Correlation functions in Liouville theory

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Abstract
We show that after integrating over the zero mode in Liouville correlation functions, the remaining functional integral resembles a free theory and may be evaluated by formally continuing the central charge. We apply this technique to the unitary minimal models coupled to gravity on the sphere, computing a number of three-point functions. After taking into account the normalizations of operators and the functional integral, we find exact agreement between the Liouville three-point functions and the results from matrix models.