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Landau-Ginzburg model

Contents

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–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)(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. (…)

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

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=2N=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 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 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 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:

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 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):

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:

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

Last revised on March 30, 2021 at 14:45:46. See the history of this page for a list of all contributions to it.