Tohoku Mathematical Journal
2017
September
SECOND SERIES VOL. 69, NO. 3

Tohoku Math. J.
69 (2017), 327-368

Title SEIDEL ELEMENTS AND POTENTIAL FUNCTIONS OF HOLOMORPHIC DISC COUNTING

Author Eduardo González and Hiroshi Iritani

(Received May 28, 2014, revised April 1, 2015)
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 \emph{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.

Mathematics Subject Classification. Primary 53D45; Secondary 53D12, 53D37.
Key words and phrases. Lagrangian torus fibres, potential functions, holomorphic discs, mirror symmetry, Jacobian ring.

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