nLab
Landau-Ginzburg model

Idea

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

$$S_{\mathrm{LB}}:(\varphi :\Sigma \to X)\mapsto {\int }_{\Sigma }(\mid (\nabla \Phi {\mid }^2+\mid (\nabla W)(\varphi ){\mid }^2+\mathrm{fermionic}\mathrm{terms})d\mu ,$$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 $W:X\to ℂ$ – 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-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.

(…)

$(\mathrm{\infty },1)$-Categories of branes

(…) curved dg-algebra

(…) CaldararuTu

References

For instance

• Anton Kapustin, Yi Li, D-Branes in Landau-Ginzburg models and algebraic geometry, arXiv:hep-th/0210296

The derived category of D-branes in type B LG-models is discussed in

• Dmitri Orlov, Derived categories of coherent sheaves and triangulated categories of singularities, arXiv:math.ag/0503632

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