From CFT to Ramond super-quantum curves

  • Pawel Ciosmak
  • Leszek HadaszEmail author
  • Zbigniew Jaskólski
  • Masahide Manabe
  • Piotr Sulkowski
Open Access
Regular Article - Theoretical Physics


As we have shown in the previous work, using the formalism of matrix and eigenvalue models, to a given classical algebraic curve one can associate an infinite family of quantum curves, which are in one-to-one correspondence with singular vectors of a certain (e.g. Virasoro or super-Virasoro) underlying algebra. In this paper we reformulate this problem in the language of conformal field theory. Such a reformulation has several advantages: it leads to the identification of quantum curves more efficiently, it proves in full generality that they indeed have the structure of singular vectors, it enables identification of corresponding eigenvalue models. Moreover, this approach can be easily generalized to other underlying algebras. To illustrate these statements we apply the conformal field theory formalism to the case of the Ramond version of the super-Virasoro algebra. We derive two classes of corresponding Ramond super-eigenvalue models, construct Ramond super-quantum curves that have the structure of relevant singular vectors, and identify underlying Ramond super-spectral curves. We also analyze Ramond multi-Penner models and show that they lead to supersymmetric generalizations of BPZ equations.


Conformal Field Theory Matrix Models 


Open Access

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  1. [1]
    M. Aganagic, R. Dijkgraaf, A. Klemm, M. Mariño and C. Vafa, Topological strings and integrable hierarchies, Commun. Math. Phys. 261 (2006) 451 [hep-th/0312085] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  2. [2]
    R. Dijkgraaf, L. Hollands, P. Sulkowski and C. Vafa, Supersymmetric gauge theories, intersecting branes and free fermions, JHEP 02 (2008) 106 [arXiv:0709.4446] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    R. Dijkgraaf, L. Hollands and P. Sulkowski, Quantum curves and D-modules, JHEP 11 (2009) 047 [arXiv:0810.4157] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    M. Aganagic, M.C.N. Cheng, R. Dijkgraaf, D. Krefl and C. Vafa, Quantum geometry of refined topological strings, JHEP 11 (2012) 019 [arXiv:1105.0630] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    C. Kozcaz, S. Pasquetti and N. Wyllard, A & B model approaches to surface operators and Toda theories, JHEP 08 (2010) 042 [arXiv:1004.2025] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  6. [6]
    R. Dijkgraaf, H. Fuji and M. Manabe, The volume conjecture, perturbative knot invariants and recursion relations for topological strings, Nucl. Phys. B 849 (2011) 166 [arXiv:1010.4542] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  7. [7]
    G. Borot and B. Eynard, All-order asymptotics of hyperbolic knot invariants from non-perturbative topological recursion of A-polynomials, arXiv:1205.2261 [INSPIRE].
  8. [8]
    H. Fuji, S. Gukov and P. Sulkowski, Super-A-polynomial for knots and BPS states, Nucl. Phys. B 867 (2013) 506 [arXiv:1205.1515] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  9. [9]
    P. Dunin-Barkowski, M. Mulase, P. Norbury, A. Popolitov and S. Shadrin, Quantum spectral curve for the Gromov-Witten theory of the complex projective line, arXiv:1312.5336 [INSPIRE].
  10. [10]
    A. Schwarz, Quantum curves, Commun. Math. Phys. 338 (2015) 483 [arXiv:1401.1574] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  11. [11]
    P. Norbury, Quantum curves and topological recursion, Proc. Symp. Pure Math. 93 (2015) 41 [arXiv:1502.04394] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  12. [12]
    M. Mariño, Spectral Theory and Mirror Symmetry, arXiv:1506.07757 [INSPIRE].
  13. [13]
    O. Dumitrescu and M. Mulase, Lectures on the topological recursion for Higgs bundles and quantum curves, arXiv:1509.09007 [INSPIRE].
  14. [14]
    V. Bouchard and B. Eynard, Reconstructing WKB from topological recursion, arXiv:1606.04498 [INSPIRE].
  15. [15]
    V. Bouchard, N.K. Chidambaram and T. Dauphinee, Quantizing Weierstrass, arXiv:1610.00225 [INSPIRE].
  16. [16]
    R. Belliard, B. Eynard and O. Marchal, Integrable differential systems of topological type and reconstruction by the topological recursion, Annales Henri Poincaré 18 (2017) 3193 [arXiv:1610.00496] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  17. [17]
    H. Fuji, K. Iwaki, M. Manabe and I. Satake, Reconstructing GKZ via topological recursion, arXiv:1708.09365 [INSPIRE].
  18. [18]
    S. Gukov and P. Sulkowski, A-polynomial, B-model and Quantization, JHEP 02 (2012) 070 [arXiv:1108.0002] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  19. [19]
    B. Eynard and N. Orantin, Invariants of algebraic curves and topological expansion, Commun. Num. Theor. Phys. 1 (2007) 347 [math-ph/0702045] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  20. [20]
    M. Manabe and P. Sulkowski, Quantum curves and conformal field theory, Phys. Rev. D 95 (2017) 126003 [arXiv:1512.05785] [INSPIRE].ADSMathSciNetGoogle Scholar
  21. [21]
    M. Fukuma, H. Kawai and R. Nakayama, Continuum Schwinger-dyson Equations and Universal Structures in Two-dimensional Quantum Gravity, Int. J. Mod. Phys. A 6 (1991) 1385 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    R. Dijkgraaf, H.L. Verlinde and E.P. Verlinde, Loop equations and Virasoro constraints in nonperturbative 2 − D quantum gravity, Nucl. Phys. B 348 (1991) 435 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  23. [23]
    H. Awata, Y. Matsuo, S. Odake and J. Shiraishi, Collective field theory, Calogero-Sutherland model and generalized matrix models, Phys. Lett. B 347 (1995) 49 [hep-th/9411053] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  24. [24]
    P. Ciosmak, L. Hadasz, M. Manabe and P. Sulkowski, Super-quantum curves from super-eigenvalue models, JHEP 10 (2016) 044 [arXiv:1608.02596] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  25. [25]
    L. Álvarez-Gaumé, H. Itoyama, J.L. Manes and A. Zadra, Superloop equations and two-dimensional supergravity, Int. J. Mod. Phys. A 7 (1992) 5337 [hep-th/9112018] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  26. [26]
    K. Becker and M. Becker, Nonperturbative solution of the superVirasoro constraints, Mod. Phys. Lett. A 8 (1993) 1205 [hep-th/9301017] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    I.N. McArthur, The Partition function for the supersymmetric Eigenvalue model, Mod. Phys. Lett. A 8 (1993) 3355 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  28. [28]
    J.C. Plefka, Supersymmetric generalizations of matrix models, Ph.D. Thesis, Hannover University, Hannover Germany (1996) [hep-th/9601041] [INSPIRE].
  29. [29]
    G.W. Semenoff and R.J. Szabo, Fermionic matrix models, Int. J. Mod. Phys. A 12 (1997) 2135 [hep-th/9605140] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  30. [30]
    H. Itoyama and H. Kanno, Supereigenvalue model and Dijkgraaf-Vafa proposal, Phys. Lett. B 573 (2003) 227 [hep-th/0304184] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  31. [31]
    J.M. Rabin and P.G.O. Freund, Supertori are algebraic curves, Commun. Math. Phys. 114 (1988) 131 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  32. [32]
    J.M. Rabin, Superelliptic curves, J. Geom. Phys. 15 (1995) 252 [hep-th/9302105] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  33. [33]
    P. Ciosmak, L. Hadasz, M. Manabe and P. Sulkowski, Singular vector structure of quantum curves, in Proceedings of the 2016 AMS von Neumann Symposium, Charlotte U.S.A. (2017) [arXiv:1711.08031] [INSPIRE].
  34. [34]
    S. Chiantese, A. Klemm and I. Runkel, Higher order loop equations for A(r) and D(r) quiver matrix models, JHEP 03 (2004) 033 [hep-th/0311258] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  35. [35]
    H. Awata and Y. Yamada, Five-dimensional AGT Relation and the Deformed beta-ensemble, Prog. Theor. Phys. 124 (2010) 227 [arXiv:1004.5122] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory, Nucl. Phys. B 241 (1984) 333 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  37. [37]
    J. Teschner, Quantization of the Hitchin moduli spaces, Liouville theory and the geometric Langlands correspondence I, Adv. Theor. Math. Phys. 15 (2011) 471 [arXiv:1005.2846] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  38. [38]
    M. Kato and S. Matsuda, Null field construction in conformal and superconformal algebras, in Proceedings of Conformal Field Theory and solvable lattice models, Kyoto Japan (1986), pg. 205, Tsukuba KEK-TH-151 (1987).Google Scholar
  39. [39]
    R.H. Poghossian, Structure constants in the N = 1 superLiouville field theory, Nucl. Phys. B 496 (1997) 451 [hep-th/9607120] [INSPIRE].ADSCrossRefGoogle Scholar
  40. [40]
    D. Chorazkiewicz, L. Hadasz and Z. Jaskolski, Braiding properties of the N = 1 super-conformal blocks (Ramond sector), JHEP 11 (2011) 060 [arXiv:1108.2355] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  41. [41]
    A.B. Zamolodchikov and R.G. Poghossian, Operator algebra in two-dimensional superconformal field theory. (In Russian), Sov. J. Nucl. Phys. 47 (1988) 929 [INSPIRE].
  42. [42]
    O. Blondeau-Fournier, P. Mathieu, D. Ridout and S. Wood, The super-Virasoro singular vectors and Jack superpolynomials relationship revisited, Nucl. Phys. B 913 (2016) 34 [arXiv:1605.08621] [INSPIRE].ADSCrossRefGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  • Pawel Ciosmak
    • 1
  • Leszek Hadasz
    • 2
    Email author
  • Zbigniew Jaskólski
    • 3
  • Masahide Manabe
    • 4
  • Piotr Sulkowski
    • 5
    • 6
  1. 1.Faculty of Mathematics, Informatics and MechanicsUniversity of WarsawWarsawPoland
  2. 2.M. Smoluchowski Institute of PhysicsJagiellonian UniversityKrakówPoland
  3. 3.Institute of Theoretical PhysicsUniversity of WroclawWroclawPoland
  4. 4.Max-Planck-Institut für MathematikBonnGermany
  5. 5.Faculty of PhysicsUniversity of WarsawWarsawPoland
  6. 6.Walter Burke Institute for Theoretical Physics, California Institute of TechnologyPasadenaU.S.A.

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