Seidel elements and potential functions of holomorphic disc counting

TL;DR

This paper explores the relationship between Seidel elements and potential functions in the context of holomorphic disc counting, providing a degeneration approach that confirms a conjecture for toric manifolds.

Contribution

It establishes a conjectural link between potential functions and Seidel elements, and proves it for Lagrangian torus fibers in semi-positive toric manifolds.

Findings

Confirmed the conjecture relating Seidel elements and potential functions in toric cases.

Derived a degeneration method to connect holomorphic disc counts with algebraic invariants.

Reproduced known results as special cases of the general theory.

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 relating certain correction terms appearing in the Seidel elements with the potential function.