Rolling Tachyons from Liouville theory

TL;DR

The paper constructs an exact solution for Liouville theory at central charge $c=1$, modeling homogeneous decay of a closed-string tachyon. By analytic continuation from the well-understood $c\ge 25$ regime, it yields two distinct $c=1$ theories, Euclidean and Lorentzian, with non-analytic momentum dependence that prevents Wick rotation between them. The Euclidean theory reproduces the interacting $c=1$ model of Runkel and Watts, while the Lorentzian theory provides new three-point couplings relevant to rolling tachyon backgrounds, offering a concrete CFT framework for time-dependent string backgrounds. The construction relies on Barnes' double Gamma functions, their $\Upsilon$ and $Y_\beta$ relatives, and careful limits of theta-functions to define the $c=1$ couplings, with implications for boundary Liouville and open-string tachyon condensation.

Abstract

In this work we propose an exact solution of the c=1 Liouville model, i.e. of the world-sheet theory that describes the homogeneous decay of a closed string tachyon. Our expressions are obtained through careful extrapolation from the correlators of Liouville theory with c > 25. In the c=1 limit, we find two different theories which differ by the signature of Liouville field. The Euclidean limit coincides with the interacting c=1 theory that was constructed by Runkel and Watts as a limit of unitary minimal models. The couplings for the Lorentzian limit are new. In contrast to the behavior at c > 1, amplitudes in both c=1 models are non-analytic in the momenta and consequently they are not related by Wick rotation.