Maximilian Kreuzer (Vienna):

Non-commutativity and effective actions in curved space-time

Abstract:

The non-commutative product of deformation quantization can be derived from string theory in a topological limit where the space-time metric is small as compared to the anti-symmetric B-field (the ancestor of the Poisson bi-vector). In terms of string physics the non-commutative product thus amounts to a summation of the leading B-field contributions to the effective action. In the (non-symplectic) Poisson case this interpretation is spoiled, however, by the absence of a canonical measure. For general string backgrounds associativity is lost, but the Born-Infeld action provides a canonical measure. We show that the concept of effective actions does not require associativity, but rather cyclicity, i.e. commutativity and associativity up to surface terms. Cyclicity thus implies a compatibility condition between the star product and the measure, which for Born-Infeld turns out to be equivalent to the generalized Maxwell equation for the gauge field on the D-brane. We show that cyclicity requires a gauge modification of the Kontsevich product at second derivative order in a derivative expansion and we discuss the physics related to these mathematical structures.

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