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The chiral critical locus and topological structures

Emile Bouaziz

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, $\mathbf{crit}^{ch}_{f}$, 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 $\mathbf{crit}^{ch}_{f}$ 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 $$d=\Big(D-\frac{2b}{a}\Big).$$ We conclude with some remarks about BV quantization and a simple computation of characters.

The chiral critical locus and topological structures

Abstract

We study a differential graded VOA associated to the derived critical locus of a function on a smooth oriented -dimensional variety . Informally, this VOA, , is just the algebra of chiral differential operators on the derived critical locus . We prove, using a generalization of a physical construction of Witten, the admits a \emph{topological structure} if is homogeneous for a action on . If has weight and has weight , we compute the rank of the topological structure in terms of the discrete invariants of the theory to be We conclude with some remarks about BV quantization and a simple computation of characters.
Paper Structure (11 sections, 12 theorems, 50 equations)

This paper contains 11 sections, 12 theorems, 50 equations.

Table of Contents

  1. Introduction and Statement of Results
  2. Preparatory Notions
  3. Vertex Algebras
  4. BRST Reduction
  5. The Chiral de Rham Complex and topological vertex algebras
  6. Topological structure on the Chiral Critical Locus
  7. Chiral Polyvectors
  8. The Chiral Critical Locus
  9. Additional Remarks and Computations
  10. Characters
  11. BV Quantization

Key Result

Theorem 1.1

Let $f$ be homogeneous of weight $a$ for a $\mathbf{G}_{m}$ action on $(X,\mathbf{vol})$, with $\mathbf{vol}$ of weight b. Then there are $\partial_{f}^{ch}$-closed currents $\{^{f}L,\,^{f}J,\,^{f}Q,\,^{f}G\}$, with $^{f}L=L,\,^{f}G=G,$ generating an action of $\mathbf{top}\{d\}$ on $\mathbf{crit}^{

Theorems & Definitions (45)

  • Theorem 1.1
  • Remark
  • Remark
  • Definition 2.1
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Definition 2.3
  • Remark
  • Definition 2.4
  • ...and 35 more