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Communications in Mathematical Physics

, Volume 357, Issue 2, pp 597–629 | Cite as

N=2 Minimal Conformal Field Theories and Matrix Bifactorisations of x d

  • Alexei Davydov
  • Ana Ros Camacho
  • Ingo RunkelEmail author
Article
  • 65 Downloads

Abstract

We establish an action of the representations of N =  2-superconformal symmetry on the category of matrix factorisations of the potentials x d and x d y d , 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 x d y d . 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.

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References

  1. Ada.
    Adamović, D.: Representations of N=2 superconformal vertex algebra Int. Math. Res. Not., 61–79 (1999). arXiv:math/9809141
  2. Ade.
    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)ADSCrossRefGoogle Scholar
  3. BHLS.
    Brunner I., Herbst M., Lerche W., Scheuner B.: Landau–Ginzburg realization of open string TFT. JHEP 0611, 043 (2003)ADSGoogle Scholar
  4. BF.
    Behr, N., Fredenhagen, S.: Matrix factorisations for rational boundary conditions by defect fusion. JHEP 1505, 055 (2015). arXiv:1407.7254 [hep-th]
  5. BK.
    Bakalov B., Kirillov A.A.: Lectures on Tensor Categories and Modular Functors. American Mathematical Society, Providence (2001)zbMATHGoogle Scholar
  6. BR.
    Brunner, I., Roggenkamp, D.: B-type defects in Landau–Ginzburg models. JHEP 08, 093 (2007). arXiv:0707.0922 [hep-th]
  7. Bu.
    Buchweitz, R.-O.: Maximal Cohen–Macaulay modules and Tate-cohomology over Gorenstein rings; unpublished manuscript.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. Ca.
    Carpi S., Hillier R., Kawahigashi Y., Longo R., Xu F.: N=2 superconformal nets. Commun. Math. Phys. 336, 1285–1328 (2015) arXiv:1207.2398 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. CM1.
    Carqueville N., Murfet D.: Computing Khovanov–Rozansky homology and defect fusion. Topology 14, 489–537 (2014) arXiv:1108.1081 MathSciNetzbMATHGoogle Scholar
  10. CM2.
    Carqueville N., Murfet D.: Adjunctions and defects in Landau–Ginzburg models. Adv. Math. 289, 480–566 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. CR1.
    Carqueville N., Runkel I.: On the monoidal structure of matrix bifactorisations. J. Phys. A 43, 275401 (2010) arXiv:1208.1481 MathSciNetCrossRefzbMATHGoogle Scholar
  12. CR2.
    Carqueville N., Runkel I.: Rigidity and defect actions in Landau–Ginzburg models. Commum. Math. Phys. 310, 135–179 (2012) arXiv:0909.4381 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. CR3.
    Carqueville N., Runkel I.: Orbifold completion of defect bicategories. Quantum Topol. 7 203, 203–279 (2016) arXiv:1006.5609 MathSciNetCrossRefzbMATHGoogle Scholar
  14. CRCR.
    Carqueville N., Ros Camacho A., Runkel I.: Orbifold equivalent potentials. J. Pure Appl. Algebra 220, 759–781 (2016) arXiv:1210.6363 MathSciNetCrossRefzbMATHGoogle Scholar
  15. Da.
    Davydov, A.: Lectures on Temperley-Lieb categories, in preparation.Google Scholar
  16. De.
    Deligne, P.: Catégories Tannakiennes. In: Grothendieck Festschrift, v. II, Birkhäuser Progress in Mathematics, vol. 87, pp. 111–195 (1990)Google Scholar
  17. DM.
    Dyckerhoff T., Murfet D.: Pushing forward matrix factorisations. Duke Math. J. 162, 1249–1311 (2013) [arXiv:1311.3354]MathSciNetCrossRefzbMATHGoogle Scholar
  18. DPYZ.
    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 ADSMathSciNetCrossRefGoogle Scholar
  19. EG.
    Eholzer W., Gaberdiel R.M.: Unitarity of rational N = 2 superconformal theories. Commun. Math. Phys. 186, 61–85 (1997) ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. Ei.
    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 MathSciNetCrossRefzbMATHGoogle Scholar
  21. EO.
    Etingof P., Ostrik V.: Module categories over representations of SL q(2) and graphs. Math. Res. Lett. 11, 103–114 (2004) MathSciNetCrossRefzbMATHGoogle Scholar
  22. FFRS1.
    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 MathSciNetCrossRefzbMATHGoogle Scholar
  23. FFRS2.
    Fröhlich J., Fuchs J., Runkel I., Schweigert C.: Correspondences of ribbon categories. Adv. Math. 199, 192–329 (2006) arXiv:hep-th/0309269 MathSciNetCrossRefzbMATHGoogle Scholar
  24. FFRS3.
    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 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. FRS.
    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 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. GW.
    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. HL.
    Herbst M., Lazaroiu C.I.: Localization and traces in open-closed topological Landau–Ginzburg models. JHEP 0505, 044 (2005) arXiv:hep-th/0204148 ADSMathSciNetCrossRefGoogle Scholar
  28. Ho.
    Hori, K., Katz, S., Klemm, A., Pandharipande, R., Thomas, R., Vafa, C., Vakil, R., Zaslow, E.: Mirror symmetry. In: Clay Mathematics Monographs. 1 (2003)Google Scholar
  29. HW.
    Hori, K., Walcher, J.: F-term equations near Gepner points. JHEP 0501, 008 (2005). arXiv:hep-th/0404184
  30. JS1.
    Joyal, A., Street, R.: Braided tensor categories. Adv. Math. 102, 20–78 (1993). arXiv:hep-th/0404196
  31. JS2.
    Joyal A., Street R.: Tortile Yang–Baxter operators in tensor categories. J. Pure Appl. Algebra 71, 43–51 (1991)arXiv:hep-th/0404184 MathSciNetCrossRefzbMATHGoogle Scholar
  32. KL.
    Kapustin A., Li Y.: D-branes in Landau–Ginzburg Models and algebraic geometry. JHEP 0312, 005 (2003) ADSMathSciNetCrossRefGoogle Scholar
  33. KO.
    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 MathSciNetCrossRefzbMATHGoogle Scholar
  34. KR.
    Khovanov M., Rozansky L.: Matrix factorizations and link homology. Fundamenta Mathematicae 199, 1–91 (2008) arXiv:math/0101219 MathSciNetCrossRefzbMATHGoogle Scholar
  35. LVW.
    Lerche W., Vafa C., Warner N.P.: Chiral rings in N = 2 superconformal theories. Nucl. Phys. B 324, 427–474 (1989)arXiv:math/0401268 ADSMathSciNetCrossRefGoogle Scholar
  36. Ma.
    Martinec E.J.: Algebraic geometry and effective lagrangians. Phys. Lett. B 217, 431–437 (1989)ADSMathSciNetCrossRefGoogle Scholar
  37. Mu.
    Murfet, D.: Cut systems and matrix factorisations I. arXiv:1402.4541
  38. Mü.
    Müger M.: On the structure of modular categories. Proc. Lond. Math. Soc. 87, 291–308 (2003) MathSciNetCrossRefzbMATHGoogle Scholar
  39. Pa.
    Pareigis B.: On braiding and dyslexia. J. Algebra 171, 413–425 (1995)arXiv:math/0201017 MathSciNetCrossRefzbMATHGoogle Scholar
  40. Tu.
    Turaev V.: Quantum Invariants of Knots and 3-Manifolds, de Gruyter Studies in Mathematics, 18. Walter de Gruyter & Co, Berlin (1994)Google Scholar
  41. VW.
    Vafa C., Warner N.: Catastrophes and the classification of conformal theories. Phys. Lett. B 218, 51–58 (1989)ADSMathSciNetCrossRefGoogle Scholar
  42. Wu.
    Wu, H.: A colored sl(N)-homology for links in S 3. arXiv:0907.0695
  43. Yo1.
    Yoshino, Y.: Cohen–Macaulay modules over Cohen–Macaulay rings. London Mathematical Society Lecture Note Series, vol. 146. Cambridge University Press, CambridgeGoogle Scholar
  44. Yo2.
    Yoshino Y.: Tensor product of matrix factorisations. Nagoya Math. J. 152, 39–56 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  45. Za.
    Zamolodchikov A.B.: “Irreversibility” of the flux of the renormalization group in a 2d field theory. JETP Lett. 43, 730–732 (1986)ADSMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsOhio UniversityAthensUSA
  2. 2.Mathematisch InstituutUniversiteit UtrechtUtrechtThe Netherlands
  3. 3.Fachbereich MathematikUniversität HamburgHamburgGermany

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