Path integral for quantum Mabuchi K-energy

We construct a measure (of finite total mass) on the space of metrics fixed 1-dimensional complex manifold genus greater than or equal to 2) which corresponds in physics literature path integral based Gibbs type with energy given by sum Liouville action and Mabuchi K-energy. To best our knowledge, this is first rigorous construction such an object done means probabilistic tools. Both functionals (the functional K-energy functional) play important role, respectively, Riemannian geometry (in case surfaces) Kähler geometry. As output, we obtain whose Weyl anomaly displays standard plus additional Motivations come from theoretical where these integrals arise as models for fluctuating surfaces when coupling certain nonconformal matter fields (mathematically noncritical statistical models) quantum gravity advocated A. Bilal, F. Ferrari, S. Klevtsov, Zelditch. On discrete side, expected describe scaling limit large planar maps decorated some models. Interestingly, computations show that corrections perturb classical produce K-energy: are reminiscent theory. Our relies variant theory Gaussian multiplicative chaos (GMC) derivative GMC (DGMC short). The technical backbone consists two estimates (derivative standard) independent interest probability First, DGMC random variables possess negative exponential moments, second, derive optimal small deviations associated recentered free field (GFF).