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}.Comment: 39+1 pages, no figure; v2 a reference added, some points are
clarified, typos corrected, version to appear in JHE