Non-holomorphic Modular Forms and SL(2,R)/U(1) Superconformal Field Theory

TL;DR

This paper analyzes the modular properties of the SL(2,R)/U(1) superconformal field theory's partition function, revealing a non-holomorphic dependence that aligns with the theory of mock theta functions and suggests a general holomorphic anomaly in non-compact string backgrounds.

Contribution

It demonstrates the non-holomorphic dependence in the partition function's coefficients and connects it to real analytic Jacobi forms and mock theta functions, highlighting a novel holomorphic anomaly.

Findings

Non-holomorphic dependence in continuous representation coefficients.

Discrete characters combined with non-holomorphic continuous characters form real analytic Jacobi forms.

The observed phenomena suggest a general holomorphic anomaly in string theories on non-compact manifolds.

Abstract

We study the torus partition function of the SL(2,R)/U(1) SUSY gauged WZW model coupled to N=2 U(1) current. Starting from the path-integral formulation of the theory, we introduce an infra-red regularization which preserves good modular properties and discuss the decomposition of the partition function in terms of the N=2 characters of discrete (BPS) and continuous (non-BPS) representations. Contrary to our naive expectation, we find a non-holomorphic dependence (dependence on \bar{\tau}) in the expansion coefficients of continuous representations. This non-holomorphicity appears in such a way that the anomalous modular behaviors of the discrete (BPS) characters are compensated by the transformation law of the non-holomorphic coefficients of the continuous (non-BPS) characters. Discrete characters together with the non-holomorphic continuous characters combine into real analytic Jacobi forms and these combinations exactly agree with the "modular completion" of discrete characters known in the theory of Mock theta functions \cite{Zwegers}. We consider this to be a general phenomenon: we expect to encounter "holomorphic anomaly" (\bar{\tau}-dependence) in string partition function on non-compact target manifolds. The anomaly occurs due to the incompatibility of holomorphy and modular invariance of the theory. Appearance of non-holomorphicity in SL(2,R)/U(1) elliptic genus has recently been observed by Troost \cite{Troost}.