Noncommutative topological theories of gravity

Abstract

The possibility of noncommutative topological gravity arising in the same manner as Yang-Mills theory is explored. We use the Seiberg-Witten map to construct such a theory based on a SL(2,C) complex connection, from which the Euler characteristic and the signature invariant are obtained. This gives us a way towards the description of noncommutative gravitational instantons as well as noncommutative local gravitational anomalies. * Electronic address: compean@fis.cinvestav.mx † Electronic address: octavio@ifug3.ugto.mx; Permanent adress:

Noncommutative deformation of four-dimensional Einstein gravity

“…Thus we look for the simplest unitary groups which contain SO (1,4) and SO (1,3) with the aim to deform their algebra with the ⋆-product. The appropriate groups turn out to be U ⋆ (2, 2) and U ⋆ (1, 1) × U ⋆ (1, 1) respectively [2].…”

“…We still have to evaluate b (1) µ and e a+(1) µ in terms of e a+ µ . We do this via the SeibergWitten map: in addition to the relations in (36) and (37) we have correspondingly…”

Non-commutativity in gravity, topological gravity and cosmology

“…This results can be applied to other gravitational systems. These kind of ideas are being applied in connection to BF theories [19], topological invariants, and black hole thermodynamics [20], with the hope of better understanding the quantum behavior.…”

“…However, as mentioned at the end of the preceding section, this term can be obtained from Eq. (19). Thus, in general we will consider as the noncommutative topological action of gravity, the SL(2, C) invariant action,…”

“…In the case of a Riemannian manifold X, the signature and the Euler topological invariants of X, are the real and imaginary parts of (19) …”