By Douglas M.R., Nekrasov N.A.

This text experiences the generalization of box thought to space-time with noncommuting coordinates,starting with the fundamentals and overlaying lots of the energetic instructions of study. Such theories are nowknown to emerge from limits of M idea and string thought and to explain quantum corridor states. Inthe previous few years they've been studied intensively, and plenty of qualitatively new phenomena havebeen chanced on, on either the classical and the quantum point.

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**Sample text**

The string theory limits are still much better understood than the others, because the string is by far the most tractable fundamental object. One can use them to make a microscopic definition of certain branes, the Dirichlet branes. A Dirichlet brane is simply an allowed end point for open strings. The crucial generalization beyond the original definition of open string theory is that one allows Dirichlet boundary conditions for some of the world-sheet coordinates and this fixes the end point to live on a submanifold in space-time.

Rev. Mod. , Vol. 73, No. 4, October 2001 VI. MATHEMATICAL ASPECTS As we mentioned in the Introduction, noncommutative gauge theory was first clearly formulated by mathematicians to address questions in noncommutative geometry. Limitations on length would not permit more than the most cursory introduction to this subject here, and since so many introductions are already available, starting with the excellent Connes (1994), much of which is quite readable by physicists, and including Connes (1995, 2000a, 2000b), Douglas (1999), Gracia-Bondia et al.

Letting U i ϭ ␥ (g i ) for a set of generators of Zn , and taking AϭMatn (C), this leads to Eqs. : Noncommutative field theory j j j U Ϫ1 i X U i ϭX ϩ ␦ i 2 R i . These are solved by the connection Eq. (144), and substituting into Eq. (149) leads to MSYM on T n ϫR. This construction admits a natural generalization, namely, one can impose the relations U i U j ϭe i ij U j U i . Again as discussed in Sec. C, Eq. (143) now defines a twisted crossed product, and its solutions (146) are connections on the noncommutative torus.