Rigorous results for timelike liouville field theory

Liouville field theory has long been a cornerstone of two-dimensional quantum field theory and quantum gravity, which has attracted much recent attention in the mathematics literature. Timelike Liouville field theory is a version of Liouville field theory where the kinetic term in the action appears with a negative sign, which makes it closer to a theory of quantum gravity than ordinary (spacelike) Liouville field theory. Making sense of this “wrong sign” requires a theory of Gaussian random variables with negative variance. Such a theory is developed in this paper, and is used to prove the timelike DOZZ formula for the $3$ -point correlation function when the parameters satisfy the so-called “charge neutrality condition.” Expressions are derived also for the k-point correlation functions for all $k\ge 3$ , and it is shown that these functions approach the correct semiclassical limits as the coupling constant is sent to zero.