"Analytic continuation" of N = 2 minimal model

Abstract

In this paper we discuss what theory should be identified as the “analytic continuation” with |$N \rightarrow -N$| 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 holomorphic linear combinations of the “modular completions” introduced in [T. Eguchi and Y. Sugawara, JHEP 1103, 107 (2011)] in the |$SL(2)_{N+2}/U(1)$| supercoset theory. We further discuss how this model could be interpreted as a kind of model of the |$SL(2)_{N+2}/U(1)$| supercoset in the |$(\widetilde {{\mathrm {R}}},\widetilde {{\rm R}})$| sector, in which only the discrete spectrum appears in the torus partition function and the potential IR divergence due to the non-compactness of the target space is removed. We also briefly discuss possible definitions of the sectors with other spin structures.