Comments on non-holomorphic modular forms and non-compact superconformal field theories

Abstract

We extend our previous work [1] on the non-compact $\mathcal{N} = {2}$ SCF T ₂ defined as the supersymmetric SL(2, $\mathbb{R}$)/U(1)-gauged WZW model. Starting from path-integral calculations of torus partition functions of both the axial-type (‘cigar’) and the vector-type (‘trumpet’) models, we study general models of the ${\mathbb{Z}_M}$ -orbifolds and M -fold covers with an arbitrary integer M. We then extract contributions of the degenerate representations (‘discrete characters’) in such a way that good modular properties are preserved. The ‘modular completion’ of the extended discrete characters introduced in [1] are found to play a central role as suitable building blocks in every model of orbifolds or covering spaces. We further examine a large M-limit (the ‘continuum limit’), which ‘deconstructs’ the spectral flow orbits while keeping a suitable modular behavior. The discrete part of partition function as well as the elliptic genus is then expanded by the modular completions of irreducible discrete characters, which are parameterized by both continuous and discrete quantum numbers modular transformed in a mixed way. This limit is naturally identified with the universal cover of trumpet model. We finally discuss a classification of general modular invariants based on the modular completions of irreducible characters constructed above.