Elliptic Genera and N=2 Superconformal Field Theory

TL;DR

This paper explores the properties and computation of elliptic genera in N=2 superconformal field theories, especially for Landau-Ginzburg orbifolds and manifolds with SU(N) holonomy, linking them to minimal models.

Contribution

It introduces a general method to compute elliptic genera for orbifold theories and compares these results with minimal model constructions and sigma model expressions.

Findings

Confirmed fundamental properties of elliptic genera in N=2 theories.

Developed a procedure to compute elliptic genera for specific orbifold theories.

Established connections between Landau-Ginzburg orbifolds, minimal models, and manifolds with SU(N) holonomy.

Abstract

Recently Witten proposed to consider elliptic genus in $N=2$ superconformal field theory to understand the relation between $N=2$ minimal models and Landau-Ginzburg theories. In this paper we first discuss the basic properties satisfied by elliptic genera in $N=2$ theories. These properties are confirmed by some fundamental class of examples. Then we introduce a generic procedure to compute the elliptic genera of a particular class of orbifold theories, {\it i.e.\/} the ones orbifoldized by $e^{2\pi iJ_0}$ in the Neveu-Schwarz sector. This enables us to calculate the elliptic genera for Landau-Ginzburg orbifolds. When the Landau-Ginzburg orbifolds allow an interpretation as target manifolds with $SU(N)$ holonomy we can compare the expressions with the ones obtained by orbifoldizing tensor products of $N=2$ minimal models. We also give sigma model expressions of the elliptic genera for manifolds of $SU(N)$ holonomy.