Correlators in Timelike Bulk Liouville Theory

TL;DR

The work investigates timelike Liouville theory as a time-dependent background modeling closed-string tachyon condensation, using analytic continuation from spacelike Liouville and renormalization to define correlators. A minisuperspace analysis reveals an exponential divergence in the closed-string pair production rate, motivating a search for exact two-point and three-point functions. The authors obtain a timelike two-point function by continuing the spacelike result, finding $|d(\omega)|=e^{-\pi\omega/\beta}$ and, at $\beta\to1$, $d(\omega)\to-(\pi\mu_R)^{i\omega}e^{-\pi\omega}$ with a renormalized coupling $\mu_R=|\mu\gamma(\beta^2)|$, which also cures related divergences. The timelike three-point function, however, remains puzzling due to an accumulation of poles in the continuation and a potential mismatch with conformal invariance, signaling subtleties in defining timelike Liouville CFT and highlighting ongoing challenges in characterizing tachyon condensation dynamics in time-dependent string backgrounds.

Abstract

Liouville theory with a negative norm boson and no screening charge corresponds to an exact classical solution of closed bosonic string theory describing time-dependent bulk tachyon condensation. A simple expression for the two point function is proposed based on renormalization/analytic continuation of the known results for the ordinary (positive-norm) Liouville theory. The expression agrees exactly with the minisuperspace result for the closed string pair-production rate (which diverges at finite time). Puzzles concerning the three-point function are presented and discussed.