Compact Formulas for the Completed Mock Modular Forms

TL;DR

This paper derives compact, modularly manifest expressions for elliptic genera and mock modular forms in superconformal theories, enhancing understanding of their spectral flow and modular properties.

Contribution

It introduces new compact formulas for elliptic genera and mock modular forms using spectral flow and Poincare series representations.

Findings

New compact formulas for elliptic genus of SL(2)/U(1) supercoset

Explicit modular completions of mock modular forms

Generalization to arbitrary spin-structures with twisted boundary conditions

Abstract

In this paper we present a new compact expression of the elliptic genus of SL(2)/U(1)-supercoset theory by making use of the `spectral flow method' of the path-integral evaluation. This new expression is written in a form like a Poincare series with a non-holomorphic Gaussian damping factor, and manifestly shows the modular and spectral flow properties of a real analytic Jacobi form. As a related problem, we present similar compact formulas for the modular completions of various mock modular forms which appear in the representation theory of N=2,4 superconformal algebras. We further discuss the generalization to the cases of arbitrary spin-structures, that is, the world-sheet fermions with twisted boundary conditions parameterized by a continuous parameter. This parameter is naturally identified with the `u-variable' in the Appell-Lerch sum.