nLab
B-model

Context

\infty-Chern-Simons theory

Ingredients

Definition

Examples

  • For targets

  • For targets

  • For discrete targets

  • For targets

    • coupled to
  • For targets extending the

    (such as the , the )

    • Chern-Simons-

  • for higher abelian targets

  • for targets

        • ,
  • for the L L_\infty-structure on the of the closed :

Quantum field theory

Contents

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  • FQFT and

String theory

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Critical string models

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Extended objects

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Topological strings

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Backgrounds

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Phenomenology

Physics

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Surveys, textbooks and lecture notes

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Contents

Idea

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.

Properties

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.

induced via

perspective via perspective
/
\;\;\;\;\downarrow on S 4S^4compactificationon followed by
\;\;\;\;\downarrow topological sector
\;\;\;\;\downarrow
on the with worldvolume theory
\;\;\;\; \downarrow on on /
with invariance; worldvolume theory with type IIB
\;\;\;\;\; \downarrow
\;\;\;\; \downarrow on
on Bun GBun_G,

\,

induced via
\;\;\;\;\downarrow on S 5S^5
\;\;\;\; \downarrow topological sector
\;\;\;\;\downarrow
\;\;\;\;\; \downarrow
\;\;\;\; \downarrow on
on Bun GBun_G and on Loc GLoc_G,

References

General

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 October 20, 2017 at 18:31:22. See the history of this page for a list of all contributions to it.