Isomonodromic Deformations Along the Caustic of a Dubrovin-Frobenius Manifold
We study the family of ordinary differential equations associated with a Dubrovin-Frobenius manifold along its caustic. Upon just losing an idempotent at the caustic and under a non-degeneracy condition, we write down a normal form for this family and prove that the corresponding fundamental matrix...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2023 |
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Інститут математики НАН України
2023
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| Цитувати: | Isomonodromic Deformations Along the Caustic of a Dubrovin-Frobenius Manifold. Felipe Reyes. SIGMA 19 (2023), 092, 21 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860269042324996096 |
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| author | Reyes, Felipe |
| author_facet | Reyes, Felipe |
| citation_txt | Isomonodromic Deformations Along the Caustic of a Dubrovin-Frobenius Manifold. Felipe Reyes. SIGMA 19 (2023), 092, 21 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We study the family of ordinary differential equations associated with a Dubrovin-Frobenius manifold along its caustic. Upon just losing an idempotent at the caustic and under a non-degeneracy condition, we write down a normal form for this family and prove that the corresponding fundamental matrix solutions are strongly isomonodromic. It is shown that the exponent of formal monodromy is related to the multiplication structure of the Dubrovin-Frobenius manifold along its caustic.
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| first_indexed | 2026-03-21T11:00:05Z |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 19 (2023), 092, 21 pages
Isomonodromic Deformations Along the Caustic
of a Dubrovin–Frobenius Manifold
Felipe REYES
SISSA, via Bonomea 265, Trieste, Italy
E-mail: llopezre@sissa.it
URL: https://lfelipe-lr.github.io/
Received May 03, 2023, in final form November 06, 2023; Published online November 16, 2023
https://doi.org/10.3842/SIGMA.2023.092
Abstract. We study the family of ordinary differential equations associated to a Dubrovin–
Frobenius manifold along its caustic. Upon just loosing an idempotent at the caustic and
under a non-degeneracy condition, we write down a normal form for this family and prove
that the corresponding fundamental matrix solutions are strongly isomonodromic. It is
shown that the exponent of formal monodromy is related to the multiplication structure of
the Dubrovin–Frobenius manifold along its caustic.
Key words: Dubrovin–Frobenius manifolds; isomonodromic deformations; differential equa-
tions
2020 Mathematics Subject Classification: 53D45; 34M56
1 Introduction
Dubrovin–Frobenius manifolds were invented by Boris Dubrovin to geometrize the study of
certain 2D topological field theories [6, 8]. The primary free energy F of a family of such theories
satisfies the so called WDVV equations. Given a quasi-homogeneous solution to these equations
one constructs a Dubrovin–Frobenius manifold structure on the domain of definition M of the
solution.
The first condition a Dubrovin–Frobenius manifold must satisfy is that the tangent sheaf TM
carries OM -bilinear multiplication ◦ : TM×TM → TM , this multiplication is required to be unital,
associative and commutative. The multiplication is required to satisfy an integrability condition
(equation (2.1)); a manifold satisfying these conditions is called an F -manifold. This integrability
condition ensures that, the decomposition of each tangent space TpM into irreducible subalgebras
extends to a decomposition of M into irreducible F -manifolds. Next up in the definition comes
the Euler vector field E, a global vector field required to satisfy LE◦ = ◦. Lastly one requires
the existence of a flat metric η satisfying some extra conditions (Definition 2.3).
As vector spaces, each tangent space TpM of a manifold is isomorphic to Cn; on a Dubrovin–
Frobenius manifold each tangent space is a C-algebra and as a C-algebra it may no longer be
isomorphic to Cn (the multiplication in Cn is done entry by entry). If as algebras TpM ∼= Cn
then the point p is called semisimple. In this case, there exists a neighborhood W̄ of p such
that all points in W̄ are semisimple; in W̄ there exist n-linearly independent vector fields π̄i
such that π̄i ◦ π̄j = δij π̄i and [π̄i, π̄j ] = 0, these vectors are called orthogonal idempotents. The
points which are not semisimple form an hypersurface K called the caustic which may be empty
(see [10, Proposition 2.6]). It is the purpose of this article to study the structure of a Dubrovin–
Frobenius manifold in a neighborhood of a non semisimple point p ∈ K. In particular, we are
interested in the restriction to the caustic of a family of differential equations associated to the
Dubrovin–Frobenius manifold.
mailto:llopezre@sissa.it
https://lfelipe-lr.github.io/
https://doi.org/10.3842/SIGMA.2023.092
2 F. Reyes
Closely related to a Dubrovin–Frobenius manifold is a family of meromorphic differential
equations on P1 (on a small semisimple domain the structure of a Dubrovin–Frobenius manifold
is equivalent to this family of equations, see [8]). Using the η-symmetric endomorphism E◦ and
the η-antisymmetric one µ := 2−d
2 Id−∇E, for each point p ∈ M the corresponding differential
equation reads
dy
dz
=
(
−(E◦)p +
1
z
µp
)
y,
where ∇ is the Levi-Civita connection of η. Each member of the family has a regular singularity
at z = 0 and an irregular at z = ∞. This family is induced by a vector bundle with a meromor-
phic flat connection over P1 ×M and as such one expects that the corresponding monodromy
data are constant.
At the regular singularity z = 0, one gets that the Jordan form of the principal part (the
endomorphism µ) of the family of meromorphic differential equations does not depend on the
point p ∈ M . As such, choosing a branch of the logarithm one writes a fundamental matrix
solution for all p ∈ M whose monodromy matrix is independent of p ∈ M .
At the irregular singularity z = ∞, the normal form of the principal part (the endomor-
phism E◦) depends on the point p ∈ M , in particular, it is diagonal for all semisimple points
and the basis that diagonalizes it, the orthogonal idempotents, ceases to exist in the caustic.
In a neighborhood of a semisimple point, after choosing an “admissible line” in the z-plane, we
can obtain holomorphic fundamental matrix solutions with prescribed asymptotic expansions
in certain sectors and compute the Stokes matrices and the exponent of formal monodromy
with respect to them; in a neighborhood of a semisimple point these data are also constant.
Lastly, the central connection matrix relating the solutions found at z = 0 and z = ∞ is also
constant. In this way, in a neighborhood of a semisimple point, the family of meromorphic dif-
ferential equations is strongly isomonodromic (i.e., the monodromy matrix, exponent of formal
monodromy, Stokes matrices and the central connection matrix are constant [6, 8]).
The fact that the monodromy data is constant on the semisimple domain depends only on
the flatness and singularities of a connection on a vector bundle over P1×M ; the semisimplicity
of the multiplication is nowhere used. Therefore, if one is able to define monodromy data for
the non-semisimple points one expects it to be constant. It is the purpose of this paper to
study this family of equations when only one idempotent is lost along the caustic, under one
non-degeneracy assumption we define monodromy data and prove that it is constant.
Under our assumptions, when restricting to the caustic the principal part at z = ∞ will
still be diagonalizable but it will have a repeated eigenvalue. Isomonodromic deformations of
meromorphic differential equations on P1 with irregular singularities are studied in the seminal
paper [11]. In their work, it is required that the principal parts at the irregular singularities have
pairwise distinct eigenvalues. This condition is not satisfied by the system we are interested in.
More recently, in [9] isomonodromic deformations of systems of the type we are interested are
studied.
There are examples of Frobenius manifolds such that at the caustic more than one idempotent
is lost (see [4]). For our purpose, it is worth to have in mind the Frobenius manifolds coming
from hypersurface singularities (see [12, 13]). In these examples it can be shown that at the
caustic only one idempotent is lost.
Main assumption and notations. On the caustic K the multiplication always has less
than n orthogonal idempotents. Throughout this article we will work under the following as-
sumptions:
Assumption 1. Generically along the caustic we have n− 1 idempotents.
Assumption 2. The metric η when pulled back to the caustic is non-degenerate so that this
hypersurface has a well defined normal direction.
Isomonodromic Deformations Along the Caustic of a Dubrovin–Frobenius Manifold 3
Under the first assumption for a point p ∈ K, we have that as algebras TpM ∼= C[z]/
(
z2
)
⊕
Cn−2 correspondingly, the integrability condition (2.1) tells us that, as germs of F -manifolds
(M,p) ∼= F 2 × (A1)
n−2 where F 2 is a two-dimensional F manifold and the Euler vector field
of M decomposes as a sum of Euler vector fields on the corresponding manifolds (see [10,
Theorem 2.11]). The F -manifold A1 is one-dimensional and we can choose a coordinate u
such that e = ∂u and E = u∂u. Germs of two-dimensional F -manifolds were classified in [10,
Theorem 4.7]. Each of these germs must be isomorphic to one of the F -manifolds I2(m) for
some m ∈ N≥2. On these F -manifolds one can choose coordinates (t, u2) such that π2 := ∂u2 is
the identity, ∂t ◦ ∂t = tm−2∂u2 and the Euler vector field is
E = 2
m t∂t + u2π2.
Thus, under the first assumption, for a point p ∈ K there exists a neighborhood W of p and
a system of local coordinates (t, u2, . . . , un) on W such that if we denote πi := ∂ui for i ≥ 2,
then
∂t ◦ ∂t = tm−2π2, ∂t ◦ π2 = ∂t, π2 ◦ π2 = π2,
and for i, j ≥ 3, we have
π2 ◦ πi = 0, ∂t ◦ πi = 0, πi ◦ πj = δijπi.
On these coordinates, the caustic has the equation t = 0 and the Euler vector field is
E =
2
m
t∂t +
n∑
i=2
uiπi.
For a point q ∈ W \K, there exists a neighborhood W̄ of q and canonical coordinates (ū1, . . . , ūn)
on W̄ such that π̄i ◦ π̄j = δij π̄i where π̄i := ∂ūi . The canonical coordinates ūi are the eigenvalues
of the operator of multiplication by the Euler vector field hence, on the overlap W̄ ∩W we have
ū1 = u2 +
2
m t
m
2 , ū2 = u2 − 2
m t
m
2 , ūi = ui for i ≥ 3. (1.1)
The corresponding basis of the tangent space are related by
∂t =
(
m
4
)m−2
m (ū1 − ū2)
m−2
m (π̄1 − π̄2), π2 = π̄1 + π̄2, πi = π̄i for i ≥ 3. (1.2)
From now on, we will denote by ui and πi for i ≥ 3 the coordinates and tangent vectors defined
on W̄ ∪W .
Organization. In Section 2, we recall the definition of an F -manifold and of a Dubrovin–
Frobenius manifold. We prove that under the main assumption, the caustic K of a massive
(generically semisimple) F -manifold M is always a massive F -manifold (Proposition 2.2). This
result was previously known (see [15, Theorem 2.10 and Example 2.5]) but we provide a different
proof. Moreover, if M has an Euler vector field then it is tangent to the caustic. This geometric
fact has the consequence that multiplication by the Euler vector field E is diagonalizable along
the caustic; but the basis that diagonalizes it outside the caustic does not coincide with the
one that diagonalizes it inside the caustic. As a corollary, we obtain that the caustic, with the
induced structures, satisfies all the axioms of a Dubrovin–Frobenius manifold except for the
flatness and in particular, the caustic of a three-dimensional Dubrovin–Frobenius manifold is
always a Dubrovin–Frobenius manifold.
In Section 3, we recall the definition of the deformed connection of a Dubrovin–Frobenius
manifold and of the associated family of meromorphic ordinary differential equations on P1 that
4 F. Reyes
it determines. We then pullback this family to the caustic and, if the metric η is non-degenerate
along the caustic, we write down this family in a convenient basis.
In Section 4, we recall the monodromy data at z = 0. As shown in [8], these data does not
depend on the point p ∈ M .
Outside the caustic the exponent of formal monodromy is identically zero, but this is no
longer true inside the caustic. In Section 5, we write down formal solutions for the family of
differential equations, compute the exponent of formal monodromy, show that it is constant and
that it is related to the decomposition (M,p) ∼= I2(m)× (A1)
n−2 (Proposition 5.3).
In Section 6, we invoke Sibuya’s theorem to find holomorphic solutions having the asymp-
totics of the formal solutions found in the previous section. Using them, we define the Stokes
matrices and show that they are constant. We also prove that the connection matrix relating
the particular solutions found at z = 0 and z = ∞ is constant. Combining these facts, we obtain
that the pullback of the family of differential equation is strongly isomonodromic (Theorem 6.6).
Section 7 provides some three-dimensional examples.
It is natural to ask if Proposition 5.3 and Theorem 6.6 can be generalized when one looses more
than one idempotent. As we will see, the classification of germs of 2-dimensional F -manifolds
will play a crucial role. Recently significant progress has been made in the classification of
germs of 3-dimensional F -manifolds; but as seen in [3] this classification is much more vast and
complicated than the two-dimensional case.
2 The caustic of an F -manifold
In this section, we prove that under the main assumption the caustic of a massive F -manifold
is again a massive F -manifold. If the starting F -manifold was a Dubrovin–Frobenius manifold
then, with the induced structures, the caustic satisfies all the axioms of a Dubrovin–Frobenius
manifold except for the flatness of the metric. For 3-dimensional Dubrovin–Frobenius manifolds,
the caustic is always a Dubrovin–Frobenius manifold.
Definition 2.1. An F -manifold is a triple (M, ◦, e) where M is a complex manifold of dimen-
sion n, ◦ : TM ⊗ TM → TM is a commutative, associative and OM -bilinear multiplication, e is
a global unit field for ◦ and the multiplication satisfies for any two local vector fields X, Y
LX◦Y (◦) = X ◦ LY (◦) + Y ◦ LX(◦). (2.1)
An Euler vector field for the F -manifold M is a global vector field E such that
LE(◦) = ◦. (2.2)
A point p ∈ M is called semisimple if the tangent space TpM has no nilpotents. In this case, it
can be shown that TpM decomposes as a sum of one-dimensional algebras
∑n
i=1C · π̄i where the
vectors π̄i satisfy π̄i ◦ π̄j = δij π̄i and [π̄i, π̄j ] = 0; these vectors are called orthogonal idempotents.
The caustic K ⊂ M is the set of points which are not semisimple. In [10, Proposition 2.6],
it is shown that the caustic is either empty or an hypersurface. Here we prove the following
proposition (see [15]).
Proposition 2.2. Let (M, ◦, e) be a massive F -manifold of dimension n and let K ̸= ∅ be its
caustic. Denote by i : K → M the inclusion. Suppose there exists a codimension 1 subvariety
(in K) K̃ ⊂ K such that there exist n − 1 vector fields π2, . . . , πn ∈ Γ
(
K \ K̃, i∗TM
)
such
that πi ◦ πj = δijπi. Then the caustic K is a massive F -manifold of dimension n − 1 and the
vectors πi are tangent to it. Moreover, if E is an Euler vector field for the F -manifold M then E
is tangent to K and it is an Euler vector field for it. The endomorphism i∗E◦ : i∗TM → i∗TM is
diagonalizable along K \ K̃.
Isomonodromic Deformations Along the Caustic of a Dubrovin–Frobenius Manifold 5
Proof. The existence of n − 1 orthogonal idempotents πi tells us that as an algebra TpM
decomposes as V ⊕
(⊕n
i=3C · πi
)
, where V is a 2-dimensional algebra and π2 is the unit on V .
We will use two results of [10]. According to [10, Theorem 2.11], the above decomposition
extends to a decomposition of the germ of the F -manifold M at the point p and the Euler
vector field decomposes as a sum of Euler vector fields of the corresponding F -manifolds. In
this case, the decomposition is F 2 ×
(∏n−2
i=1 A1
)
and the Euler vector field decomposes as
E = v +
n∑
i=3
uiπi,
with v ∈ V . The second result we will use is the classification of two-dimensional germs of
F -manifolds (see [10, Example 2.12 (iv) and Theorem 4.7 (a)]). There it is proven that F 2 must
be isomorphic to one of the germs I2(m). These germs admit local coordinates (t, u2) (here
we use a different notation from the one on [10]) such that ∂u2 = π2 is the identity on TI2(m),
∂t ◦ ∂t = tm−2π2 and the Euler vector field is v = 2
m t∂t + u2π2. Since on the hypersurface t = 0
the vector ∂t is nilpotent this hypersurface is contained in K \ K̃. The vectors tangent to this
hypersurface are π2, . . . , πn. The Euler vector field is
E =
2
m
t∂t +
n∑
i=2
uiπi.
Along the caustic, the basis ∂t, π2, . . . , πn of i∗TM diagonalizes the endomorphism E◦ and the
eigenvalues are u3, . . . , un of multiplicity one and u2 which has multiplicity two. ■
Definition 2.3. A Dubrovin–Frobenius manifold is a tuple (M, ◦, e, E, η), where (M, ◦, e) is an
F -manifold with Euler vector field E and η is a metric on M satisfying
1. For any vector fields X, Y , Z we have η(X ◦ Y, Z) = η(X,Y ◦ Z).
2. The unit e is flat, namely ∇e = 0 where ∇ is the Levi-Civita connection of η.
3. The Euler vector field satisfies LEη = (2− d)η.
4. The metric η is flat.
Corollary 2.4. Let (M, ◦, e, E, η) be a Dubrovin–Frobenius manifold and suppose that the caus-
tic K satisfies the hypothesis of Proposition 2.2. Then (K, ◦, e, E, i∗η) satisfies all the axioms
of Dubrovin–Frobenius manifold except possibly for the flatness of i∗η. Moreover, if M is 3-
dimensional then (K, ◦, e, E, i∗η) is a Dubrovin–Frobenius manifold.
Proof. The only thing that needs to be proven is the statement about the 3-dimensional
Dubrovin–Frobenius manifold. Let g = i∗η and let ∇̃ denote the Levi-Civita connection of g.
By hypothesis, ∇e = 0 so projecting to the tangent space of the caustic gives ∇̃e = 0. Using
this, we get Leg = ∇̃eg = 0. Call ∂1 = e and pick a vector field ∂2 such that [∂1, ∂2] = 0. Then
Leg = 0 implies ∂1gij = 0 so that the components of the metric in this basis are constant in the
direction of the unit vector field. Since the Christoffel symbols are functions of the metric and
its derivatives, they are also constant along the unit vector field. Now
[
∇̃∂1 , ∇̃∂2
]
∂1 = 0 because
∇̃e = 0. Finally,[
∇̃∂1 , ∇̃∂2
]
∂2 = Γ1
22∇∂1∂1 + Γ2
22∇∂1∂2 = 0. ■
Example 2.5. Let us consider the Dubrovin–Frobenius manifold M associated with the singu-
larity An. This manifold consists of the polynomials of the form
F (a; z) = zn+1 + an−1z
n−1 + · · ·+ a1z + a0,
6 F. Reyes
where a = (a0, . . . , an−1) ∈ Cn. This manifold is an affine space modeled on the vector space
of polynomials of degree at most n − 1. This means that we can identify the tangent space to
any point a ∈ M with the space of polynomials of degree at most n− 1. Given two polynomials
f, g ∈ TaM the multiplication is defined by
f ◦ g := fg mod
∂F
∂z
∣∣∣∣
a
.
If we write ∂F
∂z = (n+ 1)
∏n
i=1(z − αi), then one can easily check that the polynomials
ei :=
1
z − αi
∂F
∂z
satisfy ei ◦ ej = δijλiei with λi = ei(αi) and therefore they are multiples of the orthogonal
idempotents. Hence the caustic K consist of the points a such that the polynomial ∂F
∂z has
a double root. The set of points where ∂F
∂z has only a double root and all other roots simple
is an open set inside the caustic. In this open set the polynomials ei, with αi a simple root,
still are multiples of the orthogonal idempotents πi; we have n− 2 of them, say π3, . . . , πn. But
we have another orthogonal idempotent given by e − π3 − · · · − πn. By Proposition 2.2, the
caustic is a massive F -manifold. Note that we can apply the proposition again, indeed, the
caustic contains the locus of points K̃ such that the polynomial ∂F
∂z has a triple root and all the
other roots simple. The same argument as before shows that along K̃ we have n− 2 orthogonal
idempotents. Continuing in this way, we arrive at a 2-dimensional F -manifold, the locus of
points where ∂F
∂z has a root of multiplicity n−1 and a simple root. By the corollary, this surface
is a Dubrovin–Frobenius manifold. Dubrovin–Frobenius surfaces are classified by their charge.
Since the charge of the An Dubrovin–Frobenius manifold is d = n−1
n+1 (see [7]), the corresponding
surface is isomorphic to I2(n+ 1).
On the future, we will use the following statement.
Lemma 2.6. Let (M, ◦, e, E, η) be a Dubrovin–Frobenius manifold. Then the 1-form η(e,−) is
closed.
Proof. By torsion freeness of ∇, we have
d(η(e,−))(u, v) = uη(e, v)− vη(e, u)− η(e, [u, v])
= uη(e, v)− vη(e, u)− η(e,∇uv −∇vu).
By compatibility of the metric, we get
d(η(e,−))(u, v) = η(∇ue, v)− η(u,∇ve).
So by flatness of e, we get the result. ■
3 The deformed connection
In this section, we describe the deformed connection of a Dubrovin–Frobenius manifold M .
This consists of a family of connections parametrized by z ∈ C and at z = 0, we recover the
Levi-Civita connection of η. Thanks to the properties of a Dubrovin–Frobenius manifold, for
any z ∈ C the corresponding connection is flat. Moreover, this connection can be extended to
a flat connection with singularities on a vector bundle over P1 × M . By considering only the
derivative in direction of P1, every Dubrovin–Frobenius manifold determines a family of ordinary
differential equations on P1, this family is parametrized by the points of the Dubrovin–Frobenius
manifold. By pulling this connection to P×K, we obtain a new family of ordinary differential
equations parametrized by the caustic.
Isomonodromic Deformations Along the Caustic of a Dubrovin–Frobenius Manifold 7
Definition 3.1. Let (M, ◦, e, E, η) be a Dubrovin–Frobenius manifold, let ∇ be the Levi-Civita
connection of η and let z be a global coordinate on C. The deformed connection is a 1-parameter
family of connections on TM . For z ∈ C, it is defined as
∇̄ := ∇+ z ◦ .
Thanks to the commutativity of ◦, the deformed connection is torsionless. Moreover, the
flatness of ∇, properties (1) and (2) of Definition 2.3 and the associativity of ◦, imply that the
deformed connection is flat for every z ∈ C.
Consider the projections π : P1 × M → M and π1 : P1 × M → P1. We now extend the
deformed connection to a connection with singularities on the bundle π∗TM . First note that
thanks to property (4) of Definition 2.3, the endomorphism of TM defined by
µ :=
2− d
2
−∇E
is η-antisymmetric. The OM -linear tensors ◦ and µ on TM induce OP1×M -linear tensors on π∗TM .
Abusing notation we will denote them by the same symbols. Recall that TP1×M
∼= π∗
1TP1⊕π∗TM .
The connection ∇̄ on π∗TM is defined as
∇̄uv := π∗ (∇̄)
u
v = ∇uv + zu ◦ v
and
∇̄∂zv :=
∂v
∂z
+ E ◦ v − 1
z
µv,
where u, v ∈ π∗TM and ∂z is the vector field associated with the global coordinate z. The
equality (2.2) guarantees that ∇̄ is a flat connection on π∗TM . This means that for any point
(z, p) ∈ P1 × M we can find n linearly independent sections vi ∈ π∗TM that satisfy ∇̄vi = 0.
In particular, fixing a basis and putting the components of these n sections as columns of
a matrix Y , we get that, for each p ∈ M , Y satisfies the ordinary differential equation
dY
dz
=
(
1
z
µ− E◦
)
Y. (3.1)
Now take a semisimple point q ∈ M and let
f̄i :=
π̄i
|π̄i|
, i = 1, . . . , n,
denote the normalized orthogonal idempotents on M \ K, where the length | · | is computed
using the metric η. We have that E ◦ f̄i = ūif̄i so that the matrix Ū representing E◦ is diagonal
and the matrix V̄ representing µ is antisymmetric. In this basis, the system (3.1) is written as
dY
dz
=
(
1
z
V̄ − Ū
)
Y.
In [6], it is shown that around a semisimple point q ∈ M such that E◦ has different eigenvalues
the family of ordinary differential equations (3.1) is isomonodromic. Moreover, in [5] this re-
sult was extended to points where E◦ has repeated eigenvalues but the multiplication remains
semisimple. In this article, we show that under the conditions of Proposition 2.2 the family (3.1)
remains isomonodromic when pulled back to the caustic.
Consider the inclusion j : P1 ×K → P1 ×M (j = idP1 × i) and the vector bundle (π ◦ j)∗TM
over P1 × K. This vector bundle has a flat connection j∗∇̄. Moreover, if the metric i∗η is
8 F. Reyes
non-degenerate on every point of the caustic K we can find a unitary normal vector N to K.
Consider the vectors ∂t, π2, . . . , πn provided by Hertling’s decomposition (see equalities (1.1)
and (1.2)); by the compatibility of the multiplication ◦ and the metric η we must have that N
is a linear combination of the vectors ∂t and π2. As shown in Proposition 2.2, in the subspace
generated by these two vectors E◦ acts by multiplication by u2. Therefore, on the basis
N, f2 :=
π2
|π2|
, fi := f̄i, i = 3, . . . , n,
the matrix U representing E◦ is diagonal and the matrix V representing µ is antisymmetric.
System (3.1) takes the form
dY
dz
=
(
1
z
V − U
)
Y.
Hence we see that pulling back the family (3.1) to the caustic, we get a family of the same kind
but with one parameter less and the operator E◦ has an eigenvalue of multiplicity 2, namely u2.
For later use, let us write down the connection matrices of the flat connection j∗∇̄ on the
vector bundle (π ◦ j)∗TM over P1 × K. We will use the frame ∂z, π2, . . . , πn of TP1×K and the
frame N, f2, . . . , fn of (π ◦ j)∗TM . By the above discussion, the z-component is
ωz = U − 1
z
V.
Now ∇̄π2 = ∇π2 + zπ2◦ and since π2◦ is the identity on the subspace generated by N and π2,
and zero on the subspace generated by f3, . . . , fn, we have
ω̄2 = ω2 + zE2,
where ω2 is the connection matrix of the flat connection i∗∇ and E2 has a 2× 2 identity matrix
on the highest leftmost block and all the other entries are zero, i.e.,
(E2)
α
β = δα1 δ
1
β + δα2 δ
2
β.
Analogously, for i > 2 we have
ω̄i = ωi + zEi,
where the matrices ωi are the connection matrices of the flat connection i∗∇ and (Ei)
α
β = δαi δ
i
β.
Notice that since the first two eigenvalues of the matrix U are u2 we have dU =
∑n
i=2Eidui and
hence the connection form of the connection j∗∇̄ can be written as
ω̄ = zdU +
n∑
i=2
ωidui. (3.2)
The following lemma will be useful for some computations.
Lemma 3.2. We have the following identities:
[Ei, ωj ] = [Ej , ωi], (3.3)
∂V
∂ui
= [V, ωi], (3.4)
[U, ωi] = −[Ei, V ]. (3.5)
Isomonodromic Deformations Along the Caustic of a Dubrovin–Frobenius Manifold 9
Proof. Since the connection ∇̄ on π∗TM is flat, the connection j∗∇ on (π ◦ j)∗TM is also flat.
Flatness of j∗∇ in the plane generated by ∂i and πj gives (3.3) and flatness in the plane generated
by ∂z and πi gives (3.4) and (3.5). ■
Recall that for any point of the caustic (M,p) ∼= I2(m)× (A1)
n−2. We now introduce a con-
nection on the subbundle i∗TI2(m) of i
∗TM . This bundle is generated by the vectors N , f2. The
following connection will be useful when studying isomonodromic deformations of equation (3.1),
it is defined as
(id⊗ π2◦)i∗∇ : Ω0
K ⊗ i∗TI2(m)
→ Ω1
K ⊗ i∗TI2(m)
,
where id is the identity on Ω1
K . Since π2◦ is the identity on the subbundle generated by the
vectors N , f2, the past expression does define a connection. Indeed, C-linearity is clear and if
h ∈ OK is a holomorphic function and v ∈ ⟨N, f2⟩, we have
(id⊗ π2◦)i∗∇hv = (id⊗ π2◦)(dh⊗ v + hi∗∇v)
= dh⊗ π2 ◦ v + h(id⊗ π2◦)i∗∇v
= dh⊗ v + h(id⊗ π2◦)i∗∇v
because (id⊗π2◦) is OK-linear. Let us compute the connection matrices of this connection. We
have
i∗∇N =
n∑
i=2
(
(ωi)
1
1dui ⊗N +
n∑
s=2
(ωi)
s
1dui ⊗ fs
)
,
i∗∇f2 =
n∑
i=2
(
(ωi)
1
2dui ⊗N +
n∑
s=2
(ωi)
s
2dui ⊗ fs
)
,
so that
(id⊗ π2◦)i∗∇N =
n∑
i=2
(ωi)
1
1dui ⊗N + (ωi)
2
1dui ⊗ f2,
(id⊗ π2◦)i∗∇f2 =
n∑
i=2
(ωi)
1
2dui ⊗N + (ωi)
2
2dui ⊗ f2.
Therefore, for i = 2, . . . , n,
(id⊗ π2◦)i∗∇πiN = (ωi)
1
1N + (ωi)
2
1f2,
(id⊗ π2◦)i∗∇πif2 = (ωi)
1
2N + (ωi)
2
2f2.
Hence the connection matrices 2ωi of the connection (id⊗ π2◦)i∗∇ on i∗TI2(m) are
2ωi = E2ωiE2.
In other words, the connection matrices 2ωi of the connection (id⊗π2◦)i∗∇ are the highest left-
most 2×2 block of the connection matrices ωi of the connection i∗∇ on the vector bundle i∗TM .
Proposition 3.3. The connection (id⊗ π2◦)i∗∇ is flat.
Proof. The connection ∇ on TM is flat. Hence the connection i∗∇ on i∗TM is flat, in particular
for α, β ∈ {1, 2} and i, j ∈ {2, . . . , n}, i ̸= j, we have
0 =
∂(ωi)
α
β
∂uj
−
∂(ωj)
α
β
∂ui
− [ωi, ωj ]
α
β .
10 F. Reyes
Let us compute
[ωi, ωj ]
α
β =
∑
s=1,2
(
(ωi)
α
s (ωj)
s
β − (ωj)
α
s (ωi)
s
β
)
+
n∑
s=3
(
(ωi)
α
s (ωj)
s
β − (ωj)
α
s (ωi)
s
β
)
.
Suppose i = 2 and j > 2, for s > 2 from equation (3.3), we obtain (ωj)
α
s = −δjs(ω2)
α
s and
(ωj)
s
β = −δjs(ω2)
s
β. Hence,
n∑
s=3
(ω2)
α
s (ωj)
s
β − (ωj)
α
s (ω2)
s
β = 0.
Now suppose i, j > 2, again from equation (3.3) and s > 2, we have (ωi)
α
s = δis(ωs)
α
s and
(ωj)
s
β = δjs(ωs)
s
β so again the above sum vanishes.
Therefore, for α, β ∈ {1, 2} and i, j ∈ {2, . . . , n}, i ̸= j, we obtain
0 =
∂(ωi)
α
β
∂uj
−
∂(ωj)
α
β
∂ui
−
∑
s=1,2
(ωi)
α
s (ωj)
s
β − (ωj)
α
s (ωi)
s
β.
But this last expression is nothing else than the curvature tensor of connection (id⊗π2◦)i∗∇. ■
Remark 3.4. In the proof of the previous proposition, we only used equality (3.3). This equality
also holds true for the so called flat F -manifolds (see [1, 2]), these F -manifolds are equipped
with a flat connection ∇ on the tangent bundle such that the deformed connection ∇̄ = ∇+ z◦
is flat for every z ∈ C.
4 Monodromy data at z = 0
It this section, we describe the monodromy data of equation (3.1) at z = 0. We refer to [6],
where it is shown that the monodromy matrix of the Levelt form solution is constant.
The singularity of equation (3.1) at z = 0 is Fuchsian, and therefore there exists a holomorphic
gauge transformation
Y = T Ỹ =
(
T0 +
∞∑
k=1
Tkz
k
)
Ỹ ,
which transforms it to a simpler equation
dỸ
dz
= z−1
(
J +R1z + · · ·+Rpz
p
)
,
where J = T−1
0 µT0 is the Jordan form of the matrix µ and the entries (Rk)
i
j are different from
zero only if the eigenvalues µl of µ satisfy µi − µj = k ∈ N>0.
Writing µi = di + sii, where di ∈ Z and 0 ≤ Re(sii) < 1, we can write J = D + S with D
a diagonal matrix with di as eigenvalues (S is the only part of J which contributes to the mono-
dromy). If we also set R :=
∑p
k=1Rk, then a fundamental matrix solution of equation (3.1) is
YL = TzDzR+S . (4.1)
This particular kind of solution is called Levelt fundamental matrix solution. The monodromy
around z = 0 is the matrix
M̃ := e2πi(R+S).
In [6], it is shown that this matrix is (locally) constant for all points p of the Dubrovin–Frobenius
manifold M .
Isomonodromic Deformations Along the Caustic of a Dubrovin–Frobenius Manifold 11
5 The exponent of formal monodromy
In this section, we write a formal solution at z = ∞ of equation (3.1) and compute its exponent
of formal monodromy. On Proposition 5.3, we show that the exponent of formal monodromy
only depends on the natural number m corresponding to the I2(m) F -manifold appearing in
Hertlings decomposition (see equality (1.1)). Recall that, along the caustic, on the orthonormal
basis N , fi, i = 2, . . . , n the matrix U of E◦ is diagonal with the first two eigenvalues equal
to u2 and the matrix V of µ is antisymmetric. System (3.1) reads
dY
dz
=
(
1
z
V − U
)
Y.
We start by doing a formal Gauge transformation
Y (z, u) = GỸ =
(
Id +
∞∑
k=1
Gk(u)z
−k
)
Ỹ (z, u) =
( ∞∑
k=0
Gkz
−k
)
Ỹ , (5.1)
where the matrices Gk are to be determined. Since the matrix U has n− 2 different eigenvalues
we wish to find an equivalent block diagonal system that should consist of a 2×2 diagonal block
and n− 2 blocks of dimension 1. Setting
dỸ
dz
=
(
−U +
∞∑
k=1
Bk(u)z
−k
)
Ỹ , (5.2)
where the matrices Bk are also to be determined, we get the recursive relations
−[U,Gk] + (k − 1)Gk−1 + V Gk−1 −
k−1∑
s=1
Gk−sBs = Bk for k ≥ 1. (5.3)
So if we already now G1, . . . , Gk−1 and B1, . . . , Bk−1, we can try to solve the above equation
and obtain Gk and Bk. For k = 1, we need to solve
−[U,G1] + V = B1.
The matrix B1 = V + [G1, U ] must have entries
(B1)
i
j = V i
j − (G1)
i
j(ui − uj),
so whenever 1 ̸= i ̸= 2 or 1 ̸= j ̸= 2 (recall u1 = u2), we choose
(G1)
i
j =
(V )ij
ui − uj
, (B1)
i
j = 0, (5.4)
and therefore all entries of B1 are zero except for the highest leftmost 2× 2 block which is(
0 V 1
2
−V 1
2 0
)
and (G1)
1
2 = (G1)
2
1 = (G1)
i
i = 0.
The equations for k > 1 can be solved in an analogous way. We obtain that the only non-zero
entries of the matrix Bk are the ones in the diagonal and the highest leftmost 2× 2 block. That
is, after the formal Gauge transformation (5.1), we obtain the block diagonal system (5.2).
12 F. Reyes
Remark 5.1. Note that the matrices Gk are defined uniquely modulo ker(adU). If we set
(Gk)
1
2, (Gk)
2
1, (Gk)
i
i = 0, then the matrices Gk are uniquely defined. We could also do another
Gauge transformation Y = DȲ with D a block-diagonal matrix and still obtain a system
with diagonal principal part. In our case, this choice is fixed by writing system (3.1) in the
orthonormal basis N , fi, i = 2, . . . , n.
We now do the gauge transformation Ỹ = e−UzȲ , since e−Uz acts by scalar multiplication
on each block, from system (5.2) we obtain a new system of the form
dȲ
dz
=
(
B1z
−1 +
∞∑
k=2
Bkz
−k
)
Ȳ . (5.5)
Since all the matrices Bk have the same block structure, this last system is a direct sum of
Fuchsian systems. Notice that the eigenvalues of B1 are zero and ±iV 1
2 . We will suppose that
2iV 1
2 /∈ Z \ {0} so that the matrix B1 is non-resonant. In Proposition 5.3, we will show that in
our case this always holds true. Since B1 is non-resonant, we can find a gauge transformation
Ȳ = HŶ = H0
(
Id +
∞∑
k=1
Hkz
−k
)
Ŷ ,
where the matricesHk have the same block structure as the matrices Bk and such that Ȳ satisfies
the equation
dŶ
dz
=
B
z
Ŷ
with
B := H−1
0 B1H0 = diag
(
iV 1
2 ,−iV 1
2 , 0, . . . , 0
)
,
B̂k := H−1
0 BkH0 for k ≥ 2,
and
[B,Hk] +
1
k
Hk = −B̂k+1 −
k−1∑
l=1
B̂k+1−lHl for k ≥ 1.
Note that after choosing the diagonalizing matrix H0 the matrices Hk are uniquely determined.
Before proceeding to write down the formal solution of equation (3.1) let us pause a bit to show
that we can choose the highest leftmost block of the matrix H0 = H0(u) in a special way that
will allow us to find isomonodromic fundamental matrix solutions of equation (3.1).
Lemma 5.2. Consider the matrix
Ṽ =
(
0 V 1
2
−V 1
2 0
)
.
Then there exists a matrix H0 = H0(u) which diagonalizes Ṽ and such that
dH0 = −
n∑
i=2
E2ωiE2dui.
That is, the columns of H0 are (id⊗ π2◦)i∗∇-flat.
Isomonodromic Deformations Along the Caustic of a Dubrovin–Frobenius Manifold 13
Proof. Let H̄0 be any invertible matrix whose columns are (id⊗π2◦)i∗∇-flat (this matrix exists
thanks to Proposition 3.3). Note that the connection (id⊗ π2◦)i∗∇ is a connection on a bundle
of rank two and therefore H̄0 is a two by two matrix. Let 2ω =
∑n
i=2 2ωidui be the connection
form of the connection (id⊗ π2◦)i∗∇, by definition of the matrix H̄0, we have
d
(
H̄−1
0 Ṽ H̄0
)
= H̄−1
0
(
dṼ +
[
2
ω, Ṽ
])
H̄0.
Let us see that dṼ +
[
2
ω, Ṽ
]
= 0. From equation (3.4) we have dV + [ω, V ] = 0. This gives
∂V 1
2
∂ui
=
n∑
s=1
V 1
s (ωi)
s
2 − (ωi)
1
sV
s
2 .
From equations (3.3), we get
n∑
s=3
V 1
s (ωi)
s
2 − (ωi)
1
sV
s
2 = V 1
i (ωi)
i
2 − (ωi)
1
iV
i
2 ,
and from equation (3.5) the above sum vanishes. Therefore,
∂Ṽ 1
2
∂ui
=
∂V 1
2
∂ui
= [V, ωi]
1
2 =
[
Ṽ , 2ωi
]1
2
.
Hence the matrix H̄−1
0 Ṽ H̄0 is constant. Therefore, we can find a constant matrix C such that
C−1H̄−1
0 Ṽ H̄0C is diagonal. Since C is constant we can take H0 = H̄0C. ■
Let us go back to the formal solution of equation (3.1). Putting together the Gauge trans-
formations Y = GỸ , Ỹ = e−UzȲ and Ȳ = HŶ , we obtain a formal solution to equation (3.1)
YF = GHzBe−Uz =
(
H0 + (H0H1 +G1H0)z
−1 +O
(
z−2
))
e−UzzB. (5.6)
The matrix B is called the exponent of formal monodromy and by the above, in order to prove
that it is constant we only need to show that V 1
2 is a constant. In the following proposition, we
compute V 1
2 explicitly.
Proposition 5.3. Let (M, ◦, e, E, η) be a Dubrovin–Frobenius manifold with non-empty caus-
tic K and suppose that for a point p ∈ K the germ of M at p as an F -manifold is isomorphic
to I2(m) × (A1)
n−2 with m ≥ 3. Then the only non-zero entries of the exponent of formal
monodromy are
V 2
1 =
i
2
m− 2
m
and V 1
2 = −V 2
1 = − i
2
m− 2
m
.
Proof. By Hertling’s decomposition, on a neighborhood of a point p of the caustic there exist
coordinates (t, u2, . . . , un) such that the Euler vector field is written as E = 2
m t∂t +
∑n
s=2 usπs.
We need to compute
V 2
1 = η(f2, µN) = −η(f2,∇NE).
On the basis ∂t, π2, . . . , πn, the metric η takes the form
η11 η12 0 . . . 0
η21 η22 0 . . . 0
0 0 η33
...
...
. . .
0 0 . . . ηnn
.
14 F. Reyes
On the caustic {t = 0}, we have η11 = tm−2η22 = 0 and the normal to it isN = −i
√
η22
η12
∂t+
i√
η22
π2.
On the other hand,
∇E =
2
m
dt⊗ ∂t +
n∑
s=2
dus ⊗ πs +
2
m
t∇∂t +
n∑
s=2
us∇πs.
Therefore, using the Christoffel Γk
ij symbols of the basis ∂t, πi, i = 2, . . . , n gives
∇∂tE =
(
2
m
+
2
m
tΓ1
11 +
n∑
s=2
usΓ
1
1s
)
∂t +
(
2
m
tΓ2
11 +
n∑
s=2
usΓ
2
1s
)
π2 + · · · ,
∇π2E =
(
2
m
tΓ1
21 +
n∑
s=2
usΓ
1
2s
)
∂t +
(
1 +
2
m
tΓ2
21 +
n∑
s=2
usΓ
2
2s
)
π2 + · · · .
With this, we get
V 2
1 = i
m− 2
m
+
2
m
t
[
i
η22
(
Γ1
21η12 + Γ2
21η22
)
− i
η12
(
Γ1
11η12 + Γ2
11η22
)]
+
n∑
s=2
us
[
i
η22
(
Γ1
2sη12 + Γ2
2sη22
)
− i
η12
(
Γ1
1sη12 + Γ2
1sη22
)]
.
Now using the form of the metric and the fact that, on the caustic {t = 0}, we have η22,s = 0
for s ≥ 2 (f,s denotes the partial derivative of the function f with respect to the s-th coordinate),
we get
i
η22
(
Γ1
22η12 + Γ2
22η22
)
=
i
2
η22,2
η22
,
− i
η12
(
Γ1
12η12 + Γ2
12η22
)
= − i
2
η22,1
η12
,
and for i ≥ s
i
η22
(
Γ1
2sη12 + Γ2
2iη22
)
=
i
2
η22,s
η22
,
− i
η12
(
Γ1
1sη12 + Γ2
1sη22
)
= − i
2
η12,s
η12
.
So on the caustic
V 2
1 = i
[
m− 2
m
+
1
2
(
u2
(
η22,2
η22
− η22,1
η12
)
+
n∑
s=3
us
(
η22,s
η22
− η12,s
η12
))]
= i
[
m− 2
m
+
1
2
(
n∑
s=2
us
(
η22,s
η22
− η12,s
η12
)
+
u2
η12
(η12,2 − η22,1)
)]
.
Along the caustic, we have E =
∑n
i=2 uiπi and the condition LEη = (2− d)η implies E(η22) =
−dη22 and E(η12) =
(
−d+ m−2
m
)
η12. This gives
V 2
1 =
i
2
(
m− 2
m
+
u2
η12
(η12,2 − η22,1)
)
.
On these coordinates, we also have η(e,−) = η12dt +
∑n
i=2 ηiidui but by Lemma 2.6 this form
is closed and therefore η12,2 − η22,1 = 0. ■
Isomonodromic Deformations Along the Caustic of a Dubrovin–Frobenius Manifold 15
6 Stokes and connection matrices at the caustic
In this section, we state Sibuya’s theorem asserting that, for all ν ∈ Z there exist appropriate
sectors Sν and holomorphic solutions Yν of (3.1) having asymptotic expansion (5.6) on Sν . Then
we show that the Stokes matrices of the fundamental matrix solutions Yν are constant. We also
show that the connection matrix C relating the matrix Y0 with the Levelt fundamental matrix
solution YL is constant and therefore system (3.1) is strongly isomonodromic.
First, we define the sectors Sν . The gauge transformation (5.1) is usually divergent, but there
are certain sectors Sν of the z-plane in which this formal power series is the asymptotic expan-
sion of a holomorphic gauge transformation which takes equation (3.1) to the block diagonal
equation (5.5).
Definition 6.1. A line ℓ through the origin of the z-plane is called admissible for the system (3.1)
if for all z ∈ ℓ\{0}, we have that Re(z(ui−uj)) ̸= 0 whenever ui−uj ̸= 0. Let ϕ be the oriented
angle between the positive real axis and an admissible line ℓ. For ϵ sufficiently small and ν ∈ Z,
we define sectors Sν of opening angle π + 2ϵ by
S0 := {z ∈ C | arg(z) ∈ (ϕ− π − ϵ, ϕ+ ϵ)}, Sν := eiνπS0.
Note that the intersection of two subsequent sectors has opening angle 2ϵ.
In the following, u denotes a parameter on a small domain D ⊂ Cn−1, for the applications
we have in mind u = (u2, . . . , un) are the canonical coordinates on a sufficiently small open set
of the caustic.
Theorem 6.2 (Sibuya [14]). Let A(z, u) =
∑∞
k=0Ak(u)z
−k with Ak ∈ Matn(OCn−1) be holo-
morphic on {z ∈ C | |z| ≥ N0 > 0} × {|u| ≤ ϵ0} for some N0 ∈ N>0, ϵ0 ∈ R+ and such that
A0(u) = Λ(u) = Λ1 ⊕ · · · ⊕ Λs is diagonal with s ≤ n distinct eigenvalues
(
each matrix Λi is
diagonal ni × ni matrix with only one eigenvalue and
∑
ni = n
)
. Then, for any proper subsec-
tor S̄(α, β) of Sν there exist positive numbers N ≥ N0, ϵ ≤ ϵ0 and a matrix G(z, u) with the
following properties:
1. G(z, u) is holomorphic in (z, u) for |z| ≥ N , z ∈ S̄(α, β) and |u| ≤ ϵ.
2. G(z, u) has uniform asymptotic expansion for |u| ≤ ϵ with holomorphic coefficients Gk(u),
G(z, u) ∼ Id +
∞∑
k=1
Gk(u)z
−k, z → ∞, z ∈ S̄(α, β),
where the matrices Gk(u) are computed from (5.3)
3. The gauge transformation Y (z, u) = G(z, u)Ỹ (z, u) reduces the system dY
dz = AY to block
diagonal form
dỸ
dz
= B̃(z, u)Ỹ , B̃(z, u) = B̃1(z, u)⊕ · · · ⊕ B̃s(z, u)
and B̃ has uniform asymptotic expansion for |u| ≤ ϵ with holomorphic coefficients Bk(u)
B̃(z, u) ∼ Λ(u) +
∞∑
k=1
Bk(u)z
−k, z → ∞, z ∈ S̄(α, β).
Now we apply this theorem to the matrix A = −U+µz−1, A0 = −U of system (3.1) restricted
to the caustic K, thus we see that the formal gauge transformation of (5.1) is asymptotic, in
16 F. Reyes
proper sectors Sν , to a holomorphic gauge transformation Gν that takes system (3.1) to the
block diagonal form (5.2). We obtain holomorphic fundamental matrix solutions of system (3.1)
of the form
Yν = GνHzBe−Uz := Ŷνz
Be−Uz (6.1)
such that in the sector Sν we have
Yν ∼ YF =
(
H0 + (H0H1 +G1H0)z
−1 +O
(
z−2
))
e−UzzB. (6.2)
Stokes matrices are defined in the usual way. On the overlap of two adjacent sectors Sν ∩ Sν+1,
we have that
Yν+1(z;u) = Yν(z;u)Sν(u).
The matrix Sν is called Stokes matrix. Using the recursive relations of Section 5, we can compute
formal solutions of equation (3.1). Now we proceed to show that if we choose H0(u) in a par-
ticular way, then the corresponding holomorphic solutions are j∗∇-flat and the matrices Sν(u)
are independent of the parameter u. In the following we denote by d the differential only with
respect to the u variable excluding the z variable.
Lemma 6.3. Consider n j∗∇̄-flat linearly independent sections y1(z;u), . . . , yn(z;u) of the bun-
dle (π ◦ j)∗TM and write them down on the basis N, π2, . . . , πn. Let Y be the n×n matrix having
the sections yi as columns. Furthermore, let YF be the formal solution (5.6) of equation (3.1)
computed with H0 as in Lemma 5.2. Then Y satisfies equation (3.1) and on each sector Sν there
exists a constant matrix Cν such that Y = YνCν . In particular, Yν is j∗∇̄-flat.
Proof. The fact that Y satisfies equation (3.1) is obvious. Therefore, there exists a holomorphic
matrix Cν = Cν(u) such that Y = YνCν so we just need to show that dCν = 0. Let ω̄u be the
connection form of the connection j∗∇̄ disregarding the dz component. By the definition of Y ,
we have −ω̄u = dY · Y −1, so
−ωu − dYν · Y −1
ν = YνdCν · C−1
ν Y −1
ν .
By Proposition 5.3, we have dB = 0. The asymptotic expansion (6.2) in the sector Sν and the
block structure of B and U tell us that
YνdCν · C−1
ν Y −1
ν = −ω̄u − dYν · Y −1
ν
∼ −ω̄u + zdU − [dU,G1]− dH0 ·H−1
0 +O
(
z−1
)
. (6.3)
Looking at equation (3.2), we get
−ω̄u + zdU = −
n∑
i=2
ωidui.
Now suppose that 1 ̸= α ̸= 2 or 1 ̸= β ̸= 2, from equation (3.5)
n∑
i=2
(ωi)
α
βdui = −
duα − duβ
uα − uβ
V α
β .
On the other hand, from equation (5.4), we get
[dU,G1]
α
β =
duα − duβ
uα − uβ
V α
β .
Isomonodromic Deformations Along the Caustic of a Dubrovin–Frobenius Manifold 17
Substituting these last three equations in (6.3) gives
YνdCν · C−1
ν Y −1
ν ∼ −
n∑
i=2
E2ωiE2dui − dH0 ·H−1
0 ,
where the matrix E2 has entries (E2)
α
β = δα1 δ
1
β + δα2 δ
2
β. By Lemma 5.2, we get that in the
sector Sν
Yν(dCν · Cν)Y
−1
ν ∼ O
(
z−1
)
.
Let us write
YνdCν · C−1
ν Y −1
ν ∼
∞∑
k=1
Fkz
−k =: Fν .
Using (6.1) on Sν , we have
e−zUzBdCν · C−1
ν z−BeUz ∼ Ŷ −1
ν Fν Ŷν .
Note that since the matrix Ŷν is holomorphic on z = ∞, the term Ŷ −1
ν Fν Ŷν vanishes as z−1
when z → ∞.
Let us denote B = diag(b1, . . . , bn) = diag
(
m−2
2m ,−m−2
2m , 0, . . . , 0
)
then, since both eUz and zB
are diagonal matrices, we have
O
(
z−1
)
∼
(
e−zUzBdCν · C−1
ν z−BeUz
)α
β
= euβ−uαzbα−bβ
(
dCν · C−1
ν
)α
β
. (6.4)
Let 1 ̸= α ̸= 2 or 1 ̸= β ̸= 2 and α ̸= β, then euβ−uα ̸= 0. Since the sector Sν has opening angle
bigger than π this sector intersects the line Re((uβ − uα)z) = 0. On one side of this line the
function e(uβ−uα)z diverges when z → ∞. But the above expression must vanish as z−1 when
z → ∞ so whenever 1 ̸= α ̸= 2 or 1 ̸= β ̸= 2 and α ̸= β we must have
(dCν · Cν)
α
β = 0.
For α = β from (6.4), we get(
dCν · C−1
ν
)
∼ O
(
z−1
)
.
Since this matrix does not depend on z, we again must have
(dCν · Cν)
α
β = 0.
Finally, for α, β ∈ {1, 2} and α ̸= β from (6.4), we get
O
(
z−1
)
∼ z±
m−2
m
(
dCν · C−1
ν
)α
β
,
and again we conclude(
dCν · C−1
ν
)α
β
= 0.
Therefore,
(
dCν · C−1
ν
)
= 0 and the matrix Cν is constant. ■
Proposition 6.4. The Stokes matrices associated to the formal solution (5.6) of equation (3.1)
are constant.
18 F. Reyes
Proof. By the previous lemma, for all ν ∈ Z we have that dYν = ωYν . We also have that
Yν+1 = YνSν , so that
dYν+1 = ωYνSν + YνdSν = ωYν+1 + YνdSν .
Since dYν+1 = ωYν+1, we conclude dSν = 0 for all ν ∈ Z. ■
Proposition 6.5. Let C = C(u) be the matrix relating the fundamental matrix solution YL
in Levelt form (4.1) around z = 0 and the fundamental matrix solution Y0 having asymptotic
expansion (5.6) on S0. Then dC = 0.
Proof. We have that YL = Y0C and both YL and Y0 satisfy dY∗ = ωY∗. Hence
ωYL = dY0 · C + Y0dC = ωYL + Y0dC.
Hence dC = 0. ■
The fact that the monodromy of the Levelt fundamental solution (4.1) is constant and the
Propositions 5.3, 6.4 and 6.5 imply the following.
Theorem 6.6. Let (M, ◦, e, E, η) be a Dubrovin–Frobenius manifold with non-empty caustic K
and suppose that for a point p ∈ K the germ of M at p as an F -manifold is isomorphic to
I2(m)×(A1)
n−2 with m ≥ 3. Then the fundamental matrix solutions YL, Yν , ν ∈ Z have constant
monodromy data.
7 Three-dimensional examples
In this section, we use Proposition 5.3 to compute the decomposition (M,p) ∼= I2(m) × A1 for
the three-dimensional polynomial massive Dubrovin–Frobenius manifolds.
Locally, using flat coordinates (x, y, z), a Dubrovin–Frobenius manifold can be described
by means of a single function F called the potential. In the three-dimensional case, if we
suppose that F is polynomial and the Dubrovin–Frobenius manifold is massive, there are only
three possibilities: the Dubrovin–Frobenius manifolds corresponding to the singularities A3, B3
and H3. The corresponding potentials are (see [6, Chapter 1, Example 1.4])
FA =
1
2
x2z +
1
2
xy2 − 1
16
y2z2 +
1
960
z5,
FB =
1
2
x2z +
1
2
xy2 +
1
6
y3z +
1
6
y2z3 +
1
210
z7,
FH =
1
2
x2z +
1
2
xy2 +
1
6
y3z2 +
1
20
y2z5 +
1
3960
z11.
In these coordinates, the metric η is given by ηij =
∂F
∂t1∂ti∂tj
(here we identify the indices x 7→ 1,
y 7→ 2, z 7→ 3) and in all three cases we get0 0 1
0 1 0
1 0 0
.
In these coordinates, the structure constants of the multiplication ◦ are ckij =
∑
s η
ks ∂F
∂ts∂ti∂tj
.
The corresponding Euler vector fields are
EA = x
∂
∂x
+
3
4
y
∂
∂y
+
1
2
z
∂
∂z
,
Isomonodromic Deformations Along the Caustic of a Dubrovin–Frobenius Manifold 19
EB = x
∂
∂x
+
2
3
y
∂
∂y
+
1
3
z
∂
∂z
,
EH = x
∂
∂x
+
3
5
y
∂
∂y
+
1
5
z
∂
∂z
.
We will explicitly compute m for the H3 Dubrovin–Frobenius manifold, the other two cases
are similar and much simpler. On the basis ∂x, ∂y, ∂z the operator of multiplication by the
Euler vector field has the form x 7
10yz
(
2y + z3
)
1
20
(
12y336y2z3 + z9
)
3
5y x+ yz2 + 1
5z
5 7
10yz
(
2y + z3
)
1
5z
5 3
5y x
.
The discriminant of the characteristic polynomial of this matrix is a multiple of the polynomial
y2
(
y − z3
)5(
27y + 5z3
)3
.
Along this surface (called the bifurcation diagram), multiplication by the Euler vector field has
a repeated eigenvalue and therefore the caustic is contained in this surface. We can divide this
surface into two parts, the caustic and the semisimple coalescence locus (for more information
about this locus see [5]). In order to identify the semisimple coalescence locus, we use the
following lemma.
Lemma 7.1. Let (M, ◦, e) be an F -manifold. Suppose that at a point p ∈ M there exists
a vector v such that the operator v◦ has different eigenvalues ui ̸= uj if i ̸= j. Then p is
a semisimple point.
Proof. Let v ◦ ei = uiei then v ◦ (ei ◦ ei) = ei ◦ v ◦ ei = uiei ◦ ei so that ei ◦ ei is an eigenvector
of v◦ with eigenvalue ui. Since all eigenvalues are different, we obtain ei ◦ ei = λiei and πi :=
ei
λi
satisfies πi ◦ πi = πi. Now ui(πi ◦ πj) = v ◦ (πi ◦ πj) = uj(πi ◦ πj) but since ui ̸= uj , we obtain
πi ◦ πj = 0. ■
Along the first component of this surface y = 0, multiplication by ∂y has three different
eigenvalues and thus y = 0 belongs to the semisimple coalescence locus. To identify the caustic
note that if a point is semisimple, then the operator of multiplication by any tangent vector
is diagonalizable, indeed the orthogonal idempotents are a basis of eigenvectors. Along the
components y = z3 and y = − 5
27z
3, the operator of multiplication by ∂y is not diagonalizable
and therefore the caustic is the union of this two components.
The component y = z3 is parametrized by x = r, y = s3, z = s and the tangent space to
this surface is generated by ∂r = e and ∂s = 3s2∂y + ∂z. In this basis, multiplication by ∂s has
matrix(
0 175
4 s8
1 9s4
)
.
The eigenvectors of this matrix are e2 = −25
2 s
4∂x + 3s2∂y + ∂z and e3 = 7
2s
4∂x + 3s2∂y + ∂z.
Along the caustic the tangent space decomposes as the direct sum of a two-dimensional and
a one-dimensional algebras. To identify the unit in each of this algebras we use the Euler vector
field. In our previous notation, the eigenvalue associated with π2 must have multiplicity two
and that of π3 has multiplicity one. Thus, we obtain e = π2 + π3 = − 1
16s2
e2 +
1
16s2
e3 so the
square lengths of π2 and π3 are − 1
16s4
and 1
16s4
, respectively. The unitary normal is the vector
N = −3s2∂x+∂y and therefore an orthonormal basis along this component of the caustic consists
of the vectors
N = −3s2∂x + ∂y, f2 = i4s2π2, f3 = 4s2π3.
20 F. Reyes
On the basis ∂x, ∂y, ∂z, the endomorphism µ has matrix diag
(
−2
5 , 0,
2
5
)
and this gives
V 1
2 = η(N,µf2) = i
3
10
.
Therefore, along the component y = z3, we have m = 5. We can parametrize the other compo-
nent y = − 5
27z
3 by x = r, y = − 5
27s
3, z = s. An identical procedure now gives
m = 3.
The cases of B3 and A3 are analogous and simpler. On the B3 Dubrovin–Frobenius manifold,
the matrix of the endomorphism µ is diag
(
−1
3 , 0,
1
3
)
and the bifurcation diagram has equation
y2
(
2y − 3z2
)4(
2y + z2
)3
.
Again y = 0 corresponds to the semisimple coalescence locus and the other two components
conform the caustic. On the component
{
2y− 3z2 = 0
}
, we have m = 4, and on the component{
2y + z2 = 0
}
, we have m = 3. Finally, the A3 manifold has bifurcation diagram
y2
(
27y2 + 8z2
)
Once again y = 0 is the semisimple coalescence locus and on the other component we have
m = 3.
Acknowledgements
I would like to thank the referees for the useful comments and corrections that helped improve
the readability and proofs of this work.
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1 Introduction
2 The caustic of an F-manifold
3 The deformed connection
4 Monodromy data at z=0
5 The exponent of formal monodromy
6 Stokes and connection matrices at the caustic
7 Three-dimensional examples
References
|
| id | nasplib_isofts_kiev_ua-123456789-212039 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T11:00:05Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Reyes, Felipe 2026-01-23T10:10:40Z 2023 Isomonodromic Deformations Along the Caustic of a Dubrovin-Frobenius Manifold. Felipe Reyes. SIGMA 19 (2023), 092, 21 pages 1815-0659 2020 Mathematics Subject Classification: 53D45; 34M56 arXiv:2209.01062 https://nasplib.isofts.kiev.ua/handle/123456789/212039 https://doi.org/10.3842/SIGMA.2023.092 We study the family of ordinary differential equations associated with a Dubrovin-Frobenius manifold along its caustic. Upon just losing an idempotent at the caustic and under a non-degeneracy condition, we write down a normal form for this family and prove that the corresponding fundamental matrix solutions are strongly isomonodromic. It is shown that the exponent of formal monodromy is related to the multiplication structure of the Dubrovin-Frobenius manifold along its caustic. I would like to thank the referees for the useful comments and corrections that helped improve the readability and proofs of this work. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Isomonodromic Deformations Along the Caustic of a Dubrovin-Frobenius Manifold Article published earlier |
| spellingShingle | Isomonodromic Deformations Along the Caustic of a Dubrovin-Frobenius Manifold Reyes, Felipe |
| title | Isomonodromic Deformations Along the Caustic of a Dubrovin-Frobenius Manifold |
| title_full | Isomonodromic Deformations Along the Caustic of a Dubrovin-Frobenius Manifold |
| title_fullStr | Isomonodromic Deformations Along the Caustic of a Dubrovin-Frobenius Manifold |
| title_full_unstemmed | Isomonodromic Deformations Along the Caustic of a Dubrovin-Frobenius Manifold |
| title_short | Isomonodromic Deformations Along the Caustic of a Dubrovin-Frobenius Manifold |
| title_sort | isomonodromic deformations along the caustic of a dubrovin-frobenius manifold |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212039 |
| work_keys_str_mv | AT reyesfelipe isomonodromicdeformationsalongthecausticofadubrovinfrobeniusmanifold |