Liouville quantum gravity on complex tori (Q2795550)

The authors rigorously construct Liouville Quantum Field Theory (LQFT) on the one-dimensional complex torus with arbitrary complex structure. LQFT is given as a random field with probability law given by a Feynman path integral-type quantization of the classical Liouville action, using Gaussian free fields and the theory of Gaussian multiplicative chaos to construct the interaction measure.NEWLINENEWLINEThe main properties of LQFT on tori are studied, especially in the regime where the theory is a Conformal Field Theory (CFT): Weyl anomaly, central charge, scaling laws for the partition function, modular invariance. In particular, the associated Liouville Quantum Gravity theory (LQG) on the torus is examined and the probability law of the random Liouville modulus of the LQG is explicitly computed.NEWLINENEWLINEThis work extends a previous construction of LQFT on the Riemann sphere [\textit{F. David} et al., Commun. Math. Phys. 342, No. 3, 869--907 (2016; Zbl 1336.83042)] and provides a further step in the program of constructing LQFT on every compact Riemann surface with arbitrary genus.