Superconformal vertex algebras in differential geometry. I

TL;DR

This paper constructs superconformal vertex algebras from Riemannian manifolds, explores their supersymmetry extensions, and connects these mathematical structures to physical theories like topological quantum field theories and elliptic genera.

Contribution

It introduces a method to build N=1, N=2, and N=4 superconformal vertex algebras from various types of manifolds, linking geometry with supersymmetry and physics.

Findings

Construction of N=1 SCVA from any Riemannian manifold.

Extension to N=2 and N=4 SCVA for special holonomy and hyperkähler manifolds.

Identification of BRST cohomology with known topological invariants.

Abstract

We show how to construct an N=1 superconformal vertex algebra (SCVA) from any Riemannian manifold. When the Riemannian manifold has special holonomy groups, we discuss the extended supersymmetry. When the manifold is complex or K\"{a}hler, we also generalize the construction to obtain N=2 SCVA's. We study the BRST cohomology groups of the topological vertex algebras obtained by the $A$ twist and the $B$ twist from these N=2 SCVA's. We show that for one of them, the BRST cohomologies are isomorphic to $H^*(M, \Lambda^*(T^*M))$ and $H^*(M, \Lambda^*(TM))$ respectively. This provides a mathematical formulation of the $A$ theory and $B$ theory in physics literature. The connection with elliptic genera is also discussed. Furthermore, when the manifold is hyperk\"{a}hler, we generalize our constructions to obtain N=4 SCVA's. A heuristic relationship with super loop space is also discussed.