Integrable Models of (1+1)-Dimensional Dilaton Gravity Coupled to Scalar Matter

Abstract

We describe a class of explicitly integrable models of (1+1)-dimensional dilaton gravity coupled to scalar fields in sufficient detail. The equations of motion of these models reduce to systems of Liouville equations with energy and momentum constraints. We construct the general solution of the equations and constraints in terms of chiral moduli fields explicitly and briefly discuss some extensions of the basic integrable model. These models can be related to higher-dimensional supergravity theories, but we mostly consider them independently of such interpretations. We also briefly review other integrable models of two-dimensional dilaton gravity.

Dimensional reduction of gravity and relation between static states, cosmologies, and waves

“…1 This theory can be integrable in two main cases: (1) when V ≡ 0 and (2) when all Z mn are constants independent of the fields, while V is a special potential (see [16], [17]). If Z mn has only negative eigenvalues, we can transform the ψ fields in the latter case such that…”

“…The last restriction may be waved if the so-called phantom fields are introduced, but we do not consider such exotic models here. The usually present dilaton kinetic term W (∇ϕ) 2 can be eliminated from the Lagrangian using a Weyl transformation, and we omit it here (see [14], [16]). 2 If theory (1) is obtained by dimensional reductions of a higher-dimensional supergravity theory on tori, then the coupling constants gn are negative.…”

“…The crucial fact that allowed solving the constraints in [13]- [16] is that they are equivalent to constraints (27) and that the pseudo-orthogonality conditions imply the identity N n=1 γ n = 0.…”

“…Substituting (34) and (35) in (29), we find q n , after which the original fields ψ n , φ, and F can be written in terms of a n and b n using the orthogonality relations (see [13]- [16]; these expressions are reproduced in the next section). As shown in [14], [16], the space of the moduli (µ 1 , .…”

“…, ν n ) can be further reduced by the two coordinate gauge-fixing conditions. In [14], [16], introducing new coordinates (U, V ) by setting…”