We propose an extended set of differential operators for local mirror symmetry. If $X$ is Calabi-Yau such that $\dim H_4(X,\Z)=0$, then we show that our operators fully describe mirror symmetry. In the process, a conjecture for intersection theory for such $X$ is uncovered. We also find new operators on several examples of type $X=K_S$ through similar techniques. In addition, open string PF systems are considered.Comment: 47 pages, 7 figures. Minor errors corrected, including the correction of the triple intersection numbers for the del Pezzo surfac

We continue our study of equivariant local mirror symmetry of curves, i.e., mirror symmetry for1 with torus action (λ 1 , λ 2 ) on the bundle. For the antidiagonal action λ 1 = −λ 2 , 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 λ 1 = λ 2 , we argue that the mirror symmetry computation is equivalent to that of the projective bundleFinally, we outline the computation of equivariant Gromov-Witten invariants for A n singularities and toric tree examples via mirror symmetry.e-print archive:

We propose a recipe for determination of the partition function of N = 4 ADE gauge theory on K3 by generalizing our previous results of the SU (N ) case. The resulting partition function satisfies Montonen-Olive duality for ADE gauge group.

We provide a straightforward computational scheme for the equivariant local mirror symmetry of curves, i.e. mirror symmetry for O(k) ⊕ O(−2 − k) → P 1 for k ≥ 1, and detail related methods for dealing with mirror symmetry of non-nef toric varieties, based on the theorems of Refs. 2 and 13. The basic tools are equivariant I functions and their Birkhoff factorization.

We give an explicit procedure which computes for degree d ≤ 3 the correlation functions of topological sigma model (A-model) on a projective Fano hypersurface X as homogeneous polynomials of degree d in the correlation functions of degree 1 (number of lines). We extend this formalism to the case of Calabi-Yau hypersurfaces and explain how the polynomial property is preserved. Our key tool is the construction of universal recursive formulas which express the structural constants of the quantum cohomology ring of X as weighted homogeneous polynomial functions in the constants of the Fano hypersurface with the same degree and dimension one more. We propose some conjectures about the existence and the form of the recursive formulas for the structural constants of rational curves of arbitrary degree. Our recursive formulas should yield the coefficients of the hypergeometric series used in the mirror calculation. Assuming the validity of the conjectures we find the recursive laws for rational curves of degree 4 and 5. 1