Gravitational coset models

Paul P Cook,Michael Fleming

doi:10.1007/jhep07(2014)115

Abstract

The algebra A − 3 + + + dimensionally reduces to the E D−1 symmetry algebra of (12 − D)-dimensional supergravity. An infinite set of five-dimensional gravitational objects embedded in D-dimensions is constructed by identifying the null geodesic motion on cosets embedded in the generalised Kac-Moody algebra A − 3 + + + . By analogy with super-gravity these are bound states of dual gravitons. The metric interpolates continuously between exotic gravitational solutions generated by the action of an affine sub-group. We investigate mixed-symmetry fields in the brane sigma model, identify actions for the full interpolating bound state and investigate the dualisation of the bound state to a solution of the Einstein-Hilbert action via the Hodge dual on multiforms. We conclude that the Hodge dual is insufficient to reconstruct solutions to the Einstein-Hilbert action from mixed-symmetry tensors.

Highlights

To simplify calculation the infinite dimensional sub-algebra may be consistently truncated to a finite dimensional algebra
The work of [7] describes a method that associates a one-dimensional solution of M-theory and string theory to a null geodesic motion on cosets SL(n, R)/H where the algebra of H is the fixed point set under some generalised involution
In this paper we have investigated the gravitational solutions associated to real roots of A+D+−+3 algebras

Summary

D-2 D-1

Dualisation technique will introduce compensating matrices and modify the metric and will not be considered in the present paper. By drawing all the tableaux formed of L(D−2) boxes and projecting out all those whose associated root length squared is greater than two one arrives at a close approximation of the algebraic content of A+D+−+3 It is, perhaps, simpler to find one Young tableau at each level whose length squared is two and construct the other Young tableaux at level L by moving the Young tableaux boxes between columns - a transformation which has a simple impact on the root length squared. The highest weight generator corresponding to the deleted node defining the level has a Young tableau containing (D − 2) boxes.

The coset model for gravitational solutions

The generalised involution Ω

Solutions as null-geodesics on cosets

The supergravity dictionary and multiforms

Discussion

C The Ricci scalar in the sigma model