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132) We next need to postulate some analog of Eq. (1). Although one can define algebras without imposing commutation relations for each pair of variables, these will be much larger than the algebra of functions on S d . For dϭ2, there is a natural choice to make, Rev. Mod. , Vol. 73, No. 4, October 2001 ͓ x i ,x j ͔ ϭi ⑀ ijk x k , which preserves SO(3) symmetry. In fact, there is a very simple way to define such an algebra, called the fuzzy two-sphere, discussed in depth by Madore (1992), in Madore (1999), and many other works.
150) ϫT where the torus has We consider space-time R metric g ij ϭR 2 ␦ ij and constant Neveu-Schwarz B field B ij . In this case, the term in Eq. (150) involving B is an integral of a total derivative, and will be nonzero either because of the nontrivial topology of the torus or in the presence of a world-sheet boundary. Contact with M(atrix) theory suggests that we study the physics of D0-branes in this theory. One way to proceed (Douglas and Hull, 1998) is to apply a T duality along one axis (say x 1 ) of the torus, which one can show turns the D0-branes into D1-branes extending along the x 1 axis, and the T 2 into another T 2 with Bϭ0 and metric defined by the identifications (0,0)ϳ(l s ,0)ϳ( ϭB, ⑀ ϭV/l s ).
These branes have tensions (energy per unit volume) c 2 /l p3 and c 5 /l p6 as is clear by dimensional analysis; arguments using supersymmetry and charge quantization determine the constants c n . This is not a constructive definition and indeed one might doubt that such a theory exists at all, were it not for its connections to superstring theory. The simplest connection is to consider a compactification of the theory on R9,1ϫS 1 with a flat metric, circumference 2 R for the S 1 , and no other background fields.