nLab Landau-Ginzburg model



Quantum field theory


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Originally, the Ginzburg–Landau model is a mesoscopic 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–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. (…)

A mathematical definition of the category of B-branes in the LG model is given by Maxim Kontsevich (elaborated in Kapustin-Li, section 7), the idea is that (at least in a certain class of cases, say affine varieties) the B-branes 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 but rather a multiplication by superpotential of the LG-model, which therefore supplies a way to twist the (derived) category of coherent sheaves. The Z\mathbf{Z}-grading breaks down to a Z 2\mathbf{Z}_2-grading. This category of D-branes of type B can be presented as the product of derived categories of singularities λD sg b(X λ)\prod_\lambda D^b_{sg}(X_\lambda) (which is a finite product) where X λX_\lambda is a fiber over λ\lambda in C\mathbf{C}, what can roughly be taken as a more general definition.


The \infty-categories of branes

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

(…) curved dg-algebra

(…) CaldararuTu


In superconductivity

Discussion as a model for superconductors:

The original article:

English translation:


  • 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=2N=2 minimal models; Quantum cohomology and Kähler manifolds, in Part IV of Quantum Fields and Strings.


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 2009 (

  • Dmitri O. Orlov, Triangulated categories of singularities and equivalences between Landau-Ginzburg models, Sbornik: Mathematics 197:12 (2006) 1827 doi

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


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

and described in terms of linear logic and the geometry of interactions:

A proposed Landau-Ginzburg model for strict 2-groups:

  • Ruizhi Liu, Ran Luo, Yi-Nan Wang. Higher-Matter and Landau-Ginzburg Theory of Higher-Group Symmetries (2024). (arXiv:2406.03974).

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 pp-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:

With emphasis on orbifold CFTs:


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:

In relation to error-correcting codes:

  • Kohki Kawabata, Shinichiro Yahagi, Elliptic genera from classical error-correcting codes [[arXiv:2308.12592]]
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 XX are the generalized cohomology of XX 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 /24\mathbb{Z}/24 \simeq tmf 3(*)=π 3(tmf) tmf^{-3}(\ast) = \pi_3(tmf) \simeq π 3(𝕊)\pi_3(\mathbb{S}) (the third stable homotopy group of spheres):

Discussion properly via (2,1)-dimensional Euclidean field theory:

See also:

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:

Proposals 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:

On the elliptic genus of E-strings as M2-branes ending on M5-branes:

Last revised on June 7, 2024 at 11:17:08. See the history of this page for a list of all contributions to it.