Two-Dimensional Twisted Sigma Models, the Mirror Chiral de Rham Complex, and Twisted Generalised Mirror Symmetry

TL;DR

This paper explores the mathematical structure and mirror symmetry properties of twisted sigma models, linking physical anomalies to sheaves of vertex superalgebras and relating mirror pairs via the chiral de Rham complex.

Contribution

It provides a novel interpretation of the B-twisted heterotic sigma model using chiral differential operators and establishes connections between mirror symmetry and sheaves of CDR on complex manifolds.

Findings

Reinterpreted physical anomalies as obstructions to sheaf definitions

Connected one-loop beta functions to holomorphic data

Linked mirror CDR sheaf cohomology to mirror symmetry conjectures

Abstract

In this paper, we study the perturbative aspects of a "B-twisted" two-dimensional $(0,2)$ heterotic sigma model on a holomorphic gauge bundle $\mathcal E$ over a complex, hermitian manifold $X$. We show that the model can be naturally described in terms of the mathematical theory of ``Chiral Differential Operators". In particular, the physical anomalies of the sigma model can be reinterpreted as an obstruction to a global definition of the associated sheaf of vertex superalgebras derived from the free conformal field theory describing the model locally on $X$. In addition, one can also obtain a novel understanding of the sigma model one-loop beta function solely in terms of holomorphic data. At the $(2,2)$ locus, one can describe the resulting half-twisted variant of the topological B-model in terms of a $\it{mirror}$ "Chiral de Rham complex" (or CDR) defined by Malikov et al. in \cite{GMS1}. Via mirror symmetry, one can also derive various conjectural expressions relating the sheaf cohomology of the mirror CDR to that of the original CDR on pairs of Calabi-Yau mirror manifolds. An analysis of the half-twisted model on a non-K\"ahler group manifold with torsion also allows one to draw conclusions about the corresponding sheaves of CDR (and its mirror) that are consistent with mathematically established results by Ben-Bassat in \cite{ben} on the mirror symmetry of generalised complex manifolds. These conclusions therefore suggest an interesting relevance of the sheaf of CDR in the recent study of generalised mirror symmetry.