From CFT to Ramond super-quantum curves

Abstract

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...

Supereigenvalue models and topological recursion

“…Firstly, it is interesting to ask whether one can define supereigenvalue models that satisfy super-Virasoro constraints corresponding to the super-Virasoro subalgebra in the Ramond sector, and then see whether the appropriate correlation functions can be computed recursively. We have found such a supereigenvalue partition function, and derived super-loop equations (see also [20], where such supereigenvalue models are also derived from the point of view of quantum curves). We are currently investigating a recursive solution to these super-loop equations [37].…”

“…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.…”

“…(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.…”

“…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.…”

Topological recursion in the Ramond sector

“…1 Another important property of Hermitian matrix models is that the partition function obeys the so-called Virasoro constraint. From this perspective, we can consider supergeneralized models, known as supereigenvalue models, such that their partition function satisfies the super Virasoro constraint (see [4-7, 10, 16, 17, 32, 34-37] for the Neveu-Schwarz (NS) sector and [18] for the Ramond sector). Then one may ask whether a similar recursive structure also appears in these models.…”