Counting Feynman-like graphs: Quasimodularity and Siegel–Veech weight

Goujard, Élise; Möller, Martin

doi:10.4171/jems/924

Cited by 18 publications

(34 citation statements)

References 19 publications

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“…, z n ) dz n · · · dz 1 . 19 The first part of Theorem 7 can also be found in work of Goujard and Möller [13].…”

Section: Re(z)mentioning

confidence: 85%

“…where ⊠ denotes gluing along the m pairs of marked points with weights ±c i , the bracket [P ] X 1 denotes taking the coefficient of X 1 in a polynomial function 13 P of X (in this case defined for all sufficiently large integers X) and the sum m,…”

Section: The Product Over All These Contributions Yields H : W(h)>0 Wmentioning

confidence: 99%

“…• σ is an ordering of the vertices of Γ ′ , and is used in the usual way to determine orientations e = {h, h ′ }; • w is a balanced weighting of the non-leg half-edges of Γ ′ , which are naturally identified with the non-leg half-edges of Γ that do not belong to edges in S; • c is a weighting of the remaining half-edges of Γ (those belonging to edges in S, and legs) such that c(l i ) = X, c(l j ) = −X for some large 13 Polynomiality here follows from the polynomiality of the double ramification cycle [18,45].…”

Section: The Product Over All These Contributions Yields H : W(h)>0 Wmentioning

confidence: 99%

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Holomorphic anomaly equations and the Igusa cusp form conjecture

2018

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Let S be a K3 surface and let E be an elliptic curve. We solve the reduced Gromov-Witten theory of the Calabi-Yau threefold S × E for all curve classes which are primitive in the K3 factor. In particular, we deduce the Igusa cusp form conjecture.The proof relies on new results in the Gromov-Witten theory of elliptic curves and K3 surfaces. We show the generating series of Gromov-Witten classes of an elliptic curve are cycle-valued quasimodular forms and satisfy a holomorphic anomaly equation. The quasimodularity generalizes a result by Okounkov and Pandharipande, and the holomorphic anomaly equation proves a conjecture of Milanov, Ruan and Shen. We further conjecture quasimodularity and holomorphic anomaly equations for the cycle-valued Gromov-Witten theory of every elliptic fibration with section. The conjecture generalizes the holomorphic anomaly equations for elliptic Calabi-Yau threefolds predicted by Bershadsky, Cecotti, Ooguri, and Vafa. We show a modified conjecture holds numerically for the reduced Gromov-Witten theory of K3 surfaces in primitive classes.

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“…, z n ) dz n · · · dz 1 . 19 The first part of Theorem 7 can also be found in work of Goujard and Möller [13].…”

Section: Re(z)mentioning

confidence: 85%

Section: The Product Over All These Contributions Yields H : W(h)>0 Wmentioning

confidence: 99%

Section: The Product Over All These Contributions Yields H : W(h)>0 Wmentioning

confidence: 99%

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Holomorphic anomaly equations and the Igusa cusp form conjecture

2018

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“…In this paper, we write down explicitly only the first few identities, but it would certainly be interesting to study whether these identities are interesting from the point of view of elliptic functions and quasi-modular forms. For instance, they may be related to the results on quasi-modular forms obtained in [28]. We hope to report on that in the near future.…”

Section: Introductionmentioning

confidence: 60%

Quantizing Weierstrass

Bouchard

Chidambaram

Dauphinee

2018

Communications in Number Theory and Physics

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We study the connection between the Eynard-Orantin topological recursion and quantum curves for the family of genus one spectral curves given by the Weierstrass equation. We construct differential operators that annihilate the perturbative and non-perturbative wave-functions. In particular, for the non-perturbative wave-function, we prove, up to orderh 5 , that the differential operator is a quantum curve. As a side result, we obtain an infinite sequence of identities relating A-cycle integrals of elliptic functions and quasi-modular forms.

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“…The statement relating Hurwitz numbers and Feynman integrals (see Theorem 4.20) was known before tropical covers played a role [23]. However, the tropical approach yields a finer relation which led to interesting new results on quasimodularity [27]. Furthermore, tropical mirror symmetry for elliptic curves provides evidence and further strategies for the famous Gross-Siebert program on mirror symmetry which aims at constructing new mirror pairs and providing an algebraic framework for SYZ-mirror symmetry [29,30,49].…”

Section: 5mentioning

confidence: 99%

Tropical Curves and Covers and Their Moduli Spaces

Markwig

2020

Jahresber. Dtsch. Math. Ver.

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Tropical geometry can be viewed as an efficient combinatorial tool to study degenerations in algebraic geometry. Abstract tropical curves are essentially metric graphs, and covers of tropical curves maps between metric graphs satisfying certain conditions. In this short survey, we offer an introduction to the combinatorial theory of abstract tropical curves and covers of curves, and their moduli spaces, and we showcase three results demonstrating how this theory can be applied in algebraic geometry.

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Counting Feynman-like graphs: Quasimodularity and Siegel–Veech weight

Cited by 18 publications

References 19 publications

Holomorphic anomaly equations and the Igusa cusp form conjecture

Holomorphic anomaly equations and the Igusa cusp form conjecture

Quantizing Weierstrass

Tropical Curves and Covers and Their Moduli Spaces

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