Sommaire:

Inspired by the relations with matrix models and gauge theories, we introduce equations in 2d conformal field theory, which are the analogs of the loop equations of matrix models. Solving a conformal field theory then amounts to solving these equations. We solve them for two values of the central charge: c=1 and c=infinity. In the case c=1, the solution is built from a Lax matrix, which is a matrix square root of the energy-momentum tensor. In the case c=infinity, the solution is built from a non-commutative spectral curve, which (via topological recursion) encodes the perturbative expansion of correlation functions. In both cases, we recover the appropriate limit of the Liouville three-point function.

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