\infty-Chern-Simons theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory




Quantum field theory

String theory


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theory (physics), model (physics)

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Witten introduced two topological twists of a supersymmtric nonlinear sigma model, which is a certain N=2 superconformal field theory attached to a compact Calabi–Yau variety XX. One of them is the B-model topological string; it is a 2-dimensional topological N=1 superconformal field theory. In Kontsevich’s version, instead of SCFT with Hilbert space, one assembles all the needed data in terms of Calabi–Yau A-infinity-category which is the A-infinity-category of coherent sheaves on the underlying variety. In fact only the structure of a derived category is sufficient (and usually quoted): it is now known (by the results of Dmitri Orlov and Valery Lunts) that under mild assumptions on the variety, a derived category of coherent sheaves has a unique enhancement.

The B-model arose in considerations of superstring-propagation on Calabi–Yau varieties: it may be motivated by considering the vertex operator algebra of the 2dSCFT given by the N=2 supersymmetric nonlinear sigma-model with target XX and then changing the fields so that one of the super-Virasoro generators squares to 0. The resulting “topologically twisted” algebra may then be read as being the BRST complex of a TCFT.

One can also define a B-model for Landau–Ginzburg models. The category of D-branes for the string – the B-branes – is given by the category of matrix factorizations (this was proposed by Kontsevich and elaborated by Kapustin-Li; see also papers of Orlov). For the genus 0 closed string theory, see the work of Saito.

By homological mirror symmetry, the B-model is dual to the A-model.


Second quantization / effective background field theory

The second quantization effective field theory defined by the perturbation series of the B-model is supposed to be “Kodaira-Spencer gravity” / “BVOC theory” in 6d (BCOV 93, Costello-Lie 12).

For more on this see at TCFT – Worldsheet and effective background theories.

gauge theory induced via AdS-CFT correspondence

M-theory perspective via AdS7-CFT6F-theory perspective
11d supergravity/M-theory
\;\;\;\;\downarrow Kaluza-Klein compactification on S 4S^4compactificationon elliptic fibration followed by T-duality
7-dimensional supergravity
\;\;\;\;\downarrow topological sector
7-dimensional Chern-Simons theory
\;\;\;\;\downarrow AdS7-CFT6 holographic duality
6d (2,0)-superconformal QFT on the M5-brane with conformal invarianceM5-brane worldvolume theory
\;\;\;\; \downarrow KK-compactification on Riemann surfacedouble dimensional reduction on M-theory/F-theory elliptic fibration
N=2 D=4 super Yang-Mills theory with Montonen-Olive S-duality invariance; AGT correspondenceD3-brane worldvolume theory with type IIB S-duality
\;\;\;\;\; \downarrow topological twist
topologically twisted N=2 D=4 super Yang-Mills theory
\;\;\;\; \downarrow KK-compactification on Riemann surface
A-model on Bun GBun_G, Donaldson theory


gauge theory induced via AdS5-CFT4
type II string theory
\;\;\;\;\downarrow Kaluza-Klein compactification on S 5S^5
\;\;\;\; \downarrow topological sector
5-dimensional Chern-Simons theory
\;\;\;\;\downarrow AdS5-CFT4 holographic duality
N=4 D=4 super Yang-Mills theory
\;\;\;\;\; \downarrow topological twist
topologically twisted N=4 D=4 super Yang-Mills theory
\;\;\;\; \downarrow KK-compactification on Riemann surface
A-model on Bun GBun_G and B-model on Loc GLoc_G, geometric Langlands correspondence



The B-model was first conceived in

An early review is in

The motivation from the point of view of string theory with an eye towards the appearance of the Calabi-Yau categories is reviewed for instance in

A summary of these two reviews is in

  • H. Lee, Review of topological field theory and homological mirror symmetry (pdf)

For survey of the literature see also

The B-model on genus-0 cobordisms had been constructed in

  • S. Barannikov, Maxim Kontsevich, Frobenius manifolds and formality of Lie algebras of polyvector fields , Internat. Math. Res. Notices 1998, no. 4, 201–215; math.QA/9710032 doi

The construction of the B-model as a TCFT on cobordisms of arbitrary genus was given in

See also the MathOverflow discussion: higher-genus-closed-string-b-model

Second quantization to Kodeira-Spencer gravity

The second quantization effective field theory defined by the B-model perturbation series (“Kodeira-Spencer gravity”/“BCOV theory”) is discussed in

Discussion of how the second quantization of the B-model yields Kodeira-Spencer gravity/BCOV theory is in

  • Si Li, BCOV theory on the elliptic curve and higher genus mirror symmetry (arXiv:1112.4063)

  • Si Li, Variation of Hodge structures, Frobenius manifolds and Gauge theory (arXiv:1303.2782)

Revised on May 27, 2014 01:48:27 by Urs Schreiber (