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I'm trying to compute the OPEs of two specific primary fields in a WZW model. The issue is that I can't apply the state-field correspondence since these fields don't belong to the vacuum module (so I don't know their expansions in terms of modes). Since conformal invariance fixes the functional form of the 2 and 3 point correlators, these correlators won't give me any information about the OPEs. Is there any way I can approach this problem? Hints are appreciated, please don't give a complete solution.

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This question was quite vague, but I will post what I have learnt in case anyone else is interested. From what I understand, one can only compute arbitrary OPEs of primary fields easily for WZW models that have a free-field realisation. For example, the $\mathfrak{s}\mathfrak{l}_2(\mathbb{C})$ WZW model at level 1 is equivalent to the free boson and hence one can compute OPEs of the WZW primary fields using free boson vertex operators. This includes primary fields of all allowed modules (not just the vacuum module) since the free-field realisation is an isomorphism of modules over the respective vertex algebras.

Even if a free-field realisation for a WZW model is not known, the two-point functions can be computed using conformal invariance (i.e. by using the KZ equations and Ward identities). I also think my statement about OPEs and correlation functions is incorrect. It seems that once you know the two-point correlation functions of the primary fields, the corresponding full OPEs (not just the singular part of OPEs) are known.

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