# On non-homogeneous tachyon condensation in closed string theory

## TL;DR

This paper derives an exact formula for 3-point correlation functions in a solvable model describing non-homogeneous tachyon condensation in closed string theory, revealing deep mathematical structures and consistency with prior proposals.

## Contribution

It provides a closed-form expression for 3-point functions in the gauged $H_+^3$ WZW model at levels between 0 and 2, connecting to timelike Liouville theory and revealing new mathematical relations.

## Key findings

- Derived a closed formula for 3-point functions at level 0<k<2.
- Found the ratio of 3-point functions expressed via Jacobi theta functions.
- Demonstrated cancellations and factorization properties in the correlation functions.

## Abstract

Lorentzian continuation of the Sine-Liouville model describes non-homogeneous rolling closed string tachyon. Via T-duality, this relates to the gauged $H_+^3$ Wess-Zumino-Witten model at subcritical level. This model is exactly solvable. We give a closed formula for the 3-point correlation functions for the model at level k within the range 0<k<2, which relates to the analogous quantity for k>2 in a similar way as how the Harlow-Maltz-Witten 3-point function of timelike Liouville field theory relates to the analytic continuation of the Dorn-Otto-Zamolodchikov-Zamolodchikov structure constants: We find that the ratio between both 3-point functions can be written in terms of the Jacobi's $\theta $-function, while their product exhibits remarkable cancellations and eventually factorizes. Our formula is consistent with previous proposals made in the literature.