Timelike Liouville CFT versus timelike Liouville gravity

Determine whether timelike Liouville conformal field theory and timelike Liouville gravity define the same theory, including whether their path-integral predictions and conformal-field-theory correlators are mutually consistent.

Background

The paper compares timelike Liouville CFT data, including timelike DOZZ structure constants, with the semiclassical and perturbative path-integral description of timelike Liouville gravity. It finds a potential mismatch: a particular CFT three-point function vanishes, whereas the corresponding derivative of the gravitational sphere partition function appears nonzero.

Although an analytic-continuation prescription is proposed to address the discrepancy, the paper does not establish that the two formulations are equivalent.