nLab B-model



\infty-Chern-Simons theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory




Quantum field theory

String theory


physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes

theory (physics), model (physics)

experiment, measurement, computable physics



Witten introduced two topological twists of a supersymmetric 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-Li 12, Costello-Li 15).

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)

That the B-model Lagrangian arises in AKSZ theory by gauge fixing the Poisson sigma-model was observed in

  • M. Alexandrov, M. Kontsevich, A. Schwarz, O. Zaboronsky, around page 28 in The geometry of the master equation and topological quantum field theory, Int. J. Modern Phys. A 12(7):1405–1429, 1997

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

Computation via topological recursion

Computation via topological recursion in matrix models and all-genus proofs of mirror symmetry is due to

Last revised on July 9, 2023 at 15:56:09. See the history of this page for a list of all contributions to it.