Exact lattice bosonization of the double-scaled $c=1$ matrix model

Abstract

In a recent paper by two of us, it has been shown that entanglement entropy (EE) of matrix quantum mechanics (or, equivalently, of a non-relativistic fermion model) has a natural interpretation in terms of an exact bosonization of the fermion theory in terms of a lattice boson with lattice spacing $\sim 1/N$ (where $N$ is the rank of the matrix, equal to the number of fermions). The theory of this lattice boson is distinct from a latticized version of the "collective variable'' theory which is formulated in terms of the fermion density $ρ(λ)$ ("collective variable''); it is, in fact, more closely related to the Wigner distribution of the fermions. In this paper, we develop this connection further for the $c=1$ double scaled matrix model. Even though $N$ becomes infinite, the double scaled theory is described by a boson on a finite and non-uniform lattice spacing, of order $\sim g_s(x) l_s$ (where $g_s(x) \equiv g_s {\rm cosech}^2(x/l_s)$ is a position dependent string coupling). We explicitly derive the $c=1$ EE, as well as the one and two-point functions of fermion density from the lattice boson. The results are explicitly finite, unlike the collective variable theory which is shown to yield divergent answers. Our results suggest that while local observables such as the EE and $\langle ρ(λ) \rangle$, are characterized by a unique short distance scale $g_s(x) l_s$, the asymptotic observables such as the S-matrix at every genus are characterized by a short distance scale $l_s$. We show how this arises and discuss a possible interpretation in terms of D0 brane physics.