LG/CY correspondence for elliptic orbifold curves via modularity

Shen, Yongjun; Zhou, Jie

doi:10.4310/jdg/1527040874

Cited by 13 publications

(46 citation statements)

References 39 publications

(81 reference statements)

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“…The functions introduced make sense from the point of view of modular forms; they are just expansions of the (quasi)modular forms x(τ), y(τ) and w(τ) at the point τ = τ 0 (see Proposition 17 in [25]). This is also a coordinate form of the Cayley transform of [23].…”

Section: Group Action Via the Modular Formsmentioning

confidence: 93%

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: Group Action Via the Modular Formsmentioning

confidence: 93%

Givental-Type Reconstruction at a Nonsemisimple Point

Basalaev¹,

Priddis²

2018

Michigan Math. J.

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“…Topological string for local P 2 at orbifold point is studied in [82] and also in [109]. We first have a review on its contents.…”

Section: Orbifold Pointmentioning

confidence: 99%

Blowup equations for refined topological strings

Huang

Sun

Wang

2018

J. High Energ. Phys.

130

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Göttsche-Nakajima-Yoshioka K-theoretic blowup equations characterize the Nekrasov partition function of five dimensional N = 1 supersymmetric gauge theories compactified on a circle, which via geometric engineering correspond to the refined topological string theory on SU (N ) geometries. In this paper, we study the K-theoretic blowup equations for general local Calabi-Yau threefolds. We find that both vanishing and unity blowup equations exist for the partition function of refined topological string, and the crucial ingredients are the r fields introduced in our previous paper. These blowup equations are in fact the functional equations for the partition function and each of them results in infinite identities among the refined free energies. Evidences show that they can be used to determine the full refined BPS invariants of local Calabi-Yau threefolds. This serves an independent and sometimes more powerful way to compute the partition function other t han the refined topological vertex in the A-model and the refined holomorphic anomaly equations in the B-model. We study the modular properties of the blowup equations and provide a procedure to determine all the vanishing and unity r fields from the polynomial part of refined topological string at large radius point. We also find that certain form of blowup equations exist at generic loci of the moduli space. To Sheldon Katz on his 60th anniversary arXiv:1711.09884v2 [hep-th] 9 Nov 2018 7 Blowup equations at generic points of moduli space 70 7.1 Modular transformation 70 7.2 Conifold point 70 7.3 Orbifold point 73 -i -8 Outlook 75 References 79

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“…the genus zero potentials in the X r , r = 4, 6 cases that [ST11] could not fully determine) which are related by a complicated system of coupled differential equations. It is also what leads to the later progress on the proof of LG/CY correspondence for these elliptic orbifold curves via the Cayley transform in representation theory [SZ16].…”

Section: Introductionmentioning

confidence: 99%

Ramanujan identities and quasi-modularity in Gromov–Witten theory

Shen

Zhou

2017

Communications in Number Theory and Physics

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We prove that the ancestor Gromov-Witten correlation functions of one-dimensional compact Calabi-Yau orbifolds are quasi-modular forms. This includes the pillowcase orbifold which can not yet be handled by using Milanov-Ruan's B-model technique. We first show that genus zero modularity is obtained from the phenomenon that the system of WDVV equations is essentially equivalent to the set of Ramanujan identities satisfied by the generators of the ring of quasi-modular forms for a certain modular group associated to the orbifold curve. Higher genus modularity then follows by using tautological relations.

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LG/CY correspondence for elliptic orbifold curves via modularity

Cited by 13 publications

References 39 publications

Givental-Type Reconstruction at a Nonsemisimple Point

Givental-Type Reconstruction at a Nonsemisimple Point

Blowup equations for refined topological strings

Ramanujan identities and quasi-modularity in Gromov–Witten theory

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