Ivancevic_Applied-Diff-Geom

1232 Applied Differential Geometry: A Modern Introductionterms is not obvious. For example another way to absorb the θ’s would havegiven the familiar term R 2 F 2g−2 where F is the graviphoton field. However,such terms do not contribute in the black hole background. It would benice to find a simple way to argue why these terms do not contribute andthat we are left with this simple absorption of the θ integrals.6.8 Application: Advanced Geometry and Topology ofString Theory6.8.1 String Theory and Noncommutative GeometryThe idea that the space–time coordinates do not commute is quite old(see [Snyder (1947a); Snyder (1947b)]). It has been studied by many authorsboth from a mathematical and a physical perspective. The theory ofoperator algebras has been suggested as a framework for physics in noncommutativespace–time (see [Connes (1994)] for an exposition of the philosophy).Yang-Mills (YM) theory on a noncommutative torus has beenproposed as an example; though this example at first sight appears to beneither covariant nor causal, it has proved to arise in string theory in adefinite limit [Connes et. al. (1997)] with the non–covariance arising fromthe expectation value of a background field. This analysis involved toroidalcompactification, in the limit of small volume, with fixed and generic valuesof the world–sheet theta angles. This limit is fairly natural in the context ofthe matrix model of M−theory [Matacz (2002)], and the original discussionwas made in this context. Part of the beauty of this analysis in [Conneset. al. (1997)] was that T −duality acts within the noncommutative YMframework, rather than mixing the modes of noncommutative YM theorywith string winding states and other stringy excitations. This makes theframework of noncommutative YM theory seem very powerful.Seiberg and Witten in [Seiberg and Witten (1999)], reexamined thequantization of open strings ending on D−branes in the presence of aB−field. They have showed that noncommutative YM theory is valid forsome purposes in the presence of any nonzero constant B−field, and thatthere is a systematic and efficient description of the physics in terms ofnoncommutative YM theory when B is large. The limit of a torus of smallvolume with fixed theta angle (that is, fixed periods of B) is an examplewith large B, but it is also possible to have large B on R n and therebymake contact with the application of noncommutative YM to instantonson R 4 . An important element in their analysis is a distinction between two