Continuum Annulus Amplitude from the Two-Matrix Model

TL;DR

This paper derives explicit continuum annulus amplitudes from the two-matrix model for unitary series, connecting discrete matrix models with continuum quantum gravity and discussing potential links to Liouville gravity.

Contribution

It provides an explicit expression for continuum annulus amplitudes from the two-matrix model in the unitary series, bridging discrete and continuum approaches.

Findings

Derived integral representation of annulus amplitudes in the vanishing cosmological constant limit.

Reproduced amplitudes using continuum quantum mechanics of loops and matter system.

Commented on possible relation to the unconventional branch of Liouville gravity.

Abstract

An explicit expression for continuum annulus amplitudes having boundary lengths $\ell_{1}$ and $\ell_{2}$ is obtained from the two-matrix model for the case of the unitary series; $(p,q) = (m + 1, m)$. In the limit of vanishing cosmological constant, we find an integral representation of these amplitudes which is reproduced, for the cases of the $m = 2~(c=0)$ and the $m \rightarrow \infty~(c=1)$, by a continuum approach consisting of quantum mechanics of loops and a matter system integrated over the modular parameter of the annulus. We comment on a possible relation to the unconventional branch of the Liouville gravity.