lemma 2.40
ST
− Y
(0)
right(z)
TY
(0)
left
(
eiπz
)︸ ︷︷ ︸
1
S−1 ≡ 1.
We conclude ST
− = S. If we consider the sector Πε
+(φ) for sufficiently small ε > 0 as in proof of
Lemma 2.40, them from the relation Y
(0)
left (z) = Y
(0)
right(z)S, we deduce that
ez(ui−uj)Sij ∼ δij , |z| → ∞, z ∈ Πε
+(φ).
So, if ui = uj we deduce Sij = δij . If i 6= j are such that ui 6= uj , then if Rij ⊂ pr(Πright(φ)) we
have ∣∣ez(ui−uj)∣∣→∞ for |z| → ∞, z ∈ Πε
+(φ),
and hence necessarily Sij = 0. For the opposite ray Rji ⊂ pr (Πleft) we have∣∣ez(ui−uj)∣∣→ 0 for |z| → ∞, z ∈ Πε
+(φ),
so Sij need not to be 0. This proves (3). �
The monodromy data must satisfy some important constraints, summarised in the following
theorem, whose proof is omitted in [25, 26].
Theorem 2.43. The monodromy data µ, R, S, C at a point p ∈ Ω as in Definition 2.41 satisfy
the identities:
1) CSTS−1C−1 = M0 = e2πiµe2πiR,
2) S = C−1e−πiRe−πiµη−1
(
CT
)−1
,
3) ST = C−1eπiReπiµη−1
(
CT
)−1
.
Proof. The first identity has a simple topological motivation: loops around the origin in C∗
are homotopic to loops around infinity. So, one easily obtains the relation using Theorem 2.42,
and the definition of central connection matrix. Using the orthogonality relations for solutions,
equation (2.9) and the fact that
zµ
T
ηzµ = η
Local Moduli of Semisimple Frobenius Coalescent Structures 35
(µ being diagonal and η-antisymmetric), we can now prove the identities (2) and (3). By
Lemma 2.40 we have that
1 = Y
(0)
right(z)
TY
(0)
left
(
eiπz
)
= Y
(0)
right(z)
TY
(0)
right
(
eiπz
)
S = CTY0(z)TY0
(
eiπz
)
CS.
Now we have
Y0(z)TY0
(
eiπz
)
= zR
T
zµ
T(
Φ(z)TΨTΨΦ(−z)
)(
eiπz
)µ(
eiπz
)R
= zR
T
zµ
T
ηzµeiπµzReiπR = ηeiπµeiπR.
This shows the first identity. For the second one, we have that
1 = Y
(0)
right(z)
TY
(0)
left
(
eiπz
)
= Y
(0)
right(z)
TY
(0)
right
(
e−iπz
)
ST = CTY0(z)TY0
(
e−iπz
)
CST.
Again, we have
Y0(z)TY0
(
e−iπz
)
= zR
T
zµ
T(
Φ(z)TΨTΨΦ(−z)
)(
e−iπz
)µ(
e−iπz
)R
= zR
T
zµ
T
ηzµe−iπµzRe−iπR = ηe−iπµe−iπR. �
It follows from point (3) of Theorem 2.37 that S and C depend holomorphically on p varying in
an `-chamber Ω`, namely they define analytic matrix valued functions S(u) and C(u), u = u(p).
Moreover, due to the compatibility conditions [Ei, V ] = [Vi, U ] and ∂iΨ = ViΨ, the system (2.16)
is isomonodromic. Therefore ∂iS = ∂iC = 0. Indeed, the following holds:
Theorem 2.44 (isomonodromy Theorem, II, [23, 25, 26]). The Stokes matrix S and the central
connection matrix C, computed w.r.t. a line `, are independent of p varying in an `-chamber.
The values of S, C in two different `-chambers are related by an action of the braid group of
Section 3.
3 Ambiguity in definition of monodromy data
and braid group action
In associating the data (µ,R, S,C) to p ∈M several choices have been done, all preserving the
constraints of Theorem 2.43
S = C−1e−iπRe−iπµη−1
(
C−1
)T
,
ST = C−1eiπReiπµη−1
(
C−1
)T
.
While the operator µ is completely fixed by the choice of flat coordinates as in Section 2,
R is determined only up to conjugacy class of the (η, µ)-parabolic orthogonal group G(η, µ)
as in Theorem 2.14. Suppose now that R has been chosen in this class. The remaining local
invariants S, C are subordinate to the following choices:
1) an oriented line `(φ) =
{
z = ρeiφ, ρ ∈ R
}
in the complex plane;
2) for given φ ∈ R, the change φ 7→ φ − 2kπ, k ∈ Z, or dually, for fixed φ, the change
Y
(0)
left/right(z) 7→ Y
(k)
left/right(z);
3) the choice of an ordering of canonical coordinates on each `-chamber Ω`;
4) the choice of the branches of the square roots (2.14) defining the matrix Ψ on each `-
chamber Ω`;
5) the choice of solution Y0 in the Levelt normal form corresponding to the same exponent R.
36 G. Cotti, B. Dubrovin and D. Guzzetti
The transformations of the data depending on the choice of ` in (1) will be studied in the next
Section. Here we describe how the freedoms (2), (3), (4) and (5) affect the data (S,C):
• Action of the additive group Z: according to formula (2.22), S remains invariant and
C 7→M−k0 · C, k ∈ Z, M0 = e2πiµe2πiR, t ∈ Ω`.
• Action of the group of permutations Sn: if τ is a permutation, we can reorder the canonical
coordinates:
(u1, . . . , un) 7→ (uτ(1), . . . , uτ(n)).
The system (2.18) is changed to U 7→ PUP−1 = diag(uτ(1), . . . , uτ(n)), V 7→ PV P−1.
The fundamental matrices change as follows: Y
(0)
left/right 7→ PY
(0)
left/rightP
−1 and Y0 7−→ PY0.
Therefore
S 7→ PSP−1, C 7→ CP−1. (3.1)
• Action of the group (Z/2Z)×n: by changing signs of the normalized idempotents (matrix Ψ)
we change the signs of the entries of the matrices S and C. If I is a diagonal matrix with 1’s
or (−1)’s on the diagonal, the system (2.18) is changed to U 7→ IUI ≡ U , V 7→ IV I.
Correspondingly, Yleft/right 7→ IYleft/rightI, Y0 7→ IY0. Therefore
S 7→ ISI, C 7→ CI.
• Action of the group C0(η, µ,R): for chosen R, the choice of a fundamental system at
the origin having the form (2.24) is defined up to Y0 7→ Y0G, where G ∈ C0(η, µ,R) of
Definition 2.8. The corresponding left action on C is
C 7−→ GC, G ∈ C0(η, µ,R).
Among all possible orderings of the canonical coordinates, a particularly useful one is the
lexicographical order w.r.t. an admissible line `(φ), defined as follows. Let us introduce the
following rays in the complex plane:
Lj(φ) :=
{
uj + ρei(π2−φ) : ρ ∈ R+
}
, j = 1, . . . , n. (3.2)
Each of them starts from the point uj and is considered to be oriented from uj to ∞.
Definition 3.1 (lexicographical order). The canonical coordinates uj ’s are in `-lexicographical
order if Lj(φ) is to the left of Lk(φ) (w.r.t. the above orientation), for any 1 ≤ j < k ≤ n.
If u1, . . . , un are in lexicographical order w.r.t. the admissible line `(φ), then:
1) the Stokes matrix is in upper triangular form,
2) Ri,j ⊆ pr (Πleft(φ)) if and only if i < j,
3) the nearest Stokes rays to the positive half-line pr(`+(φ)) are of the form
Ri,i+1 ⊆ pr(Πleft(φ)), Rj,j−1 ⊆ pr(Πright(φ)),
where 1 ≤ i ≤ n− 1 and 2 ≤ j ≤ n.
Local Moduli of Semisimple Frobenius Coalescent Structures 37
In general, condition (1) alone does not imply that the canonical coordinates are in lexico-
graphical order: it does if and only if the number of nonzero entries of the Stokes matrix S is
maximal (and equal to n(n+1)
2 ). In this case, by Theorem 2.42, necessarily ui 6= uj for i 6= j.
On the other hand, if there are some vanishing entries Sij = Sji = 0 for i 6= j, and S is upper
triangular, then also PSP−1 in (3.1) is upper triangular for any permutation exchanging ui
and uj corresponding to Sij = Sji = 0. For example, this happens at a coalescence point: by
Theorem 2.42, the entries Sij with i 6= j are 0 corresponding to coalescing values ui = uj , i 6= j.
Definition 3.2 (triangular order). We say that u1, . . . , un are in triangular order w.r.t. the line
` whenever S is upper triangular.
It follows from the preceding discussion that at a semisimple coalescence point there are
more than one triangular orders. Moreover, any of them is also lexicographical. For further
comments, see Remark 4.6.
3.1 Action of the braid group Bn
In this section, canonical coordinates are pairwise distinct, corresponding to a non-coalescence
semisimple points lying in `-chambers. The braid group is
Bn = π1
((
Cn \∆
)
/Sn
)
,
where ∆ stands for the union of all diagonals in Cn. It is generated by n− 1 elementary braids
β12, β23, . . . , βn−1,n, with the relations
βi,i+1βj,j+1 = βj,j+1βi,i+1 for i+ 1 6= j, j + 1 6= i,
βi,i+1βi+1,i+2βi,i+1 = βi+1,i+2βi,i+1βi+1,i+2.
Any braid in Bn is a product of the generators β12, β23, . . . , βn−1,n and their inverses.
The action of the braid group Bn on the monodromy data manifests whenever some Stokes
ray and the chosen line ` cross under rotation. This can happen in two ways:
• First: we let vary the point of the Frobenius manifold at which we compute the data,
keeping fixed the line `; this is the case if, starting from the data computed in an `-chamber
we want to compute the data in a neighboring `-chamber, or even more in general if we
want to analyze properties of the analytic continuation of the whole Frobenius structure
by letting varying the coordinates (u1, . . . , un) on the universal cover C̃n \∆.
• Second: we fix the point at which we compute the data and change the admissible line `
by a rotation.
In the first case the `-chambers are fixed, in the second case they change: indeed, the given point
of the Frobenius manifold is in two different chambers before and after the rotation of `. In
both cases, we will always label the canonical coordinates (u1, . . . , un) in lexicographical order
w.r.t. ` both before and after the transformation (so that, in particular, any Stokes matrix is
always in upper triangular form).
Any continuous deformation of the n-tuple (u1, . . . , un), represented as a deformation of n
points in C never colliding, can be decomposed into elementary ones. If we restrict to the case of
a continuous deformation which ends exactly with the same initially ordered pattern of points,
then we can identify an elementary deformation with a generator of the pure braid group, i.e.,
π1
(
Cn \∆
)
. Otherwise, by allowing permutations, we can identify an elementary deformation
with a generator of the braid group Bn. In particular, an elementary deformation which will
be denoted by βi,i+1 consists in a counter-clockwise rotation of ui w.r.t. ui+1, so that the two
exchange. All other points uj ’s are subjected to a sufficiently small perturbation, so that the
corresponding Stokes’ rays almost do not move. βi,i+1 corresponds to
38 G. Cotti, B. Dubrovin and D. Guzzetti
• clockwise rotation of the Stokes’ ray Ri,i+1 crossing the line `,
• or, dually, counter-clockwise rotation of the line ` crossing the Stokes’ ray Ri,i+1.
This determines the following mutations of the monodromy data, as shown in [23] and [26]. For
the deformation of ui, ui+1 relatively moving anticlockwise, associated with βi,i+1, we have
S 7→ Sβi,i+1 := Aβi,i+1(S)SAβi,i+1(S)T, (3.3)
where(
Aβi,i+1(S)
)
hh
= 1, h = 1, . . . , n h 6= i, i+ 1,(
Aβi,i+1(S)
)
i+1,i+1
= −si,i+1,
(
Aβi,i+1(S)
)
i,i+1
=
(
Aβi,i+1(S)
)
i+1,i
= 1,
and all the other entries are zero. For the inverse braid β−1
i,i+1 (ui and ui+1 move clockwise) the
mutation is Sβ
−1
i,i+1 := Aβ
−1
i,i+1(S)SAβ
−1
i,i+1(S)T, where(
Aβ
−1
i,i+1(S)
)
hh
= 1, h = 1, . . . , n, h 6= i, i+ 1,(
Aβ
−1
i,i+1(S)
)
i,i
= −si,i+1,
(
Aβ
−1
i,i+1(S)
)
i,i+1
=
(
Aβ
−1
i,i+1(S)
)
i+1,i
= 1,
and all the other entries are zero. For a generic braid β = β±1
i1,i1+1 · β
±1
i2,i2+1 · · ·β
±1
iN ,iN+1, which is
a product of N ≥ 1 elementary braids or their inverses, the action is
S 7→ Sβ = Aβ(S)S
[
Aβ(S)
]T
. (3.4)
We remark that Sβ is still upper triangular. The action on the central connection matrix (in
lexicographical order) is
C 7→ Cβ := CAβ(S)−1. (3.5)
The matrix Aβ(S) is constructed successively applying (3.3), as follows:
Aβ(S) = A
β±1
iN ,iN+1
(
S
β±1
i1,i1+1·β
±1
i2,i2+1···β
±1
iN−1,iN−1+1
)
· · ·Aβ
±1
i2,i2+1
(
S
β±1
i1,i1+1
)
·Aβ
±1
i1,i1+1(S),
where
S
β±1
i1,i1+1 = A
β±1
i1,i1+1(S)SA
β±1
i1,i1+1(S)T,
S
β±1
i1,i1+1·β
±1
i2,i2+1 =
[
A
β±1
i2,i2+1
(
S
β±1
i1,i1+1
)]
S
β±1
i1,i1+1
[
A
β±1
i2,i2+1
(
S
β±1
i1,i1+1
)]T
,
S
β±1
i1,i1+1·β
±1
i2,i2+1·β
±1
i3,i3+1 =
[
A
β±1
i3,i3+1
(
S
β±1
i1,i1+1·β
±1
i2,i2+1
)]
S
β±1
i1,i1+1·β
±1
i2,i2+1
×
[
A
β±1
i3,i3+1
(
S
β±1
i1,i1+1·β
±1
i2,i2+1
)]T
,
and so on.
Now, let us consider a complete counter-clockwise 2π-rotation of the admissible line `, and
observe the following:
1. In the generic case (i.e., when the canonical coordinates uj ’s are in general position) there
are n(n − 1) distinct projected Stokes’ rays Rjk. An elementary braid acts any time the
line ` crosses a Stokes ray. So, in total, we expect that a complete rotation of ` correspond
to the product of n(n− 1) elementary braids βi,i+1’s.
Local Moduli of Semisimple Frobenius Coalescent Structures 39
2. Since the formal monodromy is vanishing, the effect of the rotation of ` on the Stokes matrix
is trivial, while the central connection matrix C is transformed to M−1
0 C, M0 being the
monodromy at the origin (point (1) of Theorem 2.42). As a consequence, the complete
rotation of the line ` can be viewed as a deformation of points uj ’s commuting with any
other braid.
From point (2) we deduce that the braid corresponding to the complete rotation of ` is an
element of the center
Z(Bn) =
{
(β12β23 · · ·βn−1,n)kn : k ∈ Z
}
.
From point (1) and from the fact that ` rotates counter-clockwise we deduce the following
Lemma 3.3. The braid corresponding to a complete counter-clockwise 2π-rotation of ` is
(β12β23 · · ·βn−1,n)n,
and its acts on the monodromy data as follows:
• trivially on Stokes matrices,
• the central connection matrix is transformed as C 7→M−1
0 C.
4 Isomonodromy theorem at coalescence points
So far the monodromy data, S and C have been defined pointwise and then the deformation theo-
ry has been described at point (3) of Theorem 2.37 and in Theorem 2.44, away from coalescence
points. In particular, S and C are constant in any `-chamber, and the matrices Y
(k)
left/right(z, u)
are u-holomorphic in all `-chambers. In this section we generalize the deformation theory to
semisimple coalescence points. We show that the monodromy data, which are well defined at
a coalescence point, actually provide the monodromy data in a neighborhood of the point, and
can be extended to the whole Frobenius manifold through the action of the braid group. In
this section we will use the following notation for objects computed at a coalescence point:
a matrix Y , S or C will be denoted Y̊ , S̊ or C̊.
Let p0 ∈ BM \KM be a semisimple coalescence point. Consider a neighbourhood Ω ⊆M \KM
of p0, satisfying the property of Remark 2.24. An ordering for canonical coordinates (u1, . . . , un)
and a holomorphic branch of the function Ψ: Ω → GLn(C) can be chosen in Ω. We denote by
u(p) := (u1(p), . . . , un(p)) the value of the canonical coordinate map u : Ω→ Cn, and we define
∆Ω :=
{
u(p) = (u1(p), . . . , un(p)) ∈ Cn
∣∣ p ∈ Ω ∩ BM
}
.
Therefore, if u ∈ ∆Ω, then ui = uj for some i 6= j. The coordinates u(p0) of p0 will be denoted
u(0) =
(
u
(0)
1 , . . . , u
(0)
n
)
. ∆Ω is not empty and contains u(0). Let r1, . . . , rs be the multiplicities of
the eigenvalues of U
(
u(0)
)
= diag
(
u
(0)
1 , . . . , u
(0)
n
)
, with s < n, r1+· · ·+rs = n. By a permutation
of (u1, . . . , un), there is no loss in generality (cf. Section 3) if we assume that the entries of u(0)
are
u
(0)
1 = · · · = u(0)
r1 =: λ1,
u
(0)
r1+1 = · · · = u
(0)
r1+r2 =: λ2,
· · · · · · · · · · · · · · · · · · · · · · · · · · ·
u
(0)
r1+···+rs−1+1 = · · · = u
(0)
r1+···+rs−1+rs−1 = u(0)
n =: λs, (4.1)
40 G. Cotti, B. Dubrovin and D. Guzzetti
λk 6= λl for k 6= l. Let
δi :=
1
2
min
{∣∣λi − λj + ρe
√
−1(π2−φ)
∣∣, j 6= i, ρ ∈ R
}
,
and let ε0 be a small positive number such that
ε0 < min
1≤i≤s
δi. (4.2)
Notice that δi > 0, because φ is admissible. We will assume that ε0 is sufficiently small so that
the polydisc at u(0), defined by13
Uε0
(
u(0)
)
:=
s
×
i=1
B(λi; ε0)×ri ,
is completely contained in the image u(Ω) of the chart Ω.
Lemma 4.1. For ε0 satisfying (4.2), if u varies in Uε0
(
u(0)
)
, the sets
I1 := {u1, . . . , ur1}, I2 := {ur1+1, . . . , ur1+r2}, . . . ,
Is := {ur1+···+rs−1+1, . . . , ur1+···+rs−1+rs} (4.3)
do never intersect. Thus, u(0) is a point of maximal coalescence in Uε0
(
u(0)
)
. We will say that
a coordinate ua is close to λj if it belongs to Ij, which is to say that ua ∈ B(λj ; ε0).
Let us fix φ ∈ R so that the line `+(φ), `−(φ), `(φ) are admissible at p0 (Definition 2.34). For
u ∈ Uε0
(
u(0)
)
, with ε0 as in (4.2), consider the subset R(u) of Stokes rays Rab,k in the universal
covering R which are associated with all couples of eigenvalues ua and ub such that ua is close
to a λi and ub is closed to λj for some i 6= j. Then, the following holds:
Lemma 4.2. Let ε0 be as in (4.2). If ua varies in B(λi; ε0) and ub in B(λj ; ε0), then the rays
Rab,k ∈ R(u) continuously rotate, but they never cross `+(φ) and `−(φ). In other words, the
projections Rab = pr(Rab,k) never cross `(φ) in C.
The choice of the line `, admissible at p0, induces a cell decomposition of Uε0
(
u(0)
)
, according
to the following
Definition 4.3. Let ` be admissible at u(0). An `-cell of Uε0
(
u(0)
)
is any connected component
of the open dense subset of points u ∈ Uε0
(
u(0)
)
such that u1, . . . , un are pairwise distinct and `
is admissible at u.
Proposition 4.4 ([19]). An `-cell is homeomorphic to a ball.
We notice that, if u(p) is in a `-cell then p lies in an `-chamber. Thus, if D is an open subset
whose closure is contained in a cell of Uε0
(
u(0)
)
, according to Theorem 2.37(3), the system
dY
dz
=
(
U +
V (u)
z
)
Y, (4.4)
for u ∈ D admits two fundamental solutions Y
(0)
right/left(z, u) uniquely determined by the canon-
ical asymptotic representation Y
(0)
right/left(z, u) ∼ Yformal(z, u) as in (2.20) valid in the sectors
13 HereB(λi; ε0) is the closed ball in C with center λi and radius ε0. Note that if the uniform norm |u| = maxi |ui|
is used, as in [19], then Uε0
(
u(0)
)
=
{
u ∈ Cn
∣∣ ∣∣u− u(0)
∣∣ ≤ ε0}.
Local Moduli of Semisimple Frobenius Coalescent Structures 41
δi
δj
λi
λj
λi
δi
δj
ua
ub
B(λi; ε0)
B(λj ; ε0)
λj
ua
ub
B(λj ; ε0)
B(λi; ε0)
Figure 2. Points λi, λj and ua, ub are represented on the same complex plane. The thick line has slope
π/2−φ. As u varies, for values of ε0 sufficiently small (left figure) the Stokes rays Rab and Rba associated
with ua in the disk B(λi; ε0) and ub in the disk of B(λj ; ε0) do not cross the line `. If the disks have
radius exceeding min1≤i≤s δi as in (4.2) (see right figure) then the Stokes rays Rab, Rba cross the line `.
Πleft/right(φ) respectively. It follows from the proof of Theorem 2.31 that Yformal(z, u) is u-
holomorphic in Uε0
(
u(0)
)
\∆Ω. By Remark 2.39 actually the asymptotic representation is valid
in wider sectors Sleft/right(u), defined as the sectors which contain Πleft/right(φ) and extends up
to the nearest Stokes rays. By Theorem 2.44 the above system with u ∈ D is isomonodromic,
so that the Stokes matrix S defined in formula (2.25) is constant.
Let us now turn our attention to the coalescence point u(0). From the results of [19] – and
more generally in [8] – it follows that there are a unique formal solution at u(0),
Y̊formal(z) =
(
1 +
∞∑
k=1
G̊k
zk
)
ezU , (4.5)
and unique actual solutions Y̊
(0)
left (z) and Y̊
(0)
right(z), with asymptotic representation given by
Y̊formal(z) in Πleft/right, and in wider sectors Sleft
(
u(0)
)
and Sright
(
u(0)
)
respectively. The Stokes
matrices of Y̊
(0)
right(z) and Y̊
(0)
left (z) are defined by
Y̊
(0)
left (z) = Y̊
(0)
right(z)S̊, Y̊
(0)
left
(
e2πiz
)
= Y̊
(0)
right(z)S̊−, S̊− = S̊T.
A priori, the following problems could emerge.
1. The asymptotic representations
Y
(0)
left/right(z, u) ∼ Yformal(z, u), for |z| → ∞ and
z ∈
⋂
u∈D
Sleft/right(u) ) Πleft/right(φ)
do no longer hold for u outside the cell containing D.
2. The coefficients Gk(u)’s of (2.20) may be divergent at ∆Ω.
3. The locus ∆Ω is expected to be a locus of singularities for the solutions Yformal(z, u) in (2.20)
and Y
(0)
left/right(z, u). Yformal(z, u).
4. The Stokes matrix S may differ from S̊.
We notice that the system (4.4) at u(0) also has a fundamental solution in Levelt form at
z = 0,
Y̊0(z) = Ψ
(
u(0)
)
(I +O(z))zµzR̊, (4.6)
42 G. Cotti, B. Dubrovin and D. Guzzetti
with a certain exponent R̊. Hence, a central connection matrix C̊ is defined by
Y̊
(0)
right(z) = Y̊0(z)C̊.
We recall that Ψ(u) is holomorphic in the whole Uε0
(
u(0)
)
, so that Vij(u) vanishes along ∆Ω
whenever ui = uj (see Lemma 2.30). These are sufficient conditions to apply the main theo-
rem of [19], adapted and particularised to the case of Frobenius manifolds, which becomes the
following:
Theorem 4.5. Let M be a semisimple Frobenius manifold, p0 ∈ BM \ KM and Ω ⊆ Mss =
M \ KM an open connected neighborhood of p0 with the property of Remark 2.24 on which
a holomorphic branch for canonical coordinates u : Ω→ Cn and Ψ: Ω→ GLn(C) has been fixed.
Let ε0 be a real positive number as above, and consider the corresponding neighborhood Uε0
(
u(0)
)
of u(0) = u(p0). Then
1. The coefficients Gk(u), k ≥ 1, in (2.20) are holomorphic over Uε0
(
u(0)
)
,
Gk
(
u(0)
)
= G̊k and Yformal
(
z, u(0)
)
= Y̊formal(z).
2. Y
(0)
left (z, u), Y
(0)
right(z, u), can be u-analytically continued as single-valued holomorphic func-
tions14 on Uε0
(
u(0)
)
. Moreover
Y
(0)
left/right
(
z, u(0)
)
= Y̊
(0)
left/right(z).
3. For any solution Y̊0(z) as in (4.6) there exists a fundamental solution Y0(z, u) in Levelt
form (2.24) such that
Y0
(
z, u(0)
)
= Y̊0(z), R = R̊.
4. For any positive ε1 < ε0, the asymptotic relations
Y
(0)
left/right(z, u) ∼
(
1 +
∞∑
k=1
Gk(u)
zk
)
ezU , z →∞ in Πleft/right(φ), (4.7)
hold uniformly in u ∈ Uε1
(
u(0)
)
. In particular they hold also at points of ∆Ω ∩ Uε1
(
u(0)
)
and at u(0).
5. For any u ∈ Uε0
(
u(0)
)
consider the sectors Ŝright(u) and Ŝleft(u) which contain the sectors
Πright(φ) and Πleft(φ) respectively, and extend up to the nearest Stokes rays in the set R(u)
defined above. Let
Ŝleft/right =
⋂
u∈Uε0
(
u(0)
) Ŝleft/right(u).
Observe that for sufficiently small ε > 0 the sectors
Πε
right(φ) := {z ∈ R : φ− π − ε < arg z < φ+ ε},
Πε
left(φ) := {z ∈ R : φ− ε < arg z < φ+ π + ε}
are strictly contained in Ŝright and Ŝleft respectively. Then, the asymptotic relations (4.7)
actually hold in the sectors Ŝleft/right.
14Hence, they are holomoprhic on R× Uε0
(
u(0)
)
.
Local Moduli of Semisimple Frobenius Coalescent Structures 43
6. The monodromy data µ, R, C, S of system (4.4), defined and constant in an open subset D
of a cell of Uε0
(
u(0)
)
, are actually defined and constant at any u ∈ Uε1
(
u(0)
)
, namely the
system is isomonodromic in Uε1
(
u(0)
)
. They coincide with the data µ, R̊, C̊, S̊ associated
with the fundamental solutions Y̊left/right(z) and Y̊0(z) of system (4.4) at u(0). The entries
of S = (Sij)
n
i,j=1 satisfy the vanishing condition (1.14), namely
Sij = Sji = 0 for all i 6= j such that u
(0)
i = u
(0)
j . (4.8)
This Theorem allows us to obtain the monodromy data µ, R, C, S in a neighbourhood of
a coalescence point just by computing them at the coalescence point, namely just by computing
µ, R̊, C̊, S̊. Its importance has been explained in the Introduction and will be illustrated in
subsequent sections.
Remark 4.6. Suppose that S is upper triangular. By formula (4.8) it follows that in any `-cell
of Uε0
(
u(0)
)
the order of the canonical coordinates is triangular, according to Definition 3.2, and
at most in one cell the order is lexicographical (Definition 3.1). Moreover, if λ1, . . . , λs are in
lexicographical order, then the order of canonical coordinates is lexicographical in exactly one
cell.
4.1 Reconstruction of monodromy data of the whole manifold
The monodromy data of the Frobenius manifold can be obtained from those computed in The-
orem 4.5 around u(0). Without loss of generality, let us suppose that the ordering (4.1) is such
that λ1, . . . , λs are in `- lexicographical order. Then, the matrix S computed at the coalescence
point u(0) is upper triangular. Therefore, by Theorem 4.5, the matrix is constant and upper
triangular in the whole polydisc Uε0
(
u(0)
)
. In particular, it is upper triangular in every cell
of Uε1
(
u(0)
)
. This means that u1, . . . , un are in triangular order (Definition 3.2) in each such
cell, and in particular they are in lexicographical order in only one of these cells (Definition 3.1,
Remark 4.6). Note that any permutation of canonical coordinates preserving the sets I1, . . . , Is
of (4.3) maintains the upper triangular structure of S, namely the triangular order of u1, . . . , un
in each cell of Uεo
(
u(0)
)
. The permutation changes the cell where the order is lexicographical.
Now, each cell of the polydisc Uε0
(
u(0)
)
is contained in a chamber of the manifold (identifying
coordinates with points of the manifold, which is possible because of the holomorphy of canoni-
cal coordinates near semisimple coalescent points). Let us start from the cell of Uε0
(
u(0)
)
where
u1, . . . , un are in lexicographical order. The monodromy data of Theorem 4.5 in this cell are
the constant data of the chamber containing the cell (Theorems 2.16 and 2.44). Since in this
chamber u1, . . . , un are in lexicographical order (and distinct!), we can apply the action of the
braid group to S and C, as dictated by formulae (3.3), (3.5). In this way, the monodromy data
for any other chamber of the manifold are obtained, as explained in Section 3.
5 First detailed example of application of Theorem 4.5:
the A3 Frobenius manifold. Stokes phenomenon
for Pearcey-type oscillating integrals from Hankel functions
With the example of A3 Frobenius manifold below, we show how Theorem 4.5 allows the compu-
tation of monodromy data in an elementary way, by means of Hankel special functions. Moreover,
we apply the results of Section 3, especially showing how the braid group can be used to re-
construct the data for the whole manifold, starting from a coalescence point. The reader not
interested in a general introduction to Frobenius manifolds associated with singularity theory
may skip Sections 5.1 and 5.2 and go directly to Section 5.3.
44 G. Cotti, B. Dubrovin and D. Guzzetti
Singularity Versal Deformation
An f(x) = xn+1 f(x, a) = xn+1 + an−1x
n−1 + · · ·+ a1x+ a0
Dn f(x) = xn−1
1 + x1x
2
2 f(x, a) = xn−1
1 + x1x
2
2 + an−1x
n−2
1 + · · ·+ a1 + a0x2
E6 f(x) = x4
1 + x3
2 f(x, a) = x4
1 + x3
2 + a6x
2
1x2 + a5x1x2
+ a4x
2
1 + a3x2 + a2x1 + a1
E7 f(x) = x3
1x2 + x3
2 f(x, a) = x3
1x2 + x3
2 + a7x2 + a6x
2
1 + a5x1
+ a4x1x2 + a3x1x2 + a2x2 + a1
E8 f(x) = x5
1 + x3
2 f(x, a) = x5
1 + x3
2 + a8x
3
1x2 + a7x
2
1x2 + a6x
3
1
+ a5x1x2 + a4x
2
1 + a3x2 + a2x1 + a1
Table 1. Arnol’d’s classification of simple singularities, and their corresponding miniversal deformations.
5.1 Singularity theory and Frobenius manifolds
Let f be a quasi-homogeneous polynomial on Cm with an isolated simple singularity at 0 ∈ Cm.
According to V.I. Arnol’d [2] simple singularities are classified by simply-laced Dynkin dia-
grams An (with n ≥ 1), Dn (with n ≥ 4), E6, E7, E8. Denoting by (x1 . . . , xm) the coordinates
in Cm (for singularities of type An we consider m = 1), the classification of simple singularities is
summarized in Table 1. Let µ be the Milnor number of f (note that µ = n for An, Dn and En),
and
f(x, a) := f(x) +
µ∑
i=1
aiφi(x)
be a miniversal unfolding of f , where a varies in a ball B ⊆ Cµ, and (φ1(x), . . . , φµ(x)) is
a basis of the Milnor ring. Using K. Saito’s theory of primitive forms [61], a flat metric and
a Frobenius manifold structure can be defined on the base space B [12]. See also [62]. For
any fixed a ∈ B, let the critical points be xi(a) =
(
x
(1)
i , . . . , x
(m)
i
)
, i = 1, . . . , µ, defined by the
condition ∂xαf(xi, a) = 0 for any α = 1, . . . ,m, with critical values ui(a) := f(xi(a), a). The
open ball B can be stratified as follows:
1) the stratum of generic points, i.e., points where both critical points x(i)’s and critical values
ui’s are distinct;
2) the Maxwell stratum, which is the closure of the set of points with distinct critical points
x(i)’s but some coalescing critical values ui’s;
3) the caustic, where some critical points coalesce.
The union of the Maxwell stratum and the caustic is called function bifurcation diagram Ξ of
the singularity (see [4] and [5, 6]).
The complement of the caustic consists exclusively of semisimple points of the Frobenius
manifold, and the critical values ui(a) := f(xi(a), a) are the canonical coordinates.
In this section we want to show how one can reconstruct local information near semisimple
points in the Maxwell stratum, by invoking Theorem 4.5. We will focus on the simplest example
of A3.
Remark 5.1. For simplicity of exposition, here we focus only on the case of Frobenius structures
associated with simple singularities. The existence of primitive forms for arbitrary singularities
was proved by M. Saito in [64].
Local Moduli of Semisimple Frobenius Coalescent Structures 45
5.2 Frobenius structure of type An
General references for this section are [21, 23, 24, 26]. Let us consider the affine space M ∼= Cn
of all polynomials
f(x, a) = xn+1 + an−1x
n−1 + · · ·+ a1x+ a0,
where (a0, . . . , an−1) ∈ M are used as coordinates. We call bifurcation diagram Ξ of the sin-
gularity An the set of polynomials in M with some coalescing critical values. The bifurcation
diagram Ξ is an algebraic subvariety in M which consists of two irreducible components (the
derivative w.r.t. the variable x will be denoted by (·)′):
• the caustic K, which is the set of polynomials with degenerate critical points (i.e., solutions
of the system of equations f ′(x, a) = f ′′(x, a) = 0);15
• The Maxwell stratum M, defined as the closure of the set of polynomials with some
coalescing critical values but different critical points.
For more information about the topology and geometry of (the complement of) these strata, the
reader can consult the paper [57], and the monograph [71]. There is a naturally defined covering
map ρ : M̃ → M of degree n!, whose fiber over a point f(x, a) consists of total orderings of its
critical points. On M̃ , x1, . . . , xn are well defined functions such that
f ′(x, ρ(w)) = (n+ 1)
n∏
i=1
(x− xi(w)), w ∈ M̃.
The caustic K is the ramification locus of the covering ρ. For any simply connected open subset
U ⊆M \K, we can choose a connected component W of ρ−1(U). The restriction of the functions
x1, . . . , xn on W defines single-valued functions of a ∈ U , which are local branches of x1, . . . , xn.
For further details see [56].
We define on M the following structures:
1. A free sheaf of rank n of OM -algebras: this is the sheaf of Jacobi–Milnor algebras
OM [x]
f ′(x, a) · OM [x]
.
For fixed a ∈ M , the fiber of this sheaf is the algebra C[x]/〈f ′(x, a)〉. We also define an
OM -linear Kodaira–Spencer isomorphism κ : TM → OM [x]/〈f ′(x, a)〉 which associates to
a vector field ξ the class Lξ(f) = ξ(f) mod f ′. In particular, for any α = 0, . . . , n− 1 the
class ∂aif is associated with the vector field ∂ai . In this way we introduce a product ◦ of
vector fields defined by
ξ ◦ ζ := κ−1
(
ξ(f) · ζ(f) mod f ′
)
.
The product ◦ is associative, commutative and with unit ∂a0 . We call Euler vector field
the distinguished vector field E corresponding to the class f mod f ′ under the Kodaira–
Spencer map κ. An elementary computation shows that
E =
n−1∑
i=0
n+ 1− i
n+ 1
ai
∂
∂ai
, LE(◦) = ◦.
15The equation of the caustic is ∆(f ′) = 0, where ∆(f ′) := Res(f ′, f ′′) is the discriminant of the polyno-
mial f ′(x, a). The reader can consult the monograph [35, Chapter 12].
46 G. Cotti, B. Dubrovin and D. Guzzetti
2. A symmetric bilinear form η, defined at a fixed point a ∈M as the Grothendieck residue
ηa(ξ, ζ) :=
1
2πi
∫
Γa
ξ(f)(u, a) · ζ(f)(u, a)
f ′(u, a)
du, (5.1)
where Γa is a circle, positively oriented, bounding a disc containing all the roots of f ′(u, a).
It is a nontrivial fact that the bilinear form η is non-degenerate (for a proof, see [5, 6]) and
flat (explicit flat coordinates can be found in [63]: notice that the natural coordinates ai’s
are not flat). Notice that
LEη =
n+ 3
n+ 1
η.
Theorem 5.2. The manifold M , endowed with the tensors (η, ◦, ∂a0 , E), is a Frobenius manifold
of charge n−1
n+1 . The caustic KM , defined as in Definition 2.22, coincides with the caustic K of
the singularity An defined above. By analytic continuation, the semisimple Frobenius structures
extends on the unramified covering space ρ−1(M\K) ⊆ M̃ . Critical values define a system of
canonical coordinates.
The reader can find detailed proofs in [23, 26, 56, 60]. If a is a given point of M \ K, i.e.,
such that f(x, a) has n distinct Morse critical points x1, . . . , xn, then the elements
πi(a) := κ−1
(
f ′(x, a)
f ′′(xi, a)(x− xi)
)
for i = 1, . . . , n
are idempotents of (TaM, ◦a). This follows from the equality f ′(x, a) = (n + 1)
n∏
i=1
(x − xi).
Consider now the critical points x1(a), . . . , xn(a) locally well defined as functions of a varying
in a simply connected open set away from the caustic. The critical values ui(a) := f(xi(a), a)
for i = 1, . . . , n can also be considered as functions on the same set. Since det
(
∂ui
∂aj
)
is the
Vandermonde determinant of xi(a)’s, the functions ui’s define a system of local coordinates
on M . In order to see that πi ≡ ∂
∂ui
, it is sufficient to prove that κ(∂ui)(xj) = δij , i.e.,
∂f
∂ui
(xi) = δij . This follows from the equalities
∂f(xi(a), a)
∂aj
= (xi(a))j ,
∂
∂aj
=
∑
i
(xi(a))j
∂
∂ui
.
5.3 The case of A3: reduction of the system for deformed flat coordinates
We consider the space M of polynomials
f(x; a) = x4 + a2x
2 + a1x+ a0,
where a0, a1, a2 ∈ C are “natural” coordinates on M . The residue theorem implies that the
metric η, defined on M as in (5.1), can be expressed as
ηa(ξ, ζ) = − res
u=∞
ξ(f)(u, a) · ζ(f)(u, a)
f ′(u, a)
du,
and consequently
ηa(∂i, ∂j) = res
v=0
v1−i−j
4 + 2a2v2 + a1v3
dv,
Local Moduli of Semisimple Frobenius Coalescent Structures 47
where ∂i = ∂
∂ai
, ∂j = ∂
∂aj
. So we find that
ηa =
0 0
1
4
0
1
4
0
1
4
0 −a2
8
.
Note that a0, a1, a2 are not flat coordinates for η. The commutative and associative product
defined on each tangent space TaM , using the Kodaira–Spencer map, is given by the structure
constants at a generic point a ∈M :
∂0 ◦ ∂i = ∂i for all i,
∂1 ◦ ∂1 = ∂2, ∂1 ◦ ∂2 = −1
2
a2∂1 −
1
4
a1∂0, ∂2 ◦ ∂2 = −1
2
a2∂2 −
1
4
a1∂1.
The Euler vector field is
E :=
2∑
i=0
4− i
4
ai∂i = a0∂0 +
3
4
a1∂1 +
1
2
a2∂2.
With such a structure M is a Frobenius manifold. The (1,1)-tensor U of multiplication by E is
U(a) =
a0 −1
8
a1a2 − 3
16
a2
1
3a1
4
a0 −
a2
2
4
−1
2
a1a2
a2
2
3
4
a1 a0 −
a2
2
4
.
Up to a multiplicative constant, the discriminant of the characteristic polynomial of U is equal
to
a2
1
(
8a3
2 + 27a2
1
)3
and so the bifurcation set of the Frobenius manifold is the locus
B = {a1 = 0} ∪
{
8a3
2 + 27a2
1 = 0
}
.
The irreducible component {a1 = 0} is the Maxwell stratum, whereas the irreducible component{
8a3
2 + 27a2
1 = 0
}
is the caustic.
Let us focus on the set {a1 = 0}, and let us look for semisimple points on it. It is enough
to consider the multiplication by the vector field λ∂1 + µ∂2 (λ, µ ∈ C), and show that it has
distinct eigenvalues. This is a (1,1)-tensor with components at points (a0, a1, a2) equal to
0 −µ
4
a1 −λ
2
a1
λ −µ
2
a2 −λ
2
a2 −
µ
4
a1
µ λ −µ
2
a2
,
whose characteristic polynomial, at points (a0, 0, a2), has discriminant
−1
8
λ2a3
2
(
2λ2 + µ2a2
)2
.
So, the points (a0, 0, a2) with a2 6= 0 are semisimple points of the bifurcation set, namely they
belong to the Maxwell stratum. In view of Theorem 5.2, they are semisimple coalescence points
of Definition 1.3. We would like to study deeper the behavior of the Frobenius structure near
points (a0, a1, a2) = (0, 0, h) of the Maxwell stratum, with fixed a0 = 0 and with h ∈ C∗.
48 G. Cotti, B. Dubrovin and D. Guzzetti
Remark 5.3. The points (a0, 0, 0), instead, are not semisimple because we have evidently ∂2
2 = 0
on them.
Let us introduce flat coordinates t1, t2, t3 defined by
a0 = t1 +
1
8
t23,
a1 = t2,
a2 = t3,
J =
(
∂ai
∂tj
)
i,j
=
1 0
1
4
t3
0 1 0
0 0 1
.
In flat coordinates we have
η =
0 0
1
4
0
1
4
0
1
4
0 0
, U(t1, t2, t3) =
t1
−5
16
t2t3 − 3
16
t22 +
1
32
t33
3t2
4
t1 −
t23
8
−5
16
t2t3
t3
2
3t2
4
t1
,
µ =
−1
4
0
1
4
.
Thus, the second system in (2.2)
∂zξ =
(
UT − 1
z
µ
)
ξ
reads
∂zξ1 =
3
4
ξ2t2 +
1
2
ξ3t3 + ξ1
(
t1 +
1
4z
)
,
∂zξ2 = − 5
16
ξ1t2t3 + ξ2
(
t1 −
t23
8
)
+
3
4
ξ3t2,
∂zξ3 = ξ1
(
− 3
16
t22 +
1
32
t33
)
− 5
16
ξ2t2t3 + ξ3
(
t1 −
1
4z
)
.
(5.2)
We know that, if (t1, t2, t3) is a semisimple point of the Frobenius manifold then the monodromy
data are well defined, and that these are invariant under (small) deformations of t1, t2, t3 by
Theorems 2.44 and 4.5. The bifurcation set is now
{t2 = 0} ∪
{
8t33 + 27t22 = 0
}
.
Now, if we fix a0 = 0, the tensor U at (0, a1, h), i.e., (t1, t2, t3) =
(
−1
8h
2, t2, h
)
, is
U
(
−1
8
h2, t2, h
)
=
−h
2
8
−5h
16
t2
1
32
(
h3 − 6t22
)
3t2
4
−h
2
4
−5h
16
t2
h
2
3t2
4
−h
2
8
.
The bifurcation locus is reached for a1 = t2 = 0. At these points
(t1, t2, t3) =
(
−1
8
h2, 0, h
)
,
Local Moduli of Semisimple Frobenius Coalescent Structures 49
we have
U
(
−1
8
h2, 0, h
)
=
−h
2
8
0
h3
32
0 −h
2
4
0
h
2
0 −h
2
8
.
Define the function
X(a) :=
[
−9a1 +
√
3
(
27a2
1 + 8a3
2
) 1
2
] 1
3 ,
which has branch points along the caustic K = {a2 = 0}∪
{
27a2
1 + 8a3
2 = 0
}
. Fix a branch of X
on a simply connected domain in M \ K, that we also denote by X(a). The critical points x1,
x2, x3 of f(x, a) are equal to
xi(a) :=
ϑi · a2
2
√
3 ·X(a)
− ϑi ·X(a)
2 · 32/3
,
where
ϑ1 := −1, ϑ2 :=
1− i
√
3
2
, ϑ3 :=
1 + i
√
3
2
are the cubic roots of (−1). Of course, different choices of branches of X correspond to permu-
tations of the xi’s. After some computations, we find the following expression for Ψ:
Ψ(t) =
√
6x2
1+a2
2
√
2(x1−x2)(x1−x3)
− (x2+x3)
√
6x2
1+a2
2
√
2(x1−x2)(x1−x3)
−
√
6x2
1+a2(a2−4x2x3)
8
√
2(x1−x2)(x1−x3)√
6x2
2+a2
2
√
2(x1−x2)(x3−x2)
(x1+x3)
√
6x2
2+a2
2
√
2(x1−x2)(x2−x3)
√
6x2
2+a2(a2−4x1x3)
8
√
2(x1−x2)(x2−x3)√
6x2
3+a2
2
√
2(x1−x3)(x2−x3)
(x1+x2)
√
6x2
3+a2
2
√
2(x1−x3)(x3−x2)
(a2−4x1x2)
√
6x2
3+a2
8
√
2(x1−x3)(x3−x2)
∣∣∣∣∣∣∣∣∣∣∣
a=a(t)
,
where
a0 = t1 +
1
8
t23, a1 = t2, a2 = t3.
The canonical coordinates are ui(t) = f(xi(a(t)), a(t)). In a neighbourhood of the point
(t1, t2, t3) =
(
−1
8h
2, 0, h
)
, i.e., for small t2 and h 6= 0, we find
u1(t2;h) = − t
2
2
4h
+
t42
16h4
− t62
16h7
+
3t82
32h10
+O
(
|t2|10
)
,
u2(t2;h) = −h
2
4
+
i
√
ht2√
2
+
t22
8h
+
it32
16
√
2h5/2
− t42
32h4
− 21it52
512
√
2h11/2
+
t62
32h7
+
429it72
8192
√
2h17/2
− 3t82
64h10
− 46189it92
524288
√
2h23/2
+O
(
|t2|10
)
,
u3(t2;h) = −h
2
4
− i
√
ht2√
2
+
t22
8h
− it32
16
√
2h5/2
− t42
32h4
+
21it52
512
√
2h11/2
+
t62
32h7
− 429it72
8192
√
2h17/2
− 3t82
64h10
+
46189it92
524288
√
2h23/2
+O
(
|t2|10
)
,
Ψ(t2) =
1√
2
√
h
0
√
h
4
√
2
i
2
√
h
− 1
2
√
2
−1
8
(
i
√
h
)
i
2
√
h
1
2
√
2
−1
8
(
i
√
h
)
+ t2
0 − 1
2
√
2h3/2 0
− 3
8
√
2h2 − i
16h3/2 − 5
32
√
2h
3
8
√
2h2 − i
16h3/2
5
32
√
2h
50 G. Cotti, B. Dubrovin and D. Guzzetti
+ t22
− 3
4
√
2h7/2 0 1
16
√
2h5/2
− 39i
128h7/2
15
128
√
2h3 − 41i
512h5/2
− 39i
128h7/2 − 15
128
√
2h3 − 41i
512h5/2
+O
(
t32
)
.
Hence, at points (t1, t2, t3) =
(
−1
8h
2, 0, h
)
, canonical coordinates ui(0;h) are
(u1, u2, u3) =
(
0,−h
2
2
,−h
2
2
)
and the system (5.2) reduces to
∂zξ1 =
(
−h
2
8
+
1
4z
)
ξ1 +
h
2
ξ3,
∂zξ2 = −h
2
4
ξ2,
∂zξ3 =
h3
32
ξ1 −
(
h2
8
+
1
4z
)
ξ3.
(5.3)
The second equation yields
ξ2(z) = c · e−
h2
4
z, c ∈ C.
From the first equation we find that
ξ3 =
2
h
(
∂zξ1 +
h2
8
ξ1 −
1
4z
ξ1
)
, (5.4)
and so from the third equation we obtain
2
h
ξ′′1 (z) +
h
2
ξ′1(z) +
3
8z2h
ξ1 = 0.
Making the ansatz
ξ1 = z
1
2 e−
h2z
8 Λ(z),
the equation for Λ becomes the following Bessel equation:
64z2Λ′′(z) + 64zΛ′(z)−
(
4 + z2h4
)
Λ(z) = 0. (5.5)
Therefore, ξ1 is of the form
ξ1 = z
1
2 e−
h2z
8
(
c1H
(1)
1
4
(
ih2
8
z
)
+ c2H
(2)
1
4
(
ih2
8
z
))
, c1, c2 ∈ C,
where H
(1)
ν (z), H
(2)
ν (z) stand for the Hankel functions of the first and second kind of parameter
ν = 1/4. Notice that if Λ(z) is a solution of equation (5.5), then also Λ
(
e±iπz
)
is a solution.
5.4 Computation of Stokes and central connection matrices
In order to compute the Stokes matrix, let us fix the line ` coinciding with the real axis. Such
a line is admissible for all points (t1, t2, t3) =
(
−1
8h
2, 0, h
)
with
|Reh| 6= | Imh|, h ∈ C∗.
Local Moduli of Semisimple Frobenius Coalescent Structures 51
Indeed, the Stokes rays for (u1, u2, u3) =
(
0,−1
4h
2,−1
4h
2
)
are
z = ±iρh
2
=⇒ arg z =
π
2
− 2 arg h (mod π).
Thus, admissibility corresponds to 1
2π− 2 arg h 6= kπ, k ∈ Z. Let us compute the Stokes matrix
in the case
−π
4
< arg h <
π
4
.
The asymptotic expansion for fundamental solutions Ξleft, Ξright of the system (5.3), is
ηΨ−1
(
1 +O
(
1
z
))
ezU = ΨT
(
1 +O
(
1
z
))
ezU
=
(
1 +O
(
1
z
))
1√
2
√
h
i
2
√
h
e−
1
4
h2z i
2
√
h
e−
1
4
h2z
0 − 1
2
√
2
e−
1
4
h2z 1
2
√
2
e−
1
4
h2z
√
h
4
√
2
− i
8e−
1
4
h2z
√
h − i
8e−
1
4
h2z
√
h
,
being
U := ΨUΨ−1 = diag(u1, u2, u3) = diag
(
0,−h
2
4
,−h
2
4
)
.
For the admissible line ` and for the above labelling of canonical coordinates the Stokes matrix
must be of the form prescribed by Theorem 2.42:
S =
1 0 0
α 1 0
β 0 1
(5.6)
for some constants α, β ∈ C to be determined. This means that the last two columns of Ξleft
must be the analytic continuation of Ξright.
Lemma 5.4. The following asymptotic expansions hold:
• if m ∈ Z, then
H
(1)
1
4
(
eimπ ih2
8
z
)
∼
√
2
π
(
eimπ ih2
8
z
)− 1
2
e−
3iπ
8 exp
(
−eimπ h
2
8
z
)
in the sector
−3
2
π −mπ − arg
(
h2
)
< arg z <
3
2
π −mπ − arg
(
h2
)
;
• if m ∈ Z, then
H
(2)
1
4
(
eimπ ih2
8
z
)
∼
√
2
π
(
eimπ ih2
8
z
)− 1
2
e
3iπ
8 exp
(
eimπ h
2
8
z
)
in the sector
−5
2
π −mπ − arg
(
h2
)
< arg z <
π
2
−mπ − arg
(
h2
)
.
52 G. Cotti, B. Dubrovin and D. Guzzetti
Proof. These formulae easily follow from the following well-known asymptotic expansion of
Hankel functions (see [74]):
H(1)
ν (z) ∼
√
2
πz
exp
(
i
(
z − ν
2
π − π
4
))
, −π + δ ≤ arg z ≤ 2π − δ,
δ being any positive acute angle. Analogously,
H(2)
ν (z) ∼
√
2
πz
exp
(
−i
(
z − ν
2
π − π
4
))
, −2π + δ ≤ arg z ≤ π − δ. �
Using Lemma 5.4, we obtain
Ξleft(z) =
ξL(1),1 ξL(2),1 ξL(3),1
0 − e−
1
4h
2z
2
√
2
e−
1
4h
2z
2
√
2
∗ ∗ ∗
, Ξright(z) =
ξR(1),1 ξR(2),1 ξR(3),1
0 − e−
1
4h
2z
2
√
2
e−
1
4h
2z
2
√
2
∗ ∗ ∗
, (5.7)
where
ξL(2),1(z) = ξL(3),1(z) = ξR(2),1(z) = ξR(3),1(z) =
i
√
π
8
h
1
2 ei 5
8
πz
1
2 e−
zh2
8 H
(1)
1
4
(
ih2
8
z
)
,
with the required asymptotic expansion in the following sector containing both Πleft and Πright{
z ∈ R : − 3
2
π − arg
(
h2
)
< arg z <
3
2
π − arg
(
h2
)}
,
and
ξL(1),1(z) =
√
π
4
√
2
h
1
2 eiπ
8 z
1
2 e−
zh2
8 H
(1)
1
4
(
e−iπ ih2
8
z
)
,
ξR(1),1(z) =
√
π
4
√
2
h
1
2 e−iπ
8 z
1
2 e−
zh2
8 H
(2)
1
4
(
ih2
8
z
)
,
with the required expansion respectively in the sectors{
z ∈ R : − π
2
− arg
(
h2
)
< arg z <
5
2
π − arg
(
h2
)}
⊇ Πleft,{
z ∈ R : − 5
2
π − arg
(
h2
)
< arg z <
π
2
− arg
(
h2
)}
⊇ Πright.
The entries of Ξleft, Ξright denoted by ∗ are reconstructed from the first rows, by applying
equation (5.4).
From the second rows of Ξleft, Ξright we can immediately say that the entries α, β of (5.6)
must be equal. Specializing the following well-known connection formula for Hankel special
functions
sin(νπ)H(1)
ν
(
zemπi
)
= − sin((m− 1)νπ)H(1)
ν (z)− e−νπi sin(mνπ)H(2)
ν (z), m ∈ Z, (5.8)
to the case m = −1, ν = 1
4 , we easily obtain
ξL(1),1(z) = ξR(1),1(z)− ξR(2),1(z)− ξR(3),1(z),
Local Moduli of Semisimple Frobenius Coalescent Structures 53
which means that α = β = −1. So, we have obtained that, at points (t1, t2, t3) =
(
−1
8h
2, 0, h
)
with
|Reh| > | Imh|, −π
4
< arg h <
π
4
(and consequently in their neighborhood, by Theorem 4.5) the Stokes matrix is
S =
1 0 0
−1 1 0
−1 0 1
.
In order to compute the central connection matrix, we observe that the A3 Frobenius manifold
structure is non-resonant, i.e., the components of the tensor µ are such that µα − µβ /∈ Z
for α 6= β. This implies that the (η, µ)-parabolic orthogonal group is trivial, and that the
fundamental system of (5.3) near the origin z = 0 can be uniquely chosen in such a way that
Ξ0(z) = (η +O(z))zµ. (5.9)
Now, let us recall the following Mellin–Barnes integral representations of Hankel functions
(see [74])
H(1)
ν (z) = −cos(νπ)
π
5
2
ei(z−πν)(2z)ν
∫ ∞i
−∞i
Γ(s)Γ(s− 2ν)Γ
(
ν +
1
2
− s
)
(−2iz)−s ds,
H(2)
ν (z) =
cos(νπ)
π
5
2
e−i(z−πν)(2z)ν
∫ ∞i
−∞i
Γ(s)Γ(s− 2ν)Γ
(
ν +
1
2
− s
)
(2iz)−s ds,
which are valid for
• 2ν /∈ 2Z + 1,
• respectively in the sectors | arg(∓iz)| < 3
2 ,
• and where the integration path separates the poles of Γ(s)Γ(s − 2ν) from the poles of
Γ
(
ν + 1
2 − s
)
.
Specializing these integral forms to ν = 1
4 , and deforming the integration path so that it reduces
to positively oriented circles around the poles
s ∈ 1
2
− 1
2
N,
we immediately obtain the following expansion of the solution ξ
(1)
1,R, ξ
(2)
1,R, ξ
(3)
1,R for the points
(t1, t2, t3) =
(
−1
8h
2, 0, h
)
, with −π
4 < arg h < π
4 , valid for small values of |z|:
Lemma 5.5. At the points (t1, t2, t3) =
(
−1
8h
2, 0, h
)
, with −π
4 < arg h < π
4 the following
expansions hold:
ξR(1),1(z) =
(1 + i)Γ
(
5
4
)
√
π
z
1
4 +
(
1
4 −
i
4
)
hΓ
(
3
4
)
√
π
z3/4
−
(
1
32 + i
32
)
h2Γ
(
1
4
)
√
π
z5/4 −
(
1
32 −
i
32
)
h3Γ
(
3
4
)
√
π
z7/4 +O
(
|z|9/4
)
,
ξR(2),1(z) = ξR(3),1(z) =
iΓ
(
5
4
)
√
π
z
1
4 −
4ihΓ
(
11
4
)
21
√
π
z3/4 −
ih2Γ
(
5
4
)
8
√
π
z5/4 +
ih3Γ
(
11
4
)
42
√
π
z7/4
+O
(
|z|9/4
)
.
54 G. Cotti, B. Dubrovin and D. Guzzetti
Moreover, using equation (5.4) we find that
ξR(1),3(z) =
(
1
4 −
i
4
)
Γ
(
3
4
)
√
π
z−
1
4 −
(
1
32 −
i
32
)
h2Γ
(
3
4
)
√
π
z3/4 +O
(
|z|5/4
)
,
ξR(2),3(z) = ξR(3),3(z) = −
4iΓ
(
11
4
)
21
√
π
z−
1
4 +
ih2Γ
(
11
4
)
42
√
π
z3/4 +O
(
|z|5/4
)
.
Proof. These expansions are the first terms of the expressions
ξR(1),1(z) =
√
π
4
√
2
h
1
2 e−iπ
8 z
1
2 e−
zh2
8
e
−i
(
−π
4
+ ih2z
8
)(
ih2z
) 1
4
2π5/2
× 2πi
∞∑
n=0
res
s= 1
2
−n
2
(
Γ(s)Γ(s− 2ν)Γ
(
ν +
1
2
− s
)(
eiπ h
2z
4
)−s)
,
and
ξR(2),1(z) = ξR(3),1(z) =
i
√
π
8
h
1
2 ei 5
8
πz
1
2 e−
zh2
8
−ei(−π4 + 1
8
ih2z)(ih2z
) 1
4
2π5/2
× 2πi
∞∑
n=0
res
s= 1
2
−n
2
(
Γ(s)Γ(s− 2ν)Γ
(
ν +
1
2
− s
)(
−eiπ h
2z
4
)−s)
. �
By a direct comparison between these expansion of solution Ξright(z) of (5.7) and the domi-
nant term of (5.9), namely
0 0
z
1
4
4
0
1
4
0
z−
1
4
4
0 0
,
we obtain the central connection matrix
C =
1
π
1
2
(1− i)Γ
(
3
4
)
−iΓ
(
3
4
)
−iΓ
(
3
4
)
0 −
√
2π
√
2π
(1 + i)Γ
(
1
4
)
iΓ
(
1
4
)
iΓ
(
1
4
)
.
Notice that such a matrix satisfies all the constraints of Theorem 2.43.
We can put the Stokes matrix in triangular form using two different permutations of the
canonical coordinates
(
0,−h2/4,−h2/4
)
, namely
• re-labelling (u1, u2, u3) 7→ (u2, u3, u1), corresponding to the permutation matrix
P =
0 1 0
0 0 1
1 0 0
,
• or re-labelling (u1, u2, u3) 7→ (u3, u2, u1), corresponding to the permutation matrix
P =
0 0 1
0 1 0
1 0 0
.
Local Moduli of Semisimple Frobenius Coalescent Structures 55
u u1
=0
u
2
= u
3
= − h
2
2
3
u=
2
u
=01
hh e
iπ/2
Figure 3. The triple (u1, u2, u2) is represented by three points u1, u2, u3 in C. We move along h 7→ hei
π
2 ,
starting from−π4 < arg h < π
4 . The two dashed regions in the left and right figures correspond respectively
to −π4 < arg h < π
4 and π
4 < arg h < 3π
4 .
In both cases these are the lexicographical orders of two different `-cells which divide any suffi-
ciently small neighborhood of the point (t1, t2, t3) =
(
−1
8h
2, 0, h
)
, with |Reh| > | Imh| and−π
4 <
arg h < π
4 , in which Theorem 4.5 applies. Using both permutations, the Stokes matrix becomes
Slex = PSP−1 =
1 0 −1
0 1 −1
0 0 1
, (5.10)
which can be thought as in the lexicographical form in one of the `-cells. The central connection
matrix, instead, has the following lexicographical forms in the two `-cells:
Clex =
1
π
1
2
−iΓ
(
3
4
)
−iΓ
(
3
4
)
(1− i)Γ
(
3
4
)
∓
√
2π ±
√
2π 0
iΓ
(
1
4
)
iΓ
(
1
4
)
(1 + i)Γ
(
1
4
)
, (5.11)
where we take the first sign if the lexicographical order is the relabeling (u1, u2, u3) 7→ (u2, u3, u1),
the second if it is the re-labeling (u1, u2, u3) 7→ (u3, u2, u1).
5.5 A “tour” in the Maxwell stratum:
reconstruction of neighboring monodromy data
From the data (5.10) and (5.11), by an action of the braid group, we can compute S and C
in the neighborhood of all other points (t1, t2, t3) =
(
−1
8h
2, 0, h
)
with |Reh| 6= | Imh|. As an
example, let us determine the Stokes matrix for points
(t1, t2, t3) =
(
−1
8
h2, 0, h
)
, with
π
4
< arg h <
3
4
π.
Starting from a point in the region −π
4 < arg h < π
4 and moving counter-clockwise towards
the region π
4 < arg h < 3
4π, the two coalescing canonical coordinates u2 = u3 = −1
2h
2 move in
the ui’s-plane counter-clockwise w.r.t. u1 = 0. For example, in Fig. 3 we move along a curve
h 7→ heiπ
2 , starting in −π
4 < arg h < π
4 . At arg h = π
4 , the Stokes rays R12 =
{
z = −iρh
2
, ρ > 0
}
and R21 =
{
z = iρh
2
, ρ > 0
}
cross the real line `, and a braid must act on the monodromy data.
In order to determine the braid and the transformed monodromy data, we proceed according to
the prescription of Section 4.1, as follows.
(1) We split the coalescing canonical coordinates, for example by considering the point
(t1, t2, t3) =
(
−1
8
h2, εeiϕ, h
)
, with − π
4
< arg h <
π
4
(5.12)
56 G. Cotti, B. Dubrovin and D. Guzzetti
for chosen ϕ and ε, being ε small (so that ε2 � ε). The corresponding canonical coordinates
u1 = O
(
ε2
)
, (5.13)
u2 = −h
2
4
+ ε|h|
1
2 exp
[
i
(
arg h
2
+ ϕ+
π
2
)]
+O
(
ε2
)
, (5.14)
u3 = −h
2
4
+ ε|h|
1
2 exp
[
i
(
arg h
2
+ ϕ− π
2
)]
+O
(
ε2
)
, (5.15)
give a point (u1, u2, u3) which lies in one of the two cells (Definition 4.3) which divide a polydisc
centred at (u1, u2, u3) =
(
0,−1
2h
2,−1
2h
2
)
. The Stokes rays are
R12 =
{
z = −iρh
2
+O(ε), ρ > 0
}
, R13 =
{
z = −iρh
2
+O(ε), ρ > 0
}
,
R23 =
{
z = ρ exp
[
−i
(
arg h
2
+ ϕ+ π
)]
+O
(
ε2
)
, ρ > 0
}
,
and opposite ones R21, R31, R32. Notice that in order for the real line ` to remain admissible,
we choose ϕ 6= kπ − 1
2 arg h, k ∈ Z, −π
4 < arg h < π
4 . The position of R23 w.r.t. the real
line ` is determined by the sign of cos
(arg h
2 + ϕ + π
2
)
. As long as ϕ varies in such a way that
sgn cos
(arg h
2 + ϕ + π
2
)
does not change, then R23 does not cross `. See Fig. 5. This means
that (u1, u2, u3) remains inside the same cell, i.e., the point corresponding to coordinates (5.12)
remains inside an `-chamber, where the isomonodromy Theorem 2.44 applies.
(2) The Stokes matrix must be put in triangular form Slex (5.10). In particular,
• if cos
(arg h
2 +ϕ+ π
2
)
< 0, then R23 is on the left of `, and the lexicographical order is given
by the permutation (u1, u2, u3) 7→ (u′1, u
′
2, u
′
3) = (u2, u3, u1);
• if cos
(arg h
2 + ϕ + π
2
)
> 0, then R23 is on the right of `, and the lexicographical order is
given by the permutation (u1, u2, u3) 7→ (u′1, u
′
2, u
′
3) = (u3, u2, u1).
We choose the cell where the triangular order coincides with the lexicographical order. The
passage to the other `-cell is obtained by a counter-clockwise rotation of u′1 w.r.t. u′2, which
corresponds to the action of the elementary braid β12. Its action (3.3) is a permutation matrix,
since (Slex)12 = 0; it is a trivial action on Slex, but not on Clex, as (5.11) shows.
(3) We move along a curve h 7→ heiπ
2 in the h-plane from a point (5.12) up to a point
(t1, t2, t3) =
(
−1
8
h2, εeiϕ′ , h
)
, with
π
4
< arg h <
3
4
π,
for some ϕ′ 6= kπ − 1
2 arg h, k ∈ Z, π
4 < arg h < 3π
4 . The transformation in Fig. 3, due to the
splitting, can substituted by the sequence of transformations in Fig. 4, each step corresponding
to an elementary braid. Each elementary braid corresponds to a Stokes ray crossing clock-wise
the real line ` as h varies along the curve h 7→ heiπ
2 .16 The total braid is then factored into the
product of the elementary braids as in Fig. 6, namely
β12β23β12, or β12β23β12β23.
Applying formulae (3.3), (3.5), we obtain
Sβ12β23β12
lex = Sβ12β23β12β23
lex =
1 1 1
0 1 0
0 0 1
. (5.16)
16Notice that the ray R23 rotates slower than R12, R13: namely, the angular velocity of R23 is approximately
(i.e., modulo negligible corrections in powers of ε) equal to 1
4
the one of R12, R13.
Local Moduli of Semisimple Frobenius Coalescent Structures 57
u u1 1
= O(ε2)2
u
u
2
3u
u
1
u
2
u
3
splitting
u1
u
2
3u
1u
u1
u3
coalescing
ε 0
elementary braid
elementary braid
elementary braid
u3= −
2
h
2
=2u
u
2 3
= u
Figure 4. The transition in Fig. 3 by splitting and elementary steps. After the splitting, we obtain a point
(u1, u2, u3), as in (5.13)–(5.15), lying in an `-cell of the polydisc centred at (u1, u2, u3) =
(
0,− 1
2h
2,− 1
2h
2
)
of the left part of Fig. 3. The transformation of Fig. 3 is obtained by successive steps following the arrows.
The final step is the right part of Fig. 3. The first elementary braid is β12 (because u′1 = u2, u′2 = u3 in
the upper left figure). The second is β23 (after relabelling in lexicographical order, u′2 = u2 and u′3 = u1
in the upper right figure). The third is β12.
These are the monodromy data in the two `-cells of a polydisc centred at the point
(t1, t2, t3) =
(
−1
8
h2, 0, h
)
, with
π
4
< arg h <
3
4
π.
The braid β23 is responsible for the passage from one cell to the other. Its action Aβ23
(
Sβ12β23β12
lex
)
is a permutation matrix, since
(
Sβ12β23β12
lex
)
23
= 0, which explains the equality in (5.16). By the
action (3.5), the central connection matrix (5.11), instead, assumes the following two forms
(differing for a permutation of the second and third column)
Cβ12β23β12
lex =
1
π
1
2
(1 + i)Γ
(
3
4
)
−iΓ
(
3
4
)
−iΓ
(
3
4
)
0 ±
√
2π ∓
√
2π
(1− i)Γ
(
1
4
)
iΓ
(
1
4
)
iΓ
(
1
4
)
,
Cβ12β23β12β23
lex =
1
π
1
2
(1 + i)Γ
(
3
4
)
−iΓ
(
3
4
)
−iΓ
(
3
4
)
0 ∓
√
2π ±
√
2π
(1− i)Γ
(
1
4
)
iΓ
(
1
4
)
iΓ
(
1
4
)
.
58 G. Cotti, B. Dubrovin and D. Guzzetti
u2
A
B
CD
E
F
A
B
C
D
E
F
`
Figure 5. In the left picture we represent relative positions of u3 w.r.t. u2 such that the real line ` is
admissible. On the right, we represent the corresponding positions of the Stokes ray R23. Notice that
if we let vary u3, by a deformation of the parameter ϕ, starting from A, going through B up to C, the
corresponding Stokes ray does not cross the line `, and no braids act. If we continue the deformation
of ϕ from C to D, an elementary braid acts on the monodromy data.
u2 = u3
u1
One way
for splitting
another way
for splitting
u1
u2
u3 u2u3
u1
u1 u1
u2 u2
u3 u3
β12
β12β23β12
β23
β23
β12
β12β23β12
β12
1 1
1 1
2 2
2 2
3 3
3
3
Figure 6. In the picture we represent u1, u2, u3 as points in C. On the left we describe all the
braids necessary to pass from a neighborhood of (t1, t2, t3) =
(
− 1
8h
2, 0, h
)
with −π4 < arg h < π
4 to one
with π
4 < arg h < 3
4π. Different columns of this diagram correspond to different `-cells of the same
neighborhood. The passage from such one cell to the other is through an action of an elementary braid
(β12 or β23) acting as a permutation matrix. In the picture on the right, we show the decomposition of
the global transformation in elementary ones.
In Table 2 we show the monodromy data for other values of arg h, with the corresponding
braid. In Fig. 6 we represent the braid corresponding to the passage from −π
4 < arg h < π
4 to
5
4π < arg h < 7
4π.
Remark 5.6. The reader can re-obtain this result by direct computation observing that, for
points
(t1, t2, t3) =
(
−1
8
h2, 0, h
)
, with
π
4
< arg h <
3
4
π,
the left and right solutions of (5.3) defining the Stokes matrix17 are of the form (5.7) with:
ξL(1),1 = ξR(1),1 =
√
π
4
√
2
h
1
2 eiπ
8 z
1
2 e−
zh2
8 H
(1)
1
4
(
e−iπ ih2
8
z
)
,
17Notice that for the points with π
4
< arg h < 3
4
π the original labelling of canonical coordinates (u1, u2, u3) =(
0,−h
2
4
,−h
2
4
)
already put the Stokes matrix in upper triangular form.
Local Moduli of Semisimple Frobenius Coalescent Structures 59
β12
β23β12
β23
β12β23
β12
β23β12
β23ub
ua
u1
2
1
3
Figure 7. Using the diagram representation of the braid group as mapping class group of the punctured
disk, we draw the braids acting along a curve h 7→ e
3πi
2 h, starting from the chambers close to (t1, t2, t3) =(
− 1
8h
2, 0, h
)
with −π4 < arg h < π
4 , and reaching the ones with 5
4π < arg h < 7
4π. The braids in red
describe mutations of the split pair u2, u3: their action on the monodromy data is a permutation matrix.
In the central disk, the blue numbers refer to the lexicographical order w.r.t. the real axis ` (i.e., from
the left to the right). The braids are the same for both cases (a, b) = (2, 3) and (3, 2).
ξL(2),1(z) = ξL(3),1(z) =
i
√
π
8
h
1
2 ei 3
8
πz
1
2 e−
zh2
8 H
(2)
1
4
(
e−3iπ ih2
8
z
)
,
ξR(2),1(z) = ξR(3),1(z) =
i
√
π
8
h
1
2 ei 5
8
πz
1
2 e−
zh2
8 H
(1)
1
4
(
ih2
8
z
)
,
having the expected asymptotic expansions in suitable sectors containing Πleft and/or Πright by
Lemma 5.4. Thus, by some manipulation of formulae (5.8) and
sin(νπ)H(2)
ν
(
zemπi
)
= eνπi sin(mνπ)H(1)
ν (z) + sin((m+ 1)νπ)H(2)
ν (z), m ∈ Z,
one sees that
ξL(2),1(z) = ξR(1),1(z) + ξR(2),1(z), ξL(3),1(z) = ξR(1),1(z) + ξR(3),1(z),
which are equivalent to (5.16). For the computation of the central connection matrix, one can
use analogous Puiseux series expansions of the solution Ξright(z), obtained from the integral
representation of Hankel functions given above.
5.6 Monodromy data as computed outside the Maxwell stratum
In this section, we compute the Stokes matrix S at non-coalescence points in a neighbourhood
of a coalescence one, by means of oscillatory integrals. We show that S coincides with that
60 G. Cotti, B. Dubrovin and D. Guzzetti
arg h Slex Clex Braid
]
−π
4 ,
π
4
[ 1 0 −1
0 1 −1
0 0 1
1
π
1
2
−iΓ
(
3
4
)
−iΓ
(
3
4
)
(1− i)Γ
(
3
4
)
∓
√
2π ±
√
2π 0
iΓ
(
1
4
)
iΓ
(
1
4
)
(1 + i)Γ
(
1
4
)
β12
]
π
4 ,
3π
4
[ 1 1 1
0 1 0
0 0 1
1
π
1
2
(1 + i)Γ
(
3
4
)
−iΓ
(
3
4
)
−iΓ
(
3
4
)
0 ±
√
2π ∓
√
2π
(1− i)Γ
(
1
4
)
iΓ
(
1
4
)
iΓ
(
1
4
)
β12β23β12β23
]
3π
4 ,
5π
4
[ 1 0 −1
0 1 −1
0 0 1
1
π
1
2
Γ
(
3
4
)
Γ
(
3
4
)
(1 + i)Γ
(
3
4
)
∓
√
2π ±
√
2π 0
Γ
(
1
4
)
Γ
(
1
4
)
(1− i)Γ
(
1
4
)
(β12β23)3β12
]
5π
4 ,
7π
4
[ 1 1 1
0 1 0
0 0 1
1
π
1
2
(−1 + i)Γ
(
3
4
)
Γ
(
3
4
)
Γ
(
3
4
)
0 ±
√
2π ∓
√
2π
(−1− i)Γ
(
1
4
)
Γ
(
1
4
)
Γ
(
1
4
)
(β12β23)3
×β12β23β12β23
]
7π
4 ,
9π
4
[ 1 0 −1
0 1 −1
0 0 1
1
π
1
2
iΓ
(
3
4
)
iΓ
(
3
4
)
(−1 + i)Γ
(
3
4
)
∓
√
2π ±
√
2π 0
−iΓ
(
1
4
)
−iΓ
(
1
4
)
(−1− i)Γ
(
1
4
)
(β12β23)6β12
Table 2. For different values of arg h, in the open angular intervals in the left column, we tabulate
the monodromy data (Slex, Clex), in lexicographical order, in the two `-cells which divide a sufficiently
small neighborhood of the point (t1, t2, t3) =
(
− 1
8h
2, 0, h
)
. The difference of the data in the two `-cells
(just a permutation of two columns in the central connection matrix) is obtained by applying the braid
written in red: if it is not applied the sign to be read is the first one, the second one otherwise. Notice
that the central element (β12β23)3 acts trivially on the Stokes matrices, and by a left multiplication by
M−10 = diag(i, 1,−i) on the central connection matrix.
obtained at the coalescence point in the previous section. Moreover, we explicitly show that the
fundamental matrices converge to those computed at the coalescence point, exactly as prescribed
by our Theorem 4.5.
The system (5.2) admits solutions given in terms of oscillating integrals,
ξ1(z, t) = z
1
2
∫
γ
exp{z · f(x, t)} dx, (5.17)
ξ2(z, t) = z
1
2
∫
γ
x exp{z · f(x, t)} dx, (5.18)
ξ3(z, t) = z
1
2
∫
γ
(
x2 +
1
4
t3
)
exp{z · f(x, t)}dx, (5.19)
where f(x, t) = x4 + t3x
2 + t2x+ t1 + 1
8 t
2
3. Here γ is any cycle along which Re(z · f(x, t))→ −∞
for |x| → +∞, i.e., a relative cycle in H1(C,CT,z,t), with
CT,z,t := {x ∈ C : Re(zf(x, t)) < −T} , with T very large positive number.
First, we show that the Stokes matrix at points in `-chambers near the coalescence point
(t1, t2, t3) =
(
−1
8h
2, 0, h
)
coincide with the one previously computed, in accordance with Theo-
rem 4.5. In what follows we will focus on the `-chamber made of points (t1, t2, t3)=
(
−1
8h
2, εeiφ, h
)
,
Local Moduli of Semisimple Frobenius Coalescent Structures 61
R21 R31
R32
R12
R13
R23
Figure 8. Disposition of the Stokes rays for
a point in the chosen `-chamber.
π
8
I3
I1
I2
Figure 9. Integration contours Ii which define
the functions Ii’s.
where −π
4 < arg h < π
4 , and ε, φ are small positive numbers. For points in this `-chamber, the
Stokes rays are disposed as described in Fig. 8.
Notice that in order to compute the Stokes matrix at a semisimple point with distinct canon-
ical coordinates it suffices to know the first rows of Ξleft/right. Assuming that z ∈ R+, we define
the following three functions obtained from the integrals (5.17) with integration cycles Ii as in
Fig. 9:
Ii(z, t2) :=
∫
Ii
exp
(
z
(
x4 + hx2 + t2x
))
dx, i = 1, 2, 3. (5.20)
For the specified integration cycles, the integrals Ii(z, t2) are convergent in the half-plane
| arg z| < π
2 . A continuous deformation of a path Ii, which maintains its asymptotic direc-
tions in the shaded sectors, yields a convergent integral and defines the analytic continuation
of Ii(z, t2) on the whole sector | arg z| < π. If we vary z (excluding z = 0), the shaded regions
continuously rotate clockwise or counterclockwise. In order to obtain the analytic continuation
of the functions Ii(z, t2) to the whole universal cover R, we can simply rotate the integration
contours Ii. This procedure also makes it clear that the functions Ii have monodromy of order 4:
indeed as arg z increases or decreases by 2π, the shaded regions are cyclically permuted.
In order to obtain information about the asymptotic expansions of the functions Ii, we
associate to any critical point xi a relative cycle Li, called Lefschetz thimble, defined as the set
of points of C which can be reached along the downward geodesic-flow
dx
dτ
= −z̄ ∂f̄
∂x̄
,
dx̄
dτ
= −z ∂f
∂x
(5.21)
starting at the critical point xi for τ → −∞. Morse and Picard–Lefschetz theory guarantees
that the cycles Li are smooth one-dimensional submanifolds of C, piecewise smoothly dependent
on the parameters z, t, and they represent a basis for the relative homology groups H1(C,CT,z,t).
Moreover, the Lefschetz thimbles are steepest descent paths: namely, Im(zf(x, t)) is constant on
each connected component of Li\{xi} and Re(zf(x, t)) is strictly decreasing along the flow (5.21).
Thus, after choosing an orientation, the paths of integration defining the functions Ii can be
expressed as integer combinations of the thimbles Li for any value of z:
Ii = n1L1 + n2L2 + n3L3, ni ∈ Z. (5.22)
If we let z vary, the Lefschetz thimbles change. When z crosses a Stokes ray, Lefschetz thimbles
jump discontinuously, as shown in Fig. 10. In particular, for z on a Stokes ray there exists a flow
line of (5.21) connecting two critical points xi’s.
This discontinuous change of the thimbles implies a discontinuous change of the integer
coefficients ni in (5.22), and a discontinuous change of the leading term of the asymptotic
62 G. Cotti, B. Dubrovin and D. Guzzetti
x1
x2
x3
x1
x3
x2
x1
x3
x2
Figure 10. Discontinuous change of a Lefschetz thimbles. As z varies in R, we pass from the configu-
ration on the left to the one on the right. The middle configuration is realized when z is on a Stokes ray:
in this case there is a downward geodesic-flow line connecting two critical points x1 and x3.
expansions of the functions Ii’s. Using the notations introduced in Fig. 11, in each configuration
the following identities hold:
(A) :
I1 = L1,
I2 = L2,
I3 = L3,
(B) :
I1 = L1 + L2,
I2 = L2,
I3 = L3,
(C) :
I1 = L1 + L2,
I2 = L2,
I3 = −L1 + L3,
(D) :
I1 = L1 − L3,
I2 = L2,
I3 = L3,
(E) :
I1 = L1 − L3,
I2 = L1 + L2,
I3 = L3.
By a straightforward application of the Laplace method we find that, at least for sufficiently
small positive values of arg z, the following asymptotic expansions hold
Ii(z, t2) = π
1
2 iz−
1
2
(
6x2
i + h
)− 1
2 ezui
(
1 +O
(
1
z
))
.
Since the deformations of the thimbles I2, I3 happen for values of z for which the exponent ezu1
is subdominant, we immediately conclude that the functions
ξL(2),1(z, t2) = ξR(2),1(z, t2) = ±iπ−
1
2 z
1
2
6x2
2 + h
2
√
2(x1 − x2)(x3 − x2)
I2(z, t2), (5.23)
ξL(3),1(z, t2) = ξR(3),1(z, t2) = ±iπ−
1
2 z
1
2
6x2
3 + h
2
√
2(x1 − x3)(x2 − x3)
I3(z, t2) (5.24)
have asymptotic expansions
Ψ21ezu2
(
1 +O
(
1
z
))
, Ψ31ezu3
(
1 +O
(
1
z
))
,
respectively, both in Πleft and Πright. Thus, we can immediately say that the Stokes matrix
computed at a point (t1, t2, t3) =
(
−1
8h
2, εeiφ, h
)
is of the form
S =
1 0 0
∗ 1 0
∗ 0 1
.
Note that the arbitrariness of the orientations of the Lefschetz thimbles can be incorporated in
the choice of the entries of the Ψ matrix, and hence it will affect the monodromy data by the
action of the group (Z/2Z)3.
Local Moduli of Semisimple Frobenius Coalescent Structures 63
I1
I1
I1
I1
I1
I2
I2
I2I2
I2
I3
I3
I3I3
L2
L2
L2
L3
L3
L3
L1
L1
L1
L2
L2
L3
L3
I3
L1 L1
(A)
(B)
(C)(E)
(D)
Figure 11. In this figure it is shown how the Lefschetz thimbles Li’s (continuous lines), and the
integrations contours Ii’s (dotted lines) change by analytic continuation with respect to the variable z.
The configuration (A) corresponds to the case arg z = 0. Increasing arg z the configuration (B) and (C)
are reached after crossing the Stokes rays R31, and R21 respectively. Decreasing arg z, we obtain the
configurations (D) and (E) after crossing the rays R12 and R13 respectively. Note that when z crosses
the Stokes rays R32 and R23 no Lefschetz thimble changes, coherently with the detailed analysis done
in [19].
After a careful analysis of the deformations of the Lefschetz thimbles, one finds that the
solutions ξL(1),1(z, t2), ξR(1),1(z, t2) are respectively given by
ξR(1),1(z, t2) = ±iΨ11π
− 1
2 z
1
2
(
6x2
1 + h
) 1
2 (I1(z, t2) + I3(z, t2)) , (5.25)
ξL(1),1(z, t2) = ±iΨ11π
− 1
2 z
1
2
(
6x2
1 + h
) 1
2 (I1(z, t2)− I2(z, t2)) , (5.26)
having the asymptotic expansion
Ψ11ezu1
(
1 +O
(
1
z
))
in Πright and Πleft respectively. This immediately allows one to compute the remaining entries
of the Stokes matrix
S21 =
Ψ11
(
6x2
1 + h
) 1
2
Ψ21
(
6x2
2 + h
) 1
2
= ±
(
6x2
1 + h
)
(x3 − x2)
(x1 − x3)
(
6x2
2 + h
) ≡ ±1,
S31 =
Ψ11
(
6x2
1 + h
) 1
2
Ψ31
(
6x2
3 + h
) 1
2
= ±
(
6x2
1 + h
)
(x2 − x3)
(x1 − x2)
(
6x2
3 + h
) ≡ ±1.
64 G. Cotti, B. Dubrovin and D. Guzzetti
This result is independent on the point (t1, t2, t3) =
(
−1
8h
2, εeiφ, h
)
of the chosen `-chamber. It
coincides with the Stokes matrix obtained at the coalescence point (t1, t2, t3) =
(
−1
8h
2, 0, h
)
, in
complete accordance with our Theorem 4.5.
Remark 5.7. It is interesting to note that the isomonodromy condition in this context is
equivalent to the condition
f ′′(x1)
f ′′(x2)
= −x1 − x3
x2 − x3
,
a relation that the reader can easily show to be valid for any polynomial f(x) of fourth degree
with three non-degenerate critical points x1, x2, x3.
Our Theorem 4.5 also states that, as t2 → 0 the solutions (5.23), (5.24), (5.25), (5.26) must
converge to the ones computed in the previous section at the coalescence point. We show this
explicitly below. In order to do this, it suffices to set t2 = 0 in the integral (5.20). With the
change of variable x = 2−
1
4 z−
1
4 s
1
2 , we obtain
I2(z, 0) = I3(z, 0) = 2−
5
4 z−
1
4
∫
L
exp
{
s2
2
+
(
hz
1
2
√
2
)
s
}
ds
= 2−
5
4 z−
1
4 (2π)
1
2 e−
h2z
8 D− 1
2
(
hz
1
2
√
2
)
= 2−
3
2 e−
h2z
8 h
1
2K 1
4
(
h2z
8
)
= πi · 2−
5
2h
1
2 e−
h2z
8 e
πi
8 H
(1)
1
4
(
ih2z
8
)
.
Here Dν(z) is the Weber parabolic cylinder function of order ν, with integral representation [1,
p. 688]
D− 1
2
(z) = ± e
1
2
z2
(2π)
1
2
∫
L
s−
1
2 exp
(
s2
2
+ zs
)
ds,
where{
(+) if − 3π
2 + 2kπ < arg s < −π
2 + 2kπ,
(−) if π
2 + 2kπ < arg s < 3π
2 + 2kπ,
the integration contour L being the one represented in Fig. 12, together with the identities
D− 1
2
(z) =
( z
2π
) 1
2
K 1
4
(
1
4
z2
)
, Kν(z) =
{
πi
2 e
νπi
2 H
(1)
ν
(
ze
πi
2
)
,
−πi
2 e−
νπi
2 H
(2)
ν
(
ze−
πi
2
)
.
It follows that
ξL(2),1(z, 0) = ξR(2),1(z, 0) = ±iπ−
1
2 z
1
2
6x2
2 + h
2
√
2(x1 − x2)(x3 − x2)
I2(z, 0)
= ± i
√
π
8
h
1
2 e
5iπ
8 z
1
2 e−
h2z
8 H
(1)
1
4
(
ih2z
8
)
,
which coincides (up to an irrelevant sign) with the solution computed in the previous section at
the coalescence point. The computations for ξL(3),1(z, 0) = ξR(3),1(z, 0) are identical.
Local Moduli of Semisimple Frobenius Coalescent Structures 65
L
Figure 12. Integration contour L used in the
integral representation of the Weber parabolic
cylinder functions.
x2
x1
x3
I1
1
I2
1
Figure 13. For t2 = 0, we can decompose
the integration cycle I1 into two pieces, I11 , I21
used to define the functions I11 and I21. The
continuous lines represent the Lefschetz thim-
bles through the critical points xi’s.
The computations for ξR(1),1 and ξL(1),1 are a bit more laborious. First of all let us observe that
the integral
g(z) :=
∫ ∞
0
exp
(
− t
2
2
− zt
)
t−
1
2 dt
is convergent for all z ∈ C, defining an entire function.18 Moreover we have
g(z) =
√
πe
z2
4 D− 1
2
(z) = 2−
1
2 e
z2
4 z
1
2K 1
4
(
z2
4
)
.
With a change of variable t = e−iθτ that rotates the half line R+ by θ, we find the following
identity
g(z) = e−i θ
2
∫
eiθR+
exp
(
−e−2iθ τ
2
2
− e−iθzτ
)
τ−
1
2 dτ. (5.27)
For t2 = 0 the integral I1(z, 0) splits into two pieces:
I1(z, 0) = I1
1(z) + I2
1(z), Ii1(z) :=
∫
Ii1
exp
(
z
(
x4 + hx2
))
dx, i = 1, 2,
where the paths Ii1 are as in Fig. 13. Setting x = 2−
1
4 z−
1
4 s
1
2 , the image of the paths Ii1 are
in two different sheets of the Riemann surface with local coordinate s. Keeping track of this,
and of the orientations of the modified paths, using formula (5.27) for θ = 3πi
2 ,
5πi
2 and a small
deformation of the paths of integration, we find that
I1
1(z) = 2−
5
4 z−
1
4
(
−
∫
e
3πi
2 R+
exp
{
s2
2
+
hz
1
2
√
2
s
}
s−
1
2 ds
)
= −2−
5
4 z−
1
4 e
3πi
4 g
(
e
πi
2
hz
1
2
√
2
)
= −2−
5
4 z−
1
4 e
3πi
4 · 2−
1
2 e−
h2z
8
(
e
πi
2
hz
1
2
√
2
) 1
2
K 1
4
(
eπih
2z
8
)
=
1
4
e−
h2z
8 h
1
2K 1
4
(
eπih
2z
8
)
,
18This is in accordance with the expression of g in terms of the modified Bessel function K, which gives
g
(
e±πiz
)
= 2−
1
2 e
z2
4 e±
πi
2 z
1
2K 1
4
(
e±2πi z
2
4
)
.
From the symmetry K 1
4
(
e4πiz
)
= −K 1
4
(z) we deduce that g
(
e−πiz
)
= g
(
eπiz
)
.
66 G. Cotti, B. Dubrovin and D. Guzzetti
and
I2
1(z) = 2−
5
4 z−
1
4
(∫
e
5πi
2 R+
exp
{
s2
2
+
hz
1
2
√
2
s
}
s−
1
2 ds
)
= 2−
5
4 z−
1
4 e
5πi
4 g
(
e−
πi
2
hz
1
2
√
2
)
= 2−
5
4 z−
1
4 e
5πi
4 · 2−
1
2 e−
h2z
8
(
e−
πi
2
hz
1
2
√
2
) 1
2
K 1
4
(
e−πih
2z
8
)
= −1
4
e−
h2z
8 h
1
2K 1
4
(
e−πih
2z
8
)
.
Thus, in the limit t2 = 0 we find that
ξR(1),1(z, 0) = ±iΨ11π
− 1
2 z
1
2
(
6x2
1 + h
) 1
2
(
I1
1(z) + I2
1(z) + I3(z, 0)
)
= ±i2−
5
2π−
1
2 z
1
2 e−
h2z
8 h
1
2
{
K 1
4
(
eiπ h
2z
8
)
−K 1
4
(
e−iπ h
2z
8
)
+ 2
1
2K 1
4
(
h2z
8
)}
= ±π
1
2 z
1
2 2−
7
2 e−
h2z
8 h
1
2 e−
πi
8
×
{
H
(2)
1
4
(
e
iπ
2
h2z
8
)
+ e
πi
4 H
(1)
1
4
(
e−
iπ
2
h2z
8
)
+ 2
1
2H
(2)
1
4
(
e−
iπ
2
h2z
8
)}
= ±π
1
2 2−
3
2 z
1
2 e−
h2z
8 h
1
2 e−
πi
8 H
(2)
1
4
(
e
iπ
2
h2z
8
)
,
which is exactly (modulo irrelevant signs) the solution at the coalescence point as computed in
the previous section. We leave as an exercise for the reader to show that all other solutions
ξ
R/L
(i),j (z) converge to the ones computed at the coalescence point.
6 Second example of application of Theorem 4.5: quantum
cohomology of the Grassmannian G2
(
C4
)
and Γ-conjecture
In this section we prove Theorem 6.10, which is Theorem 1.5 of the Introduction, we estab-
lish a correspondence between each region of the quantum cohomology and a full exceptional
collection, obtaining an explicit refinement of the original conjecture of [25]. We also prove
Proposition 6.5, which we believe to be an important characterisation of C0(η, µ,R).
The problem is to compute the monodromy data for the Frobenius manifold QH•
(
G2
(
C4
))
,
the quantum cohomology of the Grassmannian G2
(
C4
)
. This manifold has a locus of semisimple
coalescent points, called small quantum cohomology. Moreover, the structure of the manifold is
known only at the small quantum cohomology locus. Therefore, if we want an explicit compu-
tation of monodromy data, this can be done only at coalescence points. This is what we will do:
the data will be calculated at the coalescence point t = 0 in the small cohomology locus. The
validity of the result for the whole Frobenius manifold QH•
(
G2
(
C4
))
is completely justified by
our Theorem 4.5, which is thus crucial to us. Without Theorem 4.5 our computations would be
geometrically meaningless.19
As a result of the computations, we prove Theorem 6.10 (Theorem 1.5), which clarifies and
verifies a conjecture, formulated by the second author in [25] and then refined20 in [28] and [33],
in the case the quantum cohomology of the Grassmannian G2
(
C4
)
. Our explicit computations,
using elementary analytic methods only, seems to be missing from the literature.
19Without Theorem 4.5, we would anyway have an interesting and non-trivial example of computation of
monodromy data for a 6× 6 differential system with two coinciding eigenvalues at the irregular singularity. The
purpose of this article goes beyond this; our goal is the study the monodromy data of a Frobenius manifold.
20The detailed comparison between the explicit computations of the monodromy data for complex Grassman-
nians and the Γ-classes proposed in [33], is one of the contents of our paper [18].
Local Moduli of Semisimple Frobenius Coalescent Structures 67
6.1 Notations in Gromov–Witten theory
Let X be a smooth complex projective variety with vanishing odd cohomology
H2k+1(X;C) = 0, k ≥ 0.
Let us fix a homogeneous basis (T1, T2, . . . , TN ) of H•(X;C) =
⊕
kH
2k(X;C) such that
• T1 = 1 is the unity of the cohomology ring;
• deg Tα =: 2qα;
• T2, . . . , Tr span H2(X;C).
We will denote by η : H•(X;C)×H•(X;C)→ C the Poincaré metric
η(ξ, ζ) :=
∫
X
ξ ∪ ζ, ηαβ :=
∫
X
Tα ∪ Tβ.
We also introduce the Novikov ring Λ := C[[Q2, . . . , Qr]], and the symbol
Qβ := Q
∫
β T2
2 · · ·Q
∫
β Tr
r .
Let Xg,n,β be the moduli space of stable maps with target X, of genus g, with n distinct marked
points and of degree β ∈ H2(X;Z). We will denote by
〈τd1γ1, . . . , τdnγn〉Xg,n,β :=
∫
[Xg,n,β ]vir
n∧
i=1
ev∗i (γi) ∪ ψ
di
i
the value of the Gromov–Witten invariant (with gravitational descendants, if some of the di’s is
nonzero), where
• γ1, . . . , γn ∈ H•(X;C),
• (d1, . . . , dn) ∈ Nn,
• ψ1, . . . , ψn ∈ H2(Xg,n,β;Q) are the universal cotangent line classes,
• evi : Xg,n,β → X is the evaluation map at the i-th marked point,
• [Xg,n,β]vir stands for the virtual fundamental class. Recall that degree of the virtual cycle
is equal to the virtual dimension (over R)
vir dimRXg,n,β = 2(1− g) dimCX − 2
∫
β
ωX + 2(3g − 3 + n).
It is convenient to collect Gromov–Witten invariants with descendants as coefficients of a ge-
nerating function, called genus g gravitational Gromov–Witten potential, or simply genus g free
energy
FXg (γ) :=
∞∑
n=0
∑
β∈Eff(X)
Qβ
n!
〈γ, . . . , γ︸ ︷︷ ︸
n times
〉Xg,n,β,
the set Eff(X) ⊆ H2(X;Z) being the set of effective classes of X. Introducing (infinitely many)
coordinates t := (tα,p)α,p
γ =
∑
α,p
tα,pτpTα,
68 G. Cotti, B. Dubrovin and D. Guzzetti
the free energy FXg ∈ Λ[[t]] can be seen as a function on the large phase-space, and restricting
the free energy to the small phase space (naturally identified with H•(X;C)),
FXg
(
t1,0, . . . , tN,0
)
:= FXg (t)
∣∣
tα,p=0, p>0
,
one obtains the generating function of the Gromov–Witten invariants of genus g. It will also be
convenient to introduce the genus g correlation functions defined by the derivatives
〈〈τd1Tα1 , . . . , τdnTαn〉〉g :=
∂
∂tα1,d1
· · · ∂
∂tαn,dn
FXg .
Let tα := tα,0. By the Divisor axiom, the genus 0 Gromov–Witten potential FX0 (t) can be
seen as an element of the ring C
[[
t1, Q2et
2
, . . . , Qre
tr , tr+1, . . . , tN
]]
. In what follows we will be
interested in the cases when FX0 is a convergent series expansion
FX0 ∈ C
{
t1, Q2et
2
, . . . , Qre
tr , tr+1, . . . , tN
}
. (6.1)
Without loss of generality we can put Q2 = Q3 = · · · = Qr = 1. Under the assumption (6.1),
FX0 (t) defines an analytic function in an open neighbourhood Ω ⊆ H•(X;C) of the point
ti = 0, i = 1, r + 1, . . . , N ; Re ti → −∞, i = 2, 3, . . . , r. (6.2)
The function FX0 is a solution of WDVV equations [54, 56, 69, 72], and thus it defines an
analytic Frobenius manifold structure on Ω. Using the canonical identifications of tangent
spaces TpΩ ∼= H•(X;C) : ∂tα 7→ Tα, the unit vector field is e = ∂t1 ≡ 1, and the Euler vector
field is
E := c1(X) +
N∑
α=1
(
1− 1
2
deg Tα
)
tαTα.
The resulting Frobenius structure is called quantum cohomology of X, denoted QH•(X). Notice
that at the classical limit point (6.2) the algebra structure on the tangent spaces coincides with
the classical cohomological algebra structure. Notice that, because of the divisor axiom, the
Frobenius structure is 2πi-periodic in the 2-nd cohomology directions: the structure can be
considered as defined on an open region of the quotient H•(X;C)/2πiH2(X;Z).
There are no general results characterizing smooth projective varieties with semisimple quan-
tum cohomology: however, for some classes of varieties such as
• some Fano threefolds [16],
• toric varieties [48],
• some homogeneous spaces [15],
it has been proved that the small quantum locus is made of semisimple points. Grassmannians
are among these varieties. Below, we focus on the small quantum cohomology of G2
(
C4
)
, namely
the restriction to the locus H2
(
G2
(
C4
)
;C
)
, with coordinates
(
0, t2, 0, . . . , 0
)
.
6.2 Small quantum cohomology of G2
(
C4
)
6.2.1 Generalities and proof of its semisimplicity
For simplicity, let us use the notation G := G2
(
C4
)
. From the general theory of Schubert
calculus, it is known that H•(G;C) is a complex vector space of dimension 6, and a basis is
given by Schubert classes:
σ0 := 1, σ1, σ2, σ1,1, σ2,1, σ2,2.
Local Moduli of Semisimple Frobenius Coalescent Structures 69
Each σλ ∈ H2|λ|(G;C). By posing
v1 := σ0, v2 := σ1, v3 := σ2, v4 := σ1,1, v5 := σ2,1, v6 := σ2,2,
we will denote by ti the coordinate with respect to vi. The coordinates in the small quantum
cohomology are
t =
(
0, t2, 0, . . . , 0
)
.
By Pieri and Giambelli formulas one finds that the matrix of the Poincaré pairing
η(α, β) :=
∫
G
α ∧ β
with respect to the above basis is given by
η =
0 0 0 0 0 c
0 0 0 0 c 0
0 0 c 0 0 0
0 0 0 c 0 0
0 c 0 0 0 0
c 0 0 0 0 0
, c :=
∫
G
σ2,2.
Using quantum Pieri–Bertram formula [11], we deduce that the matrix of the operator of mul-
tiplication by λσ1 + µσ1,1 is
0 0 µq 0 λq 0
λ 0 0 0 µq λq
0 λ 0 0 0 µq
µ λ 0 0 0 0
0 µ λ λ 0 0
0 0 0 µ λ 0
, q := et
2
. (6.3)
The discriminant of the characteristic polynomial of this matrix is
16777216λ4µ2q8
(
λ4 + µ4q
)6
and so, if λ 6= 0, µ 6= 0 and λ4 +qµ4 6= 0, its eigenvalues are pairwise distinct. This is a sufficient
condition to state that the quantum cohomology of G is semisimple.
Notice that the value at the point p of coordinates
(
0, t2, 0, . . . , 0
)
of the Euler field of quantum
cohomology QH•(G) is21 given by the first Chern class c1(G) = 4σ1:
E|p = 4
∂
∂t2
≡ 4σ1.
The matrix U of multiplication by E at the point p is given by posing λ = 4, µ = 0 in (6.3):
U
(
0, t2, 0, . . . , 0
)
≡ 4C2
(
0, t2, 0, . . . , 0
)
=
0 0 0 0 4q 0
4 0 0 0 0 4q
0 4 0 0 0 0
0 4 0 0 0 0
0 0 4 4 0 0
0 0 0 0 4 0
.
The characteristic polynomial is p(z) = z6 − 1024qz2, so that 0 is an eigenvalue with multipli-
city 2. Therefore, the semisimple points with coordinates
(
0, t2, 0, . . . , 0
)
are semisimple coales-
cence points in the bifurcation set.
21We identify TpH
•(G) with H•(G) in the canonical way.
70 G. Cotti, B. Dubrovin and D. Guzzetti
6.2.2 Idempotents at the points
(
0, t2, 0, . . . , 0
)
The multiplication by σ1 + σ1,1 has pairwise distinct eigenvalues, at least at points for which
t2 6= iπ(2k + 1). Putting λ = µ = 1 in (6.3), we deduce that the characteristic polynomial of
this operator is
p(z) =
(
q + z2
)(
−4q + q2 − 8qz − 2qz2 + z4
)
.
So the six eigenvalues are
iq
1
2 , −iq
1
2 ,
ε1 := −i
√
2q
1
4 − q
1
2 , ε2 := i
√
2q
1
4 − q
1
2 , ε3 := −
√
2q
1
4 + q
1
2 , ε4 :=
√
2q
1
4 + q
1
2 ,
and the corresponding eigenvectors are
π1 := −q − iq
1
2σ2 + iq
1
2σ1,1 + σ2,2, π2 := −q + iq
1
2σ2 − iq
1
2σ1,1 + σ2,2,
π2+i :=
(
q2 + qε2
i
)
+
(
−q2 + 2qεi + qε2
i
)
σ1 + (2q + 2qεi)σ2 + (2q + 2qεi)σ1,1
+
(
−2q − qεi + ε3
i
)
σ2,1 +
(
q + ε2
i
)
σ2,2.
Then,
πi · πj = 0 if i 6= j, π2
i = λiπi where λi > 0;
as a consequence, these vectors are orthogonal since, for i 6= j,
η(πi, πj) = η(πi · πj , 1) = η(0, 1) = 0.
Introducing the normalized eigenvectors
fi :=
πi
η(πi, πi)
1
2
we obtain an orthonormal frame of normalized idempotent vectors, for any choice of the sign of
the square roots.
Let us now introduce a matrix Ψ = (ψij) such that
∂
∂tα
=
∑
i
ψiαfi, α = 1, 2, . . . , n.
Note that necessarily we have
ΨTΨ = η, ψi1 =
η(πi, 1)
η(πi, πi)
1
2
.
After some computations, we obtain
Ψ =
c
1
2
2
−iq−
1
2 0 −1 1 0 iq
1
2
−iq−
1
2 0 1 −1 0 iq
1
2
1
√
2q
1
2
− i
q
1
4
− 1√
2
− 1√
2
iq
1
4
q
1
2√
2
1
√
2q
1
2
i
q
1
4
− 1√
2
− 1√
2
−iq
1
4
q
1
2√
2
1
√
2q
1
2
− 1
q
1
4
1√
2
1√
2
−q
1
4
q
1
2√
2
1
√
2q
1
2
1
q
1
4
1√
2
1√
2
q
1
4
q
1
2√
2
.
Local Moduli of Semisimple Frobenius Coalescent Structures 71
This matrix diagonalizes U as follows
U := ΨUΨ−1 =
(
ΨT
)−1ÛΨT
=
u1
u2
u3
u4
u5
u6
= 4
√
2q
1
4
0 0 0 0 0 0
0 0 0 0 0 0
0 0 −i 0 0 0
0 0 0 i 0 0
0 0 0 0 −1 0
0 0 0 0 0 1
. (6.4)
The eigenvalues ui stand for ui
(
0, t2, . . . , 0
)
. Note that
ui
(
0, t2, 0, . . . , 0
)
= q
1
4ui(0, 0, . . . , 0) = e
t2
4 ui(0, 0, . . . , 0). (6.5)
6.3 Differential system for deformed flat coordinates
Our goal is to obtain the monodromy data for the small quantum cohomology. Therefore, we
need to consider system (2.2), which we rewrite as follows:
∂zξ =
(
Û − 1
z
µ
)
ξ, (6.6)
∂2ξ = zĈ2ξ, (6.7)
where ξ is a column vector, whose components are ξi = ∂it̃(t, z) (derivatives of a deformed flat
coordinate), and the matrix coefficients are
Û := ηUη−1 =
0 4 0 0 0 0
0 0 4 4 0 0
0 0 0 0 4 0
0 0 0 0 4 0
4q 0 0 0 0 4
0 4q 0 0 0 0
, Ĉ2 ≡
1
4
Û ,
µ = diag(−2,−1, 0, 0, 1, 2), (6.8)
with eigenvalues µα = deg(∂/∂α)−4
2 , 1 ≤ α ≤ 6. As it is customary in the analysis of differential
systems, it is convenient to try a reduction to an equivalent scalar equation. To this purpose,
we introduce the scalar function φ defined by
φ(t, z) :=
ξ1(t, z)
z2
.
In this way, the first equation of system (6.6) becomes a single linear differential equation
z4∂5
zφ+ 10z3∂4
zφ+ 25z2∂3
zφ+ 15z∂2
zφ+
(
1− 1024qz4
)
∂zφ− 2048qz3φ = 0. (6.9)
A column vector solution of (6.6) can be reconstructed from the formulae
ξ1 = z2φ,
ξ2 =
1
4
z2∂zφ,
ξ3 =
1
32
(
z∂zφ+ z2∂2
zφ
)
+ h,
72 G. Cotti, B. Dubrovin and D. Guzzetti
ξ4 =
1
32
(
z∂zφ+ z2∂2
zφ
)
− h,
ξ5 =
1
128
(
∂zφ+ 3z∂2
zφ+ z2∂3
zφ
)
,
ξ6 =
1
512
(
−512qz2φ+
1
z
∂zφ+ 7∂2
zφ+ 6z∂3
zφ+ z2∂4
zφ
)
(6.10)
with a constant h ∈ C to be determined later (formulae (6.14)–(6.16)).
Remark 6.1. The third and the fourth equalities (6.10) follow from the fact that
ξ3 + ξ4 =
1
16
(
z∂zφ+ z2∂2
zφ
)
, ∂z(ξ3 − ξ4) = 0, ∂2(ξ3 − ξ4) = 0,
so that ξ3 − ξ4 = 2h is a constant.
This fact reflects the peculiarity of our systems (6.6) of being direct sum of a 1-dimensional
and 5-dimensional systems. Indeed, by the change of variable ξ̃j = ξj , j = 1, 2, 5, 6, ξ̃3 = ξ3 + ξ4
and ξ̃4 = ξ3 − ξ3, form (6.6) we obtain the systems
∂z ξ̃4 = 0 and ∂z
ξ̃1
ξ̃2
ξ̃3
ξ̃5
ξ̃6
=
0 4 0 0 0
0 0 4 0 0
0 0 0 8 0
4q 0 0 0 4
0 4q 0 0 0
+
1
z
2 0 0 0 0
0 1 0 0 0
0 0 0 0 0
0 0 0 −1 0
0 0 0 0 −2
ξ̃1
ξ̃2
ξ̃3
ξ̃5
ξ̃6
.
The leading matrix has diagonal form 4
√
2q1/4diag(0,−i, i,−1, 1), with distinct eigenvalues. Also
system (6.7) decouples into a direct sum, and
∂2ξ̃4 = 0.
Notice that this decoupling into a direct sum also confirms, in our very particular case, the
general structure (4.5) of the formal solution. However, this decoupling is a peculiarity of the
small cohomology locus of G, but does not apply in general cases at coalescing eigenvalues of U .
The peculiarity is evident when we write system (6.6)–(6.7) as
y =
(
ΨT
)−1
ξ, ∂zy =
(
U +
V
z
)
y, ∂2y =
1
4
zUy,
where U is the diagonal matrix (6.4) and
V =
0 0 i/
√
2 i/
√
2 i/
√
2 i/
√
2
0 0 i/
√
2 i/
√
2 i/
√
2 i/
√
2
−i/
√
2 −i/
√
2 0 0 −i/2 i/2
−i/
√
2 −i/
√
2 0 0 i/2 −i/2
−i/
√
2 −i/
√
2 i/2 −i/2 0 0
−i/
√
2 −i/
√
2 −i/2 i/2 0 0
.
Here y = (y1, . . . , y6)T is a column vector. Now, the aforementioned peculiarity is that the
first two columns of V are equal. Hence, we can decouple into a direct sum using the variables
ỹ1 = y1 − y2, ỹ2 = y1 + y2 and ỹj = yj for j = 3, 4, 5, 6. Moreover, the first two rows are equal,
which implies ∂zy1 = ∂2ỹ1 = 0.
We thank the anonymous referee for suggesting the above observations.
Local Moduli of Semisimple Frobenius Coalescent Structures 73
From (6.7) it follows that
∂2φ =
z
4
∂zφ,
which implies the following functional form
φ(t2, z) = ϕ
(
zq
1
4
)
,
for a scalar function ϕ of one variable. As a consequence, our problem (6.9) reduces to the
solution of a single scalar ordinary differential equation with independent variable w = zq
1
4 and
dependent variable ϕ(w):
w4ϕ(5) + 10w3ϕ(4) + 25w2ϕ(3) + 15wϕ′′ +
(
1− 1024w4
)
ϕ′ − 2048w3ϕ = 0.
Multiplying by w ∈ C∗, we can rewrite this equation in a more compact form
Θ5ϕ− 1024w4Θϕ− 2048w4ϕ = 0, (6.11)
where Θ is the Euler’s differential operator w d
dw . The fact that we have reduced the problem to
a fifth order scalar differential equation reflects the observation of Remark 6.1 that the system
is a direct sum of a trivial 1-dimensional system and a non-trivial 5-dimensional one.
6.3.1 Expected asymptotic expansions
Let Ξ be a fundamental matrix solution of system (6.6), and let Y be defined by
Ξ = ηΨ−1Y. (6.12)
Then, Y is a fundamental solution of system (2.18). The asymptotic theory has been explained
in Section 2.3, and Theorem 2.37 applies. To the formal solution (2.19) it corresponds a formal
matrix solution
Ξformal = ηΨ−1G(z)−1ezU .
To the fundamental solutions Yleft/right, there correspond solutions Ξleft/right. For fixed t2, then
Ξleft/right
(
t2, z
)
≡ Ξleft/right
(
e
t2
4 z
)
has the following asymptotic expansion for z →∞
Ξleft/right
(
t2, z
)
= ηΨ−1
(
1 +O
(
1
z
))
ezU
=
c
1
2
2
(
I +O
(
1
z
))
− iezu1
q
1
2
− iezu2
q
1
2
ezu3
√
2q
1
2
ezu4
√
2q
1
2
ezu5
√
2q
1
2
ezu6
√
2q
1
2
0 0 − iezu3
q
1
4
iezu4
q
1
4
− ezu5
q
1
4
ezu6
q
1
4
−ezu1 ezu2 − ezu3√
2
− ezu4√
2
ezu5√
2
ezu6√
2
ezu1 −ezu2 − ezu3√
2
− ezu4√
2
ezu5√
2
ezu6√
2
0 0 iezu3q
1
4 −iezu4q
1
4 −ezu5q
1
4 ezu6q
1
4
iezu1q
1
2 iezu2q
1
2
ezu3q
1
2√
2
ezu4q
1
2√
2
ezu5q
1
2√
2
ezu6q
1
2√
2
. (6.13)
Now, one of our tasks is to explicitly compute the fundamental matrix solutions Ξleft/right
with behaviour (6.13) and the Stokes matrix connecting them, by means of the formulae (6.10).
Therefore, we need fundamental systems of solutions of (6.9) with the correct asymptotic be-
haviour such that formulae (6.10) will match with (6.13). It will suffice to identify solutions
74 G. Cotti, B. Dubrovin and D. Guzzetti
φ
(
z, t2
)
of (6.9) with the asymptotic behaviour given by the first row of (6.13). Such solu-
tions will be identified in Section 6.5, making use of the analysis of equation (6.9) developed in
Section 6.4 below.
The behaviour in (6.13) also allows to find the correct values of h in (6.10). This values must
be determined in order to match with the asymptotics of the third and fourth rows of (6.13).
We find
h = −c
1
2
2
, for the first column, (6.14)
h =
c
1
2
2
, for the second column, (6.15)
h = 0, for the remaining columns. (6.16)
Indeed, for φ corresponding to the first two columns we have respectively
φ
(
z, t2
)
= −c
1
2
2
iezu1
z2q
1
2
(
1 +O
(
1
z
))
or φ(z, t2) = −c
1
2
2
iezu2
z2q
1
2
(
1 +O
(
1
z
))
.
Since u1 = u2 = 0, the above expressions become
φ
(
z, t2
)
= −c
1
2
2
i
z2q
1
2
(
1 +O
(
1
z
))
.
Then
1
32
(
z∂zφ+ z2∂2
zφ
)
= O
(
1
z
)
.
Comparing with the matrix elements (3, 1), (4, 1) and (3, 2), (4, 2) of (6.13) respectively, we
obtain (6.14) and (6.15). For the remaining columns we proceed in the same way and find (6.16).
6.4 Solutions of the differential system
Defining $ := 4w4 and writing ϕ(w) = ϕ̃($), equation (6.11) becomes
Θ5
$ϕ̃−$Θ$ϕ̃−
1
2
$ϕ̃ = 0, Θ$ := $
d
d$
=
1
4
Θw.
This belongs to the class of generalized hypergeometric differential equations (see [1, 55, 58, 59]
and references therein). By applying the Mellin transform M to it, we obtain the finite difference
equation
s5τ̃(s) =
(
s+
1
2
)
τ̃(s+ 1), τ̃(s) := M(ϕ̃)(s) :=
∫ ∞
0
ϕ̃(t)ts−1 dt,
whose solutions are of the form
τ̃(s) =
Γ(s)5
Γ
(
s+ 1
2
)ψ(s), ψ(s) = ψ(s+ 1).
Hence, we expect that solutions of (6.11) are of the form
ϕ(w) =
1
2πi
∫
Λ
Γ(s)5
Γ
(
s+ 1
2
)ψ(s)4−sw−4s ds,
for suitable chosen paths of integration Λ. Indeed, the lemma below holds:
Local Moduli of Semisimple Frobenius Coalescent Structures 75
Lemma 6.2. The following functions are solutions of the generalized hypergeometric equa-
tion (6.11):
• the function
ϕ1(w) :=
1
2πi
∫
Λ1
Γ(s)5
Γ
(
s+ 1
2
)4−sw−4s ds,
defined for −π
2 < argw < π
2 , and where Λ1 is any line in the complex plane from the point
κ− i∞ to κ+ i∞ for any 0 < κ;
• the function
ϕ2(w) :=
1
2πi
∫
Λ2
Γ(s)5Γ
(
1
2
− s
)
eiπs4−sw−4s ds,
defined for −π
2 < argw < π, and where Λ2 is any line in the complex plane from the point
κ− i∞ to κ+ i∞ for any 0 < κ < 1
2 .
For the proof see Appendix A.
Note that solutions ϕ1 and ϕ2 are C-linearly independent, since their Mellin transforms are.
However we have the following identities
Lemma 6.3. By analytic continuation of the functions ϕ1 and ϕ2, we have
ϕ2
(
weiπ
2
)
= 2πϕ1(w)− ϕ2(w), (6.17)
ϕ2
(
we−iπ
2
)
= 2πϕ1
(
we−iπ
2
)
− ϕ2(w), (6.18)
ϕ2
(
weiπ
2
)
= 2πϕ1(w) + ϕ2
(
we−iπ
2
)
− 2πϕ1
(
we−iπ
2
)
. (6.19)
Proof. We have that
Γ
(
1
2
+ s
)
Γ
(
1
2
− s
)
=
π
sin
(
π
(
1
2 + s
)) =
2πe±iπs
e±2iπs + 1
for a coherent choice of the sign. So
e±2iπs =
2πe±iπs
Γ
(
1
2 + s
)
Γ
(
1
2 − s
) − 1.
First let us choose the identity with (−): we find that
ϕ2
(
weiπ
2
)
=
1
2πi
∫
Λ2
Γ(s)5Γ
(
1
2
− s
)
eiπs
(
2πe−iπs
Γ
(
1
2 + s
)
Γ
(
1
2 − s
) − 1
)
4−sw−4s ds
= 2πϕ1(w)− ϕ2(w),
which is the first identity. The second one can be deduce analogously using the formula with (+)
sign. Finally the third identity is the difference of (6.17) and (6.18). �
The following lemma gives the asymptotic expansions of ϕ1(w) and ϕ1(w) when w → ∞ in
certain sectors. Later, in Lemma 6.6 we will be able to prove the same asymptotics in larger
sectors.
76 G. Cotti, B. Dubrovin and D. Guzzetti
Lemma 6.4. The following asymptotic expansions hold for w → ∞ in the sectors specified
below:
ϕ1(w) = (2π)
3
2
e−4
√
2w
4w2
(
1 +O
(
1
w
))
, for − π
2
< argw <
π
2
,
ϕ2(w) =
iπ
5
2
2w2
(
1 +O
(
1
w
))
, for − π
2
< argw < π.
For the proof, see Appendix A. The sectors where the asymptotics is valid will be enlarged
in Lemma 6.6.
6.5 Computation of monodromy data
6.5.1 Solution at the origin and computation of C0(η, µ,R)
Monodromy data at the origin z = 0 are determined by the action of the first Chern class
c1(G) = 4σ1 on the classical cohomology ring. So,
R =
0 0 0 0 0 0
4 0 0 0 0 0
0 4 0 0 0 0
0 4 0 0 0 0
0 0 4 4 0 0
0 0 0 0 4 0
. (6.20)
By Theorems 2.12 and 2.16, there exists a fundamental matrix solution (2.10)
Y (z) = Φ
(
t2, z
)
zµzR,
for some appropriate converging power series Φ
(
t2, z
)
= 1 +O(z) such that
ΦT
(
t2,−z
)
ηΦ
(
t2, z
)
= η.
Thus, a fundamental matrix for our problem is given by
Ξ0(z) = ηΦ
(
t2, z
)
zµzR = ΦT
(
t2,−z
)−1
ηzµzR.
By applying the iterative procedure in [26] for the proof of Theorem 2.12, at t2 = 0 one finds
the following fundamental solution
Ξ0(0, z) = S(0, z)ηzµzR, (6.21)
S(0, z) =
2z4 + 1 0 0 0 0 0
2z3 1− 4z4 0 0 0 0
z2 −z3 1 0 0 0
z2 −z3 0 1 0 0
z 0 −z3 −z3 4z4 + 1 0
z4 z −z2 −z2 2z3 1− 2z4
+O
(
z5
)
.
Notice that the leading term of the solution Ξ0 in (6.21) is exactly
ηzµzR = c
64
3 z
2 log4(z) 64
3 z
2 log3(z) 8z2 log2(z) 8z2 log2(z) 4z2 log(z) z2
64
3 z log3(z) 16z log2(z) 4z log(z) 4z log(z) z 0
8 log2(z) 4 log(z) 1 0 0 0
8 log2(z) 4 log(z) 0 1 0 0
4 log(z)
z
1
z 0 0 0 0
1
z2 0 0 0 0 0
.
Local Moduli of Semisimple Frobenius Coalescent Structures 77
From the first row, we deduce that near z = 0 any solution of the equation (6.11), i.e.,
Θ5ϕ− 1024z4Θϕ− 2048z4ϕ = 0
is of the form
ϕ(z) =
∑
n≥0
zn
(
an + bn log z + cn log2 z + dn log3 z + en log4 z
)
, (6.22)
where a0, b0, c0, d0, e0 are arbitrary constants, and successive coefficients can be obtained
recursively.
Proposition 6.5. Let R be as in (6.20). Then, C0(η, µ,R) is the algebraic abelian group of
complex dimension 3 given by
C0(η, µ,R) =
1 0 0 0 0 0
α1 1 0 0 0 0
α2 α1 1 0 0 0
α3 α1 0 1 0 0
α4 α2 + α3 α1 α1 1 0
α5 α4 α3 α2 α1 1
: αi ∈ C s.t.
{
α2
1 − α2 − α3 = 0,
α2
2 + α2
3 − 2α1α4 + 2α5 = 0
.
In particular, if F (t) ∈ C[[t]] is a formal power series of the form F (t) = 1+F1t+F2t
2 + · · · , then
the matrix (computed w.r.t. the chosen Schubert basis σ0, σ1, σ2, σ1,1, σ2,1, σ2,2) representing
the endomorphism
λF ∪ (−) : H•(G;C)→ H•(G;C),
where λF ∈ H•(G;C) is such that
F̂ (TG) ∪ λF = F̂ (T ∗G),
is an element of C0(η, µ,R). Here F̂ (V ) denotes the Hirzebruch multiplicative characteristic
class of the vector bundle V → G associated with the formal power series F (t) (see [46]).
Proof. The equations defining the group C0(η, µ,R) are obtained by direct computation from
the requirement that P (z) := zµzR · C · z−Rz−µ is a polynomial of the form P (z) = 1 + A1z +
A2z
2 + · · · , together with the orthogonality condition P (−z)TηP (z) = η. Notice that the
polynomial for the generic matrix of the above form is equal to
P (z) =
1 0 0 0 0 0
zα1 1 0 0 0 0
z2α2 zα1 1 0 0 0
z2α3 zα1 0 1 0 0
z3α4 z2(α2 + α3) zα1 zα1 1 0
z4α5 z3α4 z2α3 z2α2 zα1 1
.
We leave as an exercise to show that such a matrix group is abelian. Let δ1, . . . , δ6 be the Chern
roots of TG. Then, for some complex constants ai,j ∈ C, we have
F̂ (TG) :=
6∏
j=1
F (δj) = 1 + a1σ1 + a2σ2 + a1,1σ1,1 + a2,1σ2,1 + a2,2σ2,2,
F̂ (T ∗G) :=
6∏
j=1
F (−δj) = 1− a1σ1 + a2σ2 + a1,1σ1,1 − a2,1σ2,1 + a2,2σ2,2.
78 G. Cotti, B. Dubrovin and D. Guzzetti
Thus, if
λF = 1 + x1σ1 + x2σ2 + x3σ1,1 + x4σ2,1 + x5σ2,2,
from the condition F̂ (TG) ∪ λF = F̂ (T ∗G) we obtain the constraints
x1 = −2a1,
x2 = 2a2
1,
x3 = 2a2
1,
x4 = 2a1(a2 + a1,1)− 4a3
1 − 2a2,1,
x5 = 4a1a2,1 − 4a2
1(a2 + a1,1) + 4a4
1.
From this it is immediately seen that x2
1 − x2 − x3 = 0 and x2
2 + x2
3 − 2x1x4 + 2x5 = 0. �
6.5.2 Stokes rays and computation of Ξleft, Ξright
According to Theorem 4.5, monodromy data of QH•(G) can be computed starting from a point(
0, t2, 0, . . . , 0
)
of the small quantum cohomology. Moreover, thanks to the isomonodromy theo-
rems, it suffices to do the computation at t2 = 0, i.e., q = 1, where the canonical coordinates (6.5)
are
u1 = u2 = 0, u3 = −4i
√
2, u4 = 4i
√
2, u5 = −4
√
2, u6 = 4
√
2.
The Stokes rays (2.21) are easily seen to be
R13 = R23 = {−ρ : ρ ≥ 0}, R14 = R24 = R34 = {ρ : ρ ≥ 0},
R15 = R25 = {−iρ : ρ ≥ 0}, R16 = R26 = R56 = {iρ : ρ ≥ 0},
R35 =
{
ρe−iπ
4 : ρ ≥ 0
}
, R36 =
{
ρeiπ
4 : ρ ≥ 0
}
,
R45 =
{
−ρeiπ
4 : ρ ≥ 0
}
, R46 =
{
−ρe−iπ
4 : ρ ≥ 0
}
, Rji = −Rij .
We fix the admissible line `
` :=
{
ρeiπ
6 : ρ ∈ R
}
,
so that the sectors for the asymptotic expansion, containing Πleft/right and extending up to the
nearest Stokes rays are
Sright = {z : − π < arg z < π/4}, Sleft = {z : − 0 < arg z < π + π/4}.
For such a choice of the line, according to Theorem 2.42, the structure of the Stokes matrix is
S =
1 0 ∗ 0 0 ∗
0 1 ∗ 0 0 ∗
0 0 1 0 0 ∗
∗ ∗ ∗ 1 0 ∗
∗ ∗ ∗ ∗ 1 ∗
0 0 0 0 0 1
. (6.23)
We use the following notation for fundamental matrices
Ξright =
ξR(1),1 ξR(2),1 ξR(3),1 ξR(4),1 ξR(5),1 ξR(6),1
ξR(1),2 ξR(2),2 ξR(3),2 ξR(4),2 ξR(5),1 ξR(6),2
...
...
...
...
...
...
ξR(1),6 ξR(2),6 ξR(3),6 ξR(4),6 ξR(5),6 ξR(6),6
,
Local Moduli of Semisimple Frobenius Coalescent Structures 79
Ξleft =
ξL(1),1 ξL(2),1 ξL(3),1 ξL(4),1 ξL(5),1 ξL(6),1
ξL(1),2 ξL(2),2 ξL(3),2 ξL(4),2 ξL(5),1 ξL(6),2
...
...
...
...
...
...
ξL(1),6 ξL(2),6 ξL(3),6 ξL(4),6 ξL(5),6 ξL(6),6
.
Note, in particular, that (6.23) implies that the fifth columns of Ξright and Ξleft coincide.
Then ξL(5),1 is the analytical continuation of ξR(5),1 on Sleft. Moreover, the exponential ezu5 dom-
inates all others ezuj ’s in the sector between the rays R45 and R46, i.e., for −π − π/4 < arg z <
−π + π/4. This implies that the asymptotics
ξL(5),1 = ξR(5),1 =
c
1
2
2
√
2
ezu5
(
1 +O
(
1
z
))
is valid in the whole sector −π − π/4 < arg z < π + π/4. By Lemma 6.4,
c
1
2
2π
3
2
z2ϕ1(z) =
c
1
2
2
√
2
ezu5
(
1 +O
(
1
z
))
, for − π
2
< arg z <
π
2
.
Since the exponential ezu5 is dominated by all others exponentials ezuj in the region between R35
and R36, namely for −π/4 < arg z < π/4, we conclude necessarily that
c
1
2
2π
3
2
z2ϕ1(z) = ξ
L/R
(5),1(z).
This determines the 5-th column of Ξright and Ξleft in terms of ϕ1, using equations (6.10), (6.16).
We also obtain an improvement of Lemma 6.4:
Lemma 6.6. ϕ1 and ϕ2 have the following asymptotic behaviour for w → ∞ in the sectors
below
ϕ1(w) = (2π)
3
2
e−4
√
2w
4w2
(
1 +O
(
1
w
))
, for − π − π
4
< argw < π +
π
4
,
ϕ2(w) =
iπ
5
2
2w2
(
1 +O
(
1
w
))
, for − π
2
< argw < π.
We are ready to determine the other columns of Ξleft/right. By Lemma 6.6,
− c
1
2
2π
3
2
z2ϕ1
(
zeiπ
2
)
=
c
1
2
2
√
2
ezu3
(
1 +O
(
1
z
))
, for − 2π +
π
4
< arg z <
3π
4
, (6.24)
c
1
2
2π
3
2
z2ϕ1
(
zeiπ
)
=
c
1
2
2
√
2
ezu6
(
1 +O
(
1
z
))
, for − 2π − π
4
< arg z <
π
4
. (6.25)
We consider first (6.24). Being solutions of a differential equation, the following holds:
− c
1
2
2π
3
2
z2ϕ1
(
zeiπ
2
)
= linear combination of the ξR(1),i, 1 ≤ i ≤ 6.
On the other hand, ezu3 is dominated by all other ezui ’s in the sector −π+ π/4 < arg z < −π/2
between R45 and R35. This requires that the linear combination necessarily reduces to
− c
1
2
2π
3
2
z2ϕ1
(
zeiπ
2
)
= ξR(3),1.
80 G. Cotti, B. Dubrovin and D. Guzzetti
Now we consider (6.25). As above, since ezu6 is dominated by all the other ezui ’s in the sector
−5π/4 < arg z < −3π/4 between R46 and R45, we conclude that
c
1
2
2π
3
2
z2ϕ1
(
zeiπ
)
= ξR(6),1.
Analogously we find that
− c
1
2
2π
3
2
z2ϕ1
(
ze−iπ
2
)
=
c
1
2
2
√
2
ezu4
(
1 +O
(
1
z
))
for − 3π
4
< arg z < π +
3π
4
,
c
1
2
2π
3
2
z2ϕ1
(
ze−iπ
)
=
c
1
2
2
√
2
ezu6
(
1 +O
(
1
z
))
on − π
4
< arg z < 2π +
π
4
.
By dominance considerations as above, we conclude that
ξL(4),1 = − c
1
2
2π
3
2
z2ϕ1
(
ze−iπ
2
)
, ξL(6),1 =
c
1
2
2π
3
2
z2ϕ1
(
ze−iπ
)
.
The above results reconstruct (using identities (6.10),(6.16)) three columns of matrices Ξright
and Ξleft respectively. As far as the first two columns are concerned, we invoke again Lemma 6.6
for ϕ2, which yields
c
1
2
π
5
2
z2ϕ2
(
zeiπ
2
)
= − ic
1
2
2
(
1 +O
(
1
z
))
on − π < arg z <
π
2
,
c
1
2
π
5
2
z2ϕ2
(
ze−iπ
2
)
= − ic
1
2
2
(
1 +O
(
1
z
))
on 0 < arg z <
3π
2
.
Exactly as before, dominance relations of the exponentials ezui yield
c
1
2
π
5
2
z2ϕ2
(
zeiπ
2
)
= ξR(1),1 = ξR(2),1,
c
1
2
π
5
2
z2ϕ2
(
ze−iπ
2
)
= ξL(1),1 = ξL(2),1.
Using (6.10), (6.14), (6.15), the first two columns are constructed. Summarizing, we have
determined the following columns in terms of ϕ1 and ϕ2.
Ξright =
(
ξR(1),1 ξR(2),1 ξR(3),1 unknown ξR(5) ξR(6),1
...
...
...
...
...
...
)
,
Ξleft =
(
ξL(1),1 ξL(2),1 unknown ξL(4),1 ξL(5),1 ξL(6),1
...
...
...
...
...
...
)
.
In Section 6.5.3 we show that the above partial information and the constraint (2) in Theo-
rem 2.43 are sufficient to determine the Stokes and central connection matrices simultaneously.
Since constraint (2) holds only in case S and C are related to Frobenius manifolds, we sketch be-
low – for the sake of completeness – the general method to obtain the missing columns of Ξleft/right
and S, in a pure context of asymptotic analysis of differential equations.
We observe that
− c
1
2
2π
3
2
z2ϕ1
(
ze−iπ
2
)
=
c
1
2
2
√
2
eu4z(1 +O(1/z)), for − π +
π
4
< arg z < π +
3π
4
.
Local Moduli of Semisimple Frobenius Coalescent Structures 81
The sub-sector −π < arg z < −3π/4 of Sright is not covered by the sector where the above
asymptotic behaviour holds. On the sub-sector, the dominance relation
∣∣ezu4 | < |ezu5
∣∣ holds.
Thus,
ξR(4),1 = − c
1
2
2π
3
2
z2ϕ1
(
ze−iπ
2
)
+ vξR(5),1, (6.26)
for some complex number v ∈ C, to be determined. Analogously, we observe that
− c
1
2
2π
3
2
z2ϕ1
(
ze−i 3π
2
)
=
c
1
2
2
√
2
eu4z(1 +O(1/z)), for − 2π − 3π
4
< arg z < −π
4
.
The sub-sector −π/4 < arg z < π/4 of Sright is not covered by the sector where the asymptotic
behaviour holds. Now, the following dominance relations hold:
∣∣ezu4
∣∣ < ∣∣ezui∣∣, for i = 1, 2, 3, 6,
in 0 < arg z < π/4; for i = 6 in −π/4 < arg z < 0. Thus
ξR(4),1 = − c
1
2
2π
3
2
z2ϕ1
(
zei 3π
2
)
+ γ1ξ
R
(1),1 + γ3ξ
R
(3),1 + γ6ξ
R
(6),1 (6.27)
for some complex number γ1, γ3, γ6 ∈ C, to be determined.22 The above (6.26) and (6.27)
become a 6-terms linear relation between functions ϕ2
(
zei kπ
2
)
, as follows
−ϕ1
(
ze−iπ
2
)
+ vϕ1(z) = −ϕ1
(
zei 3π
2
)
+
γ1
π
ϕ2
(
zeiπ
2
)
− γ3ϕ1(zeiπ
2 ) + γ6ϕ1
(
zeiπ
)
,
ϕ1(z) =
1
2π
[
ϕ2(z) + ϕ2
(
zeiπ
2
)]
.
Some further information is needed in order to determine the unknown constants v, γ1, γ3, γ6,
as in the following
Lemma 6.7. The solutions of the equation (6.11) satisfy the identity
ϕ
(
zei 5π
2
)
− 5ϕ
(
ze2πi
)
+ 10ϕ
(
zei 3π
2
)
− 10ϕ
(
zeiπ
)
+ 5ϕ
(
zeiπ
2
)
− ϕ(z) = 0. (6.28)
Proof. The equation Θ5ϕ − 1024z4Θϕ − 2048z4ϕ = 0 admits the symmetry z 7→ zeiπ
2 . This
means that if ϕ is a solution of the equation then also ϕ
(
zeiπ
2
)
is. Such a symmetry defines
a linear map on the vector space of solutions of the equation defined in a neighborhood of z = 0.
Because of this symmetry, the form (6.22) can be refined as
ϕ(z) =
∑
n≥0
z4n
(
an + bn log z + cn log2 z + dn log3 z + en log4 z
)
, (6.29)
where a0, b0, c0, d0, e0 are arbitrary constants, and successive coefficients can be obtained
recursively. In the basis of solutions of the form (6.29) with (a0, b0, c0, d0, e0) = (1, 0, . . . , 0),
(0, 1, 0, . . . , 0) and so on, the matrix of the operator
(Aϕ)(z) := ϕ
(
zeiπ
2
)
is of triangular form with 1’s on the diagonal. Hence, by Cayley–Hamilton theorem we deduce
that (A− 1)5 = 0, namely
A5 − 5A4 + 10A3 − 10A2 + 5A− 1 = 0. �
22There is no need to include a term +γ2ξ
R
(2),1 in the linear combination, since ξR(1),1 = ξR(2),1.
82 G. Cotti, B. Dubrovin and D. Guzzetti
Λ1/2
deformed path
Figure 14. Deformation of the path Λ1/2, in order to apply residue theorem. Poles are represented.
The relation (6.28) applied to ϕ2 determines v, γ1, γ3, γ6. For example, v = 6. This
determines ξR(4),1 through formula (6.26). The fourth column of Ξright is then constructed with
formula (6.10) applied to ξR(4),1 (with h = 0). The value v = 6 will be determined again in
Section 6.5.3 making use of the constraint (2) of Theorem 2.43.
Proceeding in the same way, we also determine ξL(3),1. One observes that
− c
1
2
2π
3
2
z2ϕ1
(
zei 3π
2
)
=
c
1
2
2
√
2
ezu3(1 +O(1/z)), for
π
4
< arg z <
3π
2
+ 2π,
− c
1
2
2π
3
2
z2ϕ1
(
zeiπ
2
)
=
c
1
2
2
√
2
ezu3(1 +O(1/z)), for − 2π − π
2
< arg z <
3π
4
.
The first asymptotic relation does not hold in the sub-sector −π/4 < arg z < π/4 of Sleft. The
second one does not hold in 3π/4 < arg z < 5π/4. Then, the dominance relations in these
sub-sectors generate a 6-terms linear relation with unknown coefficients. The coefficients are
determined by (6.28).
Once Ξleft/right has been determined, S can be computed by direct comparison of the two
fundamental matrices (formula (6.28) need to be used at some point of the comparison). The
final result is the Stokes matrix S of formula (6.31) below with v = 6.
6.5.3 Computation of Stokes and central connection matrices,
using constraint (2) of Theorem 2.43
We start from formula (6.26):
ξR(4),1 = − c
1
2
2π
3
2
z2ϕ1
(
ze−iπ
2
)
+ vξR(5),1 ≡
c
1
2
2π
3
2
z2
(
−ϕ1
(
ze−iπ
2
)
+ vϕ1(z)
)
.
Though v has already been determined above, we show that constraint (2) of Theorem 2.43
suffices to determine the value of v and reconstruct both the Stokes and the central connection
matrices, as follows.
The definition of the central connection matrix C and the transformation (6.12) imply that
Ξright = Ξ0C.
The matrix C can be obtained by comparing the leading behaviours of Ξright and Ξ0 near z = 0.
The leading behaviour of Ξ0 in (6.21) is ηzµzR. In order to find the behaviour of Ξright, we
need to compute the behaviour of ϕ1 and ϕ2 near z = 0. To this end, we consider the integral
representations in Lemma 6.2, and deform both paths Λ1 and Λ2 to the left, as shown in Fig. 14.
By residue theorem, we obtain a representations of ϕ1 and ϕ2 as a series of residues at the poles
s = 0,−1,−2, . . . . Then, by the reconstruction dictated by equations (6.10), (6.14), (6.15),
Local Moduli of Semisimple Frobenius Coalescent Structures 83
(6.16), for each entry of the matrix Ξright we obtain an expansion in z and log z, converging for
small |z|.
For example, let us compute the first and second columns of the matrix C: by deformation
of the path Λ2 we obtain that for small z the following series expansions hold:
ξR(1),1 = ξR(2),1 =
c
1
2
π
5
2
z2ϕ2
(
zeiπ
2
)
=
c
1
2
π
5
2
z2
∞∑
n=0
res
s=−n
(
Γ(s)5Γ
(
1
2
− s
)
e−iπs4−sz−4s
)
= α1z
2 log4 z + α2z
2 log3 z + α3z
2 log2 z + α4z
2 log z + α5z
2 +O
(
z4
)
,
where αi can be explicitly computed. By comparison with the first row of ηzµzR we determine
the entries
C11 = C12 =
3
64c
α1, C21 = C22 =
3
64c
α2,
C51 = C52 =
1
4c
α4, C61 = C62 =
1
c
α5.
For the other entries we have to consider expansions of ξR(1),3, ξR(2),3, ξR(1),4, ξR(2),4. For example,
ξR(1),3 = ξR(2),4 =
c
1
2
π
5
2
· 1
32
(
zϕ′2
(
zeiπ
2
)
+ z2ϕ′′2
(
zeiπ
2
))
− c
1
2
2
= −c
1
2
2
+
c
1
2
2π
5
2
∞∑
n=0
res
s=−n
(
Γ(s)5Γ
(
1
2
− s
)
e−iπs4−ss2z−4s
)
= β1 log2 z + β2 log z + β3 +O
(
z4
)
,
where βi can be explicitly computed. So, by comparison of the third row of gzµzR we obtain
C31 = C42 =
β3
c
.
Analogously one obtains C32 = C41. Note that the other entries Cij , with j = 3, 4, 5, 6, are
uniquely determined only by the expansion of ξR(j),i because of (6.16). The compuation for all
the other entries of C can be done in the same way, so it will not be repeated here. Due to the
length of the result, we write the whole C in Appendix B. As it can be seen in Appendix B, only
the fifth column of C is expressed in terms of the constant v. This v will now be determined.
Since S and C are associated with a Frobenius manifold, the constraint (2) of Theorem 2.43
holds:
S = C−1e−πiRe−πiµη−1
(
CT
)−1
. (6.30)
Substituting C of Appendix B with an indeterminate v in the above constraint, we obtain the
Stokes matrix
S =
1 0 4 0 0 4
0 1 4 0 0 4
0 0 1 0 0 6
−4 −4 −16 1 6− v −6
4(v − 1) 4(v − 1) 16v − 26 −v (v − 6)v + 1 6v − 16
0 0 0 0 0 1
. (6.31)
By a direct comparison with the expected matrix form (6.23), which dictates that S45 = 0 and
S55 = 1, we conclude that necessarily
v = 6.
In this way we have completely determined both the Stokes and central connection matrices as
well as the fundamental matrix Ξright. See also (6.37) below.
84 G. Cotti, B. Dubrovin and D. Guzzetti
6.6 Monodromy data of the small quantum cohomology
and exceptional collections in Db(G)
The monodromy data R and C computed above can be read as characteristic classes of objects of
an exceptional collection in Db(G), as it has been conjectured by one of the authors [25], though
the formulation for the central connection matrix was not well understood then. Following [53]
where the role of the Γ̂±-classes (characteristic classes obtained by the Hirzebruch’s procedure
starting from the series expansion of the functions Γ(1 ± t) near t = 0) was pointed out, we
claim that the central connection matrix (for canonical coordinates in triangular/lexicographical
order) can be identified with the matrix of the C-linear morphism
X±G : K0(G)⊗Z C→ H•(G;C),
E 7→ 1
(2π)2c
1
2
Γ̂±(G) ∪ Ch(E),
Γ̂±(G) :=
∏
j
Γ(1± δj), where δj ’s are the Chern roots of TG,
Ch(V ) :=
∑
k
e2πixk , xk’s are the Chern roots of a vector bundle V ,
expressed w.r.t.
• an exceptional basis (εi)i ofK0(G)⊗ZC, i.e., satisfying χ(εi, εi) = 1, and the Grothendieck–
Euler–Poincaré orthogonality conditions χ(εi, εj) = 0 for i > j, obtained by projection of
a full exceptional collection (Ei)i in Db(G);
• a basis in H•(G;C) related to (σ0, σ1, σ2, σ1,1, σ2,1, σ2,2) (the Schubert basis we have fixed)
by a (η, µ)-orthogonal-parabolic G endomorphism (as described in Section 2.1) which com-
mutes with the operator of classical ∪-multiplication c1(G) ∪ − : H•(G;C)→ H•(G;C).
By application of the constraint (6.30) and the Grothendieck–Hirzebruch–Riemann–Roch theo-
rem, one can prove that the Stokes matrix (in triangular/lexicographical order) is equal to the
inverse of the Gram matrix:(
S−1
)
ij
= χ(εi, εj).
See [18] for a proof.
Remark 6.8. As it was formulated in Theorem 1.2 in the Introduction and in Section 3, some
natural transformations are allowed, such as
• the left action of the group C0(η, µ,R):
no action on S, C 7−→ GC, (6.32)
where G ∈ C0(η, µ,R) and has the form prescribed by Proposition 6.5;
• the right action of the group (Z/2Z)×6:
S 7−→ ISI, C 7−→ CI, (6.33)
where I is a diagonal matrix of 1’s and −1’s;
• the right action of the braid group B6:
S 7−→ AβS
(
Aβ
)T
, C 7−→ C
(
Aβ
)−1
, (6.34)
as in formulae (3.4) and (3.5).
Local Moduli of Semisimple Frobenius Coalescent Structures 85
The above actions naturally manifest respectively on the space H•(G;C), on the set of full
exceptional collections in the category Db(G), and/or on the set of exceptional bases of the
complexified Grothendieck group K0(G)⊗Z C. More precisely,
• C0(η, µ,R) acts on H•(G;C) as (η, µ)-orthogonal-parabolic endomorphisms commuting
with the classical ∪-product by the first Chern class c1(G);
• the action of the shift functor [1] : Db(G) → Db(G) on the objects of a full exceptional
collection projects as an action of (Z/2Z)×6 on K0(G) ⊗Z C by changing of signs of the
elements of the corresponding exceptional basis;
• the braid group B6 acts on the set of exceptional collections (and the corresponding ex-
ceptional bases) as follows: the generator βi,i+1 (1 ≤ i ≤ 5) transforms the collection
(E1, . . . , Ei−1, Ei, Ei+1, Ei+2, . . . , E6) into (E1, . . . , Ei−1, LEiEi+1, Ei, Ei+2, . . . , E6), where
the object LEiEi+1 is defined, up to unique isomorphism, by the distinguished triangle
LEiEi+1[−1]→ Hom•(Ei, Ei+1)⊗ Ei → Ei+1 → LEiEi+1.
Notice that our definition of braid mutations of exceptional objects differs from the one given,
for example, in [40] by a shift: this difference is important in order to obtain the coincidence
of the braid group action on the matrix representing the morphism X±G with the action on the
central connection matrix.
Remark 6.9. The conjecture we are discussing was also formulated in [33] in the same time
as [28] for any Fano manifold X. In [33] the authors stress the relevance of the class Γ̂+(X),
while in [28] it was conjectured that Γ̂−(X) is the relevant characteristic class. As we will show
below, Γ̂+(X) and Γ̂−(X) can be interchanged by the action (6.32) of the group C0(η, µ,R).
We now show that the monodromy data computed in Section 6.5 are of the above form for
an exceptional collection in the same orbit of the Kapranov collection, under the action of the
braid group. The Kapranov exceptional collection for G is formed by vector bundles Sλ(S∗)
(S is the tautological bundle), where Sλ denotes the Schur functor corresponding to the Young
diagram λ.23 In the general case of GC(k, n), the graded Chern character of these bundles is
given by
Ch
(
Sλ(S∗)
)
= sλ
(
e2π
√
−1x1 , . . . , e2π
√
−1xk
)
:=
det
(
e2π
√
−1xi(λj+r−j)
)
1≤i,j≤k∏
i<j
(
e2π
√
−1xi − e2π
√
−1xj
) ,
i.e., the Schur polynomial calculated at the Chern roots x1, . . . , xk of S∗. In our case we obtain
the following classes: posing a := e2πix1 and b := e2πix2 with x1 + x2 = σ1 and x1x2 = σ1,1 we
have that
for λ = 0 Ch
(
Sλ(S∗)
)
= 1,
for λ = , Ch
(
Sλ(S∗)
)
= a+ b,
for λ = , Ch
(
Sλ(S∗)
)
= (a+ b)2 − ab,
for λ = , Ch
(
Sλ(S∗)
)
= ab,
for λ = , Ch
(
Sλ(S∗)
)
= (a+ b)ab,
for λ = , Ch
(
Sλ(S∗)
)
= a2b2.
23The reader can find the definition of Schur functors as endo-functors of the category of vector spaces in [32].
The definition easily extends to the category of vector bundles.
86 G. Cotti, B. Dubrovin and D. Guzzetti
Observing that
ab = 1 + 2πiσ1 − 2π2(σ2 + σ1,1)− 8
3
iπ3σ2,1 +
4
3
π4σ2,2,
a+ b = 2 + 2πiσ1 − 2π2σ2 + 2π2σ1,1 +
4
3
iπ3σ2,1,
after some computations one obtains all graded Chern characters. Recalling the value of the
Γ̂∓-class
Γ̂∓(G) = 1± 4γσ1 +
1
6
(
48γ2 + π2
)
(σ1,1 + σ2)± 4
3
(
16γ3 + γπ2 − ζ(3)
)
σ2,1
+
1
36
(
768γ4 + 96γ2π2 − π4 − 192γζ(3)
)
σ2,2
we can explicitly compute all the classes
1
4π2c
1
2
(
Γ̂∓(G) ∪ Ch
(
Sλ(S∗)
))
.
We denote by C∓Kap the matrix obtained in this way: in Appendix B the reader can find the
entries of the matrix C−Kap.
The Stokes matrix can be put in triangular form by a suitable permutation of (u1, . . . , u6),
to which a permutation matrix P is associated, according to the transformations (3.1). There
are two permutations which yield PSP−1 in triangular form, namely
τ1 : (u1, u2, u3, u4, u5, u6) 7→ (u′1, u
′
2, u
′
3, u
′
4, u
′
5, u
′
6) := (u5, u4, u2, u1, u3, u6), (6.35)
τ2 : (u1, u2, u3, u4, u5, u6) 7→ (u′1, u
′
2, u
′
3, u
′
4, u
′
5, u
′
6) := (u5, u4, u1, u2, u3, u6). (6.36)
In both cases, the Stokes matrix S in (6.31), with v = 6, becomes
S 7−→ PSP−1 =
1 −6 20 20 70 20
0 1 −4 −4 −16 −6
0 0 1 0 4 4
0 0 0 1 4 4
0 0 0 0 1 6
0 0 0 0 0 1
. (6.37)
The matrix C in Appendix B, with v = 6, becomes
C 7→ CP−1. (6.38)
Theorem 6.10. The Stokes and connection matrices at 0 ∈ QH•(G) are related to the ex-
ceptional block collections obtained from the Kapranov block collection by mutations under the
inverse of the braid β12β56β45β23β34 or the braid β34β12β56β45β23β34 (the action of β34 acting
just as a permutation of the third and fourth elements of the block).
It is important to remark that the Kapranov 5-block exceptional collection itself appears
neither at t = 0 nor anywhere else along the locus of the small quantum cohomology, see
Corollary 6.13 below.
Proof. Consider the monodromy data of the quantum cohomology of the Grassmannian G at
0 ∈ QH•(G), as computed in Section 6.5.3 with respect to an admissible line24 ` = `(φ) of slope
24The computations have been done for φ = π/6, but nothing changes if 0 < φ < π
4
, since the sectors, where
the asymptotic behaviours are studied, are the same Sleft/right.
Local Moduli of Semisimple Frobenius Coalescent Structures 87
0 < φ < π
4 and w.r.t. the basis of solutions (6.21). These are the matrix S in formula (6.31) and
the matrix C in Appendix B, with v = 6. Arrange S in triangular form as in (6.37), with P
associated with one of the above permutations τ1 or τ2 above, and transform C as in (6.38). The
data so obtained are related to the Kapranov exceptional collection by a finite sequence of natural
transformations (6.32), (6.33), (6.34). More precisely, the following sequence transforms CP−1
into C−Kap:
(1) the change of signs in the normalised idempotents vector fields, determined by the ac-
tion (6.33) of the diagonal matrix I := diag(1,−1,−1, 1,−1, 1) (if we start from the cell
where τ1 is lexicographical), or I := diag(1,−1, 1,−1,−1, 1) (if we start from the cell
where τ2 is lexicographical),
(2) change of solution at the origin through the action (6.32), with G equal to
A =
1 0 0 0 0 0
2iπ 1 0 0 0 0
−2π2 2iπ 1 0 0 0
−2π2 2iπ 0 1 0 0
−1
3
(
8iπ3
)
−4π2 2iπ 2iπ 1 0
4π4
3 −1
3
(
8iπ3
)
−2π2 −2π2 2iπ 1
∈ C0(η, µ,R),
(3) the action (6.34) with either the braid β12β56β45β23β34 (if we start from the cell where τ1
is lexicographical), or the braid β34β12β56β45β23β34 (if we start from the cell where τ2 is
lexicographical).
Moreover, CP−1 in (6.38) is transformed into C+
Kap if, after the sequence of the above transfor-
mations (1), (2), (3) above, the following transformation is further applied:
(4) the action (6.32), with matrix G equal to
B =
1 0 0 0 0 0
−8γ 1 0 0 0 0
32γ2 −8γ 1 0 0 0
32γ2 −8γ 0 1 0 0
8
3
(
ζ(3)− 64γ3
)
64γ2 −8γ −8γ 1 0
64
3
(
16γ4 − γζ(3)
)
8
3
(
ζ(3)− 64γ3
)
32γ2 32γ2 −8γ 1
∈ C0(η, µ,R).
The inverse of the Stokes matrix obtained from PSP−1 in (6.37) by either the sequence (1),
(2), (3) or (1), (2), (3), (4) (recall that steps (2) and (4) act trivially on S) coincides with the
following Gram matrix of the Kapranov exceptional collection
GKap =
1 4 10 6 20 20
0 1 4 4 16 20
0 0 1 0 4 10
0 0 0 1 4 6
0 0 0 0 1 4
0 0 0 0 0 1
. (6.39)
�
Remark 6.11. In both cases C+
Kap and C−Kap, the relation (6.30) holds between C±Kap and G−1
Kap.
Remark 6.12. The algebro-geometric meaning of the matrices A and B of Theorem 6.10 will
be thoroughly explained in our [18].
88 G. Cotti, B. Dubrovin and D. Guzzetti
6.7 Reconstruction of monodromy data along the small quantum locus
In this section we reconstruct the monodromy data at all other points of the small quantum
cohomology of G, by applying the procedure described in Section 4.1, and already illustrated in
Section 5.
We identify the small quantum cohomology with the set of points t =
(
0, t2, 0, . . . , 0
)
. These
points can be represented on the real plane
(
Re t2, Im t2
)
. At a point
(
0, t2, 0, . . . , 0
)
, the cano-
nical coordinates are (6.5), so that the Stokes rays are
Rij
(
t2
)
= et
2/4Rij(0) ≡ e−i Im t2/4Rij(0),
where Rij(0) are the rays Rij of Section 6.5.2. Let ` be a line of slope ϕ ∈ ]0, π/4[, admissible
for t2 = 0, i.e., for the Stokes rays Rij(0). Then, whenever Im t2 ∈ π · Z− 4φ, at least a pair of
rays Rij
(
t2
)
and Rji
(
t2
)
lie along the line `, for some (i, j). This means that the small quantum
cohomology of G is split into the following horizontal bands of the
(
Re t2, Im t2
)
-plane:
Hk :=
{
t2 : kπ − 4φ < Im t2 < (k + 1)π − 4φ
}
, k ∈ Z.
If t2 varies along a curve connecting two neighbouring bands, at least a pair of opposite rays
Rij
(
t2
)
and Rji
(
t2
)
cross ` in correspondence with t2 crossing the border between the bands.
A point
(
0, t2, 0, . . . , 0
)
, such that t2 is interior to a band, is a semisimple coalescence point,
where Theorem 4.5 applies. The polydisc Uε1
(
u
(
0, t2, . . . , 0
))
is split into two `-cells. Each cell
corresponds, through the coordinate map p 7→ u(p), to the closure of an open connected subset
of an `-chamber of QH•(G), as explained in Section 4.1. Therefore, each band Hk precisely
belongs to the boundary of two `-chambers corresponding to the two cells, while each line
Im t2 = kπ − 4φ between two bands Hk−1 and Hk belongs to the intersection of the boundaries
of four neighbouring chambers of QH•(G). As explained in Section 4.1, the monodromy data
computed via Theorem 4.5 in Uε1
(
u
(
0, t2, . . . , 0
))
are the data of the two chambers sharing the
boundary Hk. In particular, as a necessary consequence of Theorem 4.5, these data are the data
at each point of Hk. This means that the monodromy data are constant in each band Hk.
In order to compute the monodromy data in every chamber of QH•(G) is sufficient to apply
the procedure of Section 4.1 starting from the data C, S computed at t = 0 in Section 6.5.3. Pre-
liminarily, by a permutation P , we have obtained upper triangular PSP−1 and the corresponding
CP−1 in (6.37) and (6.38), which are the monodromy data in the cell of Uε1(u′(0, 0, . . . , 0)) where
u′1(0, 0, . . . , 0), . . . , u′6(0, 0, . . . , 0) are in lexicographical order as in (6.35) or (6.36). Thus, they
are the data of the band H0. Then, the braid group actions (3.4) and (3.5) can be applied. In
particular, we have computed the action of those braids which allow to pass from the chamber
(with lexicographical order) whose boundary contains H0, to the chambers whose boundary
contains Hk, for k = 1, 2, . . . , 8. The values of S and C so obtained are, as explained above, the
constant monodromy data for H0, H1, . . . , H8. They are reported in Table 3. From the table,
we can read the monodromy data for the whole small quantum cohomology, since for any k ∈ Z,
the data for Hk+8 are the same as for Hk, as will be clear from the explanation below.
We need to determine the braid connecting neighbouring bandsHk’s, fromH0 toH1, fromH1
to H2, and so on. The passage from Hk to Hk+1 is achieved by increasing Im(t2), to which
a clockwise rotation of the Stokes rays corresponds. In order to identify the corresponding braid,
we have to keep track of the rays which cross a fixed ` with slope φ ∈ ]0, π/4[. Equivalently, we
can consider a fixed configuration u1
(
0, t2, . . . , 0
)
, . . . , u6
(
0, t2, . . . , 0
)
in lexicographical order,
corresponding to a fixed t =
(
0, t2, 0, . . . , 0
)
in H0, so that the corresponding rays Rij
(
t2
)
are
fixed. Then, we let ` rotate counter-clockwise increasing φ, with the consequent gliding of the
`-horizontal bands towards Im
(
t2
)
→ −∞, and we keep track of the rays which are crossed by `.
In order to apply the procedure explained in Section 4.1, we actually need to start with a fixed
configuration u1(t), . . . , u6(t) of distinct canonical coordinates. This is achieved by taking t
Local Moduli of Semisimple Frobenius Coalescent Structures 89
slightly away from H0, in the interior of one of the two chambers whose boundaries contain H0,
so that the two coalescing canonical coordinates (which equal 0 in any H0) slightly split. It is
after this splitting that we let ` rotate and keep track of the rays which are crossed by `.
This process is shown in Fig. 15. The two canonical coordinates close to 0 (the centres of
the circles) come from the splitting of the two coalescing eigenvalues 0. The rays Lj(φ) defined
in (3.2) are represented. Their clockwise rotation corresponds to the counter-clockwise rotation
of `. A Stokes ray Rj,j+1(t) is crossed by ` every time a ray Lj(φ) aligns with Lj+1(φ). This
yields the braid βj,j+1. The order uj , uj+1 is `-lexicographical with reference to ` just before the
crossing. Each time ` has just crossed a ray Rj,j+1(t) (i.e., Lj(φ) has just crossed Lj+1(φ)), the
coordinates uj ’s must be relabelled in the `-lexicographical order corresponding to ` just after
the crossing.
As it can be read in Fig. 15 and Table 3, the passage from H0 to Hk, for k = 1, 2, . . . , 8, is
obtained by composition of the braids
ω1 := β12β56, ω2 := β23β45β34β23β45, ω̂1 := β12β34β56,
in the form of products of increasing length
ω1, ω1ω2, ω1ω2ω̂1, ω1ω2ω̂1ω2, ω1ω2ω̂1ω2ω1,
ω1ω2ω̂1ω2ω1ω2, ω1ω2ω̂1ω2ω1ω2ω̂1, ω1ω2ω̂1ω2ω1ω2ω̂1ω2.
The result of Table 3 is obtained applying formula (3.4).
The last braid ω1ω2ω̂1ω2ω1ω2ω̂1ω2 corresponds to a complete 2π-rotation of the admissible
line `. It is not difficult to show that
ω1ω2ω̂1ω2ω1ω2ω̂1ω2 = (β12β23β34β45β56)6,
the right hand-side being the generator of the center of the braid group B6 in Lemma 3.3 (the
proof can be done by graphically representing the two braids in the l.h.s. and r.h.s. respectively,
and noticing that they can be deformed the one into the other). This corresponds to the cyclical
repetition of the same Stokes matrix in Hk and Hk+8 (while the central connection matrix C is
transformed to M−1
0 C).
Notice that the action (3.3) of β34 in ω̂1 is a permutation of the third and fourth rows and
columns of the Stokes matrix, because the entry (3, 4) is zero (see (4.8)).
It follows from Table 3 and the explicit Gram matrix of the Kapranov exceptional collec-
tion (6.39) that the following holds:
Corollary 6.13. The Kapranov 5-block exceptional collection is not associated with the small
quantum cohomology locus.
Remark 6.14. There is a remarkable similarity between the above cyclical repetition and the
fact that exceptional collections are organised in algebraic structures called helices, introduced
in [37, 41], and extensively developed in [38, 39, 40]. This will be thoroughly explained in our
paper [18].
7 A note on the topological solution for Fano manifolds
For quantum cohomologies of smooth projective varieties, a fundamental system of solutions of
the equation for gradients of deformed flat coordinates∂αζ = zCαζ,
∂zζ =
(
U +
1
z
µ
)
ζ,
(7.1)
can be expressed in enumerative-topological terms, namely the genus 0 correlations functions.
90 G. Cotti, B. Dubrovin and D. Guzzetti
Table 3. List of Stokes matrices for all bands decomposing the small quantum cohomology of G:
the computation is done at a point
(
0, t2, 0, . . . , 0
)
w.r.t. a line ` of slope φ ∈ ]0, π/4[, admissible for
t = 0. The starting matrix Slex in H0 is PSP−1 of formula (6.37), with signs changed by (6.33)
with I = diag(−1, 1, 1,−1, 1,−1). The braids acting on the monodromy data are ω1 := β12β56,
ω2 := β23β45β34β23β45 and ω̂1 := β12β34β56. The table is computed using (3.4), namely by successively
applying (3.3) for the elementary braids.
Band Hk, 0 ≤ k ≤ 8 Slex Braid
0 < Im
(
t2
)
+ 4φ < π
1 6 −20 20 −70 20
0 1 −4 4 −16 6
0 0 1 0 4 −4
0 0 0 1 −4 4
0 0 0 0 1 −6
0 0 0 0 0 1
id
π < Im
(
t2
)
+ 4φ < 2π
1 −6 −4 4 6 20
0 1 4 −4 −16 −70
0 0 1 0 −4 −20
0 0 0 1 4 20
0 0 0 0 1 6
0 0 0 0 0 1
ω1
2π < Im
(
t2
)
+ 4φ < 3π
1 6 20 −20 −70 20
0 1 4 −4 −16 6
0 0 1 0 −4 4
0 0 0 1 4 −4
0 0 0 0 1 −6
0 0 0 0 0 1
ω1ω2
3π < Im
(
t2
)
+ 4φ < 4π
1 −6 −4 4 6 20
0 1 4 −4 −16 −70
0 0 1 0 −4 −20
0 0 0 1 4 20
0 0 0 0 1 6
0 0 0 0 0 1
ω1ω2ω̂1
4π < Im
(
t2
)
+ 4φ < 5π
1 6 20 −20 −70 20
0 1 4 −4 −16 6
0 0 1 0 −4 4
0 0 0 1 4 −4
0 0 0 0 1 −6
0 0 0 0 0 1
ω1ω2ω̂1ω2
5π < Im
(
t2
)
+ 4φ < 6π
1 −6 4 −4 6 20
0 1 −4 4 −16 −70
0 0 1 0 4 20
0 0 0 1 −4 −20
0 0 0 0 1 6
0 0 0 0 0 1
ω1ω2ω̂1ω2ω1
Local Moduli of Semisimple Frobenius Coalescent Structures 91
Table 3. Continued from the previous page.
Band Hk, 0 ≤ k ≤ 8 Slex Braid
6π < Im
(
t2
)
+ 4φ < 7π
1 6 −20 20 −70 20
0 1 −4 4 −16 6
0 0 1 0 4 −4
0 0 0 1 −4 4
0 0 0 0 1 −6
0 0 0 0 0 1
ω1ω2ω̂1ω2ω1ω2
7π < Im
(
t2
)
+ 4φ < 8π
1 −6 4 −4 6 20
0 1 −4 4 −16 −70
0 0 1 0 4 20
0 0 0 1 −4 −20
0 0 0 0 1 6
0 0 0 0 0 1
ω1ω2ω̂1ω2ω1ω2ω̂1
8π < Im
(
t2
)
+ 4φ < 9π
1 6 −20 20 −70 20
0 1 −4 4 −16 6
0 0 1 0 4 −4
0 0 0 1 −4 4
0 0 0 0 1 −6
0 0 0 0 0 1
ω1ω2ω̂1ω2ω1ω2ω̂1ω2
Proposition 7.1. For a sufficiently small R > 0, it is defined an analytic function
Θ: BC(0;R)× Ω→ End(H•(X;C))
with series expansion
Θ(z, t) := Id +
N∑
α=0
〈〈
z · (−)
1− zψ
, Tα
〉〉
0
(t)Tα = Id +
∞∑
n=0
zn+1
N∑
α=0
〈〈τn(−), Tα〉〉0(t)Tα.
This function Θ satisfies the following properties:
1) for any φ ∈ H•(X;C), the vector field
Θφ := Θ(z, t)φ = φ+
N∑
α=0
〈〈
zφ
1− zψ
, Tα
〉〉
0
(t)Tα = φ+
∞∑
n=0
zn+1
N∑
α=0
〈〈τnφ, Tα〉〉0(t)Tα
satisfies the equations
∂αΘφ = z∂α ∗Θφ;
2) when restricted to the small quantum locus Ω∩H2(X;C), i.e., ti = 0 for i = 0, r+1, . . . , N ,
then
Θφ = ezδ ∪ φ+
∑
β 6=0
N∑
α=0
e
∫
β δ
〈
zezδ ∪ φ
1− zψ
, Tα
〉X
0,2,β
Tα, δ :=
r∑
i=1
tiTi ∈ H2(X;C);
3) for any φ1, φ2 ∈ H•(X;C) we have
η (Θ(−z, t)φ1,Θ(z, t)φ2) = η (φ1, φ2) ;
92 G. Cotti, B. Dubrovin and D. Guzzetti
1
1 1 1
1 1 1
1
1
1 1 1
1
111 2
3
4
5
6
2
3
4
5
6
2 3
4
5
6
2
3
45
6
2 34
5
6
23
4
5
6
2
3
4
5
62
3
4
562
3
4
56
2
3
4
5
6 23
45
6
2
34
5
6
2
3 4
5
6
2
3
4 5
6
2
3 4
5
62
3
4
5
6
β12β56
β12β34β56
β12β56
β12β34β56
β23β45
β23β45
β23β45
β23β45
β23β45
β23β45
β23β45
β
2
3
β
4
5
β34
β34
β34
β34
Figure 15. The picture, to be read in boustrophedon order, shows the braids corresponding to the
passage from one band Hk to Hk+1 . Starting from the configuration of the canonical coordinates at
0 ∈ QH•(G), we slightly split the coalescence as described in the first red picture in the first line. The
numbers represent the lexicographical order of the canonical coordinates w.r.t. the admissible line. Letting
the admissible line ` continuously rotate by increasing its slope, we determine all elementary braids acting
in the mutation up to the next red configuration (the counter-clockwise rotation of ` is visualised by the
clockwise rotation of the rays Lj(φ) defined in (3.2)). By coalescence of the points u3, u4 in a red picture
we obtain a configuration of canonical coordinates in the locus of small quantum cohomology (i.e., in
a strip Hk). Thus we deduce that successive bands of the small quantum cohomology are related by
alternate compositions of the braids ω1 := β12β56, ω2 := β23β45β34β23β45 and ω̂1 := β12β34β56.
4) for any φ ∈ H•(X;C), the vector field
Θ̃φ :=
(
Θ(z, t) ◦ zµzc1(X)∪(−)
)
φ
is a solution of the system (7.1), i.e.,
∂αΘ̃φ = z∂α ∗ Θ̃φ, ∂zΘ̃φ =
(
U +
1
z
µ
)
Θ̃φ.
Thus, the vector fields Θ̃Tα’s are gradients of deformed flat coordinates: if
(
Θα
β
)
α,β
,
(
Θ̃α
β
)
α,β
Local Moduli of Semisimple Frobenius Coalescent Structures 93
are the matrices representing the two End(H•(X;C))-valued functions Θ and Θ̃ w.r.t. the ba-
sis (Tα)α, i.e.,
Θ(z, t)Tβ =
N∑
α=0
Θα
β(z, t)Tα, Θ̃(z, t)Tβ =
N∑
α=0
Θ̃α
β(z, t)Tα,
then there exist analytic functions
(
t̃α(z, t)
)
α
, (hα(z, t))α on BC(0;R)× Ω such that
Θ̃α
β(z, t) =
(
grad t̃β(z, t)
)α
,
(
t̃0, t̃1, . . . , t̃N
)
= (h0, h1, . . . , hN ) · zµzR,
Θα
β(z, t) = (gradhβ(z, t))α, ΘT(−z, t)ηΘ(z, t) = η,
hα(z, t) :=
∞∑
p=0
hα,p(t)z
p, hα,0(t) = tα ≡ tληλα.
Proof. Notice that
Y (z, t) := H(z, t)zµzR, H(z, t) =
∞∑
p=0
Hp(t)z
p, H0(t) ≡ 1
is a fundamental solution of (7.1) if and only if H(z, t) satisfies the system∂αH = zCαH,
∂zH = UH +
1
z
[µ,H]−HR.
Because of the symmetry of cαβγ , the columns of H are the components w.r.t. (∂α)α of the
gradients of some functions:
hα(z, t) :=
∞∑
p=0
hα,p(t)z
p, hα,0(t) = tα,
Hα
β (z, t) = (gradhβ)α, Hα
β,p(z, t) = (gradhβ,p)
α.
The above system for H is equivalent to the following recursion relations on hα,p’s functions:
∂α∂βhγ,p(t) = cναβ∂νhγ,p−1(t), p ≥ 1, (7.2)
LE(gradhα,p) =
(
p+
dimCX − 2
2
+ µα
)
gradhα,p +
N∑
β=0
(gradhβ,p−1)Rβα, p ≥ 1.
The last equation is equivalent to the recursion relations on the differentials
LE(dhα,p) =
(
p− dimCX − 2
2
+ µα
)
dhα,p +
N∑
β=0
dhβ,p−1R
β
α, p ≥ 1. (7.3)
In our case we have
H(z, t) =
(
Θα
β(z, t)
)
α,β
, ∂αhβ,p(t) = 〈〈τp−1Tβ, Tα〉〉0(t).
The recursion relations (7.2) then reads
〈〈Tα, Tβ, Tγ〉〉0 = 〈〈Tα, Tβ, T ν〉〉0ηνγ for p = 1,
〈〈Tα, τp−1Tγ , Tβ〉〉0 = 〈〈Tα, Tβ, T ν〉〉0〈〈τp−2Tγ , Tν〉〉0 for p ≥ 2.
These are exactly the topological recursion relations in genus 0.
94 G. Cotti, B. Dubrovin and D. Guzzetti
Let us now prove that also the recursion relations (7.3) hold. K. Hori [47] (see also [30])
proved that, for any ω ∈ H2(X;C), we have the following constraint on the genus g free energy
ωα
∂FXg
∂tα,0
=
∑
β∈Eff(X)
(∫
β
ω
)
FXg,β +
∑
n,α,
σ,ν
ωσcνσαt
α,n
∂FXg
∂tν,n−1
+
δ0
g
2
∑
α,ν,σ
ωσcσανt
α,0tν,0 −
δ1
g
24
∫
X
ω ∪ cdimX−1(X), (7.4)
where FXg,β is the (g, β)-free energy
FXg,β :=
∞∑
n=0
1
n!
〈γ, . . . , γ︸ ︷︷ ︸
n times
〉Xg,n,β.
By dimensional consideration, one obtains also the selection rule∑
n,α
(n+ qα − 1)tα,n
∂FXg
∂tα,n
=
∑
β∈Eff(X)
(∫
β
ω
)
FXg,β + (3− dimX)(g − 1)FXg . (7.5)
If we introduce the perturbed first Chern class
E(t) := c1(X) +
∑
m,σ
(1− qσ −m)tσ,mτm(Tσ)−
∑
m,σ
tσ,mτm−1(c1(X) ∪ Tσ),
and using the selection rule (7.5), the Hori’s constraint (7.4) (specialized to g = 0 and ω = c1(X))
can be reformulated as
〈〈E〉〉0 = (3− dimX)FX0 +
1
2
tσ,0tρ,0
∫
X
c1(X) ∪ Tσ ∪ Tρ.
Taking the derivative w.r.t. tα,n, tβ,0 we obtain
〈〈E , τnTα, Tβ〉〉0 − (n+ qα + qβ − 2)〈〈τnTα, Tβ〉〉0 − 〈〈τn−1(c1(X) ∪ Tα), Tβ〉〉0
= (3− dimX)〈〈τnTα, Tβ〉〉0 + δn,0
∫
X
c1(X) ∪ Tα ∪ Tβ.
These recursion relations, restricted to the small phase space, are easily seen to be equivalent
to (7.3). This proves (1), (4) and the convergence of Θ(z, t) for |z| small enough, because of the
regular feature of the singularity z = 0. The proof of (2) can be found in [20]. Condition (3)
follows from WDVV and string equation, as shown in [36]. �
In the case of Fano manifolds, we have the following analytic characterization of the funda-
mental solution
(
Θ̃α
β
)
α,β
. Furthermore, because of Proposition 2.18, we obtain another proof
of (3) in the previous proposition.
Proposition 7.2. If X is a Fano manifold, among all fundamental matrix solutions of the
system (7.1) for deformed flat coordinates,25 there exists a unique solution such that, on the
small quantum locus (i.e., ti = 0 for i = 0, r+1, . . . , N) the function z−µH(z, t)zµ is holomorphic
at z = 0, with series expansion
z−µH(z, t)zµ = et∪ + zK1(t) + z2K2(t) + · · · , ti = 0 for i = 0, r + 1, . . . , N,
This solution coincides with the solution
(
Θ̃α
β(z, t)
)
α,β
.
25Throughout the paper, Y (z, t) = H(z, t)zµzR has been denoted Y (z, t) = Φ(z, t)zµzR.
Local Moduli of Semisimple Frobenius Coalescent Structures 95
Proof. We already know from Proposition 2.18 that such a solution is unique. Let us now
prove the main statement. In what follows, we will denote the degree deg Tα just by |α| for
brevity. By point (2) of Proposition 7.1, we have that
z−µ
(
Θ(zµφ)
)
= z−µ
ezδ ∪ zµφ+
∑
β 6=0
N∑
α=0
e
∫
β δ
〈
zezδ ∪ zµφ
1− zψ
, Tα
〉X
0,2,β
Tα
,
with δ :=
r∑
i=1
tiTi ∈ H2(X;C). Specialising to φ = Tσ, we have
z−µ
(
Θ(zµTσ)
)
= eδ ∪ Tσ +
∑
β 6=0
N∑
α,λ=0
∞∑
n,k=0
e
∫
β δ
k!
zn+1+k+µσ−µλ
〈
τn
(
δ∪k ∪ Tσ
)
, Tα
〉X
0,2,β
ηαλTλ.
In the second addend, we have non-zero terms only if
• |α|+ |λ| = 2 dimCX,
• 2n+ 2k + |σ|+ |α| = vir dimRX0,2,β.
By putting together these conditions, we obtain
n+ 1 + k +
1
2
(|σ| − |λ|) = −
∫
β
ωX .
The assumption of being Fano is equivalent to the requirement that the functional β 7→ −
∫
β ωX
is positive on the closure of the effective cone. This proves the proposition, the l.h.s. being
exactly the exponents of z which appear in the above series expansion. �
Example 7.3. Notice that the solution (6.21) that we considered in the previous section for
the computation of the monodromy data fo QH•(G) satisfies the condition
z−µ
(
η−1S(0, z)η
)
zµ is holomorphic near z = 0,
z−µ
(
η−1S(0, z)η
)
zµ =
1− 2z4 2z4 −z4 −z4 z4 z8
0 4z4 + 1 −z4 −z4 0 z4
0 0 1 0 −z4 z4
0 0 0 1 −z4 z4
0 0 0 0 1− 4z4 2z4
0 0 0 0 0 2z4 + 1
+O
(
z9
)
.
This means that
(
η−1S(0, z)η
)
zµzR coincides with the topological solution Θ̃(0, z).
A Proofs of Lemmata 6.2 and 6.4
In this Appendix we prove Lemmata 6.2 and 6.4. Before giving the proof of Lemma 6.2, we
recall the following well-known results (see, e.g., [55, 58, 75]).
Theorem A.1 (Stirling). The following estimate holds
log Γ(s) =
(
s− 1
2
)
log s− s+
1
2
log(2π) +O
(
1
|s|
)
for s → ∞ and | arg s| < π, and where log stands for the principal branch of the complex
logarithm.
96 G. Cotti, B. Dubrovin and D. Guzzetti
Corollary A.2. For |t| → +∞ we have
|Γ(σ + it)| =
√
2π|t|σ−
1
2 e−
π
2
|t|
(
1 +O
(
1
|t|
))
,
uniformly on any strip of the complex plane σ1 ≤ σ ≤ σ2.
Proof of Lemma 6.2. First of all let us prove that the functions ϕ1(w), ϕ2(w) are well defined
on the sectors
−π
2
< argw <
π
2
, −π
2
< argw < π,
respectively. We denote by I1 and I2 the integrands in ϕ1 and ϕ2 respectively, and s = κ+ it.
By Corollary A.2 we have that
|I1| ∼ (2π)2|t|5(κ−
1
2)e−
5π
2
|t||t|−κe
π
2
|t|e−κ log 4e−4κ log |w|+4t argw.
The dominant part is
e−
5π
2
|t|e
π
2
|t|e4t argw.
In order to have |I1| → 0 for t→ +∞ we must impose
−5π
2
+
π
2
+ 4 argw < 0, i.e., argw <
π
2
;
analogously, for t→ −∞ we have to impose
5π
2
− π
2
+ 4 argw > 0, i.e., argw > −π
2
.
Analogously, from Corollary A.2 we deduce that
|I2| ∼ (2π)3|t|5(κ− 1
2
)e−
5π
2
|t||−t|−κe−
π
2
|−t|e−πte−κ log 4e−4κ log |w|+4t argw,
and now the dominant part is
e−
5π
2
|t|e−
π
2
|−t|e−πte4t argw.
In order to have |I2| → 0 for t→ ±∞, we find
−5π
2
− π
2
− π + 4 argw < 0, i.e., argw < π,
5π
2
+
π
2
− π + 4 argw > 0, i.e., argw < −π
2
.
Let us now prove that ϕ1 and ϕ2 are solutions of equation (6.11). We have that
Θ5ϕ1(w) =
45
2πi
∫
Λ1
−s5 Γ(s)5
Γ
(
s+ 1
2
)4−sw−4s ds =
45
2πi
∫
Λ1
−
(
s+
1
2
)
Γ(s+ 1)5
Γ
(
s+ 3
2
) 4−sw−4s ds
because zΓ(z) = Γ(z + 1). Changing variable t := s + 1, and consequently shifting the line of
integration Λ1 to Λ1 + 1, we have
Θ5ϕ1(w) =
45
2πi
∫
Λ1+1
−
(
t− 1
2
)
Γ(t)5
Γ
(
t+ 1
2
)4−t · 4w−4(t−1) dt
Local Moduli of Semisimple Frobenius Coalescent Structures 97
=
45
2πi
∫
Λ1+1
(−4t)
Γ(t)5
Γ
(
t+ 1
2
)4−tw−4(t−1) dt+
2 · 45
2πi
∫
Λ1+1
Γ(t)5
Γ
(
t+ 1
2
)4−tw−4(t−1) dt.
Note that in the region between Λ1 and Λ1 + 1 the two last integrands have no poles; so∫
Λ1+1 =
∫
Λ1
by Cauchy theorem. This shows that
Θ5ϕ1 = 45w4Θϕ1 + 2 · 45w4ϕ1.
Analogously we have
Θ5ϕ2 =
45
2πi
∫
Λ2
−s5Γ(s)5Γ
(
1
2
− s
)
eiπs4−sw−4s ds
=
45
2πi
∫
Λ2
−
(
s+
1
2
)
Γ(s+ 1)5Γ
(
−1
2
− s
)
eiπ(s+1)4−sw−4s ds,
where the second identity follows from zΓ(z) = Γ(z + 1). Note that the integrand function is
holomorphic at s = −1
2 : indeed we have
lim
s→− 1
2
(
s+
1
2
)
Γ
(
−1
2
− s
)
= −1.
So in the strip of the complex plane −1 < Re s < 1
2 there are no poles, and by Cauchy theorem,
we can change path of integration by shifting Λ2 to Λ2 − 1:
Θ5ϕ2 =
45
2πi
∫
Λ2−1
−s5Γ(s)5Γ
(
1
2
− s
)
eiπs4−sw−4s ds.
Posing now t = s+ 1, we can rewrite
Θ5ϕ2 =
45
2πi
∫
Λ2
−
(
t− 1
2
)
Γ(t)5Γ
(
1
2
− t
)
eiπt4−(t−1)w−4(t−1) dt = 45w4Θϕ2 + 2 · 45w4ϕ2.
This shows that effectively ϕ1 and ϕ2 are solutions. �
Proof of Lemma 6.4. By Stirling’s formula we have that
ϕ1(w) =
(2π)2
2πi
∫
Λ1
eφ(s) ds,
where
φ(s) = −5s+ 5
(
s− 1
2
)
log s+ s+
1
2
− s log
(
s+
1
2
)
− s log 4− 4s logw +O
(
1
|s|
)
for s → ∞ and where log stands for the principal branch of logarithm. Let us find stationary
points of φ(s) for large values of |s|, |w|. The derivative φ′ is
φ′(s) = −4 + 5 log s+
10s− 5
2s
− log
(
s+
1
2
)
− s
s+ 1
2
− log 4− 4 logw +O
(
1
|s|
)
.
For |s| large enough, we have
10s− 5
2s
∼ 5− 5
2s
,
s
s+ 1
2
∼ 1− 1
2s
,
98 G. Cotti, B. Dubrovin and D. Guzzetti
log
(
s+
1
2
)
= log s+ log
(
1 +
1
2s
)
∼ log s+
1
2s
.
Substituting these identities in φ′, we find that the critical point s̄(w) in functions of w (for |w|
large)
s̄(w) =
√
2w +
5
8
+O
(
1
|w|
)
.
Note that for −π
2 < argw < π
2 , the point s̄(w) is in the half-plane Re s > 0, region in which
there are no poles of the integrand functions in ϕ1. So we can shift the line Λ1 in order that it
passes through s̄. In this way we obtain
ϕ1(w) =
(2π)2
2πi
eφ(s̄)
∫
Λ1
eφ(s)−φ(s̄) ds ∼ (2π)2
2πi
eφ(s̄)
∫
Λ1
e
φ′′(s̄)
2
(s−s̄)2
ds.
The computation of this Gaussian integral shows that
ϕ1(w) ∼ (2π)2
2π
eφ(s̄)
√
2π√
φ′′(s̄)
= (2π)
3
2
eφ(s̄)√
φ′′(s̄)
,
where Re
√
φ′′(s̄) > 0. An explicit series expansion shows that
φ(s̄(w)) ∼ −4
√
2w − 5
2
logw − 5
8
log 4 +O
(
1
|w|
)
,
whereas
φ′′(s̄(w)) ∼ 2
√
2
w
+O
(
1
|w|3
)
and from this we deduce that
ϕ1(w) ∼ (2π)
3
2
e−4
√
2w
4w2
(
1 +O
(
1
w
))
.
Let us now focus on ϕ2(w). From Theorem A.1 we deduce that
Γ(−s) = e−(s+ 1
2) log se−iπses
(
−i
√
2π
)(
1 +O
(
1
|s|
))
for s→∞ and s /∈ R+. So,
ϕ2(w) =
(2π)3
2πi
∫
Λ2
eφ(s) ds,
where
φ(s) = 5
(
s− 1
2
)
log s− 5s− s log
(
s− 1
2
)
+ s− 1
2
− s log 4− 4s logw +O
(
1
w
)
,
for w → ∞. By computations analogous to those of the previous case, we find that φ has
a critical point at
s̄(w) =
√
2w +
5
4
√
2
+O
(
1
w
)
Local Moduli of Semisimple Frobenius Coalescent Structures 99
Λ2
Λ′2
poles of Φ2
s̄(w)
Figure 16. Deformation of the path Λ2.
for large values of |w|. Note explicitly that for −π
2 < argw < π
2 this critical point is in the
half-plane Re s > 0.
By modifying the path of integration as in Fig. 16, in order that it passes through the critical
point, by Cauchy theorem we have
ϕ2(w) =
(2π)3
2πi
∫
Λ′2
eφ(s) ds−
∑
p∈P
res
s=p
I2(s),
where P stands for the set of poles in the region between Λ2 and Λ′2 and
I2(s) = Γ(s)5Γ
(
1
2
− s
)
eiπs4−sw−4s
is the integrand in ϕ2(w). For the first summand we have an asymptotic behavior as before
(2π)3
2πi
∫
Λ′2
eφ(s) ds ∼ αe−4
√
2w
w2
with α a certain constant we will not need to specify. For the second summand, keeping into
account that res
z=n
Γ(−z) = (−1)n+1/n! for n ∈ N, and that Γ
(
z + 1
2
)
= π1/22−2z+1Γ(2z)Γ(z)−1,
we receive
res
s=n+ 1
2
I2(s) =
(−1)n+1
n!
Γ
(
n+
1
2
)5
eiπ(n+ 1
2)4−n−
1
2w−4n−2
= − i
n!
(
(2n− 1)!!
2n
π
1
2
)5
4−n−
1
2w−4n−2.
Therefore,
∑
p∈P
res
s=p
I2(s) = − iπ
5
2
2w2
− iπ
5
2
256w6
+O
(
1
w10
)
.
In conclusion,
ϕ2(w) ∼ iπ
5
2
2w2
(
1 +O
(
1
w
))
, for −π
2
< argw <
π
2
.
100 G. Cotti, B. Dubrovin and D. Guzzetti
Let us now use the identity (6.17) in the following form:
ϕ2(w) = 2πϕ1
(
we−iπ
2
)
− ϕ2
(
we−iπ
2
)
, −π
2
< arg
(
we−iπ
2
)
<
π
2
.
It implies that
ϕ2(w) ∼ iπ
5
2
2w2
(
1 +O
(
1
w
))
on the whole sector − π
2
< argw < π. �
B Computation of the central connection matrix
Here we summarize the explicit values for the columns of the central connection matrix C=(Cij),
computed in Section 6.5.3, where v is indicated. The correct value is v = 6 (v was first introduced
in (6.26)).
Ci1 =
1
2
√
cπ2
4γ+iπ
2
√
cπ2
48γ2+24iγπ−5π2
12
√
cπ2
48γ2+24iγπ+7π2
12
√
cπ2
64γ3+48iγ2π+4γπ2+3iπ3−4ζ(3)
6
√
cπ2
768γ4+768iγ3π+96γ2π2+144iγπ3−π4−48(4γ+iπ)ζ(3)
72
√
cπ2
,
Ci2 =
1
2
√
cπ2
4γ+iπ
2
√
cπ2
48γ2+24iγπ+7π2
12
√
cπ2
48γ2+24iγπ−5π2
12
√
cπ2
64γ3+48iγ2π+4γπ2+3iπ3−4ζ(3)
6
√
cπ2
768γ4+768iγ3π+96γ2π2+144iγπ3−π4−48(4γ+iπ)ζ(3)
72
√
cπ2
,
Ci3 =
− 1
4
√
cπ2
−2γ−iπ
2
√
cπ2
−48γ2−48iγπ+11π2
24
√
cπ2
−48γ2−48iγπ+11π2
24
√
cπ2
2ζ(3)−(2γ+iπ)(4γ+iπ)(4γ+3iπ)
6
√
cπ2
−768γ4−1536iγ3π+1056γ2π2−23π4+96iπζ(3)+96γ(3iπ3+2ζ(3))
144
√
cπ2
,
Ci4(v) =
v−1
4
√
cπ2
2γ(v−1)+iπ
2
√
cπ2
48γ2(v−1)+48iγπ+(v+11)π2
24
√
cπ2
48γ2(v−1)+48iγπ+(v+11)π2
24
√
cπ2
32γ3(v−1)+48iγ2π+2γ(v+11)π2−3iπ3−2(v−1)ζ(3)
6
√
cπ2
768γ4(v−1)+1536iγ3π+96γ2(v+11)π2−(v+23)π4−96iπζ(3)+96γ(−3iπ3−2(v−1)ζ(3))
144
√
cπ2
,
Local Moduli of Semisimple Frobenius Coalescent Structures 101
Ci4(6) =
5
4
√
cπ2
10γ+iπ
2
√
cπ2
240γ2+48iγπ+17π2
24
√
cπ2
240γ2+48iγπ+17π2
24
√
cπ2
160γ3+48iγ2π+34γπ2−3iπ3−10ζ(3)
6
√
cπ2
3840γ4+1536iγ3π+1632γ2π2−288iγπ3−29π4−960γζ(3)−96iπζ(3)
144
√
cπ2
,
Ci5 =
1
4
√
cπ2
γ√
cπ2
48γ2+π2
24
√
cπ2
48γ2+π2
24
√
cπ2
−ζ(3)+16γ3+γπ2
3
√
cπ2
−192γζ(3)−768γ4+π4−96γ2π2
144
√
cπ2
,
Ci6 =
1
4
√
cπ2
γ+iπ√
cπ2
48γ2+96iγπ−47π2
24
√
cπ2
48γ2+96iγπ−47π2
24
√
cπ2
(γ+iπ)(4γ+3iπ)(4γ+5iπ)−ζ(3)
3
√
cπ2
768γ4+3072iγ3π−4512γ2π2−2880iγπ3+671π4−192(γ+iπ)ζ(3)
144
√
cπ2
.
We write now entries of the matrix C−Kap whose columns are given by the components of the
characteristic classes
1
4πc
1
2
Γ̂−(G) ∪ Ch
(
Sλ(S∗)
)
;
the order of the column is given by λ = 0, λ = 1, λ = 2, λ = (1, 1), λ = (2, 1) and λ = (2, 2).
(
C−Kap
)
0
=
1
4
√
cπ2
γ√
cπ2
1
24
+ 2γ2
π2√
c
1
24
+ 2γ2
π2√
c
−ζ(3)+16γ3+γπ2
3
√
cπ2
−192γζ(3)−768γ4+π4−96γ2π2
144
√
cπ2
,
(
C−Kap
)
=
1
2
√
cπ2
4γ+iπ
2
√
cπ2
2γ(2γ+iπ)
π2 − 5
12√
c
2γ(2γ+iπ)
π2 + 7
12√
c
64γ3+48iγ2π+4γπ2+3iπ3−4ζ(3)
6
√
cπ2
768γ4+768iγ3π+96γ2π2+144iγπ3−π4−48(4γ+iπ)ζ(3)
72
√
cπ2
,
102 G. Cotti, B. Dubrovin and D. Guzzetti
(
C−Kap
)
=
3
4
√
cπ2
3(2γ+iπ)
2
√
cπ2
6γ(γ+iπ)
π2 − 19
8√
c
6γ(γ+iπ)
π2 + 13
8√
c
32γ3+48iγ2π−6γπ2+5iπ3−2ζ(3)
2
√
cπ2
−6γ2+ 16γ4
π2 + 32iγ3
π
+10iγπ+ 7π2
48
− 2(2γ+iπ)ζ(3)
π2√
c
,
(
C−Kap
)
=
1
4
√
cπ2
2γ+iπ
2
√
cπ2
2γ(γ+iπ)
π2 − 11
24√
c
2γ(γ+iπ)
π2 − 11
24√
c
(2γ+iπ)(4γ+iπ)(4γ+3iπ)−2ζ(3)
6
√
cπ2
768γ4+1536iγ3π−1056γ2π2−288iγπ3+23π4−96(2γ+iπ)ζ(3)
144
√
cπ2
,
(
C−Kap
)
=
1
2
√
cπ2
4γ+3iπ
2
√
cπ2
2γ(2γ+3iπ)
π2 − 29
12√
c
2γ(2γ+3iπ)
π2 − 17
12√
c
(4γ+iπ)(4γ+3iπ)(4γ+5iπ)−4ζ(3)
6
√
cπ2
768γ4+2304vγ3π−2208γ2π2−720iγπ3+47π4−48(4γ+3iπ)ζ(3)
72
√
cπ2
,
(
C−Kap
)
=
1
4
√
cπ2
γ+iπ√
cπ2
2γ(γ+2iπ)
π2 − 47
24√
c
2γ(γ+2iπ)
π2 − 47
24√
c
(γ+iπ)(4γ+3iπ)(4γ+5iπ)−ζ(3)
3
√
cπ2
768γ4+3072iγ3π−4512γ2π2−2880iγπ3+671π4−192(γ+iπ)ζ(3)
144
√
cπ2
.
By application of the constraint
S =
(
C−Kap
)−1
e−πiRe−πiµη−1
(
(C−Kap)T
)−1
,
we find
SKap =
1 −4 6 10 −20 20
0 1 −4 −4 16 −20
0 0 1 0 −4 6
0 0 0 1 −4 10
0 0 0 0 1 −4
0 0 0 0 0 1
, S−1
Kap =
1 4 10 6 20 20
0 1 4 4 16 20
0 0 1 0 4 10
0 0 0 1 4 6
0 0 0 0 1 4
0 0 0 0 0 1
.
Now, S−1
Kap coincides with the Gram matrix GKap =
(
χ
(
SλS∗, SµS∗
))
λ,µ
of the Kapranov excep-
tional collection.
Local Moduli of Semisimple Frobenius Coalescent Structures 103
Acknowledgements
We would like to thank Marco Bertola, Ugo Bruzzo, Barbara Fantechi, Claus Hertling, Claude
Sabbah, Maxim Smirnov, Jacopo Stoppa, Ian Strachan and Di Yang for several discussions
and helpful comments. We also would like to thank the anonymous referees, whose valuable
comments have improved the paper. The first author is grateful to the Max-Planck Institut für
Mathematik in Bonn, for hospitality and support. The third author is a member of the European
Union’s H2020 research and innovation programme under the Marie Sk llodowska-Curie grant
No. 778010 IPaDEGAN.
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1 Introduction and results
1.1 Results
1.2 Plan of the paper
2 Moduli of semisimple Frobenius manifolds
2.1 Spectrum of a Frobenius manifold and its monodromy data at z=0
2.2 Semisimple Frobenius manifolds
2.3 Monodromy data for a semisimple Frobenius manifold
3 Ambiguity in definition of monodromy data and braid group action
3.1 Action of the braid group Bn
4 Isomonodromy theorem at coalescence points
4.1 Reconstruction of monodromy data of the whole manifold
5 First detailed example of application of Theorem 4.5: the A3 Frobenius manifold. Stokes phenomenon for Pearcey-type oscillating integrals from Hankel functions
5.1 Singularity theory and Frobenius manifolds
5.2 Frobenius structure of type An
5.3 The case of A3: reduction of the system for deformed flat coordinates
5.4 Computation of Stokes and central connection matrices
5.5 A ``tour'' in the Maxwell stratum: reconstruction of neighboring monodromy data
5.6 Monodromy data as computed outside the Maxwell stratum
6 Second example of application of Theorem 4.5: quantum cohomology of the Grassmannian G2(to.C4)to. and -conjecture
6.1 Notations in Gromov–Witten theory
6.2 Small quantum cohomology of G2(to.C4)to.
6.2.1 Generalities and proof of its semisimplicity
6.2.2 Idempotents at the points (to.0,t2,0,…,0)to.
6.3 Differential system for deformed flat coordinates
6.3.1 Expected asymptotic expansions
6.4 Solutions of the differential system
6.5 Computation of monodromy data
6.5.1 Solution at the origin and computation of C0(,, R)
6.5.2 Stokes rays and computation of left, right
6.5.3 Computation of Stokes and central connection matrices, using constraint (2) of Theorem 2.43
6.6 Monodromy data of the small quantum cohomology and exceptional collections in Db(G)
6.7 Reconstruction of monodromy data along the small quantum locus
7 A note on the topological solution for Fano manifolds
A Proofs of Lemmata 6.2 and 6.4
B Computation of the central connection matrix
References
|
| id | nasplib_isofts_kiev_ua-123456789-210710 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-17T12:04:33Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Cotti, Giordano Dubrovin, Boris Guzzetti, Davide 2025-12-15T15:26:46Z 2020 Local Moduli of Semisimple Frobenius Coalescent Structures. Giordano Cotti, Boris Dubrovin and Davide Guzzetti. SIGMA 16 (2020), 040, 105 pages 1815-0659 2020 Mathematics Subject Classification: 34M56;53D45;18E30 arXiv:1712.08575 https://nasplib.isofts.kiev.ua/handle/123456789/210710 https://doi.org/10.3842/SIGMA.2020.040 We extend the analytic theory of Frobenius manifolds to semisimple points with coalescing eigenvalues of the operator of multiplication by the Euler vector field. We clarify which freedoms, ambiguities, and mutual constraints are allowed in the definition of monodromy data, in view of their importance for conjectural relationships between Frobenius manifolds and derived categories. Detailed examples and applications are taken from singularity and quantum cohomology theories. We explicitly compute the monodromy data at points of the Maxwell Stratum of the A₃-Frobenius manifold, as well as at the small quantum cohomology of the Grassmannian ₂(ℂ⁴). In the latter case, we analyse in detail the action of the braid group on the monodromy data. This proves that these data can be expressed in terms of characteristic classes of mutations of Kapranov's exceptional 5-block collection, as conjectured by one of the authors. We would like to thank Marco Bertola, Ugo Bruzzo, Barbara Fantechi, Claus Hertling, Claude Sabbah, Maxim Smirnov, Jacopo Stoppa, Ian Strachan, and Di Yang for several discussions and helpful comments. We would also like to thank the anonymous referees, whose valuable comments have improved the paper. The first author is grateful to the Max-Planck Institut fur Mathematik in Bonn for hospitality and support. The third author is a member of the European Union's H2020 research and innovation programme under the Marie Sklodowska-Curie grant No. 778010 IPaDEGAN. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Local Moduli of Semisimple Frobenius Coalescent Structures Article published earlier |
| spellingShingle | Local Moduli of Semisimple Frobenius Coalescent Structures Cotti, Giordano Dubrovin, Boris Guzzetti, Davide |
| title | Local Moduli of Semisimple Frobenius Coalescent Structures |
| title_full | Local Moduli of Semisimple Frobenius Coalescent Structures |
| title_fullStr | Local Moduli of Semisimple Frobenius Coalescent Structures |
| title_full_unstemmed | Local Moduli of Semisimple Frobenius Coalescent Structures |
| title_short | Local Moduli of Semisimple Frobenius Coalescent Structures |
| title_sort | local moduli of semisimple frobenius coalescent structures |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210710 |
| work_keys_str_mv | AT cottigiordano localmoduliofsemisimplefrobeniuscoalescentstructures AT dubrovinboris localmoduliofsemisimplefrobeniuscoalescentstructures AT guzzettidavide localmoduliofsemisimplefrobeniuscoalescentstructures |