Local mirror symmetry of curves: Yukawa couplings and genus 1

We continue our study of equivariant local mirror symmetry of curves, i.e., mirror symmetry for X_k - O(k) ⊕ O(-2 - k) → ℙ¹ with torus action (λ₁,λ₂) on the bundle. For the antidiagonal action λ₁ = - λ₂, we find closed formulas for the mirror map, a rational B model Yukawa coupling and consequently Picard-Fuchs equations for all k. Moreover, we give a simple closed form for the B model genus 1 Gromov-Witten potential. For the diagonal action λ₁ = λ₂, we argue that the mirror symmetry computation is equivalent to that of the projective bundle ℙ(O ⊕ O(k) ⊕O(-2 - k)) → ℙ¹. Finally, we outline the computation of equivariant Gromov-Witten invariants for A_n singularities and toric tree examples via mirror symmetry.