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FZZT branes and non-singlets of Matrix Quantum Mechanics

We explore the non-singlet sector of matrix quantum mechanics dual to $c=1$ Liouville theory. The non-singlets are obtained by adding $N_{f}\times N$ bi-fundamental fields in the gauged matrix quantum mechanics model as well as a one dimensional Chern-Simons term. The present model is associated with a spin-Calogero model in the presence of external magnetic field. In chiral variables, the low energy excitations-currents satisfy an ${SU(2N_{f})}_{\tilde{k}}$ Kǎc-Moody algebra at large $N$. We analyse the canonical partition function as well as two and four point correlation functions, discuss a Gross-Witten-Wadia phase transition at large $N, N_f$ and study different limits of the parameters that allow us to recover the matrix model of Kazakov-Kostov-Kutasov conjectured to describe a two dimensional black hole. The grand canonical partition function is a $τ$- function obeying discrete soliton equations. We finally conjecture a possible dynamical picture for the formation of a black hole in terms of condensation of long-strings in the strongly coupled region of the Liouville direction.