The geometry of Casimir W-algebras

Belliard, Raphaël; Eynard, Bertrand; Ribault, Sylvain

doi:10.21468/scipostphys.5.5.051

Cited by 3 publications

(4 citation statements)

References 19 publications

(32 reference statements)

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“…This section is somewhat independent from the rest of the paper. We explain how the meromorphic differentials constructed by the topological recursion for the r-Airy curve are generating functions for r-spin intersection numbers; however, we postpone an explicit proof from matrix models to a future publication [7]. This result was first announced in [13], and has now also been proved using a different approach in [27].…”

Section: Introductionmentioning

confidence: 93%

“…We note that the intersection numbers are non-vanishing only if This theorem was announced in September 2014 [13]. A detailed proof of this theorem based on matrix model analysis will be provided in a future publication [7]. Meanwhile, an alternative proof was presented recently in the preprint [27] by relating the topological recursion to Dubrovin's superpotential.…”

Section: Some Examplesmentioning

confidence: 99%

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Reconstructing WKB from topological recursion

Bouchard¹,

Eynard²

2017

Journal De L’École Polytechnique — Mathématiques

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183

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We prove that the topological recursion reconstructs the WKB expansion of a quantum curve for all spectral curves whose Newton polygons have no interior point (and that are smooth as affine curves). This includes nearly all previously known cases in the literature, and many more; in particular, it includes many quantum curves of order greater than two. We also explore the connection between the choice of ordering in the quantization of the spectral curve and the choice of integration divisor to reconstruct the WKB expansion.Comment: 68 pages, 9 figures. v2: published version (improved presentation

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Section: Introductionmentioning

confidence: 93%

Section: Some Examplesmentioning

confidence: 99%

Reconstructing WKB from topological recursion

Bouchard¹,

Eynard²

2017

Journal De L’École Polytechnique — Mathématiques

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183

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“…This construction is based on the middle convolution from the Katz theory of rigid systems. Connections between Fuchsian systems and W -algebras were also studied in [5], with further links to topological recursion suggested in [6].…”

Section: Introductionmentioning

confidence: 99%

Higher-rank isomonodromic deformations and W-algebras

2019

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We construct the general solution of a class of Fuchsian systems of rank N as well as the associated isomonodromic tau functions in terms of semi-degenerate conformal blocks of W N -algebra with central charge c = N − 1. The simplest example is given by the tau function of the Fuji-Suzuki-Tsuda system, expressed as a Fourier transform of the 4-point conformal block with respect to intermediate weight. Along the way, we generalize the result of Bowcock and Watts on the minimal set of matrix elements of vertex operators of the W N -algebra for generic central charge and prove several properties of semi-degenerate vertex operators and conformal blocks for c = N − 1.

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“…Building up on a conjecture of the authors and collaborators in [3] we give the general definition of a tau-function associated to the moduli space of meromorphic connections in a holomorphic principal bundle over a compact Riemann surface. We define it as a theta-series expansion whose coefficients satisfy special geometry relation (related to hyper-Kähler structures).…”

Section: Introductionmentioning

confidence: 99%

From the quantum geometry of Fuchsian systems to conformal blocks of W-algebras

Belliard,

Eynard

2019

Preprint

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We consider the moduli space of holomorphic principal bundles for reductive Lie groups over Riemann surfaces (possibly with boundaries) and equipped with meromorphic connections. We associate to this space a point-wise notion of quantum spectral curve whose generalized periods define a new set of moduli. We define homology cycles and differential forms of the quantum spectral curve, allowing to derive quantum analogs of the form-cycle duality and Riemann bilinear identities of classical geometry. A tau-function is introduced for this system in the form of a theta-series and in such a way that the variations of its coefficients with respect to moduli, isomonodromic or not, can be computed as quantum period integrals. This lays new grounds to relate our study to that of integrable hierarchies, isomonodromic deformation of meromorphic connections and non-perturbative topological string theory. In turn, we define amplitudes on the quantum spectral curve which have an interpretation in conformal field theory. The amplitudes are moreover related by W-constraints, so-called loop equations, allowing one to compute recursively a certain asymptotic expansion of the tau-function, namely the one corresponding both to the heavy-charge regime of conformal field theory and to the weak-coupling regime of topological string theory.

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The geometry of Casimir W-algebras

Cited by 3 publications

References 19 publications

Reconstructing WKB from topological recursion

Reconstructing WKB from topological recursion

Higher-rank isomonodromic deformations and W-algebras

From the quantum geometry of Fuchsian systems to conformal blocks of W-algebras

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