I've been studying the coset construction. As far as I understand it, the Sugawara energy momentum tensor is a way of embedding the virasoro algebra inside the Lie algebra of your original WZW model. The nice thing is that the weight modules match (descending weights in the virasoro rep match descendants of primary fields in the cft).

Different choices for the original Lie algebra lead to different CFTs. Most notably, taking the diagonal coset of the affine extension of su(2): $$\frac{\widehat{\mathfrak{su}}(2)_k\oplus \widehat{\mathfrak{su}}(2)_1}{\widehat{\mathfrak{su}}(2)_{k+1}}$$

leads to the set of minimal models. However, I've also been reading, for example in di Francesco (the big yellow book), that some of these coset WZW models correspond to perturbed CFTs. For example, diagonal $\widehat{\mathfrak{su}}(2)_k$ and $\widehat{E_{(8)}}_k$ at k=1 lead to the Ising model perturbed with respectively energy density and an external magnetic field.

My confusion is the following: If these diagonal coset models lead to CFTs, how can they also describe perturbed theories. Does this perturbation not destroy conformal invariance? More concretely: When investigating the field content of the coset model by looking at the character decompositions, how do you find what the perturbation is?


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