Contents

Idea

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 $X$. 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 $X$ 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.

Related concepts

• ∞-Chern-Simons theory

• sigma-model

  • AKSZ sigma-model

    • Poisson sigma-model

      • A-model, B-model

      • half-twisted model

      • twistor string theory

    • Courant sigma-model

      • Chern-Simons theory

  • topological membrane

• topologically twisted D=4 super Yang-Mills theory

• Landau-Ginzburg model

References

General

The B-model was first conceived in

• Edward Witten, Topological sigma models, Commun. Math. Phys. 118 (1988) 411–449, euclid, MR90b:81080

An early review is in

• Edward Witten. Mirror manifolds and topological field theory, in: Essays on mirror manifolds, pp. 120–-158. Int. Press, Hong Kong, 1992. (arXiv:hep-th/9112056).

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

• Paul Aspinwall, D-Branes on Calabi-Yau Manifolds (arXiv:hep-th/0403166)

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

• Kevin H. Lin, MO comment on Matrix factorization and physics, 2009

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

• Kevin Costello, The Gromov-Witten potential associated to a TCFT (math.QA/0509264)

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

• Michael Bershadsky, Sergio Cecotti, Hirosi Ooguri, Cumrun Vafa, Kodaira-Spencer Theory of Gravity and Exact Results for Quantum String Amplitudes, Commun. Math. Phys. 165 (1994) 311-428 [arXiv:hep-th/9309140, doi:10.1007/BF02099774]

• Kevin Costello, Si Li, Quantum BCOV theory on Calabi-Yau manifolds and the higher genus B-model (arXiv:1201.4501)

• 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)

• Kevin Costello, Si Li, Quantization of open-closed BCOV theory, I (arXiv:1505.06703)

Computation via topological recursion

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

• Vincent Bouchard, Albrecht Klemm, Marcos Marino, Sara Pasquetti, Remodeling the B-model, Commun.Math.Phys.287:117-178, 2009 (arXiv:0709.1453)

• Bertrand Eynard, Amir-Kian Kashani-Poor, Olivier Marchal, A matrix model for the topological string I: Deriving the matrix model (arXiv:1003.1737)

• Bertrand Eynard, Amir-Kian Kashani-Poor, Olivier Marchal, A matrix model for the topological string II: The spectral curve and mirror geometry (arXiv:1007.2194)

• Bertrand Eynard, Nicolas Orantin, Computation of open Gromov-Witten invariants for toric Calabi-Yau 3-folds by topological recursion, a proof of the BKMP conjecture (arXiv:1205.1103)

• Bohan Fang, Chiu-Chu Melissa Liu, Zhengyu Zong, All Genus Open-Closed Mirror Symmetry for Affine Toric Calabi-Yau 3-Orbifolds (arXiv:1310.4818)