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\section*{B-model}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{chernsimons_theory}{}\paragraph*{{$\infty$-Chern-Simons theory}}\label{chernsimons_theory}

[[!include infinity-Chern-Simons theory - contents]]

\hypertarget{quantum_field_theory}{}\paragraph*{{Quantum field theory}}\label{quantum_field_theory}

[[!include functorial quantum field theory - contents]]

\hypertarget{string_theory}{}\paragraph*{{String theory}}\label{string_theory}

[[!include string theory - contents]]

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{second_quantization__effective_background_field_theory}{Second quantization / effective background field theory}\dotfill \pageref*{second_quantization__effective_background_field_theory} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{ReferencesBCOV}{Second quantization to Kodeira-Spencer gravity}\dotfill \pageref*{ReferencesBCOV} \linebreak
\noindent\hyperlink{computation_via_topological_recursion}{Computation via topological recursion}\dotfill \pageref*{computation_via_topological_recursion} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{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 \emph{B-model} [[topological string]]; it is a 2-dimensional [[topological conformal field theory|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 category|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 [[enhanced triangulated category|enhancement]].

The B-model arose in considerations of [[string theory|superstring]]-propagation on Calabi--Yau varieties: it may be motivated by considering the [[vertex operator algebra]] of the 2d[[CFT|SCFT]] given by the N=2 supersymmetric nonlinear [[sigma-model]] with target $X$ and then changing the fields so that one of the super-[[Virasoro algebra|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 model|Landau?Ginzburg models]]. The category of [[D-brane|D-branes]] for the string -- the [[B-branes]] -- is given by the category of [[matrix factorization|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]].

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{second_quantization__effective_background_field_theory}{}\subsubsection*{{Second quantization / effective background field theory}}\label{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 (\hyperlink{BCOV93}{BCOV 93}, \hyperlink{CostelloLi12}{Costello-Li 12}, \hyperlink{CostelloLi15}{Costello-Li 15}).

For more on this see at \emph{\href{http://ncatlab.org/nlab/show/TCFT#ActionFunctionals}{TCFT -- Worldsheet and effective background theories}}.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[schreiber:∞-Chern-Simons theory]]


\item [[sigma-model]]

\begin{itemize}%
\item [[AKSZ sigma-model]]

\begin{itemize}%
\item [[Poisson sigma-model]]

\begin{itemize}%
\item [[A-model]], \textbf{B-model}


\item [[half-twisted model]]



\end{itemize}

\item [[Courant sigma-model]]

\begin{itemize}%
\item [[Chern-Simons theory]]

\end{itemize}


\end{itemize}

\item [[topological membrane]]



\end{itemize}

\item [[topologically twisted D=4 super Yang-Mills theory]]


\item [[Landau-Ginzburg model]]



\end{itemize}
[[!include gauge theory from AdS-CFT -- table]]

\hypertarget{References}{}\subsection*{{References}}\label{References}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

The B-model was first conceived in

\begin{itemize}%
\item [[Edward Witten]], \emph{Topological sigma models}, Commun. Math. Phys. \textbf{118} (1988) 411--449, \href{http://projecteuclid.org/euclid.cmp/1104162092}{euclid}, \href{http://www.ams.org/mathscinet-getitem?mr=0958805}{MR90b:81080}

\end{itemize}
An early review is in

\begin{itemize}%
\item [[Edward Witten]]. \emph{Mirror manifolds and topological field theory}, in: Essays on mirror manifolds, pp. 120---158. Int. Press, Hong Kong, 1992. (\href{http://arxiv.org/abs/hep-th/9112056}{arXiv:hep-th/9112056}).

\end{itemize}
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

\begin{itemize}%
\item [[Paul Aspinwall]], \emph{D-Branes on Calabi-Yau Manifolds} (\href{http://arxiv.org/abs/hep-th/0403166}{arXiv:hep-th/0403166})

\end{itemize}
A summary of these two reviews is in

\begin{itemize}%
\item H. Lee, \emph{Review of topological field theory and homological mirror symmetry} (\href{http://people.maths.ox.ac.uk/leeh/files/CYMSmini.pdf}{pdf})

\end{itemize}
That the B-model [[Lagrangian]] arises in [[AKSZ theory]] by [[gauge fixing]] the [[Poisson sigma-model]] was observed in

\begin{itemize}%
\item M. Alexandrov, [[Maxim Kontsevich|M. Kontsevich]], [[Albert Schwarz|A. Schwarz]], O. Zaboronsky, around page 28 in \emph{The geometry of the master equation and topological quantum field theory}, Int. J. Modern Phys. A 12(7):1405--1429, 1997

\end{itemize}
For survey of the literature see also

\begin{itemize}%
\item [[Kevin H. Lin]], MO comment on \emph{\href{http://mathoverflow.net/a/9748/381}{Matrix factorization and physics}, 2009}

\end{itemize}
The B-model on [[genus]]-0 [[cobordism]]s had been constructed in

\begin{itemize}%
\item S. Barannikov, [[Maxim Kontsevich]], \emph{Frobenius manifolds and formality of Lie algebras of polyvector fields} , Internat. Math. Res. Notices 1998, no. 4, 201--215; \href{http://arxiv.org/abs/alg-geom/9710032}{math.QA/9710032} \href{http://dx.doi.org/10.1155/S1073792898000166}{doi}

\end{itemize}
The construction of the B-model as a [[TCFT]] on [[cobordisms]] of arbitrary [[genus]] was given in

\begin{itemize}%
\item [[Kevin Costello]], \emph{The Gromov-Witten potential associated to a TCFT} (\href{http://arxiv.org/abs/math/0509264}{math.QA/0509264})

\end{itemize}
See also the MathOverflow discussion: \href{http://mathoverflow.net/questions/8692/higher-genus-closed-string-b-model}{higher-genus-closed-string-b-model}

\hypertarget{ReferencesBCOV}{}\subsubsection*{{Second quantization to Kodeira-Spencer gravity}}\label{ReferencesBCOV}

The [[second quantization]] [[effective field theory|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

\begin{itemize}%
\item M. Bershadsky, S. Cecotti, [[Hirosi Ooguri]], [[Cumrun Vafa]], \emph{Kodaira-Spencer Theory of Gravity and Exact Results for Quantum String Amplitudes}, Commun.Math.Phys.165:311-428,1994 (\href{http://arxiv.org/abs/hep-th/9309140}{arXiv:hep-th/9309140})


\item [[Kevin Costello]], Si Li, \emph{Quantum BCOV theory on Calabi-Yau manifolds and the higher genus B-model} (\href{http://arxiv.org/abs/1201.4501}{arXiv:1201.4501})


\item Si Li, \emph{BCOV theory on the elliptic curve and higher genus mirror symmetry} (\href{http://arxiv.org/abs/1112.4063}{arXiv:1112.4063})


\item Si Li, \emph{Variation of Hodge structures, Frobenius manifolds and Gauge theory} (\href{http://arxiv.org/abs/1303.2782}{arXiv:1303.2782})


\item [[Kevin Costello]], Si Li, \emph{Quantization of open-closed BCOV theory, I} (\href{http://arxiv.org/abs/1505.06703}{arXiv:1505.06703})



\end{itemize}
\hypertarget{computation_via_topological_recursion}{}\subsubsection*{{Computation via topological recursion}}\label{computation_via_topological_recursion}

Computation via [[topological recursion]] in [[matrix models]] and all-[[genus of a surface|genus]] proofs of [[mirror symmetry]] is due to

\begin{itemize}%
\item [[Vincent Bouchard]], [[Albrecht Klemm]], [[Marcos Marino]], [[Sara Pasquetti]], \emph{Remodeling the B-model}, Commun.Math.Phys.287:117-178, 2009 (\href{https://arxiv.org/abs/0709.1453}{arXiv:0709.1453})


\item [[Bertrand Eynard]], [[Amir-Kian Kashani-Poor]], Olivier Marchal, \emph{A matrix model for the topological string I: Deriving the matrix model} (\href{https://arxiv.org/abs/1003.1737}{arXiv:1003.1737})


\item [[Bertrand Eynard]], [[Amir-Kian Kashani-Poor]], Olivier Marchal, \emph{A matrix model for the topological string II: The spectral curve and mirror geometry} (\href{https://arxiv.org/abs/1007.2194}{arXiv:1007.2194})


\item [[Bertrand Eynard]], [[Nicolas Orantin]], \emph{Computation of open Gromov-Witten invariants for toric Calabi-Yau 3-folds by topological recursion, a proof of the BKMP conjecture} (\href{https://arxiv.org/abs/1205.1103}{arXiv:1205.1103})


\item Bohan Fang, Chiu-Chu Melissa Liu, Zhengyu Zong, \emph{All Genus Open-Closed Mirror Symmetry for Affine Toric Calabi-Yau 3-Orbifolds} (\href{https://arxiv.org/abs/1310.4818}{arXiv:1310.4818})



\end{itemize}
[[!redirects Kodeira-Spencer gravity]] [[!redirects BCOV theory]]

[[!redirects B-models]]



\end{document}