Tropical mirror symmetry for elliptic curves

Janko Böhm; Kathrin Bringmann; Arne Buchholz; Hannah Markwig

doi:10.1515/crelle-2014-0143

Published by De Gruyter April 21, 2015

Tropical mirror symmetry for elliptic curves

Janko Böhm , Kathrin Bringmann , Arne Buchholz and Hannah Markwig

From the journal Journal für die reine und angewandte Mathematik (Crelles Journal)

https://doi.org/10.1515/crelle-2014-0143

An erratum for this article can be found here: https://doi.org/10.1515/crelle-2018-0017

Showing a limited preview of this publication:

Abstract

Mirror symmetry relates Gromov–Witten invariants of an elliptic curve with certain integrals over Feynman graphs [10]. We prove a tropical generalization of mirror symmetry for elliptic curves, i.e., a statement relating certain labeled Gromov–Witten invariants of a tropical elliptic curve to more refined Feynman integrals. This result easily implies the tropical analogue of the mirror symmetry statement mentioned above and, using the necessary Correspondence Theorem, also the mirror symmetry statement itself. In this way, our tropical generalization leads to an alternative proof of mirror symmetry for elliptic curves. We believe that our approach via tropical mirror symmetry naturally carries the potential of being generalized to more adventurous situations of mirror symmetry. Moreover, our tropical approach has the advantage that all involved invariants are easy to compute. Furthermore, we can use the techniques for computing Feynman integrals to prove that they are quasimodular forms. Also, as a side product, we can give a combinatorial characterization of Feynman graphs for which the corresponding integrals are zero. More generally, the tropical mirror symmetry theorem gives a natural interpretation of the A-model side (i.e., the generating function of Gromov–Witten invariants) in terms of a sum over Feynman graphs. Hence our quasimodularity result becomes meaningful on the A-model side as well. Our theoretical results are complemented by a Singular package including several procedures that can be used to compute Hurwitz numbers of the elliptic curve as integrals over Feynman graphs.

Funding statement: The research of the second author was supported by the Alfried Krupp Prize for Young University Teachers of the Krupp foundation and the research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007–2013)/ERC Grant agreement no. 335220 – AQSER. The fourth author is partially supported by DFG-grant MA 4797/3-2.

A Appendix: Correspondence theorem

We now prove Theorem 2.13. We do this by cutting covers of E at the preimages of the base point p0, thus producing a (possibly reducible) cover of the tropical line

𝕋⁢ℙ1:=ℝ∪{-∞}∪{∞}

as in [8]. We use the Correspondence Theorem of [8] relating the numbers of tropical covers to certain tuples of elements of the symmetric group that correspond to algebraic covers of ℙ1. We study “gluing factors” that relate the tropical multiplicity of a cover of E to the tropical multiplicity of the cut cover. These factors equal the number of ways to produce a tuple of elements of the symmetric group corresponding to a cover of an elliptic curve ℰ from a tuple corresponding to a cover of ℙ1.

By pairing a cover of ℰ with a monodromy representation, the Hurwitz number Nd,g of Definition 2.1 equals the following count of tuples of permutations.

Remark A.1 (cf. [25]).

The Hurwitz numbers Nd,g of Definition 2.1 are given by 1d! times the number of tuples (τ1,…,τ2⁢g-2,α,σ) of permutations in 𝕊d such that

the τi are transpositions for all i=1,…,2⁢g-2,
the equation τ2⁢g-2∘…∘τ1∘σ=α∘σ∘α-1 holds,
the subgroup 〈τ1,…,τ2⁢g-2,σ,α〉 acts transitively on {1,…,d}.

Condition (2) is explained by Figure 9 sketching the generators of the fundamental group π1⁢(ℰ∖p1,…,p2⁢g-2): The rectangle represents the elliptic curve as a torus, the left side is identified with the right side and the top with the bottom. Clearly, α and σ belong to the two generators of π1⁢(ℰ). The loops around the branch points p1,…,p2⁢g-2 induce the permutations τ1,…,τ2⁢g-2 of sheets on the source curve. Obviously, going around all branch points simultaneously is homotopic to the composition of the paths around each of the points in order p1,…,p2⁢g-2, on the one hand, and on the other hand it is homotopic to the path that goes clockwise around the rectangle. This identity has to “lift” to the source curve, implying that

τ2⁢g-2∘…∘τ1=α∘σ∘α-1∘σ-1.

$Figure 9 The generators of the fundamental group π1⁢(ℰ∖p1,…,p2⁢g-2){\pi_{1}(\mathcal{E}\setminus{p_{1},\ldots,p_{2g-2}})}.$

Figure 9

The generators of the fundamental group π1⁢(ℰ∖p1,…,p2⁢g-2).

Let Δ be a partition of d. A permutation σ has cycle type Δ if after decomposition into a disjoint union of cycles Δ equals the partition of the lengths of the cycles. The conjugacy classes of 𝕊d are the sets of permutations with same cycle type Δ. Thus σ and α∘σ∘α-1 have the same cycle type for any σ and α.

Given a tuple as in Remark A.1, we now construct an associated tropical cover. As before, we fix the base point p0 and the 2⁢g-2 branch points p1,…,p2⁢g-2 (ordered clockwise) in E.

Construction A.2.

Given a tuple as in Remark A.1, we construct a tropical cover of E with branch points p1,…,p2⁢g-2 as follows:

For each cycle c of σ of length m draw an edge of weight m over p0 and label it with the corresponding cycle.
For i=1,…,2⁢g-2, successively cut or join edges over pi according to the effect of τi on τi-1∘…∘τ1∘σ. Label the new edges as before.
Glue the outcoming edges attached to points over p2⁢g-2 with the edges over p0 according to the action of α on the cycles of σ. More precisely: Glue the edge with the label α∘c∘α-1 over p2⁢g-2 to the edge with label c over p0.
Forget all the labels on the edges.

Note that for a cycle c=(n1⁢…⁢nl) of length l≥2 we have

α∘c∘α-1=(α⁢(n1)⁢…⁢α⁢(nl)).

We use the same convention for cycles of length 1.

Example A.3.

Let g=2 and d=4 and consider the tuple of permutations

(τ1,τ2,τ3,τ4,α,σ)=((1 3),(2 4),(1 2),(1 3),(2 3 4),(2 3))

in 𝕊4. We see that σ=(2 3)=(1)⁢(2 3)⁢(4) has cycle type (2,1,1). Moreover,

α∘σ∘α-1=(1)⁢(2)⁢(3 4)=τ4∘…∘τ1∘σ

is fulfilled, so the tuple contributes to the count of N4,2. Figure 10 sketches the construction of Remark A.2 up to the gluing step (the object can be considered as a cover of the tropical line 𝕋⁢ℙ1 as described below).

$Figure 10 The cover of 𝕋⁢ℙ1{\mathbb{TP}^{1}} associated to a tuple of permutations.$

Figure 10

The cover of 𝕋⁢ℙ1 associated to a tuple of permutations.

In the gluing step the vertices p0 and p0′ are going to be identified. Since we have

α∘(1)∘α-1=(1),

α∘(2 3)∘α-1=(3 4),

α∘(4)∘α-1=(2),

the ends of the source curve are glued according to the red numbers in Figure 10. The result is the cover of E depicted on the left in Figure 11. Note that choosing α=(2 4) yields the same gluing, while α′=(1 2 4) also fulfills α′∘σ∘α′⁣-1=τ4∘…∘τ1∘σ, but since

α′∘(1)∘α′⁣-1=(2),

α′∘(2 3)∘α′⁣-1=(3 4),

α′∘(4)∘α′⁣-1=(1),

it provides a different gluing, sketched on the right side of Figure 11. In particular, the combinatorial types of the source curves are different.

Figure 11

Two different gluings of the same cover of the line.

Now we describe how to cut tropical covers of E in general, thus producing covers of a line. As usual, we neglect edge lengths – to be precise, they have to adapted accordingly.

Construction A.4.

To every cover π:C→E of degree d we associate a (possibly disconnected) tropical cover π~:C~→𝕋⁢ℙ1 of the line 𝕋⁢ℙ1 of the same degree, by cutting E at p0 and the source curve C at every preimage of p0.

For the definition of (connected) tropical covers of 𝕋⁢ℙ1, see [8]. This definition can easily be generalized by allowing the source curve to be disconnected, and adapting the multiplicity accordingly as follows. Notice that the multiplicity of a single edge of weight m covering 𝕋⁢ℙ1 is 1m (this case is not taken care of in [8] where it is implicitly assumed that every source curve contains at least one vertex). The multiplicity of a cover π~:C~→𝕋⁢ℙ1 is

(A.1)mult(π~):=∏K1wK⋅|Aut(π~)|-1⋅∏ewe,

where the first product goes over all connected components K of C~ that just consist of one single edge mapping to 𝕋⁢ℙ1 with weight wK and the second product goes over all bounded edges e of C~, with we denoting their weight. Note that in [8], the factor |Aut(π~)|-1 is simplified to 12l1+l2, where l1 denotes the number of balanced forks (i.e., adjacent ends of the same weight) and l2 denotes the number of wieners (i.e., pairs of bounded edges of the same weight sharing both end vertices). Since we allow disconnected covers, we will have other contributions to the automorphism group: connected components consisting of single edges of the same weight as above can be permuted. So we get a contribution of 1r! to |Aut(π~)| if for a certain weight m there are exactly r copies of connected components consisting of a single edge of weight m.

For a cover π:C→E, we denote by Δ the partition of d given by the weights of the edges over p0. For the cut cover π~ (see Construction A.4) these are exactly the ramification profiles over -∞ and ∞.

Example A.5.

The two covers of E depicted in Figure 11 cut at p0 both give a cover of the line as sketched in Figure 10 (where we dropped the labels on the edges). The multiplicity of the cover of the line equals the product of the weight of the bounded edges, i.e., 32⋅4, since there are no automorphisms.

Example A.6.

Figure 12 shows a disconnected cover π~ of the line of degree 17 with six (simple) ramifications and profiles (4,4,4,1,1,1,1,1) over the ends. Its multiplicity equals

mult⁡(π~)=(12)2⋅12⋅12!⋅13!⏟=|Aut(π~)|⋅14⋅4⋅1⋅1⋅1⏟=∏K1wK⋅4⋅2⋅2⋅2⋅2⏟=∏ewe=124,

where the first factor contributing to the automorphisms comes from the two balanced forks, the second from the wiener and the other two from two single-edge components of weight 4 and three single-edge components of weight 1 respectively.

Figure 12

A (disconnected) tropical cover of the line.

The Correspondence Theorem in [8] matches tropical covers of 𝕋⁢ℙ1 as above with algebraic covers of ℙ1 having two ramifications of profile Δ over 0 and ∞ respectively and only simple ramifications else.

Similar to Remark A.1, the associated Hurwitz numbers can be written in terms of tuples of elements of the symmetric group.

Remark A.7.

The double Hurwitz number Hd,g⁢(ℙ1,Δ,Δ) counting the number of (isomorphism classes of) covers ϕ:𝒞→ℙ1 of degree d (each weighted with |Aut(ϕ)|-1), where 𝒞 is a possibly disconnected curve such that the sum of the genera of its connected components equals g, having ramification profile Δ over 0 and ∞ and only simple ramifications else, equals 1d! times the number of tuples (τ1,…,τ2⁢g-2,σ,σ′) in 𝕊d such that

σ and σ′ are permutations of cycle type Δ,
the τi are transpositions for all i=1,…,2⁢g-2,
the equation σ′∘τ2⁢g-2∘…∘τ1∘σ=id𝕊d holds.

Note that as in Definition 2.1, it follows from the Riemann–Hurwitz formula that the number of simple ramifications is 2⁢g-2. The condition σ′∘τ2⁢g-2∘…∘τ1∘σ=id𝕊d reflects the fact that the fundamental group π1⁢(ℙ1) is trivial. We do not include a condition about transitivity here, since we allow also disconnected covers.

As in Construction A.2, we can associate a tropical cover of the line to a tuple as in Remark A.7. The procedure is the same, we just drop the gluing step (3). The statement of [8, Correspondence Theorem 5.28] is that for a fixed tropical cover π~:C~→𝕋⁢ℙ1, the tropical multiplicity equals 1d! times the number of tuples that yield π~ under the above procedure.

We now relate the tuples in Remarks A.1 and A.7, respectively the multiplicities of a tropical cover π:C→E and the cut cover π~:C~→𝕋⁢ℙ1 of Construction A.4.

Definition A.8.

Given a cover π:C→E and the cut cover π~:C~→𝕋⁢ℙ1 of Construction A.4, we choose a tuple (τ1,…,τ2⁢g-2,σ,σ′) that yields the cover π~ when applying Construction A.2 (minus the gluing in step (3)). We define nπ~,π to be the number of α∈𝕊d satisfying α∘σ∘α-1=σ′ and, when labeling π~ with cycles according to our choice of tuple (τ1,…,τ2⁢g-2,σ,σ′) and performing step (3) and (4) of Construction A.2 (gluing and forgetting the cycle labels), we obtain π.

Note that the number nπ~,π is well-defined (i.e., it does not depend on the choice of the tuple (τ1,…,τ2⁢g-2,σ,σ′)). This is true since any other representative (τ¯1,…,τ¯2⁢g-2,σ¯,σ¯′) is a conjugate of (τ1,…,τ2⁢g-2,σ,σ′) and therefore the desired α¯ are in one-to-one correspondence to the desired α.

Proposition A.9.

For a cover π:C→E with partition Δ=(m1,…,mr) over the base point, the number nπ~,π of Definition A.8 is given by

nπ~,π=m1⁢⋯⁢mr⋅|Aut(π~)||Aut(π)|.

Proof.

As in the definition of nπ~,π (see Definition A.8), fix a tuple of permutations (τ1,…,τ2⁢g-2,σ,σ′) that yields π~ when applying Construction A.2 minus the gluing step (3).

The set of α such that α∘σ∘α-1=σ′ is a coset of the stabilizer of σ with respect to the operation of 𝕊d on itself via conjugation: (α,σ)↦α∘σ∘α-1. Assume that Δ consists of ki weights wi for i=1,…,s, then this stabilizer is isomorphic to the semidirect product

∏i=1sCwiki⋊∏i=1s𝕊ki

of cyclic groups Cwi of length wi and symmetric groups 𝕊ki. This can be seen as follows: for each weight wi (i.e., length of a cycle of σ) we can choose an element of 𝕊ki permuting the cycles of length wi in σ. Assume the cycle c1 of σ is mapped to the cycle c2 by this permutation. Then we consider permutations α′ in the group of bijections of the entries of c2 to the entries of c1 that satisfy α′∘c1∘α′⁣-1=c2, there are wi such α′ (and they form a cyclic group). Since the cycles of σ are disjoint, the choices for α′ for each pair of cycles (c1,c2) where c1 is mapped to c2 under the permutations in 𝕊ki that we choose for each i can be combined to a unique α in the stabilizer of σ.

We label the edges of C~ with cycles as given by the choice of our tuple. Transferred to our situation, the argument above shows that when searching for α that satisfy both requirements of Definition A.8, we always get the contributions from the Cwi, (leading to a factor of ∏i=1swiki=m1⁢⋯⁢mr). To prove the lemma, it remains to see that |Aut(π~)||Aut(π)|-1 equals the number of ways to choose permutations of the cycles of the same length (respectively permutations of the ends of C~) that correspond to a gluing of π~ equal to π when applying Construction A.2, step (3).

So let us now analyze the automorphism groups and compare the quotient of their sizes to the possibilities to glue the cover π~ (with labeled ends) to π.

The automorphism group of π~ is, as mentioned above, a direct product of symmetric groups each corresponding to a wiener, a balanced fork or the set of connected components consisting of a single edge of fixed weight. The automorphism group of π is a direct product of symmetric groups of size two corresponding to wieners. Notice that we can have long wieners as in Figure 13, where the two edges of the same weight are curled equally. Clearly, automorphisms that come from wieners that are not cut cancel in the quotient and we can thus disregard them. Since therefore all contributions to the automorphism groups we have to consider come from ends of C~, and the possibilities to glue the cover π~ to π also only depend on the ends of C~, we can analyze the situation locally on the level of the involved ends.

Figure 13

A tropical elliptic cover with a long wiener. The two wiener-edges (i.e., the red and the green edge) have the same weight.

We say that an end of C~ is distinguishable if it is not part of a balanced fork and not an end of a component consisting of a single edge. Distinguishable ends do not contribute to the automorphisms of π~.

We have to consider several cases. We first consider cases not involving connected components consisting of a single edge.

(1) If we glue two distinguishable ends of C~ to get back C, there are no choices for different gluings. Since distinguishable ends do not contribute to the automorphisms, the equality of contributions from these ends holds.

(2) Assume that an edge of C is cut in such a way that one of the ends is part of a balanced fork and the other is distinguishable. Then obviously there are two ways to glue, see Figure 14. The balanced fork contributes with a factor 2 to |Aut(π~)|. After gluing, the fork is not part of a wiener, so the contribution to |Aut(π)| is 1. Again, we see that the contributions coming from these ends to the quotient of the sizes of the automorphism groups on the one hand and to the possibilities of gluing on the other hand coincide.

Figure 14

Two ways to glue a fork to two distinguishable ends.

(3) If two balanced forks are glued, we obtain a wiener. The contribution to |Aut(π~)| and |Aut(π)| is 4 and 2 respectively. The ways to glue the forks to a wiener is 2, as illustrated in Figure 15.

Figure 15

Gluing two balanced forks to a wiener.

Now we have to consider cases involving ends of connected components consisting of a single edge, say of weight m. Assume there are l components consisting of a single edge of weight m. These ends contribute a factor of l! to |Aut(π~)|.

(4) Assume that l0 of the components are not part of a long wiener after gluing. They do not contribute to the automorphisms of π. Note that the components nevertheless might be attached to balanced forks. In this case the fork is either part of a pseudo-wiener in π (i.e., two edges sharing the same end vertices and having the same weight, but curled differently, see Figure 16) or the two edges of the fork have different end vertices.

Figure 16

A cover with a pseudo-wiener.

Let us now determine the number of ways to glue these ends of C~ to get back C. We can choose l0 of the l single-edge-components, and distribute them to l0 distinguishable places. Also, we get a factor of 2 for each balanced fork involved. The result is (ll0)⋅l0!⋅2f, where f is the number of balanced forks involved.

The remaining l-l0 components must be part of long wieners in π. Let n be the number of long wieners in π, giving a contribution of 2n to |Aut(π)|. Then 2⁢n balanced forks from π~ are involved in the gluing process contributing with a factor of 22⁢n to |Aut(π~)|. The number of ways to glue is the number of ways to distribute the l-l0 components to l-l0 gluing places and a factor of 2 for every wiener we get, just as in Figure 15. Altogether the contribution to the number of gluings providing the desired cover equals

(ll0)⋅l0!⋅2f⋅(l-l0)!⋅2n=l!⋅2f⋅2n.

The contribution to the quotient of the sizes of the automorphism groups equals

l!⋅2f⋅22⁢n2n.

Obviously, the two expressions coincide and we are done. ∎

We are now ready to prove the Correspondence Theorem 2.13, that is, the equality of tropical and algebraic Hurwitz numbers of simply ramified covers of elliptic curves.

Proof of Theorem 2.13.

By Remark A.1,

Nd,g=1d!⋅#⁢{(τ1,…,τ2⁢g-2,α,σ)},

where α,σ,τi∈𝕊d, the τi are transpositions, the equality τ2⁢g-2∘…∘τ1∘σ=α∘σ∘α-1 holds and 〈τ1,…,τ2⁢g-2,σ〉 acts transitively on the set {1,…,d}. We can group the tuples in the set according to the tropical cover π:C→E they provide under Construction A.2 and write the sum above as

1d!⋅∑π#⁢{(τ1,…,τ2⁢g-2,α,σ)⁢ yielding the cover ⁢π}.

Observe that, for a fixed cover π, instead of counting tuples yielding π, we can count tuples (τ1,…,τ2⁢g-2,σ,σ′) yielding the cut cover π~ from Construction A.4 and then multiply with the number of appropriate α, i.e., with nπ~,π (see Definition A.8):

1d!⋅∑π#⁢{(τ1,…,τ2⁢g-2,σ,σ′)⁢ that provide the cover ⁢π~}⋅nπ~,π.

By [8] (see also Remark A.7), the count of the tuples yielding a cover π~ divided by d! coincides with its tropical multiplicity

mult⁡(π~)=1|Aut(π~)|⋅∏K1wK⋅∏e~we~,

where the first product goes over all components K consisting of a single edge of weight wK and the second product goes over all bounded edges e~ of C~ and we~ denotes their weight (see (A.1)). Using Proposition A.9, the number nπ~,π can be substituted by

∏e′we′ce′⋅|Aut(π~)||Aut(π)|,

where the product goes over all edges e′ of C that contain a preimage of the base point p0 of E and ce′ denotes the number of preimages in e′, ce′=#⁢(π-1⁢(p0)∩e′). We obtain

Nd,g=∑π1|Aut(π~)|⋅∏e~we~⋅∏K1wK⋅∏e′we′ce′⋅|Aut(π~)||Aut(π)|.

An edge e′ of C of weight we′ having ce′ preimages over the base point provides exactly ce′-1 single-edge-components of weight we′ in the cut cover π~. Vice versa, each such component comes from an edge with multiple preimages over the base point. Therefore the expression ∏K1wK⋅∏e′we′ce′ simplifies to ∏e′we′. We obtain

Nd,g=∑π1|Aut(π)|⁢∏ewe=Nd,gtrop

and the theorem is proved. ∎

Acknowledgements

We would like to thank Arend Bayer, Erwan Brugallé, Albrecht Klemm, Laura Matusevich and Rainer Schulze-Pillot for helpful discussions. Part of this work was accomplished during the third and fourth author’s stay at the Max-Planck-Institute for Mathematics in Bonn. We thank the MPI for hospitality. We thank an anonymous referee for useful comments on an earlier version of this work.

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Received: 2014-6-4

Revised: 2014-10-29

Published Online: 2015-4-21

Published in Print: 2017-11-1

Tropical mirror symmetry for elliptic curves

Abstract

A Appendix: Correspondence theorem

Remark A.1 (cf. [25]).

Construction A.2.

Example A.3.

Construction A.4.

Example A.5.

Example A.6.

Remark A.7.

Definition A.8.

Proposition A.9.

Proof.

Proof of Theorem 2.13.

Acknowledgements

References

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