Tohoku Mathematical Journal

Seidel elements and potential functions of holomorphic disc counting

Eduardo González and Hiroshi Iritani

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Abstract

Let $M$ be a symplectic manifold equipped with a Hamiltonian circle action and let $L$ be an invariant Lagrangian submanifold of $M$. We study the problem of counting holomorphic disc sections of the trivial $M$-bundle over a disc with boundary in $L$ through degeneration. We obtain a conjectural relationship between the potential function of $L$ and the Seidel element associated to the circle action. When applied to a Lagrangian torus fibre of a semi-positive toric manifold, this degeneration argument reproduces a conjecture (now a theorem) of Chan-Lau-Leung-Tseng [8, 9] relating certain correction terms appearing in the Seidel elements with the potential function.

Article information

Source
Tohoku Math. J. (2), Volume 69, Number 3 (2017), 327-368.

Dates
First available in Project Euclid: 12 September 2017

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1505181621

Digital Object Identifier
doi:10.2748/tmj/1505181621

Mathematical Reviews number (MathSciNet)
MR3695989

Zentralblatt MATH identifier
06814874

Subjects
Primary: 53D45: Gromov-Witten invariants, quantum cohomology, Frobenius manifolds [See also 14N35]
Secondary: 53D12: Lagrangian submanifolds; Maslov index 53D37: Mirror symmetry, symplectic aspects; homological mirror symmetry; Fukaya category [See also 14J33]

Keywords
Lagrangian torus fibres potential functions holomorphic discs mirror symmetry Jacobian ring

Citation

González, Eduardo; Iritani, Hiroshi. Seidel elements and potential functions of holomorphic disc counting. Tohoku Math. J. (2) 69 (2017), no. 3, 327--368. doi:10.2748/tmj/1505181621. https://projecteuclid.org/euclid.tmj/1505181621


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