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Global geometry of the supersymmetric AdS 3 / CFT 2 correspondence in M-theory

2007; American Physical Society; Volume: 76; Issue: 4 Linguagem: Inglês

10.1103/physrevd.76.046007

1550-7998

Pau Figueras, Oisín A. P. Mac Conamhna, Eoin Ó Colgáin,

Noncommutative and Quantum Gravity Theories

We study the global geometry of a general class of spacetimes of relevance to the supersymmetric three-dimensional anti--de Sitter space/two-dimensional conformal field theory (${\mathrm{AdS}}_{3}/{\mathrm{CFT}}_{2}$) correspondence in 11-dimensional supergravity. Specifically, we study spacetimes admitting a globally defined ${\mathbb{R}}^{1,1}$ frame, a globally defined frame bundle with structure group contained in $\mathrm{Spin}(7)$, and an ${\mathrm{AdS}}_{3}$ event horizon or conformal boundary. We show how the global frame bundle may be canonically realized by globally defined null sections of the spin bundle, which we use to truncate 11-dimensional supergravity to a gravitational theory of a frame with structure group $\mathrm{Spin}(7)$, $SU(4)$, or $Sp(2)$. By imposing an ${\mathrm{AdS}}_{3}$ boundary condition on the truncated supergravity equations, we define the geometry of all ${\mathrm{AdS}}_{3}$ horizons or boundaries which can be obtained from solutions of these truncations. In the most generic case we study, we reproduce the most general conditions for an ${\mathrm{AdS}}_{3}$ manifold in M-theory to admit a Killing spinor. As a consistency check on our definitions of AdS geometries we verify that they are satisfied by known gauged supergravity ${\mathrm{AdS}}_{3}$ solutions. We discuss future applications of our results.