nLab
Landau-Ginzburg model

Context

Quantum field theory

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

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)(X,g), the action functional of the LG-model is schematically of the form

S LB:(ϕ:ΣX) Σ(Φ 2+(W)(ϕ) 2+fermionicterms)dμ, 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 Σ\Sigma is the 2-dimensional worldsheet and W:XW : X \to \mathbb{C} – called the model’s superpotential – is a holomorphic function. (Usually XX is actually taken to be a Cartesian space and all the nontrivial structure is in WW.)

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

If the target space XX 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)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.

(…)

Properties

The \infty-categories of branes

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

(…) curved dg-algebra

(…) CaldararuTu

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=2N=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

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 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

(…)

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