String field theory in the temporal gauge

Abstract

We construct the string field Hamiltonian for c = 1 -6 / m ( m + 1) string theory in the temporal gauge. In order to do so, we first examine the Schwinger-Dyson equations of the matrix chain models and propose the continuum version of them. The results of boundary conformal field theory are useful in making a connection between the discrete and continuum pictures. The W constraints are derived from the continuum Schwinger-Dyson equations. We also check that these equations are consistent with other known results about noncritical string theory. The string field Hamiltonian is easily obtained from the continuum Schwinger-Dyson equations. It looks similar to the Kaku-Kikkawa Hamiltonian and may readily be generalized to c > 1 cases.

The continuum limit of the Schwinger-Dyson equations of the one and two matrix model with finite loop length

“…At first sight W 1 (n) seems to become, in the continuum limit, the operator for the loop whose spin is up on almost all the points except for an infinitesimal down region. In [6] the dimension of such a loop was found to be L −11/3 . We first expect that…”

“…(41) leads to We now go on to the two matrix model. In order to obtain the continuum SchwingerDyson equations proposed in [6] to derive the W 3 constraints, we employ the loops with (i) only the same matrix, (ii) a different matrix inserted and (iii) two different matrices inserted next to each other. We therefore define the partition function as…”

“…Motivated by the transfer matrix formalism [3] string field theory in the temporal gauge was proposed for c = 0 [4] and c = 1 − 6/m(m + 1) [5] [6]. In [4] the corresponding Virasoro constraints were derived from the Schwinger-Dyson equation reproducing the results of the matrix model.…”

“…In this paper we take the continuum limit of the one and two matrix model step by step. We directly derive the Schwinger-Dyson equations in [4] and [6]. In this approach we do not expand correlation functions in terms of the length of the loop.…”

“…In [4] the corresponding Virasoro constraints were derived from the Schwinger-Dyson equation reproducing the results of the matrix model. In [6] the Schwinger-Dyson equations for c = 1−6/m(m+1) were proposed in the continuum theory, allowing for the derivation of the W m constraints. It was checked that the continuum limit of the dynamical triangulation becomes c = 0 string field theory [7].…”