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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 , the action functional of the LG-model is schematically of the form
where is the 2-dimensional worldsheet and – called the model’s superpotential – is a holomorphic function. (Usually is actually taken to be a Cartesian space and all the nontrivial structure is in .)
Landau–Ginzburg models have gained importance as constituting one type of QFTs that are related under homological mirror symmetry:
If the target space 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 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 -grading breaks down to a -grading. This category of D-branes of type B can be presented as the product of derived categories of singularities (which is a finite product) where is a fiber over in , what can roughly be taken as a more general definition.
A brane for a LG model is given by a matrix factorization of its superpotential.
(…) CaldararuTu
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)
Discussion as a supersymmetric sigma model in string theory on Calabi-Yau manifolds:
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 (doi:10.1016/0550-3213(89)90471-9)
Paul Howe, Peter West, Landau-Ginzburg Hamiltonians, string theory and critical phenomena, in: Proceedings of 25th International Conference on High Energy Physics, Singapore (August 2-8, 1990) [inspire:301119, pdf]
Edward Witten, Phases of Theories In Two Dimensions, Nucl.Phys.B 403:159–222, 1993 (arXiv:hep-th/9301042)
Lecture notes:
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 (arXiv:math.ag/0503632)
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 -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:
222 7 (2018) 1911-1955 [arXiv:1402.4541, doi:10.1016/j.jpaa.2017.08.014]
A proposed Landau-Ginzburg model for strict 2-groups:
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)
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
A. N. Schellekens, Nicholas P. Warner, Anomalies and modular invariance in string theory, Physics Letters B 177 (3-4), 317-323, 1986 (doi:10.1016/0370-2693(86)90760-4)
A. N. Schellekens, Nicholas P. Warner, Anomalies, characters and strings, Nuclear Physics B Volume 287, 1987, Pages 317-361 (doi:10.1016/0550-3213(87)90108-8)
Wolfgang Lerche, Bengt Nilsson, A. N. Schellekens, Nicholas P. Warner, Anomaly cancelling terms from the elliptic genus, Nuclear Physics B Volume 299, Issue 1, 28 March 1988, Pages 91-116 (doi:10.1016/0550-3213(88)90468-3)
and then strictly originates with:
Edward Witten, Elliptic genera and quantum field theory, Comm. Math. Phys. Volume 109, Number 4 (1987), 525-536. (euclid:cmp/1104117076)
Edward Witten, On the Landau-Ginzburg Description of Minimal Models, Int. J. Mod. Phys.A9:4783-4800,1994 (arXiv:hep-th/9304026)
Toshiya Kawai, Yasuhiko Yamada, Sung-Kil Yang, Elliptic Genera and Superconformal Field Theory, Nucl. Phys. B414:191-212, 1994 (arXiv:hep-th/9306096, doi:10.1016/0550-3213(94)90428-6)
Sujay K. Ashok, Jan Troost, A Twisted Non-compact Elliptic Genus, JHEP 1103:067, 2011 (arXiv:1101.1059)
Matthew Ando, Eric Sharpe, Elliptic genera of Landau-Ginzburg models over nontrivial spaces, Adv. Theor. Math. Phys. 16 (2012) 1087-1144 (arXiv:0905.1285)
Review in:
Miranda Cheng, (Mock) Modular Forms in String Theory and Moonshine, lecture notes 2016 (pdf)
Katrin Wendland, Section 2.4 in: Snapshots of Conformal Field Theory, in: Mathematical Aspects of Quantum Field Theories Mathematical Physics Studies. Springer 2015 (arXiv:1404.3108, doi:10.1007/978-3-319-09949-1_4)
With emphasis on orbifold CFTs:
Formulation via super vertex operator algebras:
Hirotaka Tamanoi, Elliptic Genera and Vertex Operator Super-Algebras, Springer 1999 (doi:10.1007/BFb0092541)
Chongying Dong, Kefeng Liu, Xiaonan Ma, Elliptic genus and vertex operator algebras, Algebr. Geom. Topol. 1 (2001) 743-762 (arXiv:math/0201135, doi:10.2140/agt.2001.1.743)
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:
Tentative interpretation as indices of Dirac-Ramond operators as would-be Dirac operators on smooth loop space:
Edward Witten, The Index Of The Dirac Operator In Loop Space, in: Elliptic Curves and Modular Forms in Algebraic Topology, Lecture Notes in Mathematics 1326, Springer (1988) 161-181 [doi:10.1007/BFb0078045, spire]
originating from:
Edward Witten, p. 92-94 in: Global anomalies in string theory, in: W. Bardeen and A. White (eds.) Symposium on Anomalies, Geometry, Topology, World Scientific (1985) 61-99 [pdf, spire:214913]
Orlando Alvarez, T. P. Killingback, Michelangelo Mangano, Paul Windey, The Dirac-Ramond operator in string theory and loop space index theorems, Nuclear Phys. B Proc. Suppl., 1A:189–215, 1987, in: Nonperturbative methods in field theory, 1987 (doi"10.1016/0920-5632(87)90110-1)
Orlando Alvarez, T. P. Killingback, Michelangelo Mangano, Paul Windey, String theory and loop space index theorems, Comm. Math. Phys., 111(1):1–10, 1987 (euclid:cmp/1104159462)
Gregory Landweber, Dirac operators on loop space, PhD thesis (Harvard 1999) (pdf)
Orlando Alvarez, Paul Windey, Analytic index for a family of Dirac-Ramond operators, Proc. Natl. Acad. Sci. USA, 107(11):4845–4850, 2010 (arXiv:0904.4748)
Tentative formulation via conformal nets:
The resulting suggestion that, roughly, deformation-classes (concordance classes) of 2d SCFTs with target space are the generalized cohomology of 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 (the third stable homotopy group of spheres):
Davide Gaiotto, Theo Johnson-Freyd, Edward Witten, p. 17 of: A Note On Some Minimally Supersymmetric Models In Two Dimensions, (arXiv:1902.10249) in S. Novikov et al. Integrability, Quantization, and Geometry: II. Quantum Theories and Algebraic Geometry, Proc. Symposia Pure Math., 103(2), 2021 (ISBN: 978-1-4704-5592-7)
Davide Gaiotto, Theo Johnson-Freyd, Mock modularity and a secondary elliptic genus (arXiv:1904.05788)
Theo Johnson-Freyd, Topological Mathieu Moonshine (arXiv:2006.02922)
Discussion properly via (2,1)-dimensional Euclidean field theory:
Daniel Berwick-Evans, How do field theories detect the torsion in topological modular forms? arXiv:2303.09138
Daniel Berwick-Evans, How do field theories detect the torsion in topological modular forms?, talk at QFT and Cobordism, CQTS (Mar 2023) web, video:YT
See also:
Ying-Hsuan Lin, Du Pei, Holomorphic CFTs and topological modular forms [arXiv:2112.10724]
Jan Albert, Justin Kaidi, Ying-Hsuan Lin, Topological modularity of Supermoonshine arXiv:2210.14923
Yuji Tachikawa, Mayuko Yamashita, Kazuya Yonekura, Remarks on mod-2 elliptic genus arXiv:2302.07548
Yuji Tachikawa, Hao Y. Zhang, On a -valued discrete topological term in 10d heterotic string theories [arXiv:2403.08861]
Theo Johnson-Freyd, Mayuko Yamashita, On the 576-fold periodicity of the spectrum SQFT: The proof of the lower bound via the Anderson duality pairing [arXiv:2404.06333]
Vivek Saxena, A T-Duality of Non-Supersymmetric Heterotic Strings and an implication for Topological Modular Forms [arXiv:2405.19409]
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:
Jacques Distler, Eric Sharpe, section 8.5 of Heterotic compactifications with principal bundles for general groups and general levels, Adv. Theor. Math. Phys. 14:335-398, 2010 (arXiv:hep-th/0701244)
Matthew Ando, Equivariant elliptic cohomology and the Fibered WZW models of Distler and Sharpe, talk 2007 (lecture notes pdf)
Proposals on physics aspects of lifting the Witten genus to topological modular forms:
Yuji Tachikawa, Topological modular forms and the absence of a heterotic global anomaly, Progress of Theoretical and Experimental Physics, 2022 4 (2022) 04A107 arXiv:2103.12211, doi:10.1093/ptep/ptab060
Yuji Tachikawa, Mayuko Yamashita, Topological modular forms and the absence of all heterotic global anomalies, Comm. Math. Phys. 402 (2023) 1585-1620 arXiv:2108.13542, doi:10.1007/s00220-023-04761-2
Yuji Tachikawa, Mayuko Yamashita, Anderson self-duality of topological modular forms, its differential-geometric manifestations, and vertex operator algebras arXiv:2305.06196
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:
Davide Gaiotto, Andrew Strominger, Xi Yin, The M5-Brane Elliptic Genus: Modularity and BPS States, JHEP 0708:070, 2007 (hep-th/0607010)
Davide Gaiotto, Xi Yin, Examples of M5-Brane Elliptic Genera, JHEP 0711:004, 2007 (arXiv:hep-th/0702012)
Further discussion in:
Murad Alim, Babak Haghighat, Michael Hecht, Albrecht Klemm, Marco Rauch, Thomas Wotschke, Wall-crossing holomorphic anomaly and mock modularity of multiple M5-branes, Comm. Math. Phys. 339 (2015) 773–814 arXiv:1012.1608, doi:10.1007/s00220-015-2436-3
Sergei Gukov, Du Pei, Pavel Putrov, Cumrun Vafa, 4-manifolds and topological modular forms, J. High Energ. Phys. 2021 84 (2021) arXiv:1811.07884, doi:10.1007/JHEP05(2021)084, spire:1704312
On the elliptic genus of M-strings inside M5-branes:
Stefan Hohenegger, Amer Iqbal, M-strings, Elliptic Genera and String Amplitudes, Fortschritte der PhysikVolume 62, Issue 3 (arXiv:1310.1325)
Stefan Hohenegger, Amer Iqbal, Soo-Jong Rey, M String, Monopole String and Modular Forms, Phys. Rev. D 92, 066005 (2015) (arXiv:1503.06983)
M. Nouman Muteeb, Domain walls and M2-branes partition functions: M-theory and ABJM Theory (arXiv:2010.04233)
Kimyeong Lee, Kaiwen Sun, Xin Wang, Twisted Elliptic Genera [arXiv:2212.07341]
On the elliptic genus of E-strings as wrapped M5-branes:
J. A. Minahan, D. Nemeschansky, Cumrun Vafa, N. P. Warner, E-Strings and 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)
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.