`Analytic Continuation' of N=2 Minimal Model

Abstract

In this paper we discuss what theory should be identified as the `analytic continuation' with $N\to -N$ of the $\mathcal{N}=2$ minimal model with the central charge $\hat{c}=1-\frac{2}{N}$. We clarify how the elliptic genus of the expected model is written in terms of {\em holomorphic\/} linear combinations of the `modular completions' introduced in [arXiv:1012.5721 [hep-th]] in the $SL(2{)}_{N+2}/U(1)$-supercoset theory. We further discuss how this model could be interpreted as a kind of {\em `compactified'} model of the $SL(2{)}_{N+2}/U(1)$-supercoset in the $(\tilde{R},\tilde{R})$-sector, in which only the discrete spectrum appears in the torus partition function and the potential IR-divergence due to the non-compactness of target space is removed. We also briefly argue on possible definitions of the sectors with other spin structures.

1+29 pages, no figure

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