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Computing a categorical Gromov–Witten invariant

Part of: Homological methods Projective and enumerative geometry Abelian varieties and schemes Surfaces and higher-dimensional varieties Symplectic geometry, contact geometry

Published online by Cambridge University Press:  18 June 2020

Andrei Căldăraru  and
Junwu Tu
Andrei Căldăraru
Affiliation:
Mathematics Department, University of Wisconsin – Madison, 480 Lincoln Drive, Madison, WI 53706–1388, USA email andreic@math.wisc.edu
Junwu Tu
Affiliation:
Institute of Mathematical Sciences, ShanghaiTech University, 393 Huaxia Rd, Pudong, Shanghai, 201210, PR China email tujw@shanghaitech.edu.cn
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Abstract

We compute the $g=1$, $n=1$ B-model Gromov–Witten invariant of an elliptic curve $E$ directly from the derived category $\mathsf{D}_{\mathsf{coh}}^{b}(E)$. More precisely, we carry out the computation of the categorical Gromov–Witten invariant defined by Costello using as target a cyclic $\mathscr{A}_{\infty }$ model of $\mathsf{D}_{\mathsf{coh}}^{b}(E)$ described by Polishchuk. This is the first non-trivial computation of a positive-genus categorical Gromov–Witten invariant, and the result agrees with the prediction of mirror symmetry: it matches the classical (non-categorical) Gromov–Witten invariants of a symplectic 2-torus computed by Dijkgraaf.

Keywords

Gromov–WittenA∞ categorieselliptic curves

MSC classification

Primary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants
Secondary: 53D37: Mirror symmetry, symplectic aspects; homological mirror symmetry; Fukaya category 14J33: Mirror symmetry 14K25: Theta functions 16E40: (Co)homology of rings and algebras (e.g. Hochschild, cyclic, dihedral, etc.)
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 7 , July 2020 , pp. 1275 - 1309
Copyright
© The Authors 2020

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Footnotes

The first author is partially supported by the National Science Foundation through grant number DMS-1200721.

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