The continuum limit ofaN−1(2)spin chains

Building on our previous work for a (2) 2 and a (2) 3 we explore systematically the continuum limit of gapless a (2) N −1 vertex models and spin chains. We find the existence of three possible regimes. Regimes I and II for a (2) 2n−1 are related with a (2) 2n−1 Toda, and described by n compact bosons. Regime I for a (2) 2n is related with a (2) 2n Toda and involves n compact bosons, while regime II is related instead with B (1) (0, n) super Toda, and involves in addition a single Majorana fermion. The most interesting is regime III, where non-compact degrees of freedom appear, generalising the emergence of the Euclidean black hole CFT in the a (2) 2 case. For a (2) 2n we find a continuum limit made of n compact and n non-compact bosons, while for a (2) 2n−1 we find n compact and n − 1 non-compact bosons. We also find deep relations between a (2) N−1 in regime III and the gauged WZW models SO(N)/SO(N − 1).