Singular vector structure of quantum curves

Self Cite

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...

supporting

confidence: 70%

mentioning

confidence: 61%

From CFT to Ramond super-quantum curves

Ciosmak

Hadasz

Jaskólski

et al. 2018

Self Cite

“…The spectral curve (4.12) and the Grassmann-valued equation (4.15) together can be thought as defining a "super spectral curve" (see for instance [18][19][20]). Note that these two polynomial equations are obtained from the super-loop equations without using the relation (3.18) with Hermitian matrix models.…”

Section: Discussionmentioning

confidence: 99%

“…(4.15) can be thought of as a superpartner to the spectral curve (4.12). Together they form a super spectral curve -see [18][19][20] for more on this. Ultimately, it would be great to reformulate the recursive structure as living on this super spectral curve.…”

Section: Spectral Curvementioning

confidence: 99%

“…However, it must be supplemented with a Grassmann-valued polynomial equation, which can be thought of as a super-partner of the spectral curve. Together they form a "super spectral curve" (see for instance [18][19][20]). However it is not clear how to reformulate the topological recursion and the auxiliary equations in terms of a single recursive structure living on the super spectral curve.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Supereigenvalue models and topological recursion

Bouchard

Osuga

2018

We show that the Eynard-Orantin topological recursion, in conjunction with simple auxiliary equations, can be used to calculate all correlation functions of supereigenvalue models. Hermitian Matrix Models and Topological RecursionIn this section we introduce formal Hermitian matrix models, and review the connections between Virasoro constraints, loop equations and topological recursion. A detailed discussion can be found in [24]. Formal Hermitian Matrix ModelsThe question of convergence of matrix integrals is a rather complex one. However, for many applications of matrix models in physics and enumerative geometry, convergence is not really necessary. More precisely, in this context -even though it is not always explicitly mentioned -we are often interested in so-called formal matrix models, rather than convergent matrix models. The difference between the two is well explained in [24].In this paper we focus on formal matrix models. An important consequence of formal matrix models is that the quantities of interest, such as the partition function, the free energy and correlation functions, all possess a well-defined 1/N expansion. Let us now define formal matrix models.

Topological recursion in the Ramond sector

Osuga

2019

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.