# nLab Landau-Ginzburg model

Contents

### Context

#### Quantum field theory

functorial quantum field theory

# 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

### In superconductivity

Discussion as a model for superconductors:

The original article:

English translation:

Review:

• S. J. Chapman, Section 2 of: A Hierarchy of Models for Type-II Superconductors, IAM Review Vol. 42, No. 4 (2000), pp. 555-598 (jstor:2653134)

• Carsten Timm, Section 6 of: Theory of Superconductivity, 2020 (pdf)

### In string theory

Discussion as a supersymmetric sigma model in string theory on Calabi-Yau manifolds:

Lecture notes:

• 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.

#### 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 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, 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)

### Elliptic genera as super $p$-brane partition functions

The interpretation of elliptic genera (especially the Witten genus) as the partition function of a 2d superconformal field theory (or Landau-Ginzburg model) – and especially of the heterotic string (“H-string”) or type II superstring worldsheet theory has precursors in

and then strictly originates with:

Review in:

#### Formulations

##### Via super vertex operator algebra

Formulation via super vertex operator algebras:

and for the topologically twisted 2d (2,0)-superconformal QFT (the heterotic string with enhanced supersymmetry) via sheaves of vertex operator algebras in

based on chiral differential operators:

##### Via Dirac-Ramond operators on free loop space

Tentative interpretation as indices of Dirac-Ramond operators as would-be Dirac operators on smooth loop space:

##### Via conformal nets

Tentative formulation via conformal nets:

#### Conjectural interpretation in tmf-cohomology

The resulting suggestion that, roughly, deformation-classes (concordance classes) of 2d SCFTs with target space $X$ are the generalized cohomology of $X$ with coefficients in the spectrum of topological modular forms (tmf):

and the more explicit suggestion that, under this identification, the Chern-Dold character from tmf to modular forms, sends a 2d SCFT to its partition function/elliptic genus/supersymmetric index:

This perspective is also picked up in Gukov, Pei, Putrov & Vafa 18.

Discussion of the 2d SCFTs (namely supersymmetric SU(2)-WZW-models) conjecturally corresponding, under this conjectural identification, to the elements of $\mathbb{Z}/24$ $\simeq$ $tmf^{-3}(\ast) = \pi_3(tmf)$ $\simeq$ $\pi_3(\mathbb{S})$ (the third stable homotopy group of spheres):

• Ying-Hsuan Lin, Du Pei, Holomorphic CFTs and topological modular forms &lbrack;arXiv:2112.10724&rbrack;

• Jan Albert, Justin Kaidi, Ying-Hsuan Lin, Topological modularity of Supermoonshine $[$arXiv:2210.14923$]$

#### Occurrences in string theory

##### H-string elliptic genus

Further on the elliptic genus of the heterotic string being the Witten genus:

The interpretation of equivariant elliptic genera as partition functions of parametrized WZW models in heterotic string theory:

Speculations on physics aspects of lifting the Witten genus to topological modular forms:

##### M5-brane elliptic genus

On the M5-brane elliptic genus:

A 2d SCFT argued to describe the KK-compactification of the M5-brane on a 4-manifold (specifically: a complex surface) originates with

Discussion of the resulting elliptic genus (2d SCFT partition function) originates with:

Further discussion in:

##### M-string elliptic genus

On the elliptic genus of M-strings inside M5-branes:

##### E-string elliptic genus

On the elliptic genus of E-strings as wrapped M5-branes:

• J. A. Minahan, D. Nemeschansky, Cumrun Vafa, N. P. Warner, E-Strings and $N=4$ Topological Yang-Mills Theories, Nucl. Phys. B527 (1998) 581-623 (arXiv:hep-th/9802168)

• Wenhe Cai, Min-xin Huang, Kaiwen Sun, On the Elliptic Genus of Three E-strings and Heterotic Strings, J. High Energ. Phys. 2015, 79 (2015). (arXiv:1411.2801, doi:10.1007/JHEP01(2015)079)

Last revised on September 14, 2022 at 15:50:46. See the history of this page for a list of all contributions to it.