Into the bulk: Reconstructing spacetime from thec=1matrix model

Abstract

We write down exact solutions in the collective field theory of the c=1 matrix model and in dilaton-gravity coupled to a massless scalar. Using the known correspondence between these two theories at the null boundaries of spacetime, we make a connection between scalar fields in these two theories in the bulk of spacetime. In the process, we gain insight into how a theory containing gravity can be equivalent to one without gravity. We analyze a simple time-dependent background as an example.

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

“…So fluctuations come from infinity to finite x and again come back to the infinity. In that case under assumption µ > 0 we have [11],…”

“…In the first part of this section, we have brief review to the c = 1 matrix model and Liouville string theory of coupled dilaton-gravity to the massless scalar field [11], then in the second part, we describe relation between two theories. As we know the c = 1 matrix quantum mechanics describes a system of free fermion with the harmonic oscillator potential.…”

“…In Ref. [11] such a correspondence considered, where exact solutions of the c = 1 matrix model collective field theory and coupled dilaton-gravity to a massless scalar field (tachyon field) have written and the relation between scalar field at the lightlike boundary of space-time in both theory have shown. Also they considered an simple example of the time-dependent background with the lightlike boundary.…”

“…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. In the section 2 we review the correspondence between the c = 1 matrix model and coupled dilaton-gravity to the massless scalar field and also consider the leg-pole transformations which is relation between the tachyon fields in the string theory and fluctuations in the collective field theory [11].…”

“…[11] will obtain ratio of the outgoing to the incoming tachyon fields. In the section 2 we review the correspondence between the c = 1 matrix model and coupled dilaton-gravity to the massless scalar field and also consider the leg-pole transformations which is relation between the tachyon fields in the string theory and fluctuations in the collective field theory [11]. Then in the section 3 we consider solutions with the lightlike boundaries, so after calculation of fluctuations we can obtain the tachyon fields.…”