Contents

# Contents

## Idea

Originally, the Ginzburg–Landau 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–Ginzburg 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.

(…) CaldararuTu

• holographic superconductor?

## References

### General

Original articles are

Lecture notes include

• Edward Witten, Dynamical aspects of QFT, Lecture 15: The Landau–Ginzburg description of N=2 minimal models; Quantum cohomology and Kähler manifolds, in Part IV of Quantum Fields and Strings.

### 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.A 9:4783–4800, 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

• Dmitri Orlov, Triangulated categories of singularities and D-branes in Landau–Ginzburg models, Proc. Steklov Inst. Math. 2004, no. 3 (246), 227–248 (arXiv:math/0302304)

• Dmitri OrlovDerived 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, Birkhäuser Boston,

Inc., Boston, MA, 2009 (arXiv:math.ag/0503632)

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

### Defects

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

The graded pivotal bicategory of B-twisted affine LG models is studied in detail in

• Nils Carqueville, Daniel Murfet, Adjunctions and defects in Landau–Ginzburg models, Advances in Mathematics, Volume 289 (2016), 480–566, (arXiv:1208.1481)

Orbifolds of defects are studied in

A relation to linear logic and the geometry of interaction is in

### TCFT formulation

Discussions of topological Landau–Ginzburg B-models explicitly as open TCFTs (aka open topological string theories) are in

• Nils Carqueville, Matrix factorisations and open topological string theory, JHEP 07 (2009) 005, (arXiv:0904.0862)

• Ed Segal, The closed state space of affine Landau–Ginzburg B-models (arXiv:0904.1339)

• Nils Carqueville, Michael Kay, Bulk deformations of open topological string theory, Comm. Math. Phys. 315, Number 3 (2012), 739–769, (arXiv:1104.5438)

### Relation to Solid state physics

(…)

Last revised on February 17, 2020 at 16:29:29. See the history of this page for a list of all contributions to it.