nLab
B-model

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Context

∞-Chern-Simons theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory

Ingredients

Examples

Quantum field theory

functorial quantum field theory

Contents

String theory

Ingredients

Critical string models

Relations between the critical models

Extended objects

Topological strings

Physics

physics

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 X. One of them is the (topological) B-model; 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 open string theory 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.

References

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)

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

Overflow discussion: higher-genus-closed-string-b-model

Revised on August 24, 2011 23:22:56 by Urs Schreiber (131.211.238.84)
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