Celebratio Mathematica — Baum — Complete Bibliography

Complete Bibliography

Works connected to Greg Brian Friedman

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P. Baum: “ $K$ -homology and D-branes,” pp. 81–94 in Superstrings, geometry, topology, and $C^{*}$ -algebras (Fort Worth, TX, 18–22 May 2009). Edited by R. S. Doran, G. Friedman, and J. Rosenberg. Proceedings of Symposia in Pure Mathematics 81. American Mathematical Society (Providence, RI), 2010. MR 2681759 Zbl 1210.81079 incollection

$ K $-homology is the dual theory to $ K $-theory. In algebraic geometry [14], [7], the $ K $-homology of a (possibly singular) projective variety $ X $ is the Grothendieck group of coherent algebraic sheaves on $ X $. In topology there are three ways to define $ K $-homology. First $ K $-homology is the homology theory determined by the Bott spectrum. Second, $ K $-homology is the group of geometric $ K $-cycles introduced by Baum–Douglas [6]. Third, using funtional analysis, $ K $-homology is the group of abstract elliptic operators as in the work of M. .F. Atiyah [1], Brown–Douglas–Filmore [15], and G. Kasparov [21].

The D-branes of string theory [31] are twisted geometric $ K $-cycles which are endowed with some additional structure. The charge of a D-brane is the element in the twisted $ K $-homology of spacetime determined by the underlying twisted $ K $-cycle of the D-brane. Essentially, the Baum–Douglas theory [6] was rediscovered in terms of constraints on open strings. The aim of this expository note is to briefly describe this development.

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