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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
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-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.
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A brane for a LG model is given by a matrix factorization of its superpotential.
(…) CaldararuTu
Original articles are
Cumrun Vafa, Nicholas Warner, Catastrophes and the Classification of Conformal Theories, Phys.Lett. B218 (1989) 51 (doi:10.1016/0370-2693(89)90473-5)
Brian Greene, Cumrun Vafa, Calabi-Yau Manifolds and Renormalization Group Flows, Nucl.Phys. B324 (1989) 371
Edward Witten, Phases of $N=2$ Theories In Two Dimensions, Nucl.Phys.B403:159-222,1993 (arXiv:hep-th/9301042)
Lecture notes include
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.A9:4783-4800,1994 (arXiv:hep-th/9304026)
Toshiya Kawai, Yasuhiko Yamada, Sung-Kil Yang, Elliptic Genera and $N=2$ Superconformal Field Theory (arXiv:hep-th/9306096)
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)
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
Orbifolds of defects are studied in
Ilka Brunner, Daniel Roggenkamp, Defects and Bulk Perturbations of Boundary Landau-Ginzburg Orbifolds, JHEP 0804 (2008) 001, (arXiv:0712.0188)
Nils Carqueville, Ingo Runkel, Orbifold completion of defect bicategories, (arXiv:1210.6363)
Ilka Brunner, Nils Carqueville, Daniel Plencner, Orbifolds and topological defects, Comm. Math. Phys. 332 (2014), 669-712, (arXiv:1307.3141)
Ilka Brunner, Nils Carqueville, Daniel Plencner, Discrete torsion defects, Comm. Math. Phys. 337 (2015), 429-453, (arXiv:1404.7497)
A relation to linear logic and the geometry of interaction is in
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)
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Last revised on November 21, 2019 at 03:01:50. See the history of this page for a list of all contributions to it.