Cosmological Constant as an S-duality Goldstone Mode

For vacua in type IIB string moduli space parameterized by the coupling constant, we propose a scheme for interpretation of the cosmological term in type IIB superstrings as a Goldstone mode from the spontaneous symmetry breaking of SL(2, Z) to a Fuchsian subgroup of degree one, Γh. The moduli space Σh = Γh\H has a radial direction with a massive Goldstone mode, and the contribution to the positive cosmological term in the type IIB supergravity action is classified by the eigenvalues of the hyperbolic Laplacian that acts on the moduli space. The condition for the cancellation of the positive and negative cosmological constants suggests that the number of non-compact dimensions in physical space-time is four and predicts a moduli scalar particle that is expected to generate the observable cosmological constant. Classically, the gravitational part of the Einstein-Hilbert action of general relativity absent of matter is for the space-time metric gμν with space-time indices μ and ν, and its Riemann scalar R as (1) SEH(gμν) = ∫ √−g(R+ Λ). The term √−gΛ in the Lagrangian density would appear in the Einstein equation as Λgμν . The observed value of the cosmological constant, Λ, has not been successfully determined theoretically. Its value is usually derived from the vacuum energy and its sign is negative. In the supersymmetric universe the vacuum energy is zero, but supersymmetry breaks at the GUT scale producing a non-zero cosmological constant. However, the observed cosmological constant is negative and has a value that is 123 orders of magnitude smaller than the prediction based on the four dimensional vacuum energy. [1, 2, 3] In this article we propose a scheme for resolution of this problem by adding a contribution from a positive cosmological constant. We divide the cosmological constant into a positive one, Λ ≥ 0, and a negative one, Λ ≥ 0, derived from the vacuum energy. Then Λ = Λ+−Λ−. In this scheme we reformulate the perturbative expansion of the cosmological term as the source of the Einstein equation δgμν = Λημν for Minkowski metric ημν from the aspect of type IIB string S-duality[4, 6, 5, 7] and its vacuum moduli space M, parameterized by the coupling constant, acted upon by the discrete group Γ = SL(2, Z). This is based on the interpretation of the cosmological term as a Goldstone mode in the spontaneous symmetry breaking from Γ to a Fuchsian subgroup of degree one, Γh. The subscript h labels the genus of the Riemann surface Σh = Γh\H for Poincare upper half-plane H. The orbits of degenerate vacua on the moduli spaceM under the action of Γh are closed geodesics Date: February 26, 2008.