[Paper Review] S-brane solutions with (anti-)self-dual parallel charge density form on a Ricci-flat submanifold

H. Dehnen, В. Д. Иващук+1 more|ArXiv.org|Dec 24, 2006

Black Holes and Theoretical Physics23 references3 citations

TL;DR

This paper presents a new family of cosmological S-brane solutions in D=4m+1 dimensions (e.g., D=5,9,13) with a (2m)-form charge density Q that is parallel and (anti-)self-dual on a Ricci-flat submanifold N of dimension 4m. The solutions are constructed using a σ-model reduction and involve scalar fields and a (p+2)-form field with specific dilatonic couplings, yielding exact solutions with accelerated expansion in extra factor-spaces when generalized to a chain of Ricci-flat manifolds. The key contribution is the explicit construction of such solutions on Kähler, hyper-Kähler, or Spin(7) holonomy manifolds.

ABSTRACT

A D-dimensional cosmological model with several scalar fields and antisymmetric (p+2)-form is considered. For dimensions D = 4m+1 = 5, 9, 13, ... and p = 2m-1 = 1, 3, 5, ... we obtain a family of new cosmological type solutions with 4m-dimensional oriented Ricci-flat submanifold N of Euclidean signature. These solutions are characterized by a self-dual or anti-self-dual parallel charge density form Q of rank 2m defined on N. The (sub)manifold N may be chosen to be K\ddot{a}hler, or hyper-K\ddot{a}hler one, or 8-dimensional manifold of Spin(7) holonomy. The generalization of solutions to a chain of extra marginal) Ricci-flat factor-spaces is also presented. Solutions with accelerated expansion of extra factor-spaces are singled out. Certain examples of new solutions for IIA supergravity and for a chain of B_D-models in dimensions D = 14, 15, ... are considered.

Motivation & Objective

• To generalize composite electric S-brane solutions with maximal brane configurations in D=4m+1 dimensions to include (anti-)self-dual parallel charge density forms on Ricci-flat submanifolds.
• To construct exact cosmological-type solutions with a 4m-dimensional Ricci-flat factor-space N equipped with a non-zero parallel (2m)-form Q satisfying (anti-)self-duality conditions.
• To extend these solutions to a chain of additional Ricci-flat factor-spaces and identify configurations with accelerated expansion of extra dimensions.
• To provide explicit examples in IIA supergravity and B_D-models in higher dimensions (D=14,15,...), demonstrating the viability of the construction.

Proposed method

• The model is based on a D-dimensional action with scalar fields φ^a, a (p+2)-form field strength F, and dilatonic couplings λ_a, leading to Einstein, Klein-Fock-Gordon, and Maxwell-type field equations.
• The ansatz assumes a warped product metric with a time-dependent scale factor and a Ricci-flat internal manifold N of dimension 4m, on which a (2m)-form Q is parallel and (anti-)self-dual.
• The field equations reduce to a σ-model action with a lapse function and a potential term involving the exponential of the dilaton fields and the squared norm of the form Q.
• The generalized Maxwell equation for the form A is solved by setting dA = Φ'(u) du ∧ ω, where ω is a harmonic (anti-)self-dual form on N, leading to a conserved charge Q = qω.
• The system is reduced to a one-dimensional mechanical model with a zero-energy constraint, solvable via exact integration of the Lagrangian with a potential proportional to exp(2U(x)) and a quadratic form in the charge density.
• Solutions are derived using a time-gauge condition (γ = γ₀) and explicit integration yields x(u) = -U/(U,U) ln|f(u)| + cu + c̄, with f(u) depending on the sign of (U,U) and the energy parameter C.

Experimental results

Research questions

• RQ1 Can S-brane solutions with (anti-)self-dual parallel charge density forms be constructed on Ricci-flat submanifolds in D=4m+1 dimensions?
• RQ2 What are the conditions under which such solutions yield accelerated expansion in extra factor-spaces when extended to a chain of Ricci-flat manifolds?
• RQ3 How do the (anti-)self-duality and parallelism of the charge density form Q constrain the geometry and topology of the internal manifold N?
• RQ4 What are the explicit solutions in IIA supergravity and B_D-models in higher dimensions (D=14,15,...) that realize these configurations?
• RQ5 How do the dynamical constraints from the non-diagonal Hilbert-Einstein equations lead to (anti-)self-dual relations on the charge density, distinct from topological K-theory relations?

Key findings

• The paper constructs exact cosmological S-brane solutions in D=5,9,13,... dimensions with a 4m-dimensional Ricci-flat submanifold N supporting a non-zero, parallel, (anti-)self-dual (2m)-form charge density Q.
• The solutions are valid for N being Kähler (holonomy SU(2m)), hyper-Kähler (holonomy Sp(m)), or 8-dimensional Spin(7) holonomy manifolds, all of which admit such forms.
• When generalized to a chain of extra Ricci-flat factor-spaces, the solutions exhibit accelerated expansion in the extra dimensions, as determined by the sign and magnitude of the energy parameter C in the solution.
• The construction yields explicit solutions for IIA supergravity and B_D-models in D=14,15,..., demonstrating the applicability of the method beyond the minimal model.
• The (anti-)self-duality of Q arises as a dynamical solution to the non-diagonal part of the Einstein equations, not from topological constraints, but from quadratic constraints on charge densities.
• The solution for the scalar and moduli fields is given by x(u) = -U/(U,U) ln|f(u)| + cu + c̄, with f(u) depending on the sign of (U,U), and the form Q is fixed via Q = qω with ω harmonic and (anti-)self-dual.