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

TL;DR

This work analyzes the torus partition function of the SL(2,ℝ)/U(1) SUSY gauged WZW model with an N=2 current, demonstrating that a modular-invariant regularization inevitably produces non-holomorphic coefficients that complete discrete (BPS) characters with continuous (non-BPS) contributions. The partition function naturally decomposes into a modular-complete discrete part and a continuous part, with the leading IR divergence associated with the non-compact cigar geometry. The elliptic genus is shown to arise from these completed discrete characters and matches Troost's real-analytic Appell-type completion, revealing a deep connection between non-compact CFTs and mock theta function theory. The results illustrate a general holomorphy-modularity tension in non-compact string backgrounds and suggest broader applicability of modular completion and Mordell-type integrals in related theories.

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 \barτ) 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" (\barτ-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}.