Exact Seiberg-Witten map, induced gravity and topological invariants in non-commutative field theories

We revisit the exact Seiberg-Witten (SW) map on Dirac-Born-Infeld actions, making a connection with the deformation quantization scheme. The picture on field dependent induced gravity from non-commutativity becomes more transparent in the context of deformation quantization. We also find an exact SW map for an adjoint scalar field, consistent with that deduced from RR couplings of unstable non-BPS D-branes. The dual description via the exact SW map can again be interpreted as the ordinary field theory coupling to gravity induced by gauge fields. Using the exact SW maps, we further discuss several aspects of topological invariants in non-commutative (NC) gauge theory. Especially, it is shown that the K-theory class on NC instantons is mapped to the usual second Chern class via exact SW map and it leads to an exact SW map between commutative and NC Chern-Simons terms.