×
Kapustin, Anton

Topological strings on noncommutative manifolds. (English) Zbl 1065.81108

Int. J. Geom. Methods Mod. Phys. 1, No. 1-2, 49-81 (2004).
Summary: We identify a deformation of the \(N = 2\) supersymmetric sigma model on a Calabi–Yau manifold \(X\) which has the same effect on B-branes as a noncommutative deformation of \(X\). We show that for hyper-Kähler \(X\) such deformations allow one to interpolate continuously between the A-model and the B-model. For generic values of the noncommutativity and the B-field, properties of the topologically twisted sigma-models can be described in terms of generalized complex structures introduced by N. Hitchin. For example, we show that the path integral for the deformed sigma-model is localized on generalized holomorphic maps, whereas for the A-model and the B-model it is localized on holomorphic and constant maps, respectively. The geometry of topological D-branes is also best described using generalized complex structures. We also derive a constraint on the Chern character of topological D-branes, which includes A-branes and B-branes as special cases.

Cited in 44 Documents

MSC:

81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
81T45 Topological field theories in quantum mechanics
58B34 Noncommutative geometry (à la Connes)
81R12 Groups and algebras in quantum theory and relations with integrable systems
81S40 Path integrals in quantum mechanics
83E30 String and superstring theories in gravitational theory
32Q25 Calabi-Yau theory (complex-analytic aspects)
53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry

Keywords:

Topological sigma-models; topological D-branes; noncommutative geometry
Cite Review PDF
Full Text: DOI arXiv

References:

[1] Seiberg N., JHEP 9909 pp 032–
[2] E. Witten, Mirror Symmetry I, Mirror manifolds and topological field theory, ed. S.T. Yau (AMS, 1998) pp. 121–160. · Zbl 0904.58009
[3] DOI: 10.1063/1.1374448 · Zbl 1036.81027 · doi:10.1063/1.1374448
[4] DOI: 10.4310/ATMP.2000.v4.n1.a3 · Zbl 0992.81059 · doi:10.4310/ATMP.2000.v4.n1.a3
[5] DOI: 10.1007/s00220-002-0755-7 · Zbl 1051.17017 · doi:10.1007/s00220-002-0755-7
[6] Minasian R., JHEP 9711 pp 002–
[7] Witten E., JHEP 9812 pp 019–
[8] DOI: 10.1073/pnas.37.10.704 · Zbl 0045.43101 · doi:10.1073/pnas.37.10.704
[9] DOI: 10.1016/S0393-0440(03)00026-3 · Zbl 1029.81058 · doi:10.1016/S0393-0440(03)00026-3
[10] DOI: 10.1016/0550-3213(84)90592-3 · doi:10.1016/0550-3213(84)90592-3
[11] DOI: 10.1016/0550-3213(89)90474-4 · doi:10.1016/0550-3213(89)90474-4
[12] DOI: 10.4310/ATMP.1998.v2.n2.a9 · Zbl 0947.14017 · doi:10.4310/ATMP.1998.v2.n2.a9
[13] DOI: 10.1016/S0550-3213(98)00115-1 · Zbl 0945.81061 · doi:10.1016/S0550-3213(98)00115-1
[14] Taylor W., JHEP 0007 pp 039–
[15] DOI: 10.1016/S0370-2693(02)01569-1 · Zbl 0994.81090 · doi:10.1016/S0370-2693(02)01569-1
[16] DOI: 10.1016/0550-3213(93)90033-L · Zbl 0910.14020 · doi:10.1016/0550-3213(93)90033-L
[17] Cannas da Silva A., Geometric models for noncommutative algebras (2001) · Zbl 1135.58300
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.