Hyperbolic three-string vertex

TL;DR

This paper develops mathematical tools for computing off-shell string amplitudes using hyperbolic string vertices, connecting Liouville's equation with Fuchsian monodromy problems, and exploring their limits and conservation laws.

Contribution

It introduces a method to construct local coordinates around punctures for hyperbolic three-string vertices, linking boundary value problems with monodromy, and analyzes their limits and conservation laws.

Findings

Constructed local coordinates for hyperbolic three-string vertices.

Derived conservation laws for these vertices.

Performed sample computations demonstrating the framework.

Abstract

We begin developing tools to compute off-shell string amplitudes with the recently proposed hyperbolic string vertices of Costello and Zwiebach. Exploiting the relation between a boundary value problem for Liouville's equation and a monodromy problem for a Fuchsian equation, we construct the local coordinates around the punctures for the generalized hyperbolic three-string vertex and investigate their various limits. This vertex corresponds to the general pants diagram with three boundary geodesics of unequal lengths. We derive the conservation laws associated with such vertex and perform sample computations. We note the relevance of our construction to the calculations of the higher-order string vertices using the pants decomposition of hyperbolic Riemann surfaces.