Collective Field Description of Matrix Cosmologies

Abstract

We study the Das-Jevicki collective field description of arbitrary classical solutions in the c = 1 matrix model, which are believed to describe nontrivial spacetime backgrounds in 2d string theory. Our analysis naturally includes the case of a Fermi droplet cosmology: a finite size droplet of Fermi fluid, made up of a finite number of eigenvalues. We analyze properties of the coordinates in which the metric in the collective field theory is trivial, and comment on the form of the interaction terms in these coordinates.

TIME-DEPENDENT BACKGROUNDS OF TWO-DIMENSIONAL STRING THEORY FROM THE c = 1 MATRIX MODEL

“…In summary we considered three solutions of the Fermi sea with lightlike boundaries in equations (5), (10) and (14), and three solutions of the Fermi sea with spacelike boundaries in equations (18), (25) and (27), then obtained the tachyon fields corresponding to the fluctuation fields. It may be interesting to consider any time-dependent solutions such as x 2 −p 2 = 1+(x−p) 3 e 3t [13], which is corresponding to r = 3 in relation (4), and try to obtain tachyon field.…”

“…After the time revolution this circle changes to an moving ellipse. In here there are the condition λ > √ 2µ [13], which under assumption of µ ≪ 1 will satisfied. To write the equation (27) in the form of the equation (1) we note that the described universe by the equation (27) is in weak coupling at both early and late times.…”

“…Then by using above correspondence and the leg-pole transformations we will find tachyon fields. For this reason we will consider several time-dependent backgrounds with the lightlike and spacelike boundaries [12,13,14,15,16], and by using the same method of Ref. [11] will obtain ratio of the outgoing to the incoming tachyon fields.…”

Spacelike boundaries from thec=1matrix model

“…Here, we will focus on r = 2. Special coordinates, defined up to a conformal transformation as the coordinates for which the quadratic action for small fluctuations is of the form given in equation (2), can be found for non-static Fermi sea profiles as well [22,23]. We will refer to these special coordinates as Alexandrov coordinates, and use them to define the causal structure of the theory.…”

“…To relate this to the worldsheet theory, we need to multiply by the leg-pole transform in equation (23), obtaining…”