The chiral critical locus and topological structures

Abstract

We study a differential graded VOA associated to the derived critical locus of a function $f$ on a smooth oriented $D$-dimensional variety $(X,\mathbf{vol})$. Informally, this VOA, ${\mathbf{crit}}_f^{ch}$, is just the algebra of chiral differential operators on the derived critical locus ${\mathbf{crit}}_f$. We prove, using a generalization of a physical construction of Witten, the ${\mathbf{crit}}_f^{ch}$ admits a \emph{topological structure} if $f$ is homogeneous for a ${\mathbf{G}}_m$ action on $(X,\mathbf{vol})$. If $\mathbf{vol}$ has weight $b$ and $f$ has weight $a$, we compute the rank of the topological structure in terms of the discrete invariants of the theory to be $$d=(D-\frac{2b}{a}).$$ We conclude with some remarks about BV quantization and a simple computation of characters.