Mirror symmetry in two steps: A-I-B

TL;DR

This paper proposes a two-step interpretation of mirror symmetry for toric varieties through an intermediate model, connecting the A-model and B-model via T-duality and deformations, providing new insights into superpotentials and cohomology.

Contribution

It introduces the I-model as an intermediate step in mirror symmetry, linking the A-model and B-model for toric varieties through T-duality and deformations.

Findings

Realization of mirror symmetry in two steps via the I-model.

Natural interpretation of the superpotential as divisor components.

Relation of I-model cohomology to chiral de Rham complex and quantum cohomology.

Abstract

We suggest an interpretation of mirror symmetry for toric varieties via an equivalence of two conformal field theories. The first theory is the twisted sigma model of a toric variety in the infinite volume limit (the A-model). The second theory is an intermediate model, which we call the I-model. The equivalence between the A-model and the I-model is achieved by realizing the former as a deformation of a linear sigma model with a complex torus as the target and then applying to it a version of the T-duality. On the other hand, the I-model is closely related to the twisted Landau-Ginzburg model (the B-model) that is mirror dual to the A-model. Thus, the mirror symmetry is realized in two steps, via the I-model. In particular, we obtain a natural interpretation of the superpotential of the Landau-Ginzburg model as the sum of terms corresponding to the components of a divisor in the toric variety. We also relate the cohomology of the supercharges of the I-model to the chiral de Rham complex and the quantum cohomology of the underlying toric variety.