Spectral Flow Equivariance for 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.