2 9 M ar 2 00 0 A geometrical action for dilaton gravity

We study the gravitational interaction involving the dilaton and the antisymmetrical Bμν fields that arises in the low-energy limit of string theory. It is shown that such interaction can be derived from a geometrical action principle, with the scalar of curvature of a non-Riemannian, but metric-compatible, connection as the lagrangian, and with a non-parallel volume-element. This action is contrasted with the recently proposed geometrical action for the 4-dimensional axi-dilaton gravity. PACS Numbers: 0450, 1125 Typeset using REVTEX 1 Recently, Derely and Tucker [1] have shown that the 4-dimensional axi-dilaton gravity equations can be obtained from a geometrical action principle, where the lagrangian is given by the scalar of curvature of a non-Riemannian connection. Besides of be an elegant way of understanding the axi-dilaton gravity, such result provides us with a geometrical interpretation for the massless states of the closed bosonic string. Since the equations for all the massless states of the closed string (namely the dilaton, the graviton, and the anti-symmetrical Bμν field) arise in the same footing from the requirement of conformal invariance up to one-loop order of the closed string in background fields [2,3], such common interpretation in geometrical terms to these states is natural and pertinent. The seeds of this geometrical interpretation can be encountered in the middle 70’s works of Scherk and Schwarz [4]. In the model presented in [1], the torsion tensor is proportional to the antisymmetrical field Hαβγ = ∂αBβγ + ∂γBαβ + ∂βBγα, in agreement with previous works on sigma models [5], and the non-metricity to the derivative of the dilaton field, as in the interpretation proposed in [4]. The purpose of this Letter is to present another geometrical formulation based in a metric-compatible connection, valid for any space-time dimension. The lagrangian will be the scalar of curvature of a metric-compatible connection, and the interpretation of the Hαμν field is compatible with the previous results. We remember that the usual interpretation of the torsion tensor Sαμν as being proportional to the totally anti-symmetrical tensor Hαμν determines only the traceless part of Sαμν . In our proposal, the dilaton field will determine the trace of the torsion tensor and the volume-element. This is the key point, the action principle will be formulated by using a non-parallel volume-element. Non-standard volumeelements have appeared sometimes in the literature, as for example in the characterization of half-flats solutions of Einstein equations [6] and in the description of field theory on RiemannCartan spacetimes [7]. The proposed formulation has two advantages in comparison with the proposed in [1]. First, the space-time geometry is more simpler because it involves a metric-compatible connection, and second, and more important, our geometrical formulation applies also in the presence of matter fields, elucidating the dilaton peculiar way of coupling. 2 We recall that the gravitational interaction involving the dilaton and the Bμν fields is given by the equations β = 4DμΦD Φ − 4DΦ − R + 1 12 HαβγH αβγ = 0, β μν = Rμν − 1 4 H λρ μ Hνλρ + 2DμDνΦ = 0, (1) β μν = DλH λ μν − 2 (DλΦ) Hλμν = 0, where Rμν , R, and Dμ are respectively the Ricci tensor, the scalar of curvature, and the covariant derivative in the 26-dimensional background manifold of metric gμν , all these quantities being calculated by using the Levi-Civita connection. The following conventions are adopted: sign(gμν) = (+,−,−.), R β ανμ = ∂αΓνμ + ΓαρΓνμ − (α ↔ ν), Rνμ = R α ανμ , and g = | det gμν |. One can check that the equations (1) follow from the minimization of the action: S = − ∫ dx √ ge ( R + 4∂μΦ∂ Φ − 1 12 HαβγH αβγ ) . (2) In spite of dilaton gravity is restricted by its origin in string theory to space-times of dimension N = 26, one can define stringy inspired models by assuming that (2) is valid also for other dimensions N > 2. As to the two dimensional case, the totally anti-symmetrical third-rank tensor Hαβγ vanishes identically, and we have the following action for the two dimensional dilaton gravity S = − ∫ dx √ ge (R + 4∂μΦ∂ Φ) . (3) We will discuss latter the incorporation of matter fields in (2) and (3). We wish to expound how the actions (2) and (3) can be understood as Hilbert-Einstein actions in a manifold endowed with a metric-compatible connection and with a special volume-element. To this end, let us introduce briefly some results on Riemann Cartan (RC) manifolds. A RC manifold is a N -dimensional differentiable manifold endowed with a metric tensor gαβ(x) and with a metric-compatible connection Γ μ αβ, which is non-symmetrical in its lower indices. From the anti-symmetric part of the connection one can define the torsion 3 tensor S γ αβ = 1 2 (