$c=1$ strings as a matrix integral

TL;DR

This paper demonstrates a novel matrix integral formulation of the $c=1$ string's perturbative S-matrix, establishing a triality with matrix quantum mechanics and the worldsheet description, and proves unitarity and recursion relations.

Contribution

It introduces a double-scaled matrix integral based on spectral curves for the $c=1$ string, connecting it to intersection theory and matrix quantum mechanics.

Findings

Derived closed-form Feynman rules as intersection numbers.

Proved perturbative unitarity from intersection theory.

Established agreement with matrix quantum mechanics results.

Abstract

We study the perturbative $S$-matrix of the $c=1$ string and show that it admits a description in terms of a double-scaled (0+0)-dimensional matrix integral based on the spectral curve $\mathsf{x}(z) = 2\sqrt{2}\cos(z)$, $\mathsf{y}(z)=\sin(z)$. Combined with the famous duality to matrix quantum mechanics, this establishes a triality between three formulations of the theory: the worldsheet description, matrix quantum mechanics, and a matrix integral. Starting from the intersection number expressions for the complex Liouville string, we derive closed-form Feynman rule expressions for the $c=1$ amplitudes as intersection numbers on the moduli space of Riemann surfaces. The intersection theory naturally computes amplitudes corresponding to a discretized target space where momentum is conserved only modulo an integer. The physical $S$-matrix elements are recovered by restriction to the first `Brillouin zone' and analytic continuation to Lorentzian kinematics. We prove that these amplitudes satisfy perturbative spacetime unitarity directly from the intersection theory expressions, and show that they satisfy a Mirzakhani-type recursion relation. We show detailed agreement with the known matrix quantum mechanics results, providing strong evidence for the triality.