Singular vector structure of quantum
                    curves

Ciosmak, Paweł; Hadasz, Leszek; Manabe, Masahide; Sułkowski, Piotr

doi:10.1090/pspum/100/01766

Cited by 5 publications

(10 citation statements)

References 13 publications

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“…• In this spirit, for a large class of spectral curves, the CEO topological recursion can be used to reconstruct the quantum curve and its associated wave-function [15,46]. Can our supersymmetric generalization of topological recursion be used to study super quantum curves [26][27][28] ?…”

Section: Conclusion and Open Questionsmentioning

confidence: 99%

“…However, such relations are obscure, even though corresponding supersymmetric structures in matrix models are known. Indeed, supersymmetric generalizations of matrix models, referred to supereigenvalue models, have been introduced and discussed some time ago [3,8], and also more recently [20,[26][27][28]61]. By construction, loop equations for such supereigenvalue models can be rewritten in the form of super-Virasoro constraints.…”

Section: Introductionmentioning

confidence: 99%

“…On the one hand, for large classes of spectral curves it was shown that the topological recursion can be used to reconstruct the wave-function associated to a given classical curve, and at the same time the corresponding quantum curve [15,46,58]. On the other hand, various types of supersymmetric quantum curves (also called super quantum curves) have been constructed recently in the formalism of supereigenvalue models, and reinterpreted from the conformal field theory point of view [26][27][28]. Therefore one might expect that super quantum Airy structures provide a more general framework to construct super quantum curves, and to develop their theory further.…”

Section: Introductionmentioning

confidence: 99%

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Super Quantum Airy Structures

Bouchard

Ciosmak

Hadasz

et al. 2020

Commun. Math. Phys.

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We introduce super quantum Airy structures, which provide a supersymmetric generalization of quantum Airy structures. We prove that to a given super quantum Airy structure one can assign a unique set of free energies, which satisfy a supersymmetric generalization of the topological recursion. We reveal and discuss various properties of these supersymmetric structures, in particular their gauge transformations, classical limit, peculiar role of fermionic variables, and graphical representation of recursion relations. Furthermore, we present various examples of super quantum Airy structures, both finite-dimensional—which include well known superalgebras and super Frobenius algebras, and whose classification scheme we also discuss—as well as infinite-dimensional, that arise in the realm of vertex operator super algebras.

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Section: Conclusion and Open Questionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Super Quantum Airy Structures

Bouchard

Ciosmak

Hadasz

et al. 2020

Commun. Math. Phys.

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“…In this section we shall define correlation functions and derive loop equations for supereigenvalue models in the Ramond sector. See [10,16,17] and references therein for analogous arguments in the NS sector. [18] also discuss loop equations in the Ramond sector in the context of super quantum curves.…”

Section: Jhep10(2019)286 3 Loop Equationsmentioning

confidence: 99%

“…Since the explicit definition is given, similar computational techniques described in this paper might be applicable to these models. Also, [16][17][18] discussed supereigenvalue models in length in the context of super quantum curves. It is certainly worth investigating more on those aspects.…”

Section: ?mentioning

confidence: 99%

Topological recursion in the Ramond sector

Osuga

2019

J. High Energ. Phys.

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We investigate supereigenvalue models in the Ramond sector and their recursive structure. We prove that the free energy truncates at quadratic order in Grassmann coupling constants, and consider super loop equations of the models with the assumption that the 1/N expansion makes sense. Subject to this assumption, we obtain the associated genus-zero algebraic curve with two ramification points (one regular and the other irregular) and also the supersymmetric partner polynomial equation. Starting with these polynomial equations, we present a recursive formalism that computes all the correlation functions of these models. Somewhat surprisingly, correlation functions obtained from the new recursion formalism have no poles at the irregular ramification point due to a supersymmetric correction-the new recursion may lead us to a further development of supersymmetric generalizations of the Eynard-Orantin topological recursion.

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From CFT to Ramond super-quantum curves

Ciosmak

Hadasz

Jaskólski

et al. 2018

J. High Energ. Phys.

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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. CALT-2017-070 7. Ramond-R super-eigenvalue model and super-quantum curves 56 A. Proofs and computations 63 A.1 Computations in the Ramond-NS sector: the supercurrent S(y) 63 A.2 Computations in the Ramond-NS sector: the energy-momentum tensor T (y) 66 A.3 Computations in the Ramond-NS super-eigenvalue model 67 -1 -The above results have been generalized to a supersymmetric case in [24], by considering (β-deformed) super-eigenvalue models for the Neveu-Schwarz sector [25][26][27][28][29][30]. These models generalize eigenvalue representation of hermitian matrix models in such a way, that the underlying algebra takes form of the Neveu-Schwarz version of the super-Virasoro algebra; in particular corresponding loop equations can be rewritten as super-Virasoro constraints. Consequently, to a super-eigenvalue model one can associate an infinite family of super-quantum curves, which have the structure of Neveu-Schwarz singular vectors of the super-Virasoro algebra. In the classical limit, such super-quantum curves reduce to supersymmetric algebraic curves, which are interesting in their own right [31,32].To sum up, to a given classical (possibly supersymmetric) curve one can associate an infinite family of quantum curves, which have the structure of singular vectors of the underlying algebra. This result was found in [20,24] upon the analysis of eigenvalue models, which provide a representation (or generalization) of matrix models; for a summary see also [33].The aim of the present paper is twofold. First, we clarify the role of conformal field theory in the description of quantum curves. In particular, we rederive (in Virasoro and Neveu-Schwarz case) quantum curves using only conformal field theory techniques (instead of eigenvalue models). The main feature of this approach is the fact, that the singular vector structure of qu...

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Singular vector structure of quantum curves

Cited by 5 publications

References 13 publications

Super Quantum Airy Structures

Super Quantum Airy Structures

Topological recursion in the Ramond sector

From CFT to Ramond super-quantum curves

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