"Analytic continuation" of N = 2 minimal model

In this paper we discuss what theory should be identified as the "analytic continuation" with N→−N of the N=2 minimal model with the central charge c-^=1−(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 (R-tilde, R-tilde) 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