Quantizing Weierstrass

Bouchard, Vincent; Chidambaram, Nitin Kumar; Dauphinee, Tyler

doi:10.4310/cntp.2018.v12.n2.a2

Cited by 9 publications

(7 citation statements)

References 54 publications

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“…This is only to be expected: the former corresponds to the unrefined topological string theory, while the latter corresponds to the NS limit of the refined string. In line with this observation, a recent study of the Weierstrass curve, which has genus one, indicates that ψ top is not annihilated by the quantum curve obtained by the quantization scheme (5.1) [68], in other words, it is not a formal WKB wavefunction associated to the natural quantization of the curve. The connection unveiled in this paper between the standard WKB method and the holomorphic anomaly equations is potentially interesting to further clarify the quantum mechanical meaning of the general equations (2.19).…”

Section: Discussionmentioning

confidence: 71%

Holomorphic anomaly and quantum mechanics

Codesido¹,

Mariño²

2017

J. Phys. A: Math. Theor.

107

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We show that the all-orders WKB periods of one-dimensional quantum mechanical oscillators are governed by the refined holomorphic anomaly equations of topological string theory. We analyze in detail the double-well potential and the cubic and quartic oscillators, and we calculate the WKB expansion of their quantum free energies by using the direct integration of the anomaly equations. We reproduce in this way all known results about the quantum periods of these models, which we express in terms of modular forms on the WKB curve. As an application of our results, we study the large order behavior of the WKB expansion in the case of the double well, which displays the double factorial growth typical of string theory.

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Section: Discussionmentioning

confidence: 71%

Holomorphic anomaly and quantum mechanics

Codesido¹,

Mariño²

2017

J. Phys. A: Math. Theor.

107

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“…In [8], they generalize the techniques employed in [9] to find the quantum curves for admissible curves to apply them to the family of genus one spectral curves given by the Weierstrass equation. They find an order two differential operator that annihilates the perturbative wave function .…”

Section: Introductionmentioning

confidence: 99%

“…In [23], the first author slightly generalizes the same results to the case of any elliptic curve, that is he considers not only the degenerate case of Painlevé I but tori where none of the two cycles are pinched. In both papers, they show that the corrections from [8] can be controlled by a derivative with respect to a deformation parameter. The quantum curves still contain infinitely many -correction terms, but in this case, these corrections are given by the asymptotic expansion of the solution of Painlevé I around .…”

Section: Introductionmentioning

confidence: 99%

From topological recursion to wave functions and PDEs quantizing hyperelliptic curves

Eynard,

Garcia-Failde

2023

Forum of Mathematics, Sigma

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Starting from loop equations, we prove that the wave functions constructed from topological recursion on families of degree $2$ spectral curves with a global involution satisfy a system of partial differential equations, whose equations can be seen as quantizations of the original spectral curves. The families of spectral curves can be parametrized with the so-called times, defined as periods on second type cycles, and with the poles. These equations can be used to prove that the WKB solution of many isomonodromic systems coincides with the topological recursion wave function, which proves that the topological recursion wave function is annihilated by a quantum curve. This recovers many known quantum curves for genus zero spectral curves and generalizes this construction to hyperelliptic curves.

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“…Quantum curves are intriguing objects, identified originally in various problems related to string theory and supersymmetric gauge theories [1][2][3][4][5], and analyzed from various mathematical perspectives e.g. in [6][7][8][9][10][11][12][13][14][15][16][17]. In general, quantum curves take form of differential operators A( x, y) imposing Schroedinger-like equations on appropriately defined wave-functions Ψ(x) A( x, y)Ψ(x) = 0.…”

Section: Introductionmentioning

confidence: 99%

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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Quantizing Weierstrass

Cited by 9 publications

References 54 publications

Holomorphic anomaly and quantum mechanics

Holomorphic anomaly and quantum mechanics

From topological recursion to wave functions and PDEs quantizing hyperelliptic curves

From CFT to Ramond super-quantum curves

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