Superconformal Algebras and Mock Theta Functions

TL;DR

This paper enhances the understanding of superconformal algebra characters by applying Zwegers' method to improve their modular properties, with applications to elliptic genera of hyper-Kahler manifolds.

Contribution

It introduces a novel application of Zwegers' method to decompose superconformal characters into modular components, advancing the analysis of BPS representations.

Findings

Decomposition of BPS characters into Jacobi forms and non-BPS series.

Explicit elliptic genera for K^{[2]} and A^{[[3]]} hyper-Kahler manifolds.

Improved modular properties of superconformal characters.

Abstract

It is known that characters of BPS representations of extended superconformal algebras do not have good modular properties due to extra singular vectors coming from the BPS condition. In order to improve their modular properties we apply the method of Zwegers which has recently been developed to analyze modular properties of mock theta functions. We consider the case of N=4 superconformal algebra at general levels and obtain the decomposition of characters of BPS representations into a sum of simple Jacobi forms and an infinite series of non-BPS representations. We apply our method to study elliptic genera of hyper-Kahler manifolds in higher dimensions. In particular we determine the elliptic genera in the case of complex 4 dimensions of the Hilbert scheme of points on K3 surfaces K^{[2]} and complex tori A^{[[3]]}.