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Topological open \(p\)-branes. (English) Zbl 1024.81043

Fukaya, K. (ed.) et al., Symplectic geometry and mirror symmetry. Proceedings of the 4th KIAS annual international conference, Seoul, South Korea, August 14-18, 2000. Singapore: World Scientific. 311-384 (2001).
This paper provides insight into the mathematical structures underlying the connection between string theory and noncommutative geometry. This connection was first established in the case of D-branes, whose world volume becomes noncommutative in the decoupling limit of the bulk degrees of freedom, i.e. when the fundamental open string becomes topological in the sense that it is coupled only to a 2-form field. A similar mechanism has been conjectured for the M-theory 5-brane interpreted as a D-brane of the open supermembrane whose decoupling leaves only a 3-form field.
Starting with a review of Batalin-Vilkovisky (BV) quantization and of the role of the topological open string in deformation quantization of the point particle the author arrives at the following basic picture: The BV quantization of a topological open p-brane has the structure of a “(\(p+1\)-algebra” in the bulk and induces the structure of a “\(p\)-algebra” on the boundary. These algebras and an analogous correspondence appear in the generalized Deligne conjecture proven by M. Kontsevich and Y. Soibelman [Math. Phys. Stud. 21, 255-307 (2000; Zbl 0972.18005), see also math.QA/0001151], and the quantization can be interpreted as a deformation quantization of the \((p-1)\)-brane. This interpretaion is suggested by the direct relation between the BV master equation and the Ward identity of the bulk topological theory, although not yet made explicit by the calculation of the path integral. Finally the author conjectures that homological mirror symmetry has generalizations to the categories of “\(p\)-algebras”.
For the entire collection see [Zbl 0980.00036].

MSC:

81T70 Quantization in field theory; cohomological methods
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
81T75 Noncommutative geometry methods in quantum field theory
81S10 Geometry and quantization, symplectic methods
53D55 Deformation quantization, star products

Citations:

Zbl 0972.18005
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