<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi mathvariant="script">W</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mo>∞</mml:mo></mml:mrow></mml:msub></mml:math>algebra as a symmetry behind the Alday-Gaiotto-Tachikawa relation

Kanno, Shoichi; Matsuo, Yutaka; Shiba, Shotaro

doi:10.1103/physrevd.84.026007

Cited by 22 publications

(19 citation statements)

References 45 publications

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“…A similar basis of bosonic fields was used recently in [50] to study the appearance of the W 1+∞ algebra in the context of the su(3) AGT relationship.…”

Section: Bosonisation Of the Wa K−1 Theoriesmentioning

confidence: 99%

Conformal blocks in Virasoro and W theories: Duality and the Calogero–Sutherland model

et al. 2012

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We study the properties of the conformal blocks of the conformal field theories with Virasoro or W-extended symmetry. When the conformal blocks contain only second-order degenerate fields, the conformal blocks obey second order differential equations and they can be interpreted as ground-state wave functions of a trigonometric Calogero-Sutherland Hamiltonian with nontrivial braiding properties. A generalized duality property relates the two types of second order degenerate fields. By studying this duality we found that the excited states of the CalogeroSutherland Hamiltonian are characterized by two partitions, or in the case of WA k−1 theories by k partitions. By extending the conformal field theories under consideration by a u(1) field, we find that we can put in correspondence the states in the Hilbert state of the extended CFT with the excited non-polynomial eigenstates of the Calogero-Sutherland Hamiltonian. When the action of the Calogero-Sutherland integrals of motion is translated on the Hilbert space, they become identical to the integrals of motion recently discovered by Alba, Fateev, Litvinov and Tarnopolsky in Liouville theory in the context of the AGT conjecture. Upon bosonisation, these integrals of motion can be expressed as a sum of two, or in general k, bosonic Calogero-Sutherland Hamiltonian coupled by an interaction term with a triangular structure. For special values of the coupling constant, the conformal blocks can be expressed in terms of Jack polynomials with pairing properties, and they give electron wave functions for special Fractional Quantum Hall states.

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“…A similar basis of bosonic fields was used recently in [50] to study the appearance of the W 1+∞ algebra in the context of the su(3) AGT relationship.…”

Section: Bosonisation Of the Wa K−1 Theoriesmentioning

confidence: 99%

Conformal blocks in Virasoro and W theories: Duality and the Calogero–Sutherland model

et al. 2012

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“…It is equivalent to a Schrödinger equation, i.e. a linear differential equation of second order.2 Such a structure should be related to the invariance of Nekrasov partition functions under transformations representing the SHc algebra uncovered in [23] (see also [24,25]). 3 In the case of real singularities, a prescription should be given to move away the poles from the contour by a small imaginary shift.…”

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confidence: 99%

Notes on Mayer expansions and matrix models

Bourgine

2014

Nuclear Physics B

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Mayer cluster expansion is an important tool in statistical physics to evaluate grand canonical partition functions. It has recently been applied to the Nekrasov instanton partition function of N = 2 4d gauge theories. The associated canonical model involves coupled integrations that take the form of a generalized matrix model. It can be studied with the standard techniques of matrix models, in particular collective field theory and loop equations. In the first part of these notes, we explain how the results of collective field theory can be derived from the cluster expansion. The equalities between free energies at first orders is explained by the discrete Laplace transform relating canonical and grand canonical models. In a second part, we study the canonical loop equations and associate them to similar relations on the grand canonical side. It leads to relate the multi-point densities, fundamental objects of the matrix model, to the generating functions of multi-rooted clusters. Finally, a method is proposed to derive loop equations directly on the grand canonical model. It is crucial for our considerations that f is independent of . In this way, we exclude a class of models more relevant to the study of Nekrasov partition functions. For instance, setting = γ 2 , one recovers the model proposed by J. Hoppe in [27]. This model is a one-parameter version of the Nekrasov partition function that depends on two Ω-background equivariant deformation parameters 1 and 2 [28,29]. As → 0, it exhibits a phenomenon referred 1 Except for the first (planar) equation, which is a Ricatti equation, therefore non-linear. It is equivalent to a Schrödinger equation, i.e. a linear differential equation of second order.2 Such a structure should be related to the invariance of Nekrasov partition functions under transformations representing the SHc algebra uncovered in [23] (see also [24,25]). 3 In the case of real singularities, a prescription should be given to move away the poles from the contour by a small imaginary shift.

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“…In our previous paper [10], we pointed out that W 1+∞ algebra may exist as a symmetry behind AGT-W relation, by showing that Toda correlation function plus the U (1) factor can be simply written in terms of this algebra. In this algebra, U (1) generator naturally coexists with W N generators of A N −1 Toda theory.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

Notes on 3-point functions of A N−1 Toda theory and AGT-W relation for SU(N) quiver

Shiba

2011

J. High Energ. Phys.

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W1+∞algebra as a symmetry behind the Alday-Gaiotto-Tachikawa relation

Abstract: We give some evidences which imply that W 1+∞ algebra describes the symmetry behind AGT(-W) conjecture: a correspondence between the partition function of N = 2 supersymmetric quiver gauge theories and the correlators of Liouville (Toda) field theory.

Cited by 22 publications

References 45 publications

Conformal blocks in Virasoro and W theories: Duality and the Calogero–Sutherland model

Conformal blocks in Virasoro and W theories: Duality and the Calogero–Sutherland model

Notes on Mayer expansions and matrix models

Notes on 3-point functions of A N−1 Toda theory and AGT-W relation for SU(N) quiver

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