The quantum mechanics of one $N\times N$ Hermitian matrix in the infinite $N$ limit is well known to define a two-dimensional string theory when the quartic coupling constant $g$ is tuned to a special negative value $g_*$. We study the thermodynamics of this Matrix Quantum Mechanics (MQM), treating the SU$(N)$ symmetry as a global symmetry. Crucially, the non-singlets, i.e. states in non-trivial representations of SU$(N)$, implement the effects of vortices on the discretized worldsheets. This random lattice model has the low-temperature $c=1$ and high-temperature $c=0$ phases, with the Berezinskii-Kosterlitz-Thouless (BKT) transition separating them. We map out the phase diagram in the coupling--temperature plane, which is separated by the critical stability curve $T_c(g)$ extending from $g_*< g <0$. We study this curve using the non-perturbative matrix bootstrap and low- and high-temperature perturbation theory, which explain the thermal instability. At low temperatures, thermally excited non-singlets backreact on the Fermi sea of matrix eigenvalues, causing it to spill over. At high temperatures, the theory is described by an effective matrix integral obtained by dimensional reduction on the thermal circle; the eigenvalue distribution of the zero mode becomes unstable at the critical coupling. We conjecture that the BKT transition occurs at a special point on the critical curve and estimate its position. Finally, we switch to the regular square lattice and study the MQM-inspired Villain model, where the link factor is the one-dimensional propagator on a circle. Using Monte Carlo simulations, we map out the BKT transition in this model.
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