Contents

Idea

Originally, the Ginzburg-Laundau model is a 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 Landau-Ginzburg model (LG-model) is a 2-dimensional supersymmetric sigma model QFT characterized by the fact that its Lagrangian contains a potential term: given a complex Riemannian target space $(X,g)$, the action functional of the LG-model is schematically of the form

$S_{LB} : (\phi : \Sigma \to X) \mapsto \int_\Sigma \left( \vert \nabla \Phi \vert^2 + \vert (\nabla W)(\phi) \vert^2 + fermionic\;terms \right) d \mu \,,$

where $\Sigma$ is the 2-dimensional worldsheet and $W : X \to \mathbb{C}$ – called the model’s 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 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. (…)

As suggested by Maxim Kontsevich (see 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 twisted complexes : for these the square of the differential is in general non-vanishing and identified with the superpotential of the LG-model.

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Properties

The $\infty$-categories of branes

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

(…) CaldararuTu

• holographic superconductor?

References

General

Original articles are

• Cumrun Vafa Nicholas P. Warner, Catastrophes and the Classification of Conformal Theories, Phys.Lett. B218 (1989) 51

• Brian Greene, Cumrun Vafa, Calabi-Yau Manifolds and Renormalization Group Flows, Nucl.Phys. B324 (1989) 371

• Edward Witten, Phases of $N=2$ Theories In Two Dimensions, Nucl.Phys.B403:159-222,1993 (arXiv:hep-th/9301042)

Lecture notes include

Partition function and elliptic genera

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

• Edward Witten, On the Landau-Ginzburg Description of $N=2$ Minimal Models, Int.J.Mod.Phys.A9:4783-4800,1994 (arXiv:hep-th/9304026)

• Toshiya Kawai, Yasuhiko Yamada, Sung-Kil Yang, Elliptic Genera and $N=2$ Superconformal Field Theory (arXiv:hep-th/9306096)

Branes

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

The derived category of D-branes in type B LG-models is discussed in

• Andrei Caldararu, Junwu Tu, Curved $A_\infty$-algebras and Landau-Ginzburg models (pdf)

A formulation in terms of linear logic and the geometry of interaction is in

TCFT formulation

Discussion of topological Landau-Ginzburg B-models explicitly as TCFTs is in

Relation to Solid state physics

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Revised on February 28, 2014 09:53:20 by Urs Schreiber (89.204.139.212)