Advances in Theoretical and Mathematical Physics

Volume 9 (2005)

Number 4

Topological M-theory as unification of form theories of gravity

Pages: 603 – 665



Robbert Dijkgraaf

Sergei Gukov

Andrew Neitzke

Cumrun Vafa


We introduce a notion of topological M-theory and argue that it provides a unification of form theories of gravity in various dimensions. Its classical solutions involve $G_2$ holonomy metrics on 7-manifolds, obtained from a topological action for a 3-form gauge field introduced by Hitchin. We show that by reductions of this 7-dimensional theory, one can classically obtain 6-dimensional topological A and B models, the self-dual sector of loop quantum gravity in four dimensions, and Chern--Simons gravity in 3 dimensions. We also find that the 7-dimensional M-theory perspective sheds some light on the fact that the topological string partition function is a wavefunction, as well as on S-duality between the A and B models. The degrees of freedom of the A and B models appear as conjugate variables in the 7-dimensional theory. Finally, from the topological M-theory perspective, we find hints of an intriguing holographic link between non-supersymmetric Yang--Mills in four dimensions and A model topological strings on twistor space.

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