Tropical Mirror Symmetry in Dimension One
We prove a tropical mirror symmetry theorem for descendant Gromov-Witten invariants of the elliptic curve, generalizing the tropical mirror symmetry theorem for Hurwitz numbers of the elliptic curve, Theorem 2.20 in [Böhm J., Bringmann K., Buchholz A., Markwig H., J. Reine Angew. Math. 732 (2017), 2...
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| description | We prove a tropical mirror symmetry theorem for descendant Gromov-Witten invariants of the elliptic curve, generalizing the tropical mirror symmetry theorem for Hurwitz numbers of the elliptic curve, Theorem 2.20 in [Böhm J., Bringmann K., Buchholz A., Markwig H., J. Reine Angew. Math. 732 (2017), 211-246, arXiv:1309.5893]. For the case of the elliptic curve, the tropical version of mirror symmetry holds on a fine level and easily implies the equality of the generating series of descendant Gromov-Witten invariants of the elliptic curve to Feynman integrals. To prove tropical mirror symmetry for elliptic curves, we investigate the bijection between graph covers and sets of monomials contributing to a coefficient in a Feynman integral. We also soup up the traditional approach in mathematical physics to mirror symmetry for the elliptic curve, involving operators on a Fock space, to give a proof of tropical mirror symmetry for Hurwitz numbers of the elliptic curve. In this way, we shed light on the intimate relation between the operator approach on a bosonic Fock space and the tropical approach.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 18 (2022), 046, 30 pages
Tropical Mirror Symmetry in Dimension One
Janko BÖHM a, Christoph GOLDNER b and Hannah MARKWIG b
a) Fachbereich Mathematik, TU Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany
E-mail: boehm@mathematik.uni-kl.de
b) Universität Tübingen, Fachbereich Mathematik, 72076 Tübingen, Germany
E-mail: christoph.goldner@math.uni-tuebingen.de, hannah@math.uni-tuebingen.de
Received January 24, 2022, in final form June 17, 2022; Published online June 25, 2022
https://doi.org/10.3842/SIGMA.2022.046
Abstract. We prove a tropical mirror symmetry theorem for descendant Gromov–Witten
invariants of the elliptic curve, generalizing the tropical mirror symmetry theorem for Hur-
witz numbers of the elliptic curve, Theorem 2.20 in [Böhm J., Bringmann K., Buchholz A.,
Markwig H., J. Reine Angew. Math. 732 (2017), 211–246, arXiv:1309.5893]. For the case
of the elliptic curve, the tropical version of mirror symmetry holds on a fine level and easily
implies the equality of the generating series of descendant Gromov–Witten invariants of the
elliptic curve to Feynman integrals. To prove tropical mirror symmetry for elliptic curves,
we investigate the bijection between graph covers and sets of monomials contributing to
a coefficient in a Feynman integral. We also soup up the traditional approach in mathema-
tical physics to mirror symmetry for the elliptic curve, involving operators on a Fock space,
to give a proof of tropical mirror symmetry for Hurwitz numbers of the elliptic curve. In
this way, we shed light on the intimate relation between the operator approach on a bosonic
Fock space and the tropical approach.
Key words: mirror symmetry; elliptic curves; Feynman integral; tropical geometry; Hurwitz
numbers; quasimodular forms; Fock space
2020 Mathematics Subject Classification: 14J33; 14N35; 14T05; 81T18; 11F11; 14H30;
14N10; 14H52; 14H81
1 Introduction
1.1 Context: Tropical mirror symmetry of elliptic curves
Mirror symmetry is a duality relation involving algebraic resp. symplectic varieties and their
invariants. Its main motivation comes from string theory, but it is also at the base of many
interesting developments in mathematics. We focus on statements relating generating series of
Gromov–Witten invariants of a variety X with certain integrals on its mirror X∨.
Tropical geometry becomes a new tool to prove such relations, largely due to the well-known
Gross–Siebert program, which aims at constructing new mirror pairs and providing an algebraic
framework for SYZ-mirror symmetry [21, 22, 39].
The philosophy how tropical geometry can be exploited is illustrated in the following triangle:
This paper is a contribution to the Special Issue on Enumerative and Gauge-Theoretic Invariants
in honor of Lothar Göttsche on the occasion of his 60th birthday. The full collection is available at
https://www.emis.de/journals/SIGMA/Gottsche.html
mailto:boehm@mathematik.uni-kl.de
mailto:christoph.goldner@math.uni-tuebingen.de
mailto:hannah@math.uni-tuebingen.de
https://doi.org/10.3842/SIGMA.2022.046
https://www.emis.de/journals/SIGMA/Gottsche.html
2 J. Böhm, C. Goldner and H. Markwig
tropical
GW-invariants
Gromov–Witten
invariants
Feynman
integrals
C
orrespondence
T
heorem
Mirror symmetry
In many situations, correspondence theorems relating Gromov–Witten invariants resp. enu-
merative invariants to their tropical counterparts are known [5, 12, 31, 33]. If we can relate the
generating function of tropical invariants to integrals, we obtain a proof of the desired mirror
symmetry relation using a detour via tropical geometry [20, 37].
In [9], we investigated the triangle above for the case of Hurwitz numbers of the elliptic curve
and Feynman integrals. Correspondence theorems for Hurwitz numbers existed already, tropical
Hurwitz numbers essentially count certain decorated graphs. The mirror symmetry relation in
this case was known, there is a proof in mathematical physics involving operators on a Fock
space.
The tropical approach revealed that the relation holds on an even finer level: tropically, we
can relate Feynman integrals and generating series of (labeled) tropical covers graph by graph
and order by order. As a consequence, one obtains interesting new quasimodularity statements
for graph generating series [19].
The mirror symmetry theorem (the top arrow) is an easy corollary of the more general tropical
version. The tropical mirror symmetry theorem for Hurwitz numbers of the elliptic curve can
be viewed as a support for the strategy of the Gross–Siebert program, or more generally for the
philosophy of using tropical geometry as a tool in mirror symmetry.
1.2 Part I: Generating series of tropical descendant
Gromov–Witten invariants of E
From Okounkov–Pandharipande’s Gromov–Witten/Hurwitz (GW/H) correspondence [36, The-
orem 1], it is known that Hurwitz numbers are a special case of descendant Gromov–Witten
invariants.
The central result of this article is a general tropical mirror symmetry theorem for elliptic
curves, involving descendant Gromov–Witten invariants (Theorem 2.19):
Theorem 1.1. Generating series of tropical descendant Gromov–Witten invariants of an elliptic
curve can be expressed in terms of Feynman integrals. In particular, they are quasimodular
forms, also when restricted to a certain combinatorial type of source.
Together with the suitable correspondence Theorem 2.6 (see [13, Theorem 3.1.2]) relating
tropical descendant Gromov–Witten invariants to their counterparts in algebraic geometry, it
can be applied to prove the mirror symmetry theorem for elliptic curves involving descendants
(see Theorem 2.15, [26, resp. Theorem 6.1 and Proposition 6.7] or [27, Theorem 1.2(2) and
Proposition 3.4]).
The most important new tool of [9] in the study of mirror symmetry for elliptic curves was
a bijection between tropical covers (i.e., the above mentioned decorated graphs) satisfying fixed
discrete data and sets of monomials contributing to a coefficient in a Feynman integral [9,
Theorem 2.30]. For our purpose, we generalize this in two directions, both involving the source
curves of the tropical covers in question:
Tropical Mirror Symmetry in Dimension One 3
(a) we need to allow vertices of valency different from 3, and
(b) we need to allow genus at vertices.
The task in (a) is a major extension of the bijection, formulated in Theorem 2.23, and will have
further applications. The task in (b) involves the multiplicities with which covers are counted
and is more a question of bookkeeping.
The tasks (a) and (b) are necessary, as psi-classes in the tropical world impose higher valency
resp. higher genus on vertices, see, e.g., [11, 25, 29, 30, 32]. For example, for the tropical count
of the descendant Gromov–Witten invariant
⟨τ2(pt)τ0(pt)τ0(pt)⟩E,3,trop
2,3
(see Example 2.5), the two graphs depicted in Figure 1 are needed as combinatorial types of
source curves for tropical covers.
g = 1
Figure 1. Two graphs that appear as combinatorial types of source curves of tropical covers contributing
to ⟨τ2(pt)τ0(pt)τ0(pt)⟩E,3,trop
2,3 . The left has a vertex of genus one, the right has a 4-valent vertex.
1.3 Part II: Relation to the Fock space approach
The traditional approach to mirror symmetry of an elliptic curve involves operators on Fock
spaces. There are two Fock spaces, a fermionic and a bosonic Fock space, and an isomorphism
between them called the boson–fermion correspondence. The latter is usually viewed as the
essence of mirror symmetry. The generating function of Gromov–Witten invariants can be
interpreted on the fermionic side, via the correspondence we then obtain an expression in terms
of matrix elements on the bosonic Fock space, and the latter can be related to Feynman integrals
[23, 26, 27, 36].
We reveal the close connection between the Fock space approach to mirror symmetry of
elliptic curves and the new tropical approach.
Here, tropical geometry hands us a shortcut: by passing to tropical Gromov–Witten invariants
on the tropical elliptic curve ET, we can directly relate the generating series of Gromov–Witten
invariants of the elliptic curve E to a matrix element on the bosonic Fock space (see Figure 2),
supporting the slogan “tropicalization is bosonification” from [13], and the intuition underlying
the Gross–Siebert program that tropical geometry is a natural language in the context of mirror
symmetry.
We soup up the traditional Fock space approach to give an alternative proof of the tropical
mirror symmetry relation Theorem 2.19, for simplicity restricting to the case of Hurwitz num-
bers. The main ingredient is a version of Wick’s theorem which encodes matrix elements in
a bosonic Fock space as weighted sums of graphs, which can then directly be related to tropical
Hurwitz covers (see Theorem 3.8).
Since the first version of this paper appeared as preprint, other researchers have continued
working on related topics. We would like to point out in particular a series of papers by Blomme
who studies the enumerative geometry of line bundles over elliptic curves and generalized further
to the enumerative geometry of abelian surfaces [7, 8], and the papers on enumerative geometry
of elliptic fibrations by Oberdieck and Pixton [34, 35].
Section 2 focuses on tropical mirror symmetry and its direct proof via a bijection involving
graph covers and monomials in a Feynman integral. Again, the fact that tropical mirror sym-
metry holds on a fine level has interesting implications for graph summands of the generating
4 J. Böhm, C. Goldner and H. Markwig
tropical
GWI of ET
GWI of E
Feynman
integrals
Fermionic
Fock space
Bosonic
Fock space
C
orrespondence
T
heorem
Mirror symmetry
Tr
op
ic
al
m
irr
or
sy
m
m
et
ry
boson–fermion
Correspondence
Figure 2. Boson–fermion correspondence and tropical geometry as a shortcut.
series of descendant Gromov–Witten invariants. The quasimodularity of these graph summands
is shown in Section 2.6 relying on [19, Theorem 6.1]. Section 3 is devoted to the Fock space
approach.
2 Tropical mirror symmetry for elliptic curves
2.1 Descendant Gromov–Witten invariants
Gromov–Witten invariants are virtually enumerative intersection numbers on moduli spaces of
stable maps. Let E be an elliptic curve. Gromov–Witten invariants of E do not depend on its
complex structure. A stable map of degree d from a curve of genus g to E with n markings is
a map f : C → E, where C is a connected projective curve with at worst nodal singularities,
and with n distinct nonsingular marked points x1, . . . , xn ∈ C, such that f∗([C]) = d[E] and f
has a finite group of automorphism. The moduli space of stable maps, denoted Mg,n(E, d),
is a proper Deligne–Mumford stack of virtual dimension 2g − 2 + n [3, 4]. The ith evaluation
morphism is the map evi : Mg,n(E, d) → E that sends a point [C, x1, . . . , xn, f ] to f(xi) ∈ E.
The ith cotangent line bundle Li → Mg,n(E, d) is defined by a canonical identification of its
fiber over a moduli point (C, x1, . . . , xn, f) with the cotangent space T ∗
xi
(C). The first Chern
class of the cotangent line bundle is called a psi class (ψi = c1(Li)).
Definition 2.1. Fix g, n, d and let k1, . . . , kn be non-negative integers with
k1 + · · ·+ kn = 2g − 2.
The stationary descendant Gromov–Witten invariant ⟨τk1(pt) · · · τkn(pt)⟩
E,d
g,n is defined by
⟨τk1(pt) · · · τkn(pt)⟩E,d
g,n =
∫
[Mg,n(E,d)]vir
n∏
i=1
ev∗i (pt)ψ
ki
i ,
where pt denotes the class of a point in E.
In Section 3, we use degeneration techniques to relate the proof of mirror symmetry for
elliptic curves in mathematical physics to the tropical approach. For this purpose, we also
Tropical Mirror Symmetry in Dimension One 5
need to introduce relative Gromov–Witten invariants: they are constructed using moduli spaces
of relative stable maps Mg,n
(
P1, µ, ν, d
)
, where part of the data specified are the ramification
profiles µ and ν which we fix over 0 resp. ∞ ∈ P1. The preimages of 0 and ∞ are marked.
A detailed discussion of spaces of relative stable maps to P1 and their boundary is not necessary
for our purpose, we refer to [40]. We use operator notation and denote
⟨µ|τk1(pt) · · · τkn(pt)|ν⟩P
1,d
g,n =
∫
[Mg,n(P1,µ,ν,d)]vir
n∏
i=1
ev∗i (pt)ψ
ki
i .
One can allow source curves to be disconnected, and introduce disconnected Gromov–Witten
invariants. We will add the superscript • anytime we wish to refer to the disconnected theory.
Remark 2.2. It follows from the GW/H correspondence [36, Theorem 1] that a stationary
descendant Gromov–Witten invariant with ki = 1 for all i is a Hurwitz number counting covers
of the resp. degree and genus and with n fixed simple branch points.
2.2 Tropical descendant Gromov–Witten invariants
An abstract tropical cur-ve is a connected metric graph Γ, such that edges leading to leaves
(called ends) have infinite length, together with a genus function g : Γ → Z≥0 with finite support.
Locally around a point p, Γ is homeomorphic to a star with r halfrays. The number r is called the
valence of the point p and denoted by val(p). We identify the vertex set of Γ as the points where
the genus function is nonzero, together with points of valence different from 2. The vertices of
valence greater than 1 are called inner vertices. Besides edges, we introduce the notion of flags
of Γ. A flag is a pair (V, e) of a vertex V and an edge e incident to it (V ∈ ∂e). Edges that are
not ends are required to have finite length and are referred to as bounded or internal edges.
A marked tropical curve is a tropical curve whose leaves are labeled. An isomorphism of
a tropical curve is an isometry respecting the leaf markings and the genus function. The genus
of a tropical curve Γ is given
g(Γ) = h1(Γ) +
∑
p∈Γ
g(p).
A curve of genus 0 is called rational and a curve satisfying g(v) = 0 for all v is called explicit.
The combinatorial type is the equivalence class of tropical curves obtained by identifying any
two tropical curves which differ only by edge lengths.
A tropical cover π : Γ1 → Γ2 is a surjective harmonic map of metric graphs in the sense of [1,
Section 2]. The map π is piecewise integer affine linear, the slope of π on a flag or edge e is
a nonnegative integer called the expansion factor ω(e) ∈ N.
The expansion factor of e can be 0 only if e is an end. We fix the convention that the ends
marked 1, . . . , n are the ones with expansion factor 0.
For a point v ∈ Γ1, the local degree of π at v is defined as follows. Choose a flag f ′ adjacent
to π(v), and add the expansion factors of all flags f adjacent to v that map to f ′:
dv =
∑
f 7→f ′
ω(f).
We define the harmonicity or balancing condition to be the fact that for each point v ∈ Γ1, the
local degree at v is well defined (i.e., independent of the choice of f ′).
The degree of a tropical cover is the sum over all local degrees of preimages of a point a,
d =
∑
p 7→a dp (here, we consider the map locally around a vertex of the source graph). By the
balancing condition, this definition does not depend on the choice of a ∈ Γ2. For a flag f of the
6 J. Böhm, C. Goldner and H. Markwig
image graph Γ2, let µf be the partition of expansion factors of the flags of the source graph Γ1
mapping onto f . We call µf the ramification profile above f .
The tropical projective line, P1
T, equals R∪{±∞}, a (nondegenerate) tropical elliptic curve ET
is a circle with a fixed length.
Definition 2.3 (psi- and point conditions). We say that a tropical cover π : Γ1 → Γ2 with
a marked end i satisfies a psi-condition with power k at i, if the vertex V to which the marked
end i is adjacent has valency k + 3 − 2g(V ). We say π : Γ1 → Γ2 satisfies the point conditions
p1, . . . , pn ∈ Γ2 if
{π(1), . . . , π(n)} = {p1, . . . , pn}.
Fix g, n, d and let k1, . . . , kn be non-negative integers with
k1 + · · ·+ kn = 2g − 2.
Let π : Γ → ET be a tropical cover such that Γ is of genus g and has n marked ends. Fix n
distinct points p1, . . . , pn ∈ ET. Assume that at the marked end i, a psi-condition with power ki
is satisfied, and that the point conditions are satisfied. The marked ends must be adjacent
to different vertices, since they satisfy different point conditions. It follows from an Euler
characteristic argument incorporating the valencies imposed by the psi-conditions that Γ has
exactly n vertices, each adjacent to one marked end.
Locally at the marked end i, the cover sends the vertex to an interval consisting of two
flags f and f ′. We define the local vertex multiplicity multi(π) to be a combinatorial factor
times a one-point relative descendant Gromov–Witten invariant:
multi(π) = ⟨µf |τki(pt)|µf ′⟩P
1,di
gi,1
, (2.1)
where gi denotes the genus of the vertex adjacent to the marked end i, di its local degree, and µf
resp. µf ′ the ramification profiles above the two flags of the image interval.
We define the multiplicity of π to be
1
|Aut(π)|
·
∏
i
multi(π) ·
∏
e
ω(e). (2.2)
Note that all ends of a tropical cover of ET are contracted ends, with image points the
points pi we fix as conditions in ET.
Definition 2.4 (tropical stationary descendant Gromov–Witten invariant of ET). For g, n, d,
k1, . . . , kn as above, define the tropical stationary descendant Gromov–Witten invariant
⟨τk1(pt) · · · τkn(pt)⟩E,d,trop
g,n
to be the weighted count of tropical genus g degree d covers of ET with nmarked points satisfying
point and psi-conditions as above, each counted with its multiplicity as defined in (2.2).
Note that the metric structure of the source curves of covers contributing to a tropical des-
cendant Gromov–Witten invariant is implicit in the metric data of ET and the chosen point
conditions. We can thus neglect length data in the source curve.
Example 2.5. As an example, fix three different points p1, p2, p3 on ET and let d = 3, g = 2,
k1 = 2, k2 = 0, k3 = 0. Note that
∑
i ki = 2g − 2 is satisfied. We list all covers contributing
to ⟨τ2(pt)τ0(pt)τ0(pt)⟩E,3,trop
2,3 in Figure 3 below. Figure 3 shows schematic representations of
the source curves of all covers contributing, where we assume that the top vertex of each such
Tropical Mirror Symmetry in Dimension One 7
representation is mapped to p1, the right vertex is mapped to p2 and the left one is mapped
to p3. This convention gives us one choice out of 3! choices of an order of labeled vertices of the
source curve mapping to p1, p2, p3 on ET. A green number indicates that there is a nonzero
genus gi at a vertex i. The other numbers are the weights of the edges that are greater than 1.
Note that the valency of a vertex i is given by ki+3− 2gi when taking the contracted ends into
account. When neglecting marked ends, the underlying graph is either a figure 8 or a loop (see
Example 2.12). In each case, every loop is mapped to ET. When we draw a curl in an edge, it
means that the edge is mapped once around ET.
2
2
2
2
3 3
3
2
2
2
2
1 1 1 1
1 1 1
1 1 1
Figure 3. Schematic representations of source curves.
The multiplicity with which each curve is contributing is give by (2.2). The local multiplicities
multi(π), which are one-point relative descendant Gromov–Witten invariants, can be calculated
explicitly using the one-point series (2.5). Each entry of the tabular below corresponds to one
source curve of Figure 3 in the obvious way. An entry is the multiplicity of the corresponding
cover π, where the first factor equals |Aut(π)|−1, the second factor equals
∏
imulti(π) and the
third factor equals
∏
e ω(e).
1 · 1 · 2 1 · 1 · 8 1 · 1 · 1 1 · 1 · 1 1 · 1 · 1
1 · 1 · 1 1 · 17
24 · 27 1 · 1
24 · 1 1 · 1
24 · 1 1 · 1
24 · 1
1 · 1
24 · 1 1 · 1
24 · 1 1 · 1
24 · 1 1 · 1 · 4 1 · 1 · 4
1 · 1
24 · 1 1 · 1
24 · 1 1 · 1
24 · 1 1 · 1 · 1 1 · 1 · 1
1 · 1 · 1 1 · 1 · 1 1 · 1 · 1
8 J. Böhm, C. Goldner and H. Markwig
Summing over all entries and considering the factor 3! yields
⟨τ2(pt)τ0(pt)τ0(pt)⟩E,3,trop
2,3 = 3! · 93
2
= 279.
Theorem 2.6 (correspondence Theorem I). A stationary descendant Gromov–Witten invariant
of E coincides with its tropical counterpart:
⟨τk1(pt) · · · τkn(pt)⟩E,d
g,n = ⟨τk1(pt) · · · τkn(pt)⟩E,d,trop
g,n .
For a proof, see [13, Theorem 3.2.1].
To define tropical relative stationary descendant Gromov–Witten invariants of P1
T, we fix two
partitions µ and ν of the degree d. We consider tropical covers of P1
T such that the ramification
profile over −∞ equals µ and the ramification profile over ∞ equals ν. That is, in addition to
the contracted ends that we use to impose point conditions, the source curve Γ has ℓ(µ) + ℓ(ν)
marked ends which map to ±∞ with expansion factors imposed by µ and ν. We assume that
a cover π : Γ → P1
T meets point and psi-conditions as above. Local vertex multiplicities are
defined as in equation (2.1), and the multiplicity is
1
|Aut(π)|
·
∏
i
multi(π) ·
∏
e
ω(e),
where the last product goes over the bounded edges e of Γ. Tropical relative stationary descen-
dant Gromov–Witten invariants of P1
T, ⟨µ|τk1(pt) · · · τkn(pt)|ν⟩
P1,d,trop
g,n , are defined as counts of
tropical covers with the expansion factors of the unmarked ends imposed by µ and ν and satis-
fying the point and psi-conditions, counted with their multiplicity (see [13, Definition 3.1.1]).
Example 2.7. Choose three different points p1, p2, p3 on ET and let d = 3, g = 2, k1 = 2,
k2 = 0, k3 = 0 be as in Example 2.5. Let p0 be a base point on ET such that p0, p1, p2, p3 are
ordered this way on ET. Consider the source curve of a cover π of ET depicted in the upper left
corner of Figure 3 and cut it along π−1(p0). Stretching the cut edges to infinity yields the cover
shown below (we let i be mapped to pi). Note that this is a cover π′ to P1
T that contributes to
⟨(2, 1)|τ2(pt)τ0(pt)τ0(pt)|(2, 1)⟩P
1,3,trop
0,3 .
π′
P1
Tp1 p2 p3
1 2 3
2
1
2
1
Theorem 2.8 (correspondence Theorem II). A relative (stationary) descendant Gromov–Witten
invariant of P1 coincides with its tropical counterpart:
⟨µ|τk1(pt) · · · τkn(pt)|ν⟩P
1,d
g,n = ⟨µ|τk1(pt) · · · τkn(pt)|ν⟩P
1,d,trop
g,n .
For a proof, see [13, Theorem 3.1.2].
Just as before, we can also in the tropical world allow source curves to be disconnected, and
add the superscript • in the notation.
Tropical Mirror Symmetry in Dimension One 9
Remark 2.9 (leaking). We can tweak the definition of tropical covers of ET (resp. P1
T) satisfying
point and psi-conditions as follows: fix a direction for the target curve and specify for each end i
of the source curve an integer li. Change the balancing condition in such a way that for the
two flags f1 and f2 adjacent to π(i) ∈ {p1, . . . , pn} (where we chose the notation to match the
direction), the local degrees are not equal but differ by li:∑
f ′ 7→f1
ω(f ′) =
∑
f ′′ 7→f2
ω(f ′′)− li.
We call such covers leaky tropical covers. Leaky tropical covers show up as floor diagrams
representing counts of tropical curves in toric surfaces resp. in P1-bundles over E (see, e.g.,
[2, 7, 8, 10, 18]). We introduce them here, since they can be treated in terms of Feynman
integrals analogously to their balanced versions.
2.3 Feynman integrals
Definition 2.10 (the propagator and the S-function). We define the propagator as a (formal)
series in x and q:
P (x, q) =
∞∑
w=1
w · xw +
∞∑
a=1
(∑
w|a
w
(
xw + x−w
))
qa
and the S-function as series in z:
S(z) = sinh(z/2)
z/2
.
We also consider another formal series in q (which should be viewed as the propagator for
loop edges):
P loop(q) =
∞∑
a=1
(∑
w|a
w
)
qa.
Definition 2.11 (Feynman graphs). Fix n > 1. A Feynman graph is a (non-metrized) graph Γ
without ends with n vertices which are labeled x1, . . . , xn and with labeled edges q1, . . . , qr.
By convention, we assume that q1, . . . , qs are loop edges and qs+1, . . . , qr are non-loop edges.
We do not fix the number of edges for a Feynman graph, the index r can vary from graph to
graph. We always use the letter r for the number of edges in a fixed Feynman graph Γ.
Example 2.12. Recall Example 2.5, where we provided all covers contributing to
⟨τ2(pt)τ0(pt)τ0(pt)⟩E,3,trop
2,3 .
We can label their source curves, turning them into Feynman graphs, see Figure 4.
Definition 2.13 (Feynman integrals). Let Γ be a Feynman graph. Let Ω be an order of the n
vertices of Γ.
For k > s, denote the vertices adjacent to the (non-loop) edge qk by xk1 and xk2 , where we
assume xk1 < xk2 in Ω.
For integers l1, . . . , ln, we define the Feynman integral for Γ and Ω to be
I l1,...,lnΓ,Ω (q) = coef
[x
l1
1 ···xln
n ]
s∏
k=1
P loop(q) ·
r∏
k=s+1
P
(
xk1
xk2
, q
)
10 J. Böhm, C. Goldner and H. Markwig
and the refined Feynman integral to be
I l1,...,lnΓ,Ω (q1, . . . , qr) = coef
[x
l1
1 ···xln
n ]
s∏
k=1
P loop(qk)
r∏
k=s+1
P
(
xk1
xk2
, qk
)
.
Finally, we set
I l1,...,lnΓ (q) =
∑
Ω
I l1,...,lnΓ,Ω (q),
where the sum goes over all n! orders of the vertices of Γ, and
I l1,...,lnΓ (q1, . . . , qr) =
∑
Ω
I l1,...,lnΓ,Ω (q1, . . . , qr).
If we drop the superscript l1, . . . , ln in the notations above, then this stands for li = 0 for
all i.
q1 q2
q3
x1
x2
x3
q1
q2
q3
q4
q1
q2
q3
q4
x1
x2
x3
x1x2 x3
Figure 4.
If we assume |x| < 1 to express the (in q) constant coefficient of the (non-loop) propagator
(i.e., the first sum appearing in the propagator series in Definition 2.10) as the rational func-
tion x2
(x2−1)2
(using geometric series expansion), we can view the series from which we take the
xl11 · · ·xlnn -coefficient in the Feynman integral above as a function on a Cartesian product of
elliptic curves. If li = 0 for all i, the Feynman integral then becomes a path integral in complex
analysis (see [9, Definition 2.5 and equation (2.4)]). Note that using the change of coordinates
x = eiπu the (non-loop) propagator has the following nice form
P (u, q) = − 1
4π2
℘(u, q)− 1
12
E2
(
q2
)
in terms of the Weierstraß-P-function ℘ and the Eisenstein series
E2(q) := 1− 24
∞∑
d=1
σ(d)qd.
Here, σ denotes the sum-of-divisors function σ(d) =
∑
m|dm. The variable q above should be
considered as a coordinate of the moduli space of elliptic curves, the variable u as the complex
coordinate of a fixed elliptic curve. (More precisely, q = e2iπτ , where τ ∈ C is the parameter in
the upper half plane in the well-known definition of the Weierstraß-P-function.)
Definition 2.14 (Feynman integrals with vertex contributions). Let Γ be a Feynman graph,
and equip it with an additional genus function g associating a nonnegative integer gi to every
vertex xi. Let Ω be an order of the n vertices of Γ. We adapt our notion of propagators from
Definitions 2.10 and 2.13 to include vertex contributions: for non-loop edges, we set
P̃
(
xk1
xk2
, q
)
=
∞∑
w=1
S(wzk1)S(wzk2) · w ·
(
xk1
xk2
)w
+
∞∑
a=1
(∑
w|a
S(wzk1)S(wzk2) · w ·
((
xk1
xk2
)w
+
(
xk2
xk1
)w))
· qa.
Tropical Mirror Symmetry in Dimension One 11
For loop-edges connecting the vertex xk1 to itself, we set
P̃ loop(q) =
∞∑
a=1
(∑
w|a
S(wzk1)2 · w
)
qa.
The variables zki are new variables introduced for each vertex in order to take care of the genus
contribution.
We define the Feynman integral with vertex contributions for Γ, g and Ω to be
I l1,...,lnΓ,g,Ω (q) = coef
[z
2g1
1 ···z2gnn ]
coef
[x
l1
1 ···xln
n ]
n∏
i=1
1
S(zi)
s∏
k=1
P̃ loop(q)
r∏
k=s+1
P̃
(
xk1
xk2
, q
)
and the refined Feynman integral with vertex contributions
I l1,...,lnΓ,g,Ω (q1, . . . , qr) = coef
[z
2g1
1 ···z2gnn ]
coef
[x
l1
1 ···xln
n ]
n∏
i=1
1
S(zi)
s∏
k=1
P̃ loop(qk)
r∏
k=s+1
P̃
(
xk1
xk2
, qk
)
.
Again, we set
I l1,...,lnΓ,g (q) =
∑
Ω
I l1,...,lnΓ,g,Ω (q),
where the sum goes over all n! orders of the vertices, and
I l1,...,lnΓ,g (q1, . . . , qr) =
∑
Ω
I l1,...,lnΓ,g,Ω (q1, . . . , qr).
Also here, dropping the superscript l1, . . . , ln refers to the case li = 0 for all i.
2.4 (Tropical) mirror symmetry for elliptic curves
Theorem 2.15 (mirror symmetry for E). Fix g ≥ 2, n ≥ 1 and k1, . . . , kn ≥ 1 satisfying
k1 + · · ·+ kn = 2g − 2.
We can express the series of descendant Gromov–Witten invariants of E in terms of Feynman
integrals:∑
d≥1
⟨τk1(pt) · · · τkn(pt)⟩E,d
g,n q
d =
∑
(ft(Γ),g)
1
|Aut(ft(Γ), g)|
IΓ,g(q),
where Γ is a Feynman graph (see Definition 2.11) with a genus function g, such that the vertex xi
has genus gi and valency ki + 2 − 2gi, and such that h1(Γ) +
∑
gi = g, and where we consider
automorphisms of unlabeled graphs (ft is the forgetful map that forgets all labels of a Feynman
graph Γ, see Definition 2.18) that are required to respect the genus function.
A version of Theorem 2.15 is proved in [26, Proposition 6.7] (resp. [27, Proposition 3.4])
using the Fock space approach common in mathematical physics to which we relate the tropical
approach in Section 3. In our approach, Theorem 2.15 becomes an easy corollary obtained by
combining our Tropical mirror symmetry Theorem 2.19 with the correspondence Theorem 2.6.
Example 2.16 (automorphisms). Consider the middle Feynman graph of Example 2.12, de-
note it by Γ and let its genus function be g = 0, i.e., there is no genus at the vertices. The
automorphisms appearing in Theorem 2.15 are automorphisms respecting the underlying graph
structure and the genus function of (Γ, g). In other words, we forget the labels of Γ before
12 J. Böhm, C. Goldner and H. Markwig
determining its automorphisms. In case of Γ as above, the automorphism group is Z2×Z2×Z2,
because we can exchange the edges q1 and q2 (see Example 2.12) which gives a factor of Z2, we
can exchange the edges q3 and q4 and we can exchange the vertices x2 and x3 in such a way
that the edge q1 maps to q3 and the edge q2 maps to q4, see also the left side of Figure 5.
In Section 2.6, we deal with unlabeled tropical covers, but with fixed order. That is, we
fix which end i maps to which point pj on the elliptic curve. In such a case, on the Feynman
integral side, we deal with automorphisms of the underlying Feynman graph with vertex labels
(see Corollary 2.27). If we choose (Γ, g) as above, then the automorphism group of the graph
with vertex labels is Z2×Z2 since we cannot exchange the vertices x2 and x3 anymore, they are
now distinguishable (see also the right side of Figure 5).
x1
x2 x3
Figure 5. A non-labeled and a partially labeled graph.
Remark 2.17. If ki = 1 for all i, then the valency condition implies that the genus at each vertex
is 0 and the vertices are 3-valent. When forming the integral, the in the zi constant coefficient
is just 1, so we can neglect the zi and obtain Feynman integrals without vertex contributions in
this case.
By Remarks 2.2 and 2.17, the equality in Theorem 2.15 specializes to the well-known re-
lation involving the generating series of Hurwitz numbers and Feynman graphs, see, e.g., [15,
Theorem 9] and [9, Theorem 2.6].
Using the correspondence Theorem 2.6, we can formulate a version of the mirror symmetry re-
lation in Theorem 2.15, where instead of the generating function of des-cendant Gromov–Witten
invariants we use the generating function of tropical des-cendant Gromov–Witten invariants.
It turns out however that a finer version of a mirror symmetry relation naturally holds in the
tropical world, which uses labeled tropical covers, multidegrees and refined Feynman integrals:
Definition 2.18 (labeled tropical cover). Let π be a tropical cover satisfying given psi-conditions
with powers k1, . . . , kn and denote the genus of a vertex of the source curve which is adjacent
to end i by gi, where gi is given by ki via the psi-conditions (see Definition 2.3). We can shrink
the ends of the source curve and label the vertex that used to be adjacent to end i with xi.
The cover π is called labeled tropical cover if there is an isomorphism of multigraphs sending
a Feynman graph (Γ, g′) with a genus function (see Definition 2.11) to the combinatorial type
of the source curve of π, where the ends of the source curve are shrunk, such that g′i = gi for all
vertices. We say that π is of type Γ.
Shortly, a labeled tropical cover is a tropical cover for which we label the vertices and edges
of the source (vertices of different genus are distinguishable) according to a Feynman graph.
We fix a point p0 ∈ ET. For a labeled tropical cover of ET of type Γ, we introduce its
multidegree as the vector a in Nr with k-th entry ak =
∣∣π−1(p0)∩qk
∣∣ ·ω(qk), where ω(qk) denotes
the expansion factor of the edge qk. We define a labeled tropical descendant invariant
⟨τk1(pt) · · · τkn(pt)⟩
E,a,trop
Γ,n
as a count of labeled tropical covers of type Γ and with multidegree a satisfying the prescribed
point- and psi-conditions, again counted with multiplicity as in equation (2.2). (Note that
there are no nontrivial automorphism for a labeled tropical cover since all edges and vertices are
distinguishable by their labeling.)
Tropical Mirror Symmetry in Dimension One 13
Theorem 2.19 (tropical mirror symmetry for ET). Fix g ≥ 2, n ≥ 1 and k1, . . . , kn ≥ 1
satisfying k1 + · · · + kn = 2g − 2. Fix a Feynman graph Γ such that the vertex xi has valency
ki + 2− 2gi, and record the numbers gi in a genus vector g.
Then we can express the series of descendant Gromov–Witten invariants of ET of type Γ in
terms of a Feynman integral:∑
a∈Nr
⟨τk1(pt) · · · τkn(pt)⟩
E,a,trop
Γ,n qa11 · · · qarr = IΓ,g(q1, . . . , qr).
Theorem 2.15 follows from Theorem 2.19 using the correspondence Theorem 2.6, summing
over all Feynman graphs Γ such that h1(Γ) +
∑n
i=1 gi = g (where the gi are defined by the
valencies of the vertices as above), setting the qk equal to q again for all k and keeping track of
automorphisms (as in the proof of Theorem 2.14 using Theorem 2.20 in [9]).
We prove Theorem 2.19 in Section 2.5 using Theorem 2.23, which establishes a bijection
between labeled tropical covers contributing to a descendant Gromov–Witten invariant and
monomials contributing to a term of the series used for the Feynman integrals.
Example 2.20. Fix g = 2. We want to use Theorem 2.19 to calculate contributions to
⟨τ2(pt)τ0(pt)τ0(pt)⟩E,a,trop
Γ,3 for two cases, where in the first case the covers contributing have
a source curve with a nonzero genus function and in the second case the source curves have
a loop.
First case: we choose a = (0, 0, 3) and Γ as the left Feynman graph of Example 2.12. So
Theorem 2.19 tells us that we need to calculate the q01q
0
2q
3
3-coefficient of IΓ,g(q1, q2, q3) with
g = (1, 0, 0). We fix an order Ω, namely the identity as we did in Example 2.5. That is,
we require that end i is mapped to the point pi. Notice that the covers contributing to
⟨τ2(pt)τ0(pt)τ0(pt)⟩E,(3,0,0),trop
Γ,3 for Ω as above are the ones corresponding to the entries (2, 2)
and (3, 1) in the table given in Example 2.5. So we expect
1
24
+
17 · 27
24
=
115
6
(2.3)
as the contribution. We start by calculating terms of the propagators that contribute to the
q01q
0
2q
3
3-coefficient (we first let w = 1 for a3) in the product of the propagators such that their
product is constant in the xi, i.e., l1 = l2 = l3 = 0,
P̃
(
x1
x3
, q3
)
=
4 sinh
(
z1
2
)
sinh
(
z3
2
)(
x1
x3
+ x3
x1
)
q3
3
z1z3
+ · · · ,
P̃
(
x2
x3
, q2
)
=
4 sinh
(
z2
2
)
sinh
(
z3
2
)
x2
z2z3x3
+ · · · ,
P̃
(
x1
x2
, q1
)
=
4 sinh
(
z1
2
)
sinh
(
z2
2
)
x1
z1z2x2
+ · · · .
Therefore,
coef [q01q02q33 ]
P̃ (x1
x3
, q3)P̃ (
x2
x3
, q2)P̃ (
x1
x2
, q1)
S(z3)S(z2)S(z1)
=
8 sinh
(
z1
2
)
sinh
(
z2
2
)
sinh
(
z3
2
)
z1z2z3
= · · ·+ 1
1920
z1
4 +
1
576
z1
2z2
2 +
1
576
z1
2z3
2
+
1
1920
z2
4 +
1
576
z2
2z3
2 +
1
1920
z3
4 +
1
24
z1
2 +
1
24
z2
2 +
1
24
z3
2 + 1
14 J. Böhm, C. Goldner and H. Markwig
and, hence, the z21z
0
2z
0
3-coefficient is 1
24 , which is precisely the first summand of (2.3). The second
summand is obtained by letting w = 3 for a3 such that
P̃
(
x1
x3
, q3
)
=
4 sinh
(
3z1
2
)
sinh
(
3z3
2
)(x3
1
x3
3
+
x3
3
x3
1
)
q3
3
3z1z3
+ · · · ,
P̃
(
x2
x3
, q2
)
=
4 sinh
(
3z2
2
)
sinh
(
3z3
2
)
x32
3z2z3x33
+ · · · ,
P̃
(
x1
x2
, q1
)
=
4 sinh
(
3z1
2
)
sinh
(
3z2
2
)
x31
3z1z2x32
+ · · ·
and therefore
coef [q01q02q33 ]
P̃ (x1
x3
, q3)P̃ (
x2
x3
, q2)P̃ (
x1
x2
, q1)
S(z3)S(z2)S(z1)
=
8
(
sinh
(
3z1
2
))2(
sinh
(
3z2
2
))2(
sinh
(
3z3
2
))2
27 sinh
(
z1
2
)
sinh
(
z2
2
)
sinh
(
z3
2
)
z1z2z3
= · · · 3369
640
z1
4 +
867
64
z1
2z2
2 +
867
64
z1
2z3
2
+
3369
640
z2
4 +
867
64
z2
2z3
2 +
3369
640
z3
4 +
153
8
z1
2 +
153
8
z2
2 +
153
8
z3
2 + 27,
where the z21z
0
2z
0
3-coefficient is 153
8 which equals the second summand of (2.3).
Second case: we choose a = (2, 0, 0, 1) and Γ as the right Feynman graph of Example 2.12.
By Theorem 2.19, we need to calculate the q21q
0
2q
0
3q
1
4-coefficient of IΓ,g(q1, q2, q3, q4) with g = 0.
Again, we pick Ω as the order given by the identity. As before, we calculate the terms of the
propagators that contribute to the q21q
0
2q
0
3q
1
4-coefficient in the product of the propagators such
that their product is constant in the xi, i.e., l1 = l2 = l3 = l4 = 0, and let w = 2 for a1, then
P̃ loop(q1) =
2(sinh(z1))
2q1
2
z12
,
P̃
(
x1
x2
, q2
)
=
4 sinh
(
z1
2
)
sinh
(
z2
2
)
x1
z1z2x2
+ · · · ,
P̃
(
x2
x3
, q3
)
=
4 sinh
(
z2
2
)
sinh
(
z3
2
)
x2
z2z3x3
+ · · · ,
P̃
(
x1
x3
, q4
)
=
4 sinh
(
z1
2
)
sinh
(
z3
2
)(
x1
x3
+ x3
x1
)
q4
z1z3
+ · · ·
and
coef [q21q02q03q14 ]
P̃ loop(q1)P̃
(
x1
x3
, q4
)
P̃
(
x2
x3
, q3
)
P̃
(
x1
x2
, q2
)
S(z3)S(z2)S(z1)
=
16
(
sinh
(
z1
))2
sinh
(
z1
2
)
sinh
(
z2
2
)
sinh
(
z3
2
)
z13z2z3
= 2 +
3
4
z1
2 +
1
12
z2
2 +
1
12
z3
2 +
113
960
z1
4
+
1
32
z1
2z2
2 +
1
32
z3
2z1
2 +
1
960
z2
4 +
1
288
z3
2z2
2 +
1
960
z3
4 + · · · ,
where the constant coefficient in the zi is 2. If we let w = 1 for a1, we get 1. This makes 3
in total, which is the number we expect when using the table from Example 2.5 again (entries
(1, 1) and (2, 1)).
Tropical Mirror Symmetry in Dimension One 15
2.5 The bijection
This subsection is devoted to the proof of the tropical mirror symmetry Theorem 2.19. The main
ingredient is a bijection of graph covers and monomials contributing to a Feynman integral.
Let Γ be a Feynman graph (see Definition 2.11). Fix a multidegree a and an order Ω.
We can view Ω as an element in the symmetric group on n elements, associating to i the
place Ω(i) that the vertex xi takes in the order Ω.
Fix an orientation of ET and points p0, p1, . . . , pn ordered in this way when going around ET
in the fixed orientation starting at p0.
Definition 2.21 (graph covers of fixed order and multidegree). A graph cover of type Γ, order Ω
and multidegree a is a (possibly leaky w.r.t. (l1, . . . , ln), see Remark 2.9) tropical cover π:
Γ′ → ET, where Γ′ is a metrization of Γ, such that the multidegree of π at p0 is a and such that
π−1(pΩ(i)) contains xi. (Since there are n point conditions and n vertices, it follows that there
is precisely one vertex of Γ in each preimage π−1(pj)).)
We define N l1,...,ln
Γ,a,Ω to be the weighted count of (l1, . . . , ln)-leaky graph covers of type Γ,
order Ω and multidegree a, where we count each with the product of the expansion factors of
the edges.
If there is no mentioning of l1, . . . , ln, we refer to the case of no leaking as usual.
Fix g ≥ 2, n ≥ 1 and k1, . . . , kn ≥ 1 satisfying k1 + · · · + kn = 2g − 2. Let Γ be a Feynman
graph. Fix a multidegree a and an order Ω. Assume that for each vertex xi of Γ, ki + val(xi) is
even.
Lemma 2.22 (graph covers and labeled tropical covers). There is a bijection between graph
covers of type Γ, order Ω and multidegree a and labeled tropical covers π : Γ′ → ET contribu-
ting to
⟨τk1(pt) · · · τkn(pt)⟩
E,a,trop
Γ,n
and satisfying π(i) = pΩ(i).
Proof. Let π : Γ′ → ET be such a labeled tropical cover. We can describe the bijection as
the map sending π to a graph cover π̃ by shrinking marked ends of Γ′, labeling the vertex
that used to be adjacent to end i by xi, and forgetting the genus at vertices. By definition of
⟨τk1(pt) · · · τkn(pt)⟩
E,a,trop
Γ,n , the graph cover is of type Γ. The multidegree is the same for the
tropical cover and the graph cover. The set π−1(pΩ(i)) contains xi, since the marked end i is
mapped to pΩ(i) by π. The inverse map associates the genus ki+2−val(xi)
2 to the vertex xi (which
is an integer by our assumption), and attaches the end marked i. Then the valence is ki+3−2gi
and the psi-condition is satisfied. ■
Theorem 2.23 (bijection of graph covers and tuples in Feynman integrals). Let Γ be a Feynman
graph as in Definition 2.11. Fix a multidegree a satisfying ak > 0 for all k ≤ s, an order Ω, and
integers l1, . . . , ln. As in Definition 2.13, we use the notation xk1 and xk2 for the two vertices
adjacent to the edge qk, where we assume xk1 < xk2 in Ω.
There is a bijection between (l1, . . . , ln)-leaky graph covers of type Γ, order Ω and multide-
gree a, and tuples(
(wk)k=1,...,s,
((
ak, wk ·
(
xki
xkj
)wk
))
k=s+1,...,r
)
, (2.4)
where i = 1 and j = 2 if ak = 0, and {i, j} = {1, 2} otherwise, where wk divides ak if ak ̸= 0,
and where the product of the fractions has exponent li in xi.
16 J. Böhm, C. Goldner and H. Markwig
Moreover, the weighted count of graph covers equals the qa11 · · · qann -coefficient of the refined
Feynman integral:
N l1,...,ln
Γ,a,Ω = coef [qa11 ···qarr ] I
l1,...,ln
Γ,Ω (q1, . . . , qr).
Remark 2.24 (tuples and Feynman integrals). Note that the products of second entries for
k > s of a tuple as in (2.4) are precisely the contributions showing up in the series
r∏
k=s+1
P
(
xk1
xk2
, qk
)
with the exponents of the xi given by the li, and the exponents of the qi given by the ai. By defi-
nition of the refined Feynman integral (see Definition 2.13), adding a choice of summand wk for
each loop-edge qk, k ≤ s, each tuple contributes exactly w1 · · ·wr to the qa11 · · · qann -coefficient of
the refined Feynman integral I l1,...,lnΓ,Ω (q1, . . . , qr).
In particular, if ak = 0 for some k ≤ s, the qa11 · · · qann -coefficient of the refined Feynman
integral I l1,...,lnΓ,Ω (q1, . . . , qr) is zero, and there are no tuples.
Proof of Theorem 2.23. Given a tuple as in (2.4), we construct a graph cover as follows.
We keep track of the cover by drawing the vertices and edges projecting onto their images.
To ease the drawing, we think of ET as being cut off at p0 (see Example 2.7).
We start by drawing vertices xi above the points pΩ(i) in ET.
For k > s and for an entry wk ·
( xki
x
kj
)wk , we draw an edge with expansion factor wk connecting
vertex xki to vertex xkj . If ak = 0, we draw this edge in our cut picture direct, without passing
over p0. If ak > 0, we “curl it”, passing over p0
ak
wk
times before it reaches its end vertex.
We assume in our tuple that i = 1 and j = 2 if ak = 0. By Definition 2.13, xk1 < xk2 in Ω,
which implies that in our picture, the vertex xk1 is drawn before xk2 (in the orientation of ET),
which makes it possible to draw the edge qk directly without passing p0.
Since wk divides ak, it is possible to “curl” the edges qk with ak > 0 as required.
For k ≤ s and an entry wk, we draw a loop-edge of weight wk connecting the vertex of qk to
itself, “curled” over p0
ak
wk
times.
In the drawing we created for the tuple (2.4), we have obviously drawn a graph cover with
source curve of type Γ, since we connected the vertices xk1 and xk2 with the edge qk. Further-
more, the multidegree is a because of our curling requirement. The order Ω is respected by the
way we have drawn the vertices. It remains to show that the graph cover is (l1, . . . , ln)-leaky.
To see this, notice that the edges adjacent to vertex xi correspond to tuples whose fraction
contains a power of xi, and that the exponent equals ± the expansion factor of the edge, where
the sign is positive if the edge leaves xi and negative if it enters xi (w.r.t. the orientation of ET).
Since we require the total power in xi to be li, the cover leaks li at vertex xi.
Clearly, the process can be reversed associating a tuple to a graph cover, and using the same
arguments as before, the tuple satisfies the requirements from above. In particular, the entry ak
of the multidegree of a cover with a loop-edge qk is nonzero. Thus, we have a bijection between
graph covers and tuples.
The equality follows from Remark 2.24, taking into account that a graph cover is counted
with multiplicity the product of its expansion factors (which are the wi) in N
l1,...,ln
Γ,a,Ω . ■
Example 2.25. We illustrate the proof of Theorem 2.23 by constructing a graph cover from
a given tuple as in (2.4). We let Γ be the right graph of Example 2.12, Ω the identity, and li = 0
for all i. We choose the tuple(
1,
(
2, 1 ·
(
x1
x3
)−1)
,
(
0, 1 ·
(
x1
x2
)1)
,
(
0, 1 ·
(
x2
x3
)1))
Tropical Mirror Symmetry in Dimension One 17
and the order Ω given by the identity. Note that this tuple is not leaky. See Figure 6 for the
following: We start by drawing the vertices x1, x2, x3 above p1, p2, p3. After that we add the
non-curled edges q3, q4 which are given by the third and fourth entry of our tuple above. The
edge q2 is obtained by starting at x1 and going left (we have a negative exponent in the second
entry of our tuple), curling once (we want to pass p0 twice with q2) and ending at x3. There
is also one loop edge (the first entry of the tuple) adjacent to x1 which does not curl. Since all
weights of edges are 1, we can also see from the graph that it is not leaky as we expected. The
upper graph in Figure 6 inherits a metrization from downstairs. Thus we obtain a graph cover.
P1
Tp1 p2 p3
x1
x2 x3
q3 q4
q2
q2
q2
q1 q1
Figure 6. A graph cover constructed from a tuple. Note that this graph cover arises from cutting
(and labeling) the upper right source curve in Figure 3 along the preimages of a point p0 (see also
Example 2.7).
As we have seen in Lemma 2.22, graph covers are closely related to tropical covers showing
up in a tropical descendant Gromov–Witten invariant. However, the multiplicity of a tropical
cover contains, besides the expansion factors for edges which already appear in the bijection in
Theorem 2.23, also factors for each vertex which can be viewed as local descendant Gromov–
Witten invariants (see equation (2.1)). The need to take these into account takes us to Feynman
integrals with vertex contributions (see Definition 2.14).
We use Okounkov–Pandharipande’s one-point series, i.e., the following nice form for the
generating series of relative one-point descendant Gromov–Witten invariants, resp. for the local
vertex multiplicities of tropical covers satisfying point and psi-conditions:∑
g≥0
⟨µ|τ2g−2+ℓ(µ)+ℓ(ν)(pt)|ν⟩
P1,d
g,1 · z2g =
∏
S(µiz) ·
∏
S(νiz)
S(z)
. (2.5)
Here, the S-function is as in Definition 2.10, and the product goes over all entries µi resp. νi of
the two fixed partitions (see [36, Theorem 2], note that we consider Gromov–Witten invariants
with the preimages of 0 and ∞ marked). Note that the S-function is an even function (i.e.,
S(−z) = S(z)), since sinh is an odd function and quotients of odd functions are even.
Proof of Theorem 2.19. We prove the equality by restricting to the qa11 · · · qarr -coefficient on
each side. It follows from Lemma 2.22 that we can expand the left side as a sum over orders Ω,
which we can do by definition of Feynman integral also on the right. We thus have to show
that the weighted count of labeled tropical covers contributing to ⟨τk1(pt) · · · τkn(pt)⟩
E,a,trop
Γ,n and
satisfying π(i) = pΩ(i) equals coef [qa11 ···qarr ] IΓ,g,Ω(q1, . . . , qr).
To see this, note that by Remark 2.24 we deal with tuples as in Theorem 2.23 when com-
puting coef [qa11 ···qarr ] IΓ,Ω(q1, . . . , qr), however since we compute a Feynman integral with vertex
contributions now each second entry wk ·
( xki
xkj
)wk showing up in a tuple meets S(wkzk1)S(wkzk2)
first. By Theorem 2.23, the tuples are in bijection with graph covers. For a fixed graph cover
18 J. Böhm, C. Goldner and H. Markwig
corresponding to a fixed tuple, the vertex contributions in the Feynman integral thus produce
factors of S(wkzk1)S(wkzk2) for an edge of expansion factor wk connecting the vertices xk1
and xk2 . Collecting those factors, sorting by zi, and adding in the factor 1
S(zi) we have in the
definition of Feynman integral with vertex contributions (see Definition 2.14), we obtain for
each vertex xi a contribution of
∏
S(µjzi)·
∏
S(νjzi)
S(zi) . Here, the notation is set up as follows: we
collect the expansion factors of all incoming edges adjacent to xi in the partition µ and those
of all outgoing edges in the partition ν. Taking the z2gii -coefficient, we obtain a local vertex
contribution of
⟨µ|τki(pt)|ν⟩
P1,|µ|
gi,1
by the one-point series from equation (2.5). By equation (2.1), this is exactly the local vertex
multiplicity we need to take into account for the labeled tropical cover. ■
Remark 2.26. Let us compare the Tropical mirror symmetry Theorem 2.19 for descendant
invariants with the version for Hurwitz numbers [9, Theorem 2.20]. As we saw in Remarks 2.2
and 2.17, in the version for Hurwitz numbers, we only have to take 3-valent graphs such that
all vertices have genus 0 into account. Adding in descendants requires us to generalize in
two ways: we need to include graphs whose vertices have other valencies, and whose vertices
have genus. The main ingredient in our proof of tropical mirror symmetry is the bijection
between graph covers and monomials contributing to a Feynman integral, see Theorem 2.23.
Graphs with vertices of valence bigger 3 fit into this context. The genus at vertices requires
us to use local vertex multiplicities for the tropical covers, which are hard to translate to the
Feynman integral world. The fact that the one-point series (2.5) can be expressed in a way
separating contributions for the edges adjacent to a vertex makes it possible to incorporate
these multiplicities in a Feynman integral with vertex contributions as in Definition 2.14.
2.6 Quasimodularity
In the case that all ki = 1, the Mirror symmetry Theorem 2.15 specializes to the well-known
relation involving the generating series of Hurwitz numbers and Feynman integrals for 3-valent
graphs without vertex contributions (see Remark 2.17). This special case of the mirror symmetry
relation was used in [15, 24] to prove that the generating function of Hurwitz numbers for g ≥ 2
is a quasimodular form of weight 6g − 6. Quasimodularity of generating functions of covers is
a phenomenon studied beyond the case considered here, other important cases are generating
functions of pillowcase covers [16] or generating functions of numbers of covers of an elliptic
curve with fixed ramifications with respect to the parity of the pullback of the trivial theta
characteristic [17].
Quasimodularity behaviour is desirable because it controls the asymptotic of the generating
function. A series in q is quasimodular if and only if it is in the polynomial ring generated by
the three Eisenstein series E2, E4 and E6 [24]. The weight of a quasimodular form refers to its
degree when viewed as a polynomial in the Eisenstein series. A series is called a quasimodular
form of weight w if it is a homogeneous polynomial of degree w in the Eisenstein series, and
it is called a quasimodular form of mixed weight if it is a non-homogeneous polynomial in the
Eisenstein series.
From the Tropical mirror symmetry Theorem 2.19, the generating function of descendant
Gromov–Witten invariants of an elliptic curve obtains a natural stratification as sum over Feyn-
man graphs, and, even finer, as sum over orders Ω for each Feynman graph (see Corollary 2.27).
If we fix a Feynman graph Γ and a suitable genus function g – if ki = 1 for all i, this means
we fix a 3-valent graph with genus 0 at each vertex – we can study quasimodularity of individ-
ual summands. We can consider summands IΓ,g(q), or we can even break the sum into finer
contributions by considering IΓ,g,Ω(q) for a fixed order Ω.
Tropical Mirror Symmetry in Dimension One 19
For the case that ki = 1 for all i, this study was initiated in [9, Theorem 3.2], where it is
conjectured that IΓ,g(q) is quasimodular of weight 6g − 6. In [19], Goujard and Möller provide
tools to study quasimodularity of generating series depending on Feynman graphs, and they
prove that if ki = 1 for all i, each summand IΓ,g,Ω(q) is a quasimodular form of mixed weight,
where the highest appearing weight is 6g− 6. They also compute examples where lower weights
appear. Since the whole sum (over all graphs, and over all orders) is quasimodular of weight
6g − 6, the contributions of lower weights must cancel in the sum. It remains an open question
whether they already cancel in a summand IΓ,g(q), i.e., when we sum over all orders Ω, but for
a fixed graph Γ.
In this section, we deduce from [19, Theorem 6.1] that IΓ,g,Ω(q) is a quasimodular form of
mixed weight also in the case of arbitrary ki. Also in the case of arbitrary ki, quasimodularity
(of mixed weight) of the whole generating series (the sum over all graphs, and all orders) was
studied before [28]. First, we interpret IΓ,g,Ω(q) as a generating function of tropical covers:
Corollary 2.27. Fix g ≥ 2, n ≥ 1 and k1, . . . , kn ≥ 1 satisfying k1 + · · · + kn = 2g − 2. Fix
a Feynman graph Γ such that the vertex xi has valency ki + 2− 2gi with gi ∈ N, and record the
numbers gi in the genus function g. Fix an order Ω. For d ∈ N, let ⟨τk1(pt) · · · τkn(pt)⟩
E,d,trop
Γ,n,Ω
denote the number of (unlabeled) tropical covers (counted with multiplicity) contributing to
⟨τk1(pt) · · · τkn(pt)⟩
E,d,trop
g,n , for which the source curve has combinatorial type Γ after shrinking
the ends and satisfying π(i) = pΩ(i).
Then we can express the generating function of these invariants in terms of the Feynman
integral∑
d∈N
⟨τk1(pt) · · · τkn(pt)⟩
E,d,trop
Γ,n,Ω qd =
1
|Aut(ftedge(Γ), g)|
IΓ,g,Ω(q),
where ftedge is the map that forgets the edge labels of a Feynman graph, and automorphisms
respect the remaining vertex labels and the genus function (see Example 2.16).
Proof. Consider Theorem 2.19 and let q1 = · · · = qr = q. With a similar argument as we
use to deduce Theorem 2.15 from Theorem 2.19, we also obtain an automorphism factor here.
Fixing the order leads to labels on the vertices of the source curves, i.e., we need to consider
automorphisms which respect partially labeled graphs as in Example 2.16. ■
Example 2.28. We want to express IΓ,g,Ω(q) as polynomial in the Eisenstein series, where Ω is
the identity, g = (0, 0, 0) or g = (1, 0, 0) and Γ is any Feynman graph as shown in Example 2.12.
So this example is a continuation of Examples 2.5 and 2.20.
First, let Γ1 be the left Feynman graph of Example 2.12 and let g
1
= (1, 0, 0). We calculate
that
IΓ1,g1,Ω
(q) =
1
20736
E6(q)−
1
13824
E2(q)E4(q) +
1
41472
E3
2(q) +
1
20736
E2
4(q)
− 1
10368
E2(q)E6(q) +
1
20736
E2
2(q)E4(q)
=
1
24
q +
5
2
q2 +
39
2
q3 +
278
3
q4 +
1025
4
q5 + 738q6 +
4165
3
q7 + 3080q8 + · · · .
Notice that IΓ1,g1,Ω
is of of mixed weight since E6 and E2E6 are of different weight. Recall
that we calculated 115
6 as contribution to the q3-coefficient. The other covers contributing are
shown in Figure 3 of Example 2.5 and are the ones corresponding to the following entries of the
table of Example 2.5: (2, 3), (2, 4), (2, 5), (3, 2), (3, 3), (4, 1), (4, 2), (4, 3). Each of these covers
contributes with 1
24 such that in total we expect (see Corollary 2.27)
coef [q3] IΓ1,g1,Ω
(q) =
115
6
+
8
24
=
39
2
,
which matches our calculation.
20 J. Böhm, C. Goldner and H. Markwig
Second, we choose Γ2 to be the right Feynman graph of Example 2.12 and let g
2
= 0.
We calculate that
IΓ2,g2,Ω
(q) = − 1
20736
E6(q) +
1
13824
E2(q)E4(q)−
1
41472
E3
2(q) +
1
20736
E2(q)E6(q)
− 1
13824
E2
2(q)E4(q) +
1
41472
E4
2(q)
= q2 + 15q3 + 76q4 + 275q5 + 720q6 + 1666q7 + 3440q8 + 6129q9 + · · · .
Notice that, again, IΓ2,g2,Ω
is of mixed weight, but IΓ1,g1,Ω
+ IΓ2,g2,Ω
is homogeneous. As above,
we can verify the q3-coefficient using Example 2.5.
Third, we choose Γ3 to be the middle Feynman graph of Example 2.12 and let g
3
= 0. In this
case, we obtain the homogeneous expression
IΓ3,g3,Ω
(q) =
1
20736
E2
4(q)−
1
10368
E2
2(q)E4(q) +
1
20736
E4
2(q)
= 4q2 + 48q3 + 240q4 + 800q5 + 2160q6 + 4704q7 + 9920q8 + 17280q9 + · · ·
= 4 · (q2 + 12q3 + 60q4 + 200q5 + 540q6 + 1176q7 + 2480q8 + 4320q9 + · · · ),
where the factor 4 in the last row is due to the automorphisms of the underlying Feynman graph,
see Corollary 2.27. Again, we can verify the q3-coefficient using Example 2.5.
Corollary 2.29. Fix g, n and k1, . . . , kn with k1+ · · ·+kn = 2g−2. Let Γ be a Feynman graph
with r edges (see Definition 2.11) and g a genus function satisfying h1(Γ)+
∑n
i=1 gi = g. Fix an
order Ω. Then the Feynman integral IΓ,g,Ω(q) – i.e., the generating function counting tropical
covers for the tropical descendant Gromov–Witten invariant ⟨τk1(pt) · · · τkn(pt)⟩
E,d,trop
Γ,n of type Γ
and order Ω, see Corollary 2.27 – is a quasimodular form of mixed weight, with highest occuring
weight 2 ·
(
r +
∑n
i=1 gi
)
.
Proof. This follows from [19, Theorem 6.1], since the local vertex contributions we have to
take into account for a vertex xi is polynomial of even degree 2gi in the expansion factors of the
adjacent edges by [19, Theorem 4.1] (see [36, 38]). ■
This statement is essentially a byproduct of [19, Corollary 6.2] which states that the generating
series of tropical covers with fixed ramification profiles (see [14, Definition 2.1.3]) and with fixed
underlying graph Γ and order Ω is a quasimodular form of mixed weight. The proof in [19]
detours by deducing the quasimodularity of the function above from the quasimodularity of
our IΓ,g,Ω(q) (without explicitely stating this). The descendant Gromov–Witten invariants we
focus on here are called Hurwitz numbers with completed cycles in [19], which is explained by
the Okounkov–Pandharipande GW/H correspondence in [36], see also [38].
3 Tropical mirror symmetry and the boson–fermion
correspondence
The purpose of this section is to reveal the close relation between the proof of Theorem 2.15 in
mathematical physics, using Fock spaces, and our tropical approach. Since the tropical setting
requires a labeling of the underlying Feynman graphs and the use of the variables q1, . . . , qr to
distinguish degree contributions from the different edges, we enrich the Fock space approach
by incorporating adequate labelings. This enlarges the set of operators, but makes it easier to
distinguish contributions for a fixed Feynman graph to a matrix element. In this way, we extend
the Fock space approach so that it gives an alternative proof of the tropical mirror symmetry
Tropical Mirror Symmetry in Dimension One 21
Theorem 2.19, which holds on a finer level. Our main ingredient is Theorem 3.8, proving the
equality of the number of labeled tropical covers with fixed underlying source graph, fixed
multidegree and order and a sum of matrix elements in a bosonic Fock space.
For the sake of explicitness, we limit our considerations to the case of Hurwitz numbers, i.e.,
ki = 1 for all i, and we do not have vertex contributions for Feynman integrals (see Remarks 2.2
and 2.17). In particular, all our graphs are 3-valent, have no loops and genus 0 at vertices.
Higher descendants resp. vertex contributions can be incorporated into our discussion also, but
would increase the amount of notation largely – we would have to consider more summands for
a bosonic vertex operator, and the tropical local vertex multiplicities would have to show up as
coefficients of the bosonic vertex operator (see [13, Section 5]).
As shown in Figure 2, tropical geometry hands us a short-cut in the Fock space setting: we
can relate the generating series of Hurwitz numbers directly to operators on the bosonic Fock
space and do not need to invoke the fermionic Fock space and the boson–fermion correspondence,
which is often viewed as the essence of mirror symmetry for elliptic curves.
3.1 Hurwitz numbers as matrix elements
We begin by shortly reviewing the bosonic Fock space approach for generating series of Hurwitz
numbers.
The bosonic Heisenberg algebra H is the Lie algebra with basis αn for n ∈ Z such that for
n ̸= 0 the following commutator relations are satisfied:
[αn, αm] = (n · δn,−m)α0,
where δn,−m is the Kronecker symbol and [αn, αm] := αnαm−αmαn. The bosonic Fock space F
is a representation of H. It is generated by a single “vacuum vector” v∅. The positive generators
annihilate v∅ : αn ·v∅ = 0 for n > 0, α0 acts as the identity and the negative operators act freely.
That is, F has a basis bµ indexed by partitions, where
bµ = α−µ1 · · ·α−µm · v∅.
We define an inner product on F by declaring ⟨v∅|v∅⟩ = 1 and αn to be the adjoint of α−n.
We write ⟨v|A|w⟩ for ⟨v|Aw⟩, where v, w ∈ F and the operator A is a product of elements inH,
and ⟨A⟩ for ⟨v∅|A|v∅⟩. The first is called a matrix element, the second a vacuum expectation.
Definition 3.1. The cut-join operator is defined by
M =
1
2
∑
k>0
∑
0<i,j
i+j=k
α−jα−iαk + α−kαiαj .
The relative invariants of P1 can be interpreted as a matrix element involving M (notice that
the invariants in questions are equal to double Hurwitz numbers by Okounkov–Pandharipande’s
GW/H correspondence, [36, Theorem 1]):
Proposition 3.2. A relative Gromov–Witten invariant of P1, resp. a double Hurwitz number,
equals a matrix element on the bosonic Fock space:
⟨µ|τ1(pt)n|ν⟩P
1,d,•
g,n =
n!∏
i µi ·
∏
j νj
⟨bµ|Mn|bν⟩.
This statement follows by combining Wick’s theorem with the correspondence Theorem 2.8:
Wick’s theorem ([13, Theorem 5.4.3], [6, Proposition 5.2], [41]) expresses a matrix element as
a weighted count of graphs that are obtained by completing local pictures. It turns out that the
22 J. Böhm, C. Goldner and H. Markwig
graphs in question are exactly the tropical covers we enumerate to obtain ⟨µ|τ1(pt)n|ν⟩P
1,d,trop
g,n ,
where n! arises from fixing a set of points to which labeled ends are mapped to (rather than
prescribing a point a labeled end should map to, see Definition 2.3).
Notice that we have to use the disconnected theory here (•), since the matrix element encodes
all graphs completing the local pictures and cannot distinguish connected and disconnected
graphs.
The local pictures are built as follows: we draw one vertex for each cut-join operator. For
an αn with n > 0, we draw an edge germ of weight n pointing to the right. If n < 0, we draw an
edge germ of weight n pointing to the left. For the two Fock space elements bµ and bν , we draw
germs of ends: of weights µi on the left pointing to the right, of weights νi on the right pointing
to the left. Wick’s theorem states that the matrix element ⟨bµ|Mn|bν⟩ equals a sum of graphs
completing all possible local pictures, where each graph contributes the product of the weights
of all its edges (including the ends). A completion of the local pictures can be interpreted as
a tropical cover of P1
T (with suitable metrization).
The cut-join operator sums over all the possibilities of the local pictures for the graphs, i.e.,
it sums over all possibilities how a vertex of a tropical cover can look like (see Figure 7).
i
j
k k
i
j
Figure 7. Local pictures of graphs with weights on the edges.
Example 3.3. Consider the local pieces shown below. There are three ways of completing them
to a graph with local pictures like in Figure 7.
2
1
2
1
The completed graphs are shown in Figure 8. The product of the upper graph’s edge weights
(including the ends) is 12, 4 for the middle graph and 4 for the lower graph. Hence Wick’s
theorem and Proposition 3.2 yield
⟨(2, 1)|τ1(pt)2|(2, 1)⟩P
1,3,•
2,2 = 2! ·
(
3 + 1 +
1
2
)
= 9,
where we have to divide the last summand by two because there is an automorphism exchanging
the two edges that connect the vertices in the lower graph of Figure 8.
Combining Proposition 3.2 with a degeneration argument, we can express Gromov–Witten
invariants, resp. Hurwitz numbers of the elliptic curve in terms of matrix elements:
Proposition 3.4. A Hurwitz number of the elliptic curve equals a weighted sum of double
Hurwitz numbers of P1:
⟨τ1(pt)n⟩E,d,•
g,n =
∑
µ ⊢d
∏
i µi
|Aut(µ)|
⟨µ|τ1(pt)n|µ⟩P
1,d,•
g−ℓ(µ),n.
Here, the sum goes over all partitions µ of d, µi denotes their entries, and ℓ(µ) the length.
Corollary 3.5. A Hurwitz number of the elliptic curve E equals a sum of matrix elements on
the bosonic Fock space:
⟨τ1(pt)n⟩E,d,•
g,n =
∑
µ ⊢d
n!
|Aut(µ)|
∏
i µi
⟨bµ|Mn|bµ⟩.
Tropical Mirror Symmetry in Dimension One 23
2
1
2
1
3
2
1
2
1
1
2
1
1
1
2
1
Figure 8. Completions of the local pieces above. Note that there is an automorphism exchanging the
two bounded edges in the lower graph.
Proposition 3.4 is a corollary from the two correspondence Theorems 2.6 and 2.8: given
a tropical cover of ET, let µ be the partition encoding the weights of the edges mapping to the
base point p0. We mark the preimages of p0, for which we have |Aut(µ)| choices. For each
choice, we cut off ET at p0 and the covering curve at the preimages of p0, obtaining a cover
of P1
T with ramification profiles µ and µ above ±∞. The cut off tropical cover contributes to
⟨µ|τ1(pt)n|µ⟩P
1,d,•,trop
g,n , but its multiplicity differs from the multiplicity of the cover of ET by
a factor of
∏
µi, since the edges we cut off are no longer bounded.
Example 3.6. We want to calculate ⟨τ1(pt)2⟩E,3,•
2,2 using Corollary 3.5 and Wick’s theorem. The
partitions of 3 are (1, 1, 1), (2, 1) and (3). The summand of (2, 1) follows from Example 3.3,
namely 9·2 = 18. The figure below shows how to complete the local pieces given by the partitions
(1, 1, 1) and (3).
1
1
1
1
3
1
2
1 1
3
2
Figure 9. More completions of local pieces. Note that there are automorphisms of the upper graph that
exchange the edges of weight one adjacent to a 3-valent vertex.
Note that there are in fact 9 choices of how to complete the local pieces of (1, 1, 1) since we can
choose which ends (in the upper graph) the straight line should connect. Thus the upper graphs
contribute (9 of them) 2! · 9·2
4 · 1
3! =
3
2 and the lower graph contributes 2! · 2 · 3 = 12. Therefore
⟨τ1(pt)2⟩E,3,•
2,2 = 63
2 .
24 J. Böhm, C. Goldner and H. Markwig
3.2 Labeled matrix elements for labeled tropical covers
Now we would like to link this Fock space language for Gromov–Witten invariants resp. Hurwitz
numbers with tropical mirror symmetry. Recall that tropical mirror symmetry holds naturally on
a fine level, giving an equality of the qa11 · · · qarr -coefficient of a Feynman integral IΓ,Ω(q1, . . . , qr)
and the number NΓ,a,Ω, which counts labeled covers of type Γ, with multidegree a and such that
the order Ω is satisfied, i.e., the contribution to ⟨τ1(pt)n⟩E,a,trop
Γ,n of covers π satisfying π(i) = pΩ(i)
(see Lemma 2.22, Theorems 2.23 and 2.19).
Fix a Feynman graph Γ, a multidegree a and an order Ω. Remember that Γ is a 3-valent
connected graph with first Betti number g, because of our restriction that ki = 1 for all i.
In particular, Γ has no loops.
Our expression for NΓ,a,Ω in terms of matrix elements (see Theorem 3.8 below) involves a sum
over all tuples (wk)k : ak>0 with wk|ak for all k with ak > 0, since we incorporate the degeneration
idea from above.
For a fixed choice of (wk)k, let Γ′ be the graph obtained from Γ by cutting the edge qk
exactly ak
wk
times. We introduce the following labels for the (cut) edges of Γ′: we denote the
pieces by qk,1, . . . , qk, ak
wk
+1. There are at most ak+1 pieces, depending on wk. For an edge which
is not cut, i.e., ak = 0, we call it qk,1 to consistently have two indices for the edge labels in Γ′.
We enlarge our set of operators in a way that allows to distinguish the edges of the cut graph
Γ′: Let the αk,j
n , for each k = 1, . . . , r, j = 1, . . . , ak+1, and n ∈ Z\{0}, satisfy the commutator
relations
[αk,j
n , αl,i
m] = (n · δk,l · δj,i · δn,−m)α0.
As before, we let the bosonic Fock space F be generated by v∅, following the rules from before:
αk,j
n · v∅ = 0 for n > 0, α0 acts as identity, and the operators with negative subscript act freely.
We let ⟨v∅|v∅⟩ = 1 and let αk,j
n be the adjoint of αk,j
−n.
Definition 3.7. Let Γ, a and (wk)k be as above. Let xi be a vertex of Γ. We denote the three
adjacent edges by qi1 , qi2 and qi3 . For l = 1, 2, 3 set cl =
ail
wil
+ 1 if ail > 0 and cl = 1 else.
We also set dml
= cl if ml > 0 and dml
= 1 otherwise.
The labeled cut-join operator for the vertex xi is
Mi =
∑
m1,m2,m3∈Z\{0}
m1+m2+m3=0
α
i1,dm1
m1 α
i2,dm2
m2 α
i3,dm3
m3 .
Since the first superscript differs for the α-operators in a summand, the commutator relations
imply that these factors can be permuted within a summand without changing the cut-join
operator.
This operator sums over all possibilities of how, locally, a vertex with its adjacent edge germs
can be arranged, as shown in Figure 10.
Notice that, compared to the (unlabeled) cut-join operator, we do not need a factor of 1
2 which
had to be there to take automorphisms into account resp. to undo overcounting by distinguishing
edges which are not distinguishable. Here, all edges are labeled and thus distingushable.
Theorem 3.8. For a fixed Feynman graph Γ, multidegree a and order Ω, the count of labeled
tropical covers of ET of type Γ and of the right multidegree and order (see Lemma 2.22, Theo-
rems 2.23 and 2.19) equals a sum of matrix elements:
NΓ,a,Ω =
∑
(wk)k
wk|ak
r∏
k=1
(
1
wk
) ak
wk ·
〈
r∏
k=1
ak
wk∏
l=1
αk,l
−wk
v∅
∣∣∣∣∣
n∏
i=1
MΩ−1(i)
∣∣∣∣∣
r∏
k=1
ak
wk
+1∏
l=2
αk,l
−wk
v∅
〉
.
Tropical Mirror Symmetry in Dimension One 25
qi1 ,m1
qi2 ,m2
qi3 ,m3 qi3 ,m3
qi1 ,m1
qi2 ,m2
qi1 ,m1
qi3 ,m3
qi2 ,m2 qi2 ,m2
qi1 ,m1
qi3 ,m3
qi2 ,m2
qi3 ,m3
qi1 ,m1 qi1 ,m1
qi2 ,m2
qi3 ,m3
Figure 10. Local pictures of graphs with weights m1, m2, m3 on the labeled edges qi1 , qi2 , qi3 .
Proof. We use Wick’s theorem: the right hand side is a sum over all possible ways to combine
the local pictures given by the cut-join operators to a graph Γ′ that covers P1
T. Our local
pictures are now vertices with labels xΩ−1(i). For each αk,j
n , we have an adjacent edge germ
with label qk,j of weight |n|, pointing to the right if n is positive and to the left otherwise.
Fix a graph Γ′ which is a completion of such local pictures. The preimages of −∞ are leaf
vertices of Γ′ whose adjacent edges are labeled qk,1, . . . , qk, ak
wk
and are of weight wk (for all k).
The preimages of ∞ are leaf vertices whose adjacent edges have labels qk,2, . . . , qk, ak
wk
+1, also of
weight wk. Since the α-operators in the cut-join operator only have the values 1 or ak
wk
+ 1 as
their second superscript, the commutator relation guarantees that the leaves of qk,2, . . . , qk, ak
wk
over −∞ have to be connected to the leaves with the corresponding label over ∞. The leaf
adjacent to qk,1 over −∞ is merged with an interior vertex adjacent to qk, by definition of the
labeled cut-join operator which depends on Γ. The same holds for the leaf adjacent to qk, ak
wk
+1.
To produce a tropical cover of ET, we glue Γ
′ as follows: for all k and for i = 1, . . . , ak
wk
, the leaf
of qk,i over −∞ is attached to the leaf of qk,i+1 over ∞. Identifying the edges qk1 , . . . , qk, ak
wk
+1
(which are subdivided by 2-valent vertices obtained from gluing) to one edge qk, we obtain
a graph cover of ET of type Γ which is of the right order and multidegree: the order is imposed
by the order in which we multiply the cut-join operators, the multidegree is given by the “curls”
of the edge qk, which has weight wk and which is curled ak
wk
times by our way of gluing.
Obviously, each tropical cover of type Γ and multidegree a with order Ω can be obtained by
gluing a graph Γ′ that arises with Wick’s theorem from the right hand side.
On the right hand side, a graph Γ′ that we produce with Wick’s theorem contributes with
the product of the weights of all edges which are connected, including the ends. For an edge qk
(which is cut into w
ak
wk
+1
k pieces in Γ′) with ak > 0, we thus obtain a factor of w
ak
wk
+1
k , where we
actually only want wk for the tropical multiplicity. This is taken care of by the pre-factor before
the summands on the right. ■
Example 3.9. Fix the multidegree a = (2, 1, 0), an order Ω on the vertices x1, x2 such that
x1 < x2 and the following Feynman graph: Fix points p1, p2 on ET. We obtain a labeled tropical
cover of P1
T that can be glued to a cover of ET of type Γ by choosing local pieces (see Figure 10).
Notice that there are two choices of the expansion factor w1, namely w1 = 1 or w1 = 2.
26 J. Böhm, C. Goldner and H. Markwig
x1
x2
q1
q2
q3
We start with w1 = 2 and obtain the following source curve of a tropical cover to P1
T, where the
local pieces are indicated by boxes. In case of w1 = 2, there are no other choices of local pieces
2
1
2
1
1
q1,1
q2,1
q1,1
q1,2
q2,2
q3,1
q3,1
q2,1
q1,2
q2,2
x1
x2
1 1
2 2
3q2,1 q2,2
q1,1 q1,2
q1,2
q2,2q3,1q3,1
x1 x2
q2,1
q1,1
that fit Γ than the ones shown above. If we choose w1 = 1, then another valid choice of local
pieces is shown below.
1
1
1
1
1 1
2q1,1
q1,2 q1,2
q1,3
q2,1 q2,2
q2,2
q1,3q3,1q3,1
x1 x2
q1,1
q2,1
Note that these two graphs are labeled version of the middle and upper graph of Figure 8
and the upper graph of Figure 9. However, we do not get all graphs of Examples 3.3 and 3.6
since we fixed Γ and a.
3.3 From matrix elements to Feynman integrals
Finally, we link the matrix elements on the right of the equation in Theorem 3.8 with Feynman
integrals.
For this purpose, we introduce formal variables for our vertices in the labeled cut-join oper-
ators:
Definition 3.10. Let Γ, a and (wk)k be as above. For l = 1, 2, 3 set cl =
ail
wil
+ 1 if ail > 0 and
cl = 1 otherwise. We also set dml
= cl if ml > 0 and dml
= 1 otherwise.
The labeled cut-join operator for the vertex xi is
M(xi) =
∑
m1,m2,m3∈Z\{0}
α
i1,dm1
m1 xm1
i α
i2,dm2
m2 xm2
i α
i3,dm3
m3 xm3
i .
Tropical Mirror Symmetry in Dimension One 27
Here, we treat the cut-join operators as formal series in x1, . . . , xn.
With this, we can rewrite the equation of Theorem 3.8 as follows:
NΓ,a,Ω = coefx0
1···x0
n
∑
(wk)k
wk|ak
r∏
k=1
(
1
wk
) ak
wk
×
〈
r∏
k=1
ak
wk∏
l=1
αk,l
−wk
v∅
∣∣∣∣∣
n∏
i=1
M(xΩ−1(i))
∣∣∣∣∣
r∏
k=1
ak
wk
+1∏
l=2
αk,l
−wk
v∅
〉
. (3.1)
Each matrix element on the right hand side is now a series in x1, . . . , xn when evaluated.
Lemma 3.11. Fix Γ, a and Ω as above. The matrix elements of equation (3.1), viewed as series
in x1, . . . , xn equals the following product:
∑
(wk)k
wk|ak
r∏
k=1
(
1
wk
) ak
wk ·
〈
r∏
k=1
ak
wk∏
l=1
αk,l
−wk
v∅
∣∣∣∣∣
n∏
i=1
M(xΩ−1(i))
∣∣∣∣∣
r∏
k=1
ak
wk
+1∏
l=2
αk,l
−wk
v∅
〉
=
∏
k : ak>0
wk ·
((
xk1
xk2
)wk
+
(
xk2
xk1
)wk
)
·
∏
k : ak=0
(∑
wk>0
wk ·
(
xk1
xk2
)wk
)
.
Here, xk1 and xk2 denote the vertices adjacent to the edge qk, where in the order Ω we have
xk1 < xk2.
Proof. Let qk be an edge with ak = 0. Since ak = 0, an α with first superscript k does not
show up in the vectors of the matrix element, only in the labeled cut-join operators. Also, the
second superscript must be 1, and it appears for exactly two cut-join operators, namely the
one for xk1 and the one for xk2 . Thus, we draw an edge germ labeled qk,1 at xk1 and an edge
germ labeled qk,1 at xk2 . These are the only edge germs with this label. To obtain a nonzero
contribution to the matrix element, the edge germ at xk1 must point to the right and the one
at xk2 must point to the left. Furthermore, they must have the same weight wk. There is
no restriction on the weight wk. (The balancing condition is imposed only after we take the
x01 · · ·x0n-coefficient in equation (3.1).) So, for any wk > 0, we have nonzero contributions to the
matrix elements above with an αk,1
wk in the cut-join operator M(xk1) and an αk,1
−wk
in the cut-join
operator M(xk2). Combining those α-operators with the respective power of the variable, we
obtain αk,1
wk · xwk
k1
· αk,1
−wk
· x−wk
k2
, which, after applying the commutator relation and simplifying
becomes wk ·
(xk1
xk2
)wk .
We have treated the sum of matrix elements as a weighted sum of graphs. Any nonzero
summand must have an edge connecting the edge germs above, and it can be of any weight.
More precisely, if we have a graph with such an edge of a certain weight, we also have all
summands that correspond to the same graph, but with the weight of the edge varying. Thus,
we can pull out a factor∑
wk>0
wk ·
(
xk1
xk2
)wk
for the edge qk.
Let us now consider an edge qk with ak > 0. The matrix elements on the left are summed over
all wk|ak. For the local pictures, we draw end germs of weight wk on the left pointing to the right,
with labels qk,1, . . . , qk, ak
wk
, and on the right, pointing to the left, with labels qk,2, . . . , qk, ak
wk
+1.
28 J. Böhm, C. Goldner and H. Markwig
We use the commutator relations for the α in charge of connecting the “curls” qk,2, . . . , qk, ak
wk
,
they produce a factor of wk which is cancelled by the pre-factor. The edge germ qk,1 must
be connected to an edge germ appearing in a cut-join operator, that can be either M(xk1)
or M(xk2). The edge germ qk, ak
wk
+1 must also be connected to an edge germ of a cut-join
operator, necessarily the other on in the choice of M(xk1) or M(xk2). Thus, we either have
αk,1
wk
· αk,1
−wk
· x−wk
k1
· α
k,
ak
wk
+1
wk · xwk
k2
· α
k,
ak
wk
+1
−wk
or
αk,1
wk
· αk,1
−wk
· x−wk
k2
· α
k,
ak
wk
+1
wk · xwk
k1
· α
k,
ak
wk
+1
wk
(notice the subscript changes sign when we let factors jump in the scalar product, by convention
of the adjoints). Taking the commutator relations into account, and realizing that one factor
of wk is again cancelled by the pre-factor, we obtain either wk ·
(xk1
xk2
)wk or wk ·
(xk2
xk1
)wk . For any
graph which produces the first factor, we can connect the edge germs differently thus obtaining
the graph which produces the second factor. Also, wk was imposed by the summand we picked on
the left hand side. But for a given graph with an edge of weight wk, we also have the analogous
graph (with fewer or more curls), where the edge has another weight which divides ak. Thus,
for the edge qk we obtain a total factor of∑
wk|ak
wk ·
((
xk1
xk2
)wk
+
(
xk2
xk1
)wk
)
. ■
With this, we can now give an alternative proof of Theorem 2.19 (in the case ki = 1 for all i),
which follows the more traditional Fock space approach, now with a larger set of operators in
charge of the labels. In the tropical world, we can however take a shortcut avoiding the fermionic
Fock space and relying on Wick’s theorem instead.
Proof of Theorem 2.19 in the case ki = 1 for all i. We prove the equality by restricting
to the qa11 · · · qarr -coefficient on each side. It follows from Lemma 2.22 that we can expand the
left side as a sum over orders Ω, which we can do by definition of Feynman integral also on the
right. We thus have to show that the weighted count NΓ,a,Ω of labeled tropical covers contribut-
ing to ⟨τk1(pt) · · · τkn(pt)⟩
E,a,trop
Γ,n and satisfying π(i) = pΩ(i) equals coef [qa11 ···qlrr ]
IΓ,gΩ(q1, . . . , qr).
By definition of a Feynman integral (see Definition 2.13), the x01 · · ·x0n-coefficient of the right
hand side of Lemma 3.11 equals the qa11 · · · qarr -coefficient of IΓ,Ω(q1, . . . , qr). Using Lemma 3.11
and equation (3.1) (which follows from Theorem 3.8 relying on Wick’s theorem), it follows that
it also equals NΓ,a,Ω. The statement is proved. ■
Acknowledgements
We would like to thank Renzo Cavalieri, Elise Goujard, Gerhard Hiss, Martin Möller and Dhruv
Ranganathan for helpful discussions. Gefördert durch die Deutsche Forschungsgemeinschaft
(DFG) – Projektnummer 286237555 – TRR 195 [Funded by the Deutsche Forschungsgemein-
schaft (DFG, German Research Foundation) – Project ID 286237555 – TRR 195]. The authors
have been supported by Project I.10 (INST 248/237-1) of TRR 195. Computations have been
made with Singular using the ellipticcovers library. Part of this work was completed during the
Mittag-Leffler programm Tropical geometry, amoebas and polytopes in spring 2018. The authors
would like to thank the institute for hospitality and excellent working conditions. We would like
to thank the anonymous referees for helpful suggestions to improve the paper.
Tropical Mirror Symmetry in Dimension One 29
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1 Introduction
1.1 Context: Tropical mirror symmetry of elliptic curves
1.2 Part I: Generating series of tropical descendant Gromov–Witten invariants of E
1.3 Part II: Relation to the Fock space approach
2 Tropical mirror symmetry for elliptic curves
2.1 Descendant Gromov–Witten invariants
2.2 Tropical descendant Gromov–Witten invariants
2.3 Feynman integrals
2.4 (Tropical) mirror symmetry for elliptic curves
2.5 The bijection
2.6 Quasimodularity
3 Tropical mirror symmetry and the boson–fermion correspondence
3.1 Hurwitz numbers as matrix elements
3.2 Labeled matrix elements for labeled tropical covers
3.3 From matrix elements to Feynman integrals
References
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| id | nasplib_isofts_kiev_ua-123456789-211622 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-12T21:57:35Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Böhm, Janko Goldner, Christoph Markwig, Hannah 2026-01-07T13:39:42Z 2022 Tropical Mirror Symmetry in Dimension One. Janko Böhm, Christoph Goldner and Hannah Markwig. SIGMA 18 (2022), 046, 30 pages 1815-0659 2020 Mathematics Subject Classification: 14J33; 14N35; 14T05; 81T18; 11F11; 14H30; 14N10; 14H52; 14H81 arXiv:1809.10659 https://nasplib.isofts.kiev.ua/handle/123456789/211622 https://doi.org/10.3842/SIGMA.2022.046 We prove a tropical mirror symmetry theorem for descendant Gromov-Witten invariants of the elliptic curve, generalizing the tropical mirror symmetry theorem for Hurwitz numbers of the elliptic curve, Theorem 2.20 in [Böhm J., Bringmann K., Buchholz A., Markwig H., J. Reine Angew. Math. 732 (2017), 211-246, arXiv:1309.5893]. For the case of the elliptic curve, the tropical version of mirror symmetry holds on a fine level and easily implies the equality of the generating series of descendant Gromov-Witten invariants of the elliptic curve to Feynman integrals. To prove tropical mirror symmetry for elliptic curves, we investigate the bijection between graph covers and sets of monomials contributing to a coefficient in a Feynman integral. We also soup up the traditional approach in mathematical physics to mirror symmetry for the elliptic curve, involving operators on a Fock space, to give a proof of tropical mirror symmetry for Hurwitz numbers of the elliptic curve. In this way, we shed light on the intimate relation between the operator approach on a bosonic Fock space and the tropical approach. We would like tothank Renzo Cavalieri, Elise Goujard, Gerhard Hiss, Martin Möller, and Dhruv Ranganathan for helpful discussions. Gefordert durch die Deutsche Forschungsgemeinschaft (DFG) – Projektnummer 286237555 – TRR195 [Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project ID 286237555 –TRR195]. The authors have been supported by Project I.10 (INST 248/237-1) of TRR 195. Computations have been made with Singular using the ellipticcovers library. Part of this work was completed during the Mittag-Leffler program Tropical geometry, amoebas and polytopes in spring 2018. The authors would like to thank the institute for its hospitality and excellent working conditions. We would like to thank the anonymous referees for their helpful suggestions to improve the paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Tropical Mirror Symmetry in Dimension One Article published earlier |
| spellingShingle | Tropical Mirror Symmetry in Dimension One Böhm, Janko Goldner, Christoph Markwig, Hannah |
| title | Tropical Mirror Symmetry in Dimension One |
| title_full | Tropical Mirror Symmetry in Dimension One |
| title_fullStr | Tropical Mirror Symmetry in Dimension One |
| title_full_unstemmed | Tropical Mirror Symmetry in Dimension One |
| title_short | Tropical Mirror Symmetry in Dimension One |
| title_sort | tropical mirror symmetry in dimension one |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211622 |
| work_keys_str_mv | AT bohmjanko tropicalmirrorsymmetryindimensionone AT goldnerchristoph tropicalmirrorsymmetryindimensionone AT markwighannah tropicalmirrorsymmetryindimensionone |