Spectral Flow Equivariance for Calabi-Yau Sigma Models

TL;DR

This paper constructs an explicit operator on the chiral de Rham complex of Calabi-Yau varieties that demonstrates spectral flow equivariance, leading to a categorification of the elliptic genus.

Contribution

It introduces a new explicit operator intertwining the $ abla=2$ superconformal algebra modules with their spectral flow twists, providing a novel categorification of elliptic genus properties.

Findings

Explicit operator on chiral de Rham complex for spectral flow

Categorification of elliptic genus via trace computations

Demonstrates spectral flow equivariance in Calabi-Yau sigma models

Abstract

We write down an explicit operator on the chiral de Rham complex of a Calabi-Yau variety $X$ which intertwines the usual $\mathcal{N}=2$ module structure with its twist by the spectral flow automorphism of the $\mathcal{N}=2$, producing the expected \emph{spectral flow equivariance}. Taking the trace of the operators $L_{0}$ and $J_{0}$ on cohomology, and using the obvious interaction of spectral flow with characters, we obtain an explicit categorification of ellipticity of the elliptic genus of $X$, which is well known by other means.