Abstract
By determining the relation between topological M-theory and the Chern-Simons actions for a gauge field constructed from the Lie algebra of either SL(2,R)x SL(2,R) or SL(2,C)x SL(2,C), depending on the sign of the space-time curvature, we show that the standard and exotic actions of 3-dimensional gravity can be recovered from topological M-theory. With this result, we provide a concrete realisation of a conjecture by Dijkgraaf et al. stating that the partition function of topological M-theory is equivalent to the partition function of a black hole in a related theory. We do this for the standard and exotic BTZ black holes in 3-dimensional gravity.
Highlights
One of the most useful tools for understanding gravitational interaction is three-dimensional gravity
By determining the relation between topological M-theory and the Chern-Simons actions for a gauge field constructed from the Lie algebra of either SLð2; RÞ × SLð2; RÞ or SLð2; CÞ × SLð2; CÞ, depending on the sign of the space-time curvature, we show that the standard and exotic actions of three-dimensional gravity can be recovered from topological M-theory
Given the different ways of writing down V7 either in terms of Vþ, V− or both, one could think that the result only applies to the extremal case, which turns out to be associated with the situation where we demand that the linear combinations of Vþ and V−, for instance Ist and Iex, preserve a given multiple of V7; but as we argue below, the partition function obtained from topological M-theory (TMT) correctly gives the BH partition function even away from the extremal case
Summary
One of the most useful tools for understanding gravitational interaction is three-dimensional gravity. Some of these form theories, including Chern-Simons (CS) three-dimensional (3d) gravity and the A and B models of topological strings can be unified in a seven-dimensional space-time, X, through the topological M-theory (TMT) proposed by Dijkgraaf et al [5] Essential in this theory is the volume form V constructed from an invariant p-form whose existence is characteristic of special holonomy manifolds. In order to give a concrete example of the relation between ZH and the black hole entropy we consider an extremal BTZ black hole, compute its volume form in terms of the 2 þ 1 dimensional standard and exotic actions for gravity, we obtain ZH, and we compare it to the norm of the wave function for the same black hole [9]. We review and formalize the derivation of the standard and exotic actions for 2 þ 1 gravity from TMT and construct the topological partition function.
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