nLab Landau-Ginzburg model

Idea

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 $(X,g)$, the action functional of the LG-model is schematically of the form

$$S_{LB} : (\phi : \Sigma \to X) \mapsto \int_\Sigma \left( \vert \nabla \Phi \vert^2 + \vert (\nabla W)(\phi) \vert^2 + fermionic\;terms \right) d \mu \,,$$

where $\Sigma$ is the 2-dimensional worldsheet and $W : X \to \mathbb{C}$ – 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–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 $\mathbf{Z}$-grading breaks down to a $\mathbf{Z}_2$-grading. This category of D-branes of type B can be presented as the product of derived categories of singularities $\prod_\lambda D^b_{sg}(X_\lambda)$ (which is a finite product) where $X_\lambda$ is a fiber over $\lambda$ in $\mathbf{C}$, what can roughly be taken as a more general definition.

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

Related concepts

• 2d TQFT, TCFT

• holographic superconductor?

References

In superconductivity

Discussion as a model for superconductors:

The original article:

• В. Л. Гинзбург, Л. Д. Ландау, К теории сверхпроводимости, ЖЭТФ 20 (1950), 1064.

English translation:

• Vitaly Ginzburg, Lev Landau, On the Theory of Superconductivity, In: On Superconductivity and Superfluidity Springer (2009) (doi:1007/978-3-540-68008-6_4)

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:

• 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 $N=2$ Theories In Two Dimensions, Nucl.Phys.B 403:159–222, 1993 (arXiv:hep-th/9301042)

Lecture notes:

• Edward Witten, Dynamical aspects of QFT, Lecture 15: The Landau–Ginzburg description of $N=2$ 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

• Anton Kapustin, Yi Li, D-Branes in Landau–Ginzburg models and algebraic geometry, J. High Energy Phys. 12 (2003), 005 arXiv:hep-th/0210296

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 $A_\infty$-algebras and Landau–Ginzburg models (pdf)

Defects

General defects of B-twisted affine LG models were first discussed in

• Ilka Brunner, Daniel Roggenkamp, B-type defects in Landau–Ginzburg models, JHEP 0708 (2007) 093, (arXiv:0707.0922)

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 289 (2016) 480-566 [doi:10.1016/j.aim.2015.03.033, arXiv:1208.1481]

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

• Daniel Murfet, The cut operation on matrix factorisations, Journal of Pure and Applied Algebra

  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:

• 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

• 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

• Daniel Murfet, Computing with cut systems (arXiv:1402.4541)

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 $p$-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

• 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 $N=2$ 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, 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:

• Sujay K. Ashok, Jan Troost, Orbifolded Elliptic Genera of Non-Compact Models [arXiv:2405.08370]

Formulations

Via super vertex operator algebra

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

• Pokman Cheung, The Witten genus and vertex algebras (arXiv:0811.1418)

based on chiral differential operators:

• Vassily Gorbounov, Fyodor Malikov, Vadim Schechtman, Gerbes of chiral differential operators, Math. Res. Lett. 7(1), 55–66 (2000) (arXiv:math/9906117, arXiv:math/0003170, arXiv:math/0005201)

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:

• 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)

Via conformal nets

Tentative formulation via conformal nets:

• Chris Douglas, André Henriques, Topological modular forms and conformal nets, in Hisham Sati, Urs Schreiber (eds.), Mathematical Foundations of Quantum Field and Perturbative String Theory, Proceedings of Symposia in Pure Mathematics, AMS (2011) (arXiv:1103.4187, doi:10.1090/pspum/083)

Conjectural interpretation in tmf-cohomology

The resulting suggestion that, roughly, deformation-classes (concordance classes) of 2d SCFTs with target space $X$ are the generalized cohomology of $X$ with coefficients in the spectrum of topological modular forms (tmf):

• Stephan Stolz, Peter Teichner, Supersymmetric field theories and generalized cohomology, in Hisham Sati, Urs Schreiber (eds.), Mathematical foundations of Quantum field theory and String theory, Proceedings of Symposia in Pure Mathematics, Volume 83, AMS 2011 (arXiv:1108.0189)

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:

• Davide Gaiotto, Theo Johnson-Freyd, Section 5 of: Holomorphic SCFTs with small index, Canadian Journal of Mathematics (arXiv:1811.00589, doi:10.4153/S0008414X2100002X)

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 $\mathbb{Z}/24$ $\simeq$ $tmf^{-3}(\ast) = \pi_3(tmf)$ $\simeq$ $\pi_3(\mathbb{S})$ (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 $\mathbb{Z}_3$-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]

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:

• 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$]$

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

• Juan Maldacena, Andrew Strominger, Edward Witten, Black Hole Entropy in M-Theory, JHEP 9712:002, 1997 (arXiv:hep-th/9711053)

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$]$

M-string elliptic genus

On the elliptic genus of M-strings inside M5-branes:

• Stefan Hohenegger, Amer Iqbal, M-strings, Elliptic Genera and $\mathcal{N}=4$ 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]

E-string elliptic genus

On the elliptic genus of E-strings as wrapped M5-branes:

• J. A. Minahan, D. Nemeschansky, Cumrun Vafa, N. P. Warner, E-Strings and $N=4$ 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:

• Joonho Kim, Seok Kim, Kimyeong Lee, Jaemo Park, Cumrun Vafa, Elliptic Genus of E-strings, JHEP 1709 (2017) 098 (arXiv:1411.2324)