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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.
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-CFT6 | F-theory perspective |
---|---|
11d supergravity/M-theory | |
$\;\;\;\;\downarrow$ Kaluza-Klein compactification on $S^4$ | compactificationon 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 invariance | M5-brane worldvolume theory |
$\;\;\;\; \downarrow$ KK-compactification on Riemann surface | double dimensional reduction on M-theory/F-theory elliptic fibration |
N=2 D=4 super Yang-Mills theory with Montonen-Olive S-duality invariance; AGT correspondence | D3-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 $Bun_G$, Donaldson theory |
$\,$
gauge theory induced via AdS5-CFT4 |
---|
type II string theory |
$\;\;\;\;\downarrow$ Kaluza-Klein compactification on $S^5$ |
$\;\;\;\; \downarrow$ 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 $Bun_G$ and B-model on $Loc_G$, 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
That the B-model Lagrangian arises in AKSZ theory by gauge fixing the Poisson sigma-model was observed in
For survey of the literature see also
The B-model on genus-0 cobordisms had been constructed in
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
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
M. Bershadsky, S. Cecotti, Hirosi Ooguri, Cumrun Vafa, Kodaira-Spencer Theory of Gravity and Exact Results for Quantum String Amplitudes, Commun.Math.Phys.165:311-428,1994 (arXiv:hep-th/9309140)
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 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)