By A.N. Parshin (editor), I.R. Shafarevich (editor), I. Rivin, V.S. Kulikov, P.F. Kurchanov, V.V. Shokurov

This two-part EMS quantity offers a succinct precis of complicated algebraic geometry, coupled with a lucid advent to the hot paintings at the interactions among the classical quarter of the geometry of advanced algebraic curves and their Jacobian types. a great significant other to the older classics at the topic.

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Using this relation, the worldvolume theory of Dbranes can be regarded as the worldvolume theory of the infinitely many lower dimensional branes. In the description in terms of the lower dimensional branes, some of the worldvolume coordinates become noncommutative. Actually this noncommutative theory can be regarded as noncommutative Yang-Mills theory. Therefore the worldvolume theory of D-branes have two equivalent descriptions, namely the usual static gauge description using ordinary Yang-Mills theory and the noncommutative description using noncommutative Yang-Mills theory.

In order to obtain the wD-1,l-1(X, y), first we introduce aD,l(y) as, aD,l(y) = LwD,l(x,y) x 1 = TIdXJLl ... L1 ... 7) where aJLI+1"',JLD(y) = I:xWJLI+l"',JLD(X,y). 4) one obtains, dxwD-1,l-1(x, y) = -d;Ox,yaD,l(x) + dxd;19 D - 1,l(x, y). 9) The final expression for wD-1,l-1(x, y) will be given soon after the following lemma. 10) where aD-1,l-1(x) is given by aD-1,l-1(x) 1 (l - I)! dX/-L2 ... dX JLI dYJL2 .. 1 a JLI +1,",JLD (x )dXJLI +1 ... 4) one can obtain Lx d;wD,I(x, y) = 0 . Changing the order of Lx and d; we get an important property of aD,I(y), d~aD,I(y) = O.

We will comment on the relation between our results and theirs. In section 2 we explain how Dp-branes can be expressed as a configuration of infinitely many D(p - 2r )-branes. In section 3 we study the worldvolume theory of the Dp-branes regarding it as a configuration of the D(p - 2r )-branes. Section 4 is devoted to discussions. This note is based on a talk presented at the "Workshop on Noncommutative Differential Geometry and its Application to Physics", Shonan-Kokusaimura, Japan, May31-June 4, 1999.