Quantum Field Theory on Noncommutative Curved Space-times and Noncommutative Gravity

Abstract

In the noncommutative gauge-theoretical formulation of Langman and Szabo, apparently it appears that the torsion generated there is a generalized one i.e. it may containvector, axial vector and tensor components. However, when we transcribe the noncommutative gauge theory in terms of the Maxwell gauge theory using the Seiberg-Wittencorrespondence, we have noted that upto the first order in the noncommutative parameter, this effectively can be taken to induce a change in chiral anomaly and hence theassociated torsion should be an axial vector one. The noncommutative gauge symmetriesgive a very natural and explicit realizations of the mixing of space-time and internal symmetries which is a characteristic feature of the conventional gauge theory of gravity. Thegauge fields of the dimensionally reduced noncommutative Yang-Mills theory map ontoaWeitzenbӧck space time and a teleparallel theory of gravity arises as the zero curvature reduction of a Poincare gauge theory which induces an Einstein-Cartan space-timecharacterized by connections with both nonvanishing torsion and curvature. However, theteleparallelism equivalent of general relativity involves all the components of torsion. Thechiral anomaly in the Einstein-Cartan space U₄ is characterized by the topological invariants like Pontryagin density as well as the Nieh-Yan density when the latter term involvesthe length scale governed by the measure of noncommutativity of space points. It is shownthat we have discussed the equivalence of this formalism with noncommutative U(1) Yang Mills theory.