Skip to main content
SpringerLink
Log in

N=2 Minimal Conformal Field Theories and Matrix Bifactorisations of xd

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We establish an action of the representations of N =  2-superconformal symmetry on the category of matrix factorisations of the potentials xd and xd−yd, for d odd. More precisely we prove a tensor equivalence between (a) the category of Neveu–Schwarz-type representations of the N =  2 minimal super vertex operator algebra at central charge 3–6/d, and (b) a full subcategory of graded matrix factorisations of the potential xd−yd. The subcategory in (b) is given by permutation-type matrix factorisations with consecutive index sets. The physical motivation for this result is the Landau–Ginzburg/conformal field theory correspondence, where it amounts to the equivalence of a subset of defects on both sides of the correspondence. Our work builds on results by Brunner and Roggenkamp [BR], where an isomorphism of fusion rules was established.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adamović, D.: Representations of N=2 superconformal vertex algebra Int. Math. Res. Not., 61–79 (1999). arXiv:math/9809141

  2. Ademollo M., Brink L., D’Adda A., D’Auria R., Napolitano E., Sciuto S., Del Giudice E., Di Vecchia P., Ferrara S., Gliozzi F., Musto R., Pettorino R., Schwarz J.H.: Dual string models with nonabelian color and flavor symmetries. Nucl. Phys. B 111, 77–110 (1976)

    Article  ADS  Google Scholar 

  3. Brunner I., Herbst M., Lerche W., Scheuner B.: Landau–Ginzburg realization of open string TFT. JHEP 0611, 043 (2003)

    ADS  Google Scholar 

  4. Behr, N., Fredenhagen, S.: Matrix factorisations for rational boundary conditions by defect fusion. JHEP 1505, 055 (2015). arXiv:1407.7254 [hep-th]

  5. Bakalov B., Kirillov A.A.: Lectures on Tensor Categories and Modular Functors. American Mathematical Society, Providence (2001)

    MATH  Google Scholar 

  6. Brunner, I., Roggenkamp, D.: B-type defects in Landau–Ginzburg models. JHEP 08, 093 (2007). arXiv:0707.0922 [hep-th]

  7. Buchweitz, R.-O.: Maximal Cohen–Macaulay modules and Tate-cohomology over Gorenstein rings; unpublished manuscript. (1987). https://tspace.library.utoronto.ca/handle/1807/16682

  8. Carpi S., Hillier R., Kawahigashi Y., Longo R., Xu F.: N=2 superconformal nets. Commun. Math. Phys. 336, 1285–1328 (2015) arXiv:1207.2398

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Carqueville N., Murfet D.: Computing Khovanov–Rozansky homology and defect fusion. Topology 14, 489–537 (2014) arXiv:1108.1081

    MathSciNet  MATH  Google Scholar 

  10. Carqueville N., Murfet D.: Adjunctions and defects in Landau–Ginzburg models. Adv. Math. 289, 480–566 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. Carqueville N., Runkel I.: On the monoidal structure of matrix bifactorisations. J. Phys. A 43, 275401 (2010) arXiv:1208.1481

    Article  MathSciNet  MATH  Google Scholar 

  12. Carqueville N., Runkel I.: Rigidity and defect actions in Landau–Ginzburg models. Commum. Math. Phys. 310, 135–179 (2012) arXiv:0909.4381

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Carqueville N., Runkel I.: Orbifold completion of defect bicategories. Quantum Topol. 7 203, 203–279 (2016) arXiv:1006.5609

    Article  MathSciNet  MATH  Google Scholar 

  14. Carqueville N., Ros Camacho A., Runkel I.: Orbifold equivalent potentials. J. Pure Appl. Algebra 220, 759–781 (2016) arXiv:1210.6363

    Article  MathSciNet  MATH  Google Scholar 

  15. Davydov, A.: Lectures on Temperley-Lieb categories, in preparation.

  16. Deligne, P.: Catégories Tannakiennes. In: Grothendieck Festschrift, v. II, Birkhäuser Progress in Mathematics, vol. 87, pp. 111–195 (1990)

  17. Dyckerhoff T., Murfet D.: Pushing forward matrix factorisations. Duke Math. J. 162, 1249–1311 (2013) [arXiv:1311.3354]

    Article  MathSciNet  MATH  Google Scholar 

  18. Di Vecchia P., Petersen J. L., Yu M., Zheng H. B.: Explicit construction of unitary representations of the N = 2 superconformal algebra. Phys. Lett. B 174, 280–284 (1986) arXiv:1102.2957

    Article  ADS  MathSciNet  Google Scholar 

  19. Eholzer W., Gaberdiel R.M.: Unitarity of rational N = 2 superconformal theories. Commun. Math. Phys. 186, 61–85 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. Eisenbud D.: Homological algebra on a complete intersection, with an application to group representations. Trans. Am. Math. Soc. 260, 35–64 (1980) arXiv:hep-th/9601163

    Article  MathSciNet  MATH  Google Scholar 

  21. Etingof P., Ostrik V.: Module categories over representations of SL q (2) and graphs. Math. Res. Lett. 11, 103–114 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  22. Fröhlich J., Fuchs J., Runkel I., Schweigert C.: Algebras in tensor categories and coset conformal field theories. Fortschr. Phys. 52, 672–677 (2004) arXiv:math/0302130

    Article  MathSciNet  MATH  Google Scholar 

  23. Fröhlich J., Fuchs J., Runkel I., Schweigert C.: Correspondences of ribbon categories. Adv. Math. 199, 192–329 (2006) arXiv:hep-th/0309269

    Article  MathSciNet  MATH  Google Scholar 

  24. Fröhlich J., Fuchs J., Runkel I., Schweigert C.: Duality and defects in rational conformal field theory. Nucl. Phys. B 763, 354–430 (2007) arXiv:math/0309465

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. Fuchs J., Runkel I., Schweigert C.: TFT construction of RCFT correlators. I: partition functions. Nucl. Phys. B 646, 353–497 (2002) arXiv:hep-th/0607247

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. Goodman, F.M., Wenzl, H.: Ideals in the Temperley-Lieb category, Appendix to A magnetic model with a possible Chern–Simons phase by M. Freedman. Commun. Math. Phys. 234, 129–183 (2003). arXiv:math/0206301

  27. Herbst M., Lazaroiu C.I.: Localization and traces in open-closed topological Landau–Ginzburg models. JHEP 0505, 044 (2005) arXiv:hep-th/0204148

    Article  ADS  MathSciNet  Google Scholar 

  28. Hori, K., Katz, S., Klemm, A., Pandharipande, R., Thomas, R., Vafa, C., Vakil, R., Zaslow, E.: Mirror symmetry. In: Clay Mathematics Monographs. 1 (2003)

  29. Hori, K., Walcher, J.: F-term equations near Gepner points. JHEP 0501, 008 (2005). arXiv:hep-th/0404184

  30. Joyal, A., Street, R.: Braided tensor categories. Adv. Math. 102, 20–78 (1993). arXiv:hep-th/0404196

  31. Joyal A., Street R.: Tortile Yang–Baxter operators in tensor categories. J. Pure Appl. Algebra 71, 43–51 (1991)arXiv:hep-th/0404184

    Article  MathSciNet  MATH  Google Scholar 

  32. Kapustin A., Li Y.: D-branes in Landau–Ginzburg Models and algebraic geometry. JHEP 0312, 005 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  33. Kirillov A.A., Ostrik V.: On q-analog of McKay correspondence and ADE classification of sl(2) conformal field theories. Adv. Math. 171, 183–227 (2002) arXiv:hep-th/0210296

    Article  MathSciNet  MATH  Google Scholar 

  34. Khovanov M., Rozansky L.: Matrix factorizations and link homology. Fundamenta Mathematicae 199, 1–91 (2008) arXiv:math/0101219

    Article  MathSciNet  MATH  Google Scholar 

  35. Lerche W., Vafa C., Warner N.P.: Chiral rings in N = 2 superconformal theories. Nucl. Phys. B 324, 427–474 (1989)arXiv:math/0401268

    Article  ADS  MathSciNet  Google Scholar 

  36. Martinec E.J.: Algebraic geometry and effective lagrangians. Phys. Lett. B 217, 431–437 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  37. Murfet, D.: Cut systems and matrix factorisations I. arXiv:1402.4541

  38. Müger M.: On the structure of modular categories. Proc. Lond. Math. Soc. 87, 291–308 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  39. Pareigis B.: On braiding and dyslexia. J. Algebra 171, 413–425 (1995)arXiv:math/0201017

    Article  MathSciNet  MATH  Google Scholar 

  40. Turaev V.: Quantum Invariants of Knots and 3-Manifolds, de Gruyter Studies in Mathematics, 18. Walter de Gruyter & Co, Berlin (1994)

    Google Scholar 

  41. Vafa C., Warner N.: Catastrophes and the classification of conformal theories. Phys. Lett. B 218, 51–58 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  42. Wu, H.: A colored sl(N)-homology for links in S 3. arXiv:0907.0695

  43. Yoshino, Y.: Cohen–Macaulay modules over Cohen–Macaulay rings. London Mathematical Society Lecture Note Series, vol. 146. Cambridge University Press, Cambridge

  44. Yoshino Y.: Tensor product of matrix factorisations. Nagoya Math. J. 152, 39–56 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  45. Zamolodchikov A.B.: “Irreversibility” of the flux of the renormalization group in a 2d field theory. JETP Lett. 43, 730–732 (1986)

    ADS  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Ohio University, Athens, OH, 45701, USA

    Alexei Davydov

  2. Mathematisch Instituut, Universiteit Utrecht, Postbus 80.010, 3508 TA, Utrecht, The Netherlands

    Ana Ros Camacho

  3. Fachbereich Mathematik, Universität Hamburg, Bundesstraße 55, 20146, Hamburg, Germany

    Ingo Runkel

Authors
  1. Alexei Davydov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ana Ros Camacho
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ingo Runkel
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ingo Runkel.

Additional information

Communicated by Y. Kawahigashi

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Davydov, A., Camacho, A.R. & Runkel, I. N=2 Minimal Conformal Field Theories and Matrix Bifactorisations of xd. Commun. Math. Phys. 357, 597–629 (2018). https://doi.org/10.1007/s00220-018-3086-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-018-3086-z

Search

Navigation