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\section*{Landau-Ginzburg model}

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

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

[[!include functorial quantum field 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{the_categories_of_branes}{The $\infty$-categories of branes}\dotfill \pageref*{the_categories_of_branes} \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{partition_function_and_elliptic_genera}{Partition function and elliptic genera}\dotfill \pageref*{partition_function_and_elliptic_genera} \linebreak
\noindent\hyperlink{branes}{Branes}\dotfill \pageref*{branes} \linebreak
\noindent\hyperlink{defects}{Defects}\dotfill \pageref*{defects} \linebreak
\noindent\hyperlink{tcft_formulation}{TCFT formulation}\dotfill \pageref*{tcft_formulation} \linebreak
\noindent\hyperlink{relation_to_solid_state_physics}{Relation to Solid state physics}\dotfill \pageref*{relation_to_solid_state_physics} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Originally, the \emph{Ginzburg-Landau model} is a [[model (physics)|model]] in [[solid state physics]] for [[superconductivity]].

Roughly this type of model has then been used as models for 2d [[quantum field theory]] in [[string theory]]. There, a \emph{Landau-Ginzburg model} (LG-model) is a 2-[[dimension|dimensional]] [[supersymmetry|supersymmetric]] [[sigma model]] [[QFT]] characterized by the fact that its [[Lagrangian]] contains a [[potential]] term: given a [[complex manifold|complex]] [[Riemannian manifold|Riemannian]] [[target space]] $(X,g)$, the [[action functional]] of the LG-model is schematically of the form

\begin{displaymath}
S_{LB} : (\phi : \Sigma \to X) \mapsto
  \int_\Sigma \left( \vert \nabla \Phi \vert^2 + \vert (\nabla W)(\phi) \vert^2 
  + fermionic\;terms \right) d \mu
  \,,
\end{displaymath}
where $\Sigma$ is the 2-[[dimension]]al [[worldsheet]] and $W : X \to \mathbb{C}$ -- called the model's \emph{superpotential} -- is a [[holomorphic function]]. (Usually $X$ is actually taken to be a [[Cartesian space]] and all the nontrivial structure is in $W$.)

Landau-Ginzburg models have gained importance as constituting one type of QFTs that are related under [[homological mirror symmetry]]:

If the [[target space]] $X$ is a [[Fano variety]], the usual [[B-model]] does not quite exist on it, since the corresponding supersymmetric [[string]] [[sigma model]] is not conformally invariant as a quantum theory, and the axial $U(1)$ [[R-current]] used to define the B-twist is [[quantum anomaly|anomalous]]. Still, there exists an analogous [[derived category]] of B-branes. A Landau-Ginburg model is something that provides the dual A-branes to this under [[homological mirror symmetry]]. Conversely, Landau-Ginzburg B-branes are homological mirror duals to the [[A-model]] on a Fano variety. (\ldots{})

As suggested by [[Maxim Kontsevich]] (see \hyperlink{KapustinLi}{Kapustin-Li, section 7}), the B-branes in the LG-model (at least in a certain class of cases) are not given by [[chain complexes]] of [[coherent sheaves]] as in the [[B-model]], but by \emph{[[twisted complexes]]} : for these the square of the [[differential]] is in general non-vanishing and identified with the \emph{superpotential} of the LG-model.

(\ldots{})

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

\hypertarget{the_categories_of_branes}{}\subsubsection*{{The $\infty$-categories of branes}}\label{the_categories_of_branes}

A [[brane]] for a LG model is given by a [[matrix factorization]] of its superpotential.

(\ldots{}) [[curved dg-algebra]]

(\ldots{}) \hyperlink{CaldararuTu}{CaldararuTu}

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

\begin{itemize}%
\item [[2d TQFT]], [[TCFT]]


\item [[holographic superconductor]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

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

Original articles are

\begin{itemize}%
\item [[Cumrun Vafa]] Nicholas P. Warner, \emph{Catastrophes and the Classification of Conformal Theories}, Phys.Lett. B218 (1989) 51


\item [[Brian Greene]], [[Cumrun Vafa]], \emph{Calabi-Yau Manifolds and Renormalization Group Flows}, Nucl.Phys. B324 (1989) 371


\item [[Edward Witten]], \emph{Phases of $N=2$ Theories In Two Dimensions}, Nucl.Phys.B403:159-222,1993 (\href{http://arxiv.org/abs/hep-th/9301042}{arXiv:hep-th/9301042})



\end{itemize}
Lecture notes include

\begin{itemize}%
\item [[Edward Witten]], \emph{Dynamical aspects of QFT}, Lecture 15: \emph{The Landau-Ginzburg description of N=2 minimal models; Quantum cohomology and K\"a{}hler manifolds}, in Part IV of \emph{[[Quantum Fields and Strings]]}.

\end{itemize}
\hypertarget{partition_function_and_elliptic_genera}{}\subsubsection*{{Partition function and elliptic genera}}\label{partition_function_and_elliptic_genera}

The [[partition function]] of LG-models and its relation to [[elliptic genera]] is disucssed in

\begin{itemize}%
\item [[Edward Witten]], \emph{On the Landau-Ginzburg Description of $N=2$ Minimal Models}, Int.J.Mod.Phys.A9:4783-4800,1994 (\href{http://arxiv.org/abs/hep-th/9304026}{arXiv:hep-th/9304026})


\item Toshiya Kawai, Yasuhiko Yamada, Sung-Kil Yang, \emph{Elliptic Genera and $N=2$ Superconformal Field Theory} (\href{http://arxiv.org/abs/hep-th/9306096}{arXiv:hep-th/9306096})



\end{itemize}
\hypertarget{branes}{}\subsubsection*{{Branes}}\label{branes}

The [[branes]] of the LG-model are discussed for instance in

\begin{itemize}%
\item [[Anton Kapustin]], Yi Li, \emph{D-Branes in Landau-Ginzburg models and algebraic geometry}, \href{http://arxiv.org/abs/hep-th/0210296}{arXiv:hep-th/0210296}

\end{itemize}
The [[derived category]] of [[D-brane]]s in type B LG-models is discussed in

\begin{itemize}%
\item [[Dmitri Orlov]], \emph{Triangulated categories of singularities and D-branes in Landau-Ginzburg models}, Proc. Steklov Inst. Math. 2004, no. 3 (246), 227--248 (\href{http://arxiv.org/abs/math/0302304}{arXiv:math/0302304})


\item [[Dmitri Orlov]]\_Derived categories of coherent sheaves and triangulated categories of singularities\_, Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. II, 503--531, Progr. Math., 270, Birkh\"a{}user Boston, Inc., Boston, MA, 2009 (\href{http://arxiv.org/abs/math.ag/0503632}{arXiv:math.ag/0503632})


\item Andrei Caldararu, Junwu Tu, \emph{Curved $A_\infty$-algebras and Landau-Ginzburg models} (\href{http://www.math.wisc.edu/~andreic/publications/AinfinityLG.pdf}{pdf})



\end{itemize}
\hypertarget{defects}{}\subsubsection*{{Defects}}\label{defects}

General defects of B-twisted affine LG models were first discussed in

\begin{itemize}%
\item [[Ilka Brunner]], [[Daniel Roggenkamp]], \emph{B-type defects in Landau-Ginzburg models}, JHEP 0708 (2007) 093, (\href{http://arxiv.org/abs/0707.0922}{arXiv:0707.0922})

\end{itemize}
The graded pivotal bicategory of B-twisted affine LG models is studied in detail in

\begin{itemize}%
\item Nils Carqueville, [[Daniel Murfet]], \emph{Adjunctions and defects in Landau-Ginzburg models}, Advances in Mathematics, Volume 289 (2016), 480-566, (\href{http://arxiv.org/abs/1208.1481}{arXiv:1208.1481})

\end{itemize}
Orbifolds of defects are studied in

\begin{itemize}%
\item [[Ilka Brunner]], [[Daniel Roggenkamp]], \emph{Defects and Bulk Perturbations of Boundary Landau-Ginzburg Orbifolds}, JHEP 0804 (2008) 001, (\href{http://arxiv.org/abs/0712.0188}{arXiv:0712.0188})


\item Nils Carqueville, Ingo Runkel, \emph{Orbifold completion of defect bicategories}, (\href{http://arxiv.org/abs/1210.6363}{arXiv:1210.6363})


\item [[Ilka Brunner]], Nils Carqueville, Daniel Plencner, \emph{Orbifolds and topological defects}, Comm. Math. Phys. 332 (2014), 669-712, (\href{http://arxiv.org/abs/1307.3141}{arXiv:1307.3141})


\item [[Ilka Brunner]], Nils Carqueville, Daniel Plencner, \emph{Discrete torsion defects}, Comm. Math. Phys. 337 (2015), 429-453, (\href{http://arxiv.org/abs/1404.7497}{arXiv:1404.7497})



\end{itemize}
A relation to [[linear logic]] and the [[geometry of interaction]] is in

\begin{itemize}%
\item [[Daniel Murfet]], \emph{Computing with cut systems} (\href{http://arxiv.org/abs/1402.4541}{arXiv:1402.4541})

\end{itemize}
\hypertarget{tcft_formulation}{}\subsubsection*{{TCFT formulation}}\label{tcft_formulation}

Discussions of topological Landau-Ginzburg [[B-models]] explicitly as open [[TCFT]]s (aka open topological string theories) are in

\begin{itemize}%
\item Nils Carqueville, \emph{Matrix factorisations and open topological string theory}, JHEP 07 (2009) 005, (\href{http://arxiv.org/abs/0904.0862}{arXiv:0904.0862})


\item [[Ed Segal]], \emph{The closed state space of affine Landau-Ginzburg B-models} (\href{http://arxiv.org/abs/0904.1339}{arXiv:0904.1339})


\item Nils Carqueville, Michael Kay, \emph{Bulk deformations of open topological string theory}, Comm. Math. Phys. 315, Number 3 (2012), 739-769, (\href{http://arxiv.org/abs/1104.5438}{arXiv:1104.5438})



\end{itemize}
\hypertarget{relation_to_solid_state_physics}{}\subsubsection*{{Relation to Solid state physics}}\label{relation_to_solid_state_physics}

(\ldots{})

[[!redirects LG-model]] [[!redirects LG-models]]

[[!redirects Landau-Ginzburg models]]



\end{document}