Seminar

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The Liouville action is a function of the conformal factor relating two conformally equivalent metrics.

I will prove the relevant properties for the Liouville action and show how they define real determinant lines on Riemann surfaces and later on loops. The determinant line characterizes how quantities associated with the Riemann surface can change in a covariant way under conformal maps. This applies to the locally conformal covariance property of measures on the space of loops on a Riemann surface, the existence and uniqueness of which has been conjectured by Kontsevich and Suhov and whose construction involves SLE.