Non-holomorphic modular forms and SL(2,R)/U(1) superconformal field theory

Abstract

We study the torus partition function of the ${{{{\text{SL}}\left( {2,\mathbb{R}} \right)}} \left/ {{{\text{U}}(1)}} \right.}$ SUSY gauged WZW model coupled to $\mathcal{N} = 2$ 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 $\mathcal{N} = 2$ 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 [11].

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 ${{{{\text{SL}}\left( {2,\mathbb{R}} \right)}} \left/ {{{\text{U}}(1)}} \right.}$ elliptic genus has recently been observed by Troost [12].