Topological recursion for hyperbolic string field theory

TL;DR

This paper develops a recursion relation for hyperbolic string vertices inspired by Mirzakhani's work, enabling the iterative construction of higher order vertices in closed string field theory using systolic volumes and hyperbolic geometry.

Contribution

It introduces a modified Mirzakhani recursion for hyperbolic string vertices and applies it to hyperbolic string field theory, showing higher vertices are determined by the cubic vertex.

Findings

Recursion relation for systolic volumes in moduli spaces.

Higher order vertices are generated iteratively from the cubic vertex.

Application demonstrated in a scalar string field theory example.

Abstract

We derive an analog of Mirzakhani's recursion relation for hyperbolic string vertices and investigate its implications for closed string field theory. Central to our construction are systolic volumes: the Weil-Petersson volumes of regions in moduli spaces of Riemann surfaces whose elements have systoles $L \geq 0$. These volumes can be shown to satisfy a recursion relation through a modification of Mirzakhani's recursion as long as $L \leq 2 \sinh^{-1} 1$. Applying the pants decomposition of Riemann surfaces to off-shell string amplitudes, we promote this recursion to hyperbolic string field theory and demonstrate the higher order vertices are determined by the cubic vertex iteratively for any background. Such structure implies the solutions of closed string field theory obey a quadratic integral equation. We illustrate the utility of our approach in an example of a stubbed scalar theory.