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Local mirror symmetry of curves: Yukawa couplings and genus 1

Brian Forbes·Masao Jinzenji
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Abstract

We continue our study of equivariant local mirror symmetry of curves, i.e. mirror symmetry for X_k=O(k)+O(-2-k) over P^1 with torus action (lambda_1,lambda_2) on the bundle. For the antidiagonal action lambda_1=-lambda_2, we find closed formulas for the mirror map and a rational B model Yukawa coupling for all k. Moreover, we give a simple closed form for the B model genus 1 Gromov-Witten potential. For the diagonal action lambda_1=lambda_2, we argue that the mirror symmetry computation is equivalent to that of the projective bundle P(O+O(k)+O(-2-k)) over P^1. Finally, we outline the computation of equivariant Gromov-Witten invariants for A_n singularities and toric tree examples via mirror symmetry.

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Brian Forbes, Masao Jinzenji. 2006-09-01. Local mirror symmetry of curves: Yukawa couplings and genus 1. https://doi.org/10.4310/atmp.2007.v11.n1.a5

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