Exact calculations beyond charge neutrality in timelike Liouville field theory

Abstract

Timelike Liouville field theory (also known as imaginary Liouville theory or imaginary Gaussian multiplicative chaos) is expected to describe two-dimensional quantum gravity in a positive-curvature regime, but its path integral is not a probability measure and rigorous exact computations are currently available only in the charge-neutral (integer screening) case. In this paper we show that at the special coupling $b=1/\sqrt{2}$, the Coulomb-gas expansion of the timelike path integral becomes explicitly computable beyond charge neutrality. The reason is that the $n$-fold integrals generated by the interaction acquire a Vandermonde/determinantal structure at $b=1/\sqrt{2}$, which allows exact evaluation in terms of classical special functions. We derive Mellin-Barnes type representations (involving the Barnes $G$-function and, in a three-point case, Gauss hypergeometric functions) for the zero- and one-point functions, for an antipodal two-point function, and for a three-point function with a resonant insertion $\alpha_2=b$. We then address the subtle zero-mode integration: after a Gaussian regularization we obtain an explicit renormalized partition function $C(1/\sqrt{2},\mu)=e(4\pi\sqrt2 \mu)^{-1}$, identify distributional limits in the physically relevant regime $\alpha_j=\frac{1}{2}Q+\mathrm{i} P_j$, and compare with the Hankel-contour prescription recently proposed in the physics literature. These results provide the first rigorously controlled family of exact calculations in timelike Liouville theory outside charge neutrality.