Genus-one correction to asymptotically free Seiberg-Witten prepotential from Dijkgraaf-Vafa matrix model

Fujita, Mitsutoshi; Hatsuda, Yasuyuki; Tai, Ta-Sheng

doi:10.1007/jhep03(2010)046

Cited by 34 publications

(21 citation statements)

References 44 publications

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“…Although these results are not new and can be found in the existing literature on integrability, we present them here for completeness. 7 We restrict ourselves to the kernel given in (1.4) for N = 2 SYM. Consider the variables u i , i = 1 · · · M , satisfying a set of M equations,…”

Section: Relation With the Tbamentioning

confidence: 99%

Confinement and Mayer cluster expansions

Bourgine

2014

Int. J. Mod. Phys. A

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In these notes, we study a class of grand-canonical partition functions with a kernel depending on a small parameter . This class is directly relevant to Nekrasov partition functions of N = 2 SUSY gauge theories on the 4d Ω-background, for which is identified with one of the equivariant deformation parameter. In the NekrasovShatashvili limit → 0, we show that the free energy is given by an on-shell effective action. The equations of motion take the form of a TBA equation. The free energy is identified with the Yang-Yang functional of the corresponding system of Bethe roots. We further study the associated canonical model that takes the form of a generalized matrix model. Confinement of the eigenvalues by the short-range potential is observed. In the limit where this confining potential becomes weak, the collective field theory formulation is recovered. Finally, we discuss the connection with the alternative expression of instanton partition functions as sums over Young

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Section: Relation With the Tbamentioning

confidence: 99%

Confinement and Mayer cluster expansions

Bourgine

2014

Int. J. Mod. Phys. A

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“…The AGT correspondence [1] implies a relation between the canonical partition function of a β-ensemble and the grand canonical partition function of a generalized matrix model. The former represents a correlator of Liouville theory, according to the proposal of Dijkgraaf and Vafa [2], further investigated in [3,4,5,6,7,8,9,10,11]. The latter describes the instanton partition function of a 4d N = 2 supersymmetric gauge theory in the Ω-background, as derived using localization techniques in [12].…”

Section: Introductionmentioning

confidence: 99%

“…The potential Q(x) and the kernel K(x) are free of singularities over the real axis. 3 We propose to study the expansion of Z GC (q) when the kernel is close to one. More precisely, we assume the form…”

Section: Introductionmentioning

confidence: 99%

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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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“…The matrix model technique, in particular large N limit, is useful to show that the gauge theory result can be reproduced [38][39][40][41]. (See also [42] for 1/N correction.) It has been discussed that direct integral calculation leads to the instanton (q-)expansion of the Nekrasov partition function (and the corresponding expansion of the conformal block) [43][44][45][46][47][48][49].…”

Section: Introductionmentioning

confidence: 99%

Seiberg-Witten curve via generalized matrix model

Maruyoshi

Yagi

2011

J. High Energ. Phys.

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Abstract:We study the generalized matrix model which corresponds to the n-point toric Virasoro conformal block. This describes four-dimensional N = 2 SU(2) n gauge theory with circular quiver diagram by the AGT relation. We first verify that the generalized matrix model is obtained from the perturbative calculation of the Liouville correlation function. We then derive the Seiberg-Witten curve for N = 2 gauge theory as a spectral curve of the generalized matrix model.

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Genus-one correction to asymptotically free Seiberg-Witten prepotential from Dijkgraaf-Vafa matrix model

Abstract: We find perfect agreements on the genus-one correction to the prepotential of SU (2) Seiberg-Witten theory with N f = 2, 3 between field theoretical and Dijkgraaf-Vafa-Penner type matrix model results.

Cited by 34 publications

References 44 publications

Confinement and Mayer cluster expansions

Confinement and Mayer cluster expansions

Notes on Mayer expansions and matrix models

Seiberg-Witten curve via generalized matrix model

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