Seiberg-Witten equations and non-commutative spectral curves in Liouville theory

Abstract

We propose that there exist generalized Seiberg-Witten equations in the Liouville conformal field theory, which allow the computation of correlation functions from the resolution of certain Ward identities. These identities involve a multivalued spin one chiral field, which is built from the stress-energy tensor. We solve the Ward identities perturbatively in an expansion around the heavy asymptotic limit, and check that the first two terms of the Liouville three-point function agree with the known result of Dorn, Otto, Zamolodchikov and Zamolodchikov. We argue that such calculations can be interpreted in terms of the geometry of non-commutative spectral curves.