Dilaton gravity (with a Gauss-Bonnet term) derived from five-dimensional Chern-Simons gravity

We study the problem of boundary terms and boundary conditions for Chern-Simons gravity in five dimensions. We show that under reasonable boundary conditions one finds an effective field theory at the four-dimensional boundary described by dilaton gravity with a Gauss-Bonnet term. The coupling of matter is also discussed. The existing link between three-dimensional Chern-Simons theory and two-dimensional conformal field theory [1] is a remarkably powerful tool in the applications of Chern-Simons theory to 2+1 gravity. Carlip [2] has shown that the number of states of a conformal field theory lying at the horizon gives the correct Bekenstein-Hawking expression for the 2+1 black hole entropy. Also, the rich asymptotic structure of anti-de Sitter 2+1 gravity [3] can be analyzed in a simple way as a WZW model lying at the boundary [4–6]. Chern-Simons theories exist in all odd-dimensional spacetimes. It is therefore a natural question to ask whether they induce a ‘conformal’ field theory in a lower dimension. Steps in that direction were taken in [7] where a generalization to four dimensions of the WZW action and its associated current algebra was constructed. The WZW4 theory was also shown to be related to a Kahler-Chern-Simons theory. Recently [8], the precise connection between pure Chern-Simons theory and the four-dimensional current algebra found in [7] was established by studying the problem of boundary terms and boundary conditions in Chern-Simons theory for groups of the form G× U(1). 1 We consider in this paper the problem of boundary conditions and boundary terms in a particular five-dimensional Chern-Simons theory which is the analog of the Chern-Simons formulation of 2+1 gravity studied in [9,10]. We shall see that by imposing natural boundary conditions on the five-dimensional problem one obtains a four-dimensional field theory described by dilaton gravity with a Gauss-Bonnet term. 1. Five-dimensional Poincare Chern-Simons gravity. We consider a five-dimensional ChernSimons theory for the group ISO(3, 2) [or ISO(4, 1)] defined by the action ICS = 1 2 ∫ M ǫABCDER̃ AB ∧R̃∧e , (1) where R̃ = dW + WC∧W CB is the five-dimensional curvature two-form. The fields W and e can be collected together to form an ISO(3, 2) [ISO(4, 1)] connection, and (1) can be shown to be an ISO(3, 2) [ISO(4, 1)] Chern-Simons action [11–13]. Indeed (1) is explicitly invariant under SO(3, 2) [SO(4, 1)] rotations. It is also invariant, up to a boundary term, under the Abelian translation δe = ∇λ, δW = 0. Hence, (1) is invariant under ISO(3, 2) [ISO(4, 1)] provided one imposes some appropriate boundary conditions. For this reason, we call (1) the Poincare Chern-Simons action. The action (1) is a natural extension of the 2+1 Chern-Simons gravity action studied in [9,10] for zero cosmological constant. Our notations are the following: ηAB = diag(−1, 1, 1, 1, σ); Capital indices A,B, ... run over SO(3, 2) if σ = −1, and over SO(4, 1) if σ = 1. M is a five-dimensional manifold with the topology Σ×R and Σ has a boundary denoted by ∂Σ (see Fig. 1). We shall call B the “cylinder” formed by the direct product of R and ∂Σ, B = ∂Σ × R. (2) According to Fig. 1, the surface B is represented as the hypersurface x = const.. Local coordinates on M are denoted by x , M = [0, 1, 2, 3, 4]; local coordinates on B are denoted by x, μ = [0, 1, 2, 3]; and local coordinates of Σ, for all times, are denoted by x, i = [1, 2, 3, 4]. The SO(3, 2) [SO(4, 1)] covariant derivative is denoted by ∇.