On the Crossroads of Enumerative Geometry and Geometric Representation Theory

Okounkov, Andreĭ

doi:10.1142/9789813272880_0030

Cited by 12 publications

(14 citation statements)

References 61 publications

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“…We state this result in Theorem [1.3.1] below. For recent mathematical surveys of the classical and quantum periodic Benjamin-Ono equations, see §5.2 in Saut [51] and §1.1.6 in Okounkov [46], respectively.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…Comments on Hilbert Schemes of Points on Surfaces. Our notation ε in (1.1), ε 1 in (1.9), ε 2 in (1.10), and a in (3.2) reflect the appearance of the quantum periodic Benjamin-Ono equation in equivariant cohomology of Hilbert schemes of points in C 2 reviewed in §1.1.6 of Okounkov [46]. To interpret our Theorem [1.3.1] in this context, note that our coefficient of classical dispersion ε = ε 1 +ε 2 is the deformation parameter of the Maulik-Okounkov Yangian [34] while our coefficient of quantization = −ε 2 ε 1 is the handle-gluing element in [34].…”

Section: 3mentioning

confidence: 99%

“…To interpret our Theorem [1.3.1] in this context, note that our coefficient of classical dispersion ε = ε 1 +ε 2 is the deformation parameter of the Maulik-Okounkov Yangian [34] while our coefficient of quantization = −ε 2 ε 1 is the handle-gluing element in [34]. These ε, appear in §1.1.2 of [46] in trading C 2 for a surface S. For other exact Bohr-Sommerfeld conditions in the related theory of Nekrasov [42], see Mironov-Morozov [35]. Note that we have related dispersive action profiles of (1.1) to profiles in Nekrasov-Shatashvili [44] and Nekrasov-Pestun-Shatashvili [43] in §2.5 of [39].…”

Section: 3mentioning

confidence: 99%

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Exact Bohr-Sommerfeld Conditions for the Quantum Periodic Benjamin-Ono Equation

Moll

2019

SIGMA

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In this paper we describe the spectrum of the quantum periodic Benjamin-Ono equation in terms of the multi-phase solutions of the underlying classical system (the periodic multi-solitons). To do so, we show that the semi-classical quantization of this system given by Abanov-Wiegmann is exact and equivalent to the geometric quantization by Nazarov-Sklyanin. First, for the Liouville integrable subsystems defined from the multi-phase solutions, we use a result of Gérard-Kappeler to prove that if one neglects the infinitely-many transverse directions in phase space, the regular Bohr-Sommerfeld conditions on the actions are equivalent to the condition that the singularities of the Dobrokhotov-Krichever multi-phase spectral curves define an anisotropic partition (Young diagram). Next, we show that the renormalization of the classical dispersion coefficient in Abanov-Wiegmann is implicit in the definition of the quantum Lax operator in Nazarov-Sklyanin. Finally, we verify that the regular Bohr-Sommerfeld conditions for the multi-phase solutions in the renormalized theory give the exact quantum spectrum determined by Nazarov-Sklyanin without any Maslov index correction.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Section: 3mentioning

confidence: 99%

Section: 3mentioning

confidence: 99%

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Exact Bohr-Sommerfeld Conditions for the Quantum Periodic Benjamin-Ono Equation

Moll

2019

SIGMA

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“…Fugacity for C * ⊂ T 2 Angular momentum refinement, q Rotations of P 1 Vortex number Quasimap degree d K-theoretic PT vertex. Quasimaps to the Hilbert scheme of N points in C 2 correspond to points in the K-theoretic PT moduli space of C 3 [52,53]. The equivariant Euler characteristics (3.22) are then identified with the bare 1-leg vertex and following the notation of e.g.…”

Section: Adhm Vorticesmentioning

confidence: 99%

Blocks and vortices in the 3d ADHM quiver gauge theory

Samuel

Dorey

Zhang

2021

J. High Energ. Phys.

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We study the hemisphere partition function of a three-dimensional $$ \mathcal{N} $$ N = 4 supersymmetric U(N) gauge theory with one adjoint and one fundamental hypermultiplet — the ADHM quiver theory. In particular, we propose a distinguished set of UV boundary conditions which yield Verma modules of the quantised chiral rings of the Higgs and Coulomb branches. In line with a recent proposal by two of the authors in collaboration with M. Bullimore, we show explicitly that the hemisphere partition functions recover the characters of these modules in two limits, and realise blocks gluing exactly to the partition functions of the theory on closed three-manifolds. We study the geometry of the vortex moduli space and investigate the interpretation of the vortex partition functions as equivariant indices of quasimaps to the Hilbert scheme of points in ℂ2. We also investigate half indices of the ADHM quiver gauge theory in the presence of a line operator and discuss their geometric interpretation. Along the way we find interesting relations between our hemisphere blocks and related quantities in topological string theory and equivariant quantum K-theory.

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“…Our ε in (1.1) is chosen to match the standard notation ε = ε 1 + ε 2 in Nekrasov's Omega background [59], a gauge theory known to be related to a quantization of (1.1) with = −ε 1 ε 2 . For a recent discussion of (1.1) from this point of view, see §1.1.6 in Okounkov [63]. We study this quantization of the classical periodic Benjamin-Ono equation in [55] where we derive exact Bohr-Sommerfeld quantization conditions on the classical multi-phase solutions (1.13).…”

Section: Comments On Dispersion Coefficientmentioning

confidence: 99%

Finite gap conditions and small dispersion asymptotics for the classical periodic Benjamin–Ono equation

Moll

2020

Quart. Appl. Math.

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In this paper we characterize the Nazarov-Sklyanin hierarchy for the classical periodic Benjamin-Ono equation in two complementary degenerations: for the multi-phase initial data (the periodic multi-solitons) at fixed dispersion and for bounded initial data in the limit of small dispersion. First, we express this hierarchy in terms of a piecewise-linear function of an auxiliary real variable which we call a dispersive action profile and whose regions of slope ±1 we call gaps and bands, respectively. Our expression uses Kerov's theory of profiles and Kreȋn's spectral shift functions. Next, for multi-phase initial data, we identify Baker-Akhiezer functions in Dobrokhotov-Krichever and Nazarov-Sklyanin and prove that multi-phase dispersive action profiles have finitely-many gaps determined by the singularities of their Dobrokhotov-Krichever spectral curves. Finally, for bounded initial data independent of the coefficient of dispersion, we show that in the small dispersion limit, the dispersive action profile concentrates weakly on a convex profile which encodes the conserved quantities of the dispersionless equation. To establish the weak limit, we reformulate Szegő's first theorem for Toeplitz operators using spectral shift functions. To illustrate our results, we identify the dispersive action profile of sinusoidal initial data with a profile found by Nekrasov-Pestun-Shatashvili and its small dispersion limit with the convex profile found by Vershik-Kerov and Logan-Shepp. CONTENTS

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On the Crossroads of Enumerative Geometry and Geometric Representation Theory

Cited by 12 publications

References 61 publications

Exact Bohr-Sommerfeld Conditions for the Quantum Periodic Benjamin-Ono Equation

Exact Bohr-Sommerfeld Conditions for the Quantum Periodic Benjamin-Ono Equation

Blocks and vortices in the 3d ADHM quiver gauge theory

Finite gap conditions and small dispersion asymptotics for the classical periodic Benjamin–Ono equation

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