New coset matrix for D = 6 self-dual supergravity

Gérard Clément,Dmitri V Gal’Tsov

doi:10.1007/jhep03(2013)143

Abstract

Abstract Toroidal reduction of minimal six-dimensional supergravity, minimal five-dimensional supergravity and four-dimensional Einstein-Maxwell gravity to three dimensions gives rise to a sequence of cosets O(4, 3)/(O(4) × O(3)) ⊃ G 2(2)/(SU(2) × SU(2)) ⊃ SU(2, 1)/S(U(2) × U(1)) which are invariant subspaces of each other. The known matrix representations of these cosets, however, are not suitable to realize these embeddings which could be useful for solution generation. We construct a new representation of the largest coset in terms of 7 × 7 real symmetric matrices and show how to select invariant subspaces corresponding to lower cosets by algebraic constraints. The new matrix representative may be also directly applied to minimal five-dimensional supergravity. Due to full O(4, 3) covariance it is simpler than the one derived by us previously for the coset G 2(2)/(SU(2) × SU(2)).

Highlights

The purpose of the present paper is to construct a new matrix representative of the coset O(4, 3)/(O(4) × O(3)) which allows for simple truncation to subspaces corresponding to MSG5 and EM4 theories
We construct a new representation of the largest coset in terms of 7 × 7 real symmetric matrices and show how to select invariant subspaces corresponding to lower cosets by algebraic constraints
The new matrix representative may be directly applied to minimal five-dimensional supergravity

Summary

Invariant subspaces

The new matrix (4.9) parameterizes the twelve-dimensional coset space of MSG6 theory. These may be selected by purely algebraic constraints on the potentials To find these constraints one has to consider dimensional reductions and consistent truncations which relate these theories to MSG6. In view of the simplicity of the matrix representation (4.9) compared to that previously known for MSG5, this procedure might be easier to implement than direct generation by G2(2) transformations The generators preserving both five-dimensional Myers-Perry (or black string) asymptotics and G2 truncation are. To identify the constraints selecting the SU(2, 1)/S(U(2) × U(1)) subspace of the G2 coset, one must first compactify MSG5 on a circle [5], since this subcoset corresponds to fourdimensional Einstein-Maxwell theory.

Outlook

Findings

B Matrix representative