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Extra resources for Encyclopedia of Mathematical Physics - Vol 2 [D-H]
Deformation Quantization, IRMA Lectures in Mathematics and Theoretical Physics, vol. 1, pp. 9–54. Berlin: Walter de Gruyter. Gutt S (2000) Variations on deformation quantization. ) Confe´rence Moshe´ Flato 1999. Quantization, Deformations, and Symmetries, Mathematical Physics Studies, vol. 21, pp. 217–254. Dordrecht: Kluwer Academic. Waldmann S (2005) States and representations in deformation quantization. Reviews in Mathematical Physics 17: 15–75. " @-Approach to Integrable Systems P G Grinevich, L D Landau Institute for Theoretical Physics, Moscow, Russia ª 2006 Elsevier Ltd.
For a set of ordinary D-branes, the description above suffices. However, more generally one would like to describe collections of D-branes and antiD-branes, and tachyons. An anti-D-brane has all the same physical properties as an ordinary Dbrane, modulo the fact that they try to annihilate each other. The open-string spectrum between coincident D-branes and anti-D-branes contains tachyons. One can give an (off-shell) vacuum expectation value to such tachyons, and then the unstable brane–antibrane–tachyon system will evolve to some other, usually simpler, configuration.
Showing such a statement directly is usually not possible – it is usually technically impractical to follow renormalization group flow explicitly. There is no symmetry reason or other basic physics reason why renormalization group flow must respect quasiisomorphism. The strongest constraint that is clearly applied by physics is that renormalization group flow must preserve D-brane charges (Chern characters, or more properly, K-theory), but objects in a derived category contain much more information than that.