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.
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].
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].
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].
I.N. McArthur, The Partition function for the supersymmetric Eigenvalue model, Mod. Phys. Lett. A 8 (1993) 3355 [INSPIRE].
J.M. Rabin and P.G.O. Freund, Supertori are algebraic curves, Commun. Math. Phys. 114 (1988) 131 [INSPIRE].
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].
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].
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).
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].
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].
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
ArXiv ePrint: 1712.07354
About this article
Cite this article
Ciosmak, P., Hadasz, L., Jaskólski, Z. et al. From CFT to Ramond super-quantum curves. J. High Energ. Phys. 2018, 133 (2018). https://doi.org/10.1007/JHEP05(2018)133
- Conformal Field Theory
- Matrix Models