The Torus One-Point Diagram in Two-Dimensional String Cosmology

TL;DR

This paper numerically computes the torus one-point string diagram in two-dimensional string cosmology, revealing a simple energy-dependent result and discussing its implications for cosmological wavefunctions.

Contribution

It introduces a numerical method to evaluate the torus one-point diagram in 2D string cosmology and uncovers a surprisingly simple functional form.

Findings

Derived a simple expression for the torus one-point diagram as a function of energy.

Connected the diagram's contribution to cosmological wavefunctions at different orders.

Provided a numerical approach for complex conformal block decompositions.

Abstract

We calculate numerically the torus one-point string diagram in the two-dimensional string cosmology background by decomposing the one-point functions in $c=1$ and $c=25$ Liouville CFT into torus one-point Virasoro conformal blocks and integrating over the fundamental domain of the torus moduli space. We find a remarkably simple result as a function of the outgoing closed string energy. This torus one-point diagram is expected to contribute to the one-point cosmological wavefunction at order $g_s$, and to the four-point cosmological wavefunction at order $g_s^2$ through the disconnected product of the torus one-point diagram and the sphere three-point diagram.