Seidel elements and potential functions of holomorphic disc counting
Eduardo Gonzalez,Hiroshi Iritani +1 more
TLDR
In this paper, the problem of counting holomorphic disc sections of the trivial $M$-bundle over a disc with boundary in a Lagrangian submanifold was studied.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.read more
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Tropical counting from asymptotic analysis on Maurer-Cartan equations
Kwokwai Chan,Ziming Nikolas Ma +1 more
TL;DR: In this paper, the authors apply asymptotic analysis to study the extended deformation theory of the LG model, and prove that semi-classical limits of Fourier modes of a specific class of Maurer-Cartan solutions naturally give rise to tropical disks in with Maslov index 0 or 2.
Journal ArticleDOI
Quantum Cohomology and Closed-String Mirror Symmetry for Toric Varieties
TL;DR: In this paper, it was shown that a class of generalised Jacobian rings associated to an arbitrary smooth toric variety are free as modules over the Novikov ring, and the quantum cohomology of such a variety can be computed using the Kodaira-Spencer map.
Journal ArticleDOI
Tropical counting from asymptotic analysis on Maurer-Cartan equations
Kwokwai Chan,Ziming Nikolas Ma +1 more
TL;DR: In this article, the authors apply asymptotic analysis to study the extended deformation theory of the LG model, and prove that semi-classical limits of Fourier modes of a specific class of Maurer-Cartan solutions naturally give rise to tropical disks in $X$ with Maslov index 0 or 2, the latter of which produces a universal unfolding of $W$.