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The Homogeneous sine-Gordon (HSG) theories are integrable perturbations of$G_k/U(1)^{r_G}$ coset CFTs, where $G$ is a simple compact Lie group of rank$r_G$ and $k>1$ is an integer. Using their T-duality symmetries, we investigatethe relationship between the different theories corresponding to a given coset,and between the different phases of a particular theory. Our results suggestthat for $G=SU(n)$ with $n\geq5$ and $E_6$ there could be two non-equivalentHSG theories associated to the same coset, one of which has not been consideredso far.

Searching for new homogeneous sine-Gordon theories using T-duality symmetries