Ramanujan identities and quasi-modularity in Gromov–Witten theory

Shen, Yongjun; Zhou, Jie

doi:10.4310/cntp.2017.v11.n2.a5

Cited by 16 publications

(32 citation statements)

References 51 publications

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“…We make a few preparations, before we prove Theorem 4.1. The following Proposition appeared first in [22] in a slightly different notation. is equivalent to the following system of equations:…”

Section: For Anymentioning

confidence: 99%

Givental-Type Reconstruction at a Nonsemisimple Point

Basalaev¹,

Priddis²

2018

Michigan Math. J.

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In this paper we consider the orbifold curve, which is a quotient of an elliptic curve E by a cyclic group of order 4. We develop a systematic way to obtain a Givental-type reconstruction of Gromov-Witten theory of the orbifold curve via the product of the Gromov-Witten theories of a point. This is done by employing mirror symmetry and certain results in FJRW theory. In particular, we present the particular Giventals action giving the CY/LG correspondence between the Gromov-Witten theory of the orbifold curve E /Z 4 and FJRW theory of the pair defined by the polynomial x 4 + y 4 + z 2 and the maximal group of diagonal symmetries. The methods we have developed can easily be applied to other finite quotients of an elliptic curve. Using Givental's action we also recover this FJRW theory via the product of the Gromov-Witten theories of a point. Combined with the CY/LG action we get a result in pure Gromov-Witten theory with the help of modern mirror symmetry conjectures. 4,4,2 39References 40 ALEXEY BASALAEV AND NATHAN PRIDDIS pairing η. A cohomological field theory (CohFT for brevity) Λ g,n on (V, η) is a system of linear maps Λ g,n

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Section: For Anymentioning

confidence: 99%

Givental-Type Reconstruction at a Nonsemisimple Point

Basalaev¹,

Priddis²

2018

Michigan Math. J.

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“…It is clear from the above formula (2.25) for the prepotential that the dependence in u are in polynomials of f 1 , f 2 , f 3 . Note also that the above expression assembles the same form as the one for the orbifold GW theory of P 1 3,3,3 given in [ST11,SZ14]. This observation is one of the motivations for finding the exact matching between the two theories.…”

Section: Cubic Casementioning

confidence: 65%

“…We recall that in [SZ14], the WDVV equations satisfied by these functions are shown to be equivalent to the Ramanujan identities (2.7) for the modular group Γ 0 (3). By matching the boundary conditions, the following equalities are obtained Proof.…”

Section: Lg/cy Correspondence For P 1 333mentioning

confidence: 93%

“…• In [SZ14], the authors proved that the GW correlation functions of any elliptic orbifold curve are q-expansions of some quasi-modular forms for certain modular group (described in Tab. 2 below).…”

Section: Lg/cy Correspondence Via Cayley Transformmentioning

confidence: 99%

“…We first recall the basics of GW theories of elliptic orbifold curves following the exposition part in [SZ14].…”

Section: Wdvv Equations In Gw Theory and Fjrw Theory 21 Quasi-modulamentioning

confidence: 99%

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LG/CY correspondence for elliptic orbifold curves via modularity

Shen

Zhou

2018

J. Differential Geom.

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We prove the Landau-Ginzburg/Calabi-Yau correspondence between the Gromov-Witten theory of each elliptic orbifold curve and its Fan-Jarvis-Ruan-Witten theory counterpart via modularity. We show that the correlation functions in these two enumerative theories are different representations of the same set of quasi-modular forms, expanded around different points on the upper-half plane. We relate these two representations by the Cayley transform. Appendix A WDVV equations in FJRW theories 32Appendix B Analytic continuation of periods and connection matrices 34

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WDVV equations and invariant bi-Hamiltonian formalism

Vašíček

Vitolo

2021

J. High Energ. Phys.

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The purpose of the paper is to show that, in low dimensions, the WDVV equations are bi-Hamiltonian. The invariance of the bi-Hamiltonian formalism is proved for N = 3. More examples in higher dimensions show that the result might hold in general. The invariance group of the bi-Hamiltonian pairs that we find for WDVV equations is the group of projective transformations. The significance of projective invariance of WDVV equations is discussed in detail. The computer algebra programs that were used for calculations throughout the paper are provided in a GitHub repository.

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Ramanujan identities and quasi-modularity in Gromov–Witten theory

Cited by 16 publications

References 51 publications

Givental-Type Reconstruction at a Nonsemisimple Point

Givental-Type Reconstruction at a Nonsemisimple Point

LG/CY correspondence for elliptic orbifold curves via modularity

WDVV equations and invariant bi-Hamiltonian formalism

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