2 , R ) / U ( 1 ) Superconformal Field Theory

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 τ̄) 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 [9]. We consider this to be a general phenomenon: we expect to encounter “holomorphic anomaly” (τ̄ -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 [10]. ∗eguchi@yukawa.kyoto-u.ac.jp †ysugawa@se.ritsumei.ac.jp