Bosonisation and BTZ Black Hole Microstates

When the boundary dynamics of $AdS_3$ gravity is governed by the collective field theory Hamiltonian proposed by Jevicki and Sakita, its asymptotic symmetry algebra becomes the centerless $U(1)$ Kac-Moody algebra. We quantize this system using the quantum bosonization of relativistic free fermions and relate these to the dynamical fields of $AdS_3$ gravity. This leads to a correspondence where different bulk configurations correspond to distinct states (particle-hole pair excitations) in the fermionic Hilbert space. This mapping allows us to construct BTZ black hole microstates, represented by Young diagrams of irreducible $U(\infty)$ representations. Notably, the logarithm of the microstate degeneracy exactly reproduces the classical entropy of the BTZ black hole.