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$$K$$-homology and D-branes. (English) Zbl 1210.81079
Doran, Robert S. (ed.) et al., Superstrings, geometry, topology, and $$C^*$$-algebras. Collected papers based on the presentations at the NDF-CBMS regional conference on mathematics on topology, $$C^*$$-algebras, and string duality, Fort Worth, TX, USA, May 18–22, 2009. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4887-6/hbk). Proceedings of Symposia in Pure Mathematics 81, 81-94 (2010).
From the introduction: The D-branes of string theory [E. Witten, J. High Energy Phys. 1998, No. 12, Paper No. 19, 41 p., electronic only (1998; Zbl 0959.81070)] 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 space-time determined by the underlying twisted $$K$$-cycle of the D-brane. Essentially, the Baum-Douglas theory [the author and R. G. Douglas, in: Operator algebras and applications, Proc. Symp. Pure Math. 38, Part 1, Kingston/Ont. 1980, 117–173 (1982; Zbl 0532.55004)] was rediscovered in terms of constraints on open strings. The aim of this expository note is to briefly describe this development.
For the entire collection see [Zbl 1201.81004].
##### MSC:
 81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory 53D40 Symplectic aspects of Floer homology and cohomology 55N15 Topological $$K$$-theory 19K33 Ext and $$K$$-homology 13D15 Grothendieck groups, $$K$$-theory and commutative rings