$c=1$ strings as a matrix integral

We study the perturbative S S -matrix of the c=1 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) 𝗑 ( z ) = 2 2 cos ⁡ ( z ) , \mathsf{y}(z)=\sin(z) 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 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 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.