Mirror symmetry in two steps: A-I-B
Edward Frenkel, Andrei Losev
TL;DR
The paper proposes a two-step mirror-symmetry framework for toric varieties by introducing an intermediate I-model that sits between the $A$-model (infinite-volume twisted sigma model) and the $B$-model (twisted Landau–Ginzburg theory). It shows that the $A$-model is equivalent to the I-model in the infinite-volume limit, while the I-model’s BPS sector mirrors the LG $B$-model, with the superpotential arising as a sum over irreducible divisors. By constructing the I-model for ${f P}^1$ and then generalizing to arbitrary Fano toric varieties, the work links the chiral de Rham complex to the quantum cohomology via cohomology of the right-moving supercharge, and connects Gromov–Witten invariants with LG oscillatory integrals through T-duality and holomortex operator techniques. The framework offers a concrete operator-algebraic realization of mirror symmetry, clarifying the role of divisors in generating the superpotential and highlighting a path toward extending the construction to broader classes of toric and non-toric varieties.
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.