Seidel elements and potential functions of holomorphic disc counting

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.