Baxter’sT-Q equation, SU(N)/SU(2) N − 3 correspondence and Ω-deformed Seiberg-Witten prepotential

Muneyuki, Kenji; Tai, Ta-Sheng; Yonezawa, Nobuhiro; Yoshioka, Reiji

doi:10.1007/jhep09(2011)125

Cited by 11 publications

(5 citation statements)

References 33 publications

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“…In recent years the connection between the gauge theories and the spin chains, and the inspired duality between the Gaudin-like integrable systems and the Heisenberg spin chain was discussed in [126][127][128][129], building on the earlier work in [21,70,[130][131][132].…”

Section: Periodic Monopoles and The Phase Space Of Class I Theories 113mentioning

confidence: 99%

Seiberg-Witten geometry of four dimensional N=2 quiver gauge theories

Nekrasov,

Pestun

2012

Preprint

104

258

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Seiberg-Witten geometry of mass deformed N = 2 superconformal ADE quiver gauge theories in four dimensions is determined. We solve the limit shape equations derived from the gauge theory and identify the space M of vacua of the theory with the moduli space of the genus zero holomorphic (quasi)maps to the moduli space Bun G (E) of holomorphic G C -bundles on a (possibly degenerate) elliptic curve E defined in terms of the microscopic gauge couplings, for the corresponding simple ADE Lie group G. The integrable systems P underlying, or, rather, overlooking the special geometry of M are identified. The moduli spaces of framed G-instantons on R 2 × T 2 , of G-monopoles with singularities on R 2 × S 1 , the Hitchin systems on curves with punctures, as well as various spin chains play an important rôle in our story. We also comment on the higher dimensional theories. In the companion paper the quantum integrable systems and their connections to the representation theory of quantum affine algebras will be discussed. ContentsAcknowledgments iii Chapter 1. Introduction 1.1. Organization of the material 1.2. Notations Chapter 2. Supersymmetric quiver theories 2.1. Gauge group, matter fields, couplings, parameters Chapter 3. The ADE classifications of superconformal N = 2 theories 3.1. Beta functions and Cartan matrix 3.2. Class I theories 3.3. Class II theories 3.4. Class II* theories Chapter 4. Low-energy effective theory 4.1. Special Kähler geometry 4.2. Extended moduli space 4.3. Finite size effects 4.4. The appearance of an integrable system 4.5. Integrable systems from classical gauge theories 4.6. Extended moduli space as a complex integrable system Chapter 5. The limit shape equations 5.1. The amplitude functions 5.2. The limit shape prepotential 5.3. The iWeyl group. 5.4. Moduli of vacua and mass parameters Chapter 6. Solution of the limit shape equations 6.1. From invariants to the curve 6.2. The cameral curve as a modular object 6.3. Spectral curves 6.4. Obscured curve iv CONTENTS v Chapter 7. The Seiberg-Witten curves in some detail 7.1. Class I theories of A type 7.2. Class I theories of D type 7.3. Class I theories of E type 7.4. Class II theories of A type and class II* theories 7.5. Class II theories of D type 7.6. Class II theories of E type Chapter 8. The integrable systems of monopoles and instantons 8.1. Periodic Monopoles and the phase space of Class I theories105 8.2. Double-periodic instantons and Class II theories 8.3. Noncommutative instantons and Class II* theories Chapter 9. Higher dimensional theories Appendix A.

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Section: Periodic Monopoles and The Phase Space Of Class I Theories 113mentioning

confidence: 99%

Seiberg-Witten geometry of four dimensional N=2 quiver gauge theories

Nekrasov,

Pestun

2012

Preprint

104

258

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“…For the quiver which is a Dynkin graph of (affine) ADE Lie algebra g Γ , we find that the ǫ-deformed limit shape partition profile equations of [21] can be mapped to the Bethe equations of a formal trigonometric XXZ g Γ -spin chain. Despite the similarity to the problems (for finite A r quivers in four dimensions) studied in [29][30][31] and also in [2,3,5,[32][33][34][35][36][37][38][39][40][41][42][43][44][45][46], there are important conceptual differences which we shall explain below. In fact, the universal part of the gauge theory twisted superpotential does not correspond to any spin chain.…”

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confidence: 92%

Quantum Geometry and Quiver Gauge Theories

Nekrasov

Pestun

Shatashvili

2018

Commun. Math. Phys.

148

273

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We study macroscopically two dimensional N = (2, 2) supersymmetric gauge theories constructed by compactifying the quiver gauge theories with eight supercharges on a product T d × R 2 ǫ of a d-dimensional torus and a two dimensional cigar with Ω-deformation. We compute the universal part of the effective twisted superpotential. In doing so we establish the correspondence between the gauge theories, quantization of the moduli spaces of instantons on R 2−d × T 2+d and singular monopoles on R 2−d × T 1+d , for d = 0, 1, 2, and the Yangian Y ǫ (g Γ ), quantum affine algebra U aff q (g Γ ), or the quantum elliptic algebra U ell q,p (g Γ ) associated to Kac-Moody algebra g Γ for quiver Γ.

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“…The studies of the interrelationships between conformal field theory in two dimensions, supersymmetric N = 2 quiver gauge theories and integrable systems are recently attracting a great attention in the scientific community [1][2][3][4][5][6][7][8][9][10][11][12]. This is mainly due to the discovery of the so-called AGT [13] and Bethe/gauge [14][15][16] correspondences.…”

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confidence: 99%

Liouville theory, $ \mathcal{N} = 2 $ gauge theories and accessory parameters

Ferrari

Piątek

2012

J. High Energ. Phys.

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The correspondence between the semiclassical limit of the DOZZ quantum Liouville theory and the Nekrasov-Shatashvili limit of the N = 2 (Ω-deformed) U(2) superYang-Mills theories is used to calculate the unknown accessory parameter of the Fuchsian uniformization of the 4-punctured sphere. The computation is based on the saddle point method. This allows to find an analytic expression for the N f = 4, U(2) instanton twisted superpotential and, in turn, to sum up the 4-point classical block. It is well known that the critical value of the Liouville action functional is the generating function of the accessory parameters. This statement and the factorization property of the 4-point action allow to express the unknown accessory parameter as the derivative of the 4-point classical block with respect to the modular parameter of the 4-punctured sphere. It has been found that this accessory parameter is related to the sum of all rescaled column lengths of the so-called 'critical' Young diagram extremizing the instanton 'free energy'. It is shown that the sum over the 'critical' column lengths can be rewritten in terms of a contour integral in which the integrand is built out of certain special functions closely related to the ordinary Gamma function.

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Baxter’sT-Q equation, SU(N)/SU(2) N − 3 correspondence and Ω-deformed Seiberg-Witten prepotential

Cited by 11 publications

References 33 publications

Seiberg-Witten geometry of four dimensional N=2 quiver gauge theories

Seiberg-Witten geometry of four dimensional N=2 quiver gauge theories

Quantum Geometry and Quiver Gauge Theories

Liouville theory, $ \mathcal{N} = 2 $ gauge theories and accessory parameters

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