Ε-Corrected SEIBERG–WITTEN PREPOTENTIAL OBTAINED FROM HALF-GENUS EXPANSION IN Β-Deformed MATRIX MODEL

Itoyama, H.; Yonezawa, Nobuhiro

doi:10.1142/s0217751x11053882

Cited by 18 publications

(16 citation statements)

References 144 publications

(155 reference statements)

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“…We have thus verified that (3.15) coincide, in the sense of [22], with the collective field action of the canonical model. 12 To treat the kernel, we also need to assume 11) i.e. to neglect the term 2 pG.…”

Section: Collective Field Theory and α → 0 Limitmentioning

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: Collective Field Theory and α → 0 Limitmentioning

confidence: 99%

Confinement and Mayer cluster expansions

Bourgine

2014

Int. J. Mod. Phys. A

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“…This technique, originally developed for matrix models, has recently been extended to a large class of models to which Z C belongs [14]. 7 Our goal in this section is to map these loop equations to similar relations among objects pertaining to the cluster expansion. These objects are the n-points Y -functions, generating functions of n-rooted clusters.…”

Section: Loop Equationsmentioning

confidence: 99%

“…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%

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 analysis for the subleading orders can also be done. In particular, it was shown that the corresponding parts of the free energy agrees with F 1 in [49] and with H and G in [52,53]. For generic b, this expansion of the matrix model was compared [54] with the finite N calculation which will be explained in section 5.…”

Section: )mentioning

confidence: 97%

$$\beta $$-Deformed Matrix Models and 2d/4d Correspondence

Maruyoshi

2015

New Dualities of Supersymmetric Gauge Theories

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We review the β-deformed matrix model approach to the correspondence between fourdimensional N = 2 gauge theories and two-dimensional conformal field theories. The βdeformed matrix model equipped with the log-type potential is obtained as a free field (Dotsenko-Fateev) representation of the conformal block of chiral conformal algebra in two dimensions, with the precise choice of integration contours. After reviewing various matrix models related to the conformal field theories in two-dimensions, we study the large N limit corresponding to turning off the Omega-background 1 , 2 → 0. We show that the large N analysis produces the purely gauge theory results. Furthermore we discuss the Nekrasov-Shatashvili limit ( 2 → 0) by which we see the connection with the quantum integrable system. We then perform the explicit integration of the matrix model. With the precise choice of the contours we see that this reproduces the expansion of the conformal block and also the Nekrasov partition function. This is a contribution to the special volume on the 2d/4d correspondence, edited by J. Teschner. * k.maruyoshi@imperial.ac.uk * The matrix model with a logarithmic potential was first studied by Penner [20] related to the Eular characteristic of a Riemann surface.

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Ε-Corrected SEIBERG–WITTEN PREPOTENTIAL OBTAINED FROM HALF-GENUS EXPANSION IN Β-Deformed MATRIX MODEL

Cited by 18 publications

References 144 publications

Confinement and Mayer cluster expansions

Confinement and Mayer cluster expansions

Notes on Mayer expansions and matrix models

$$\beta $$-Deformed Matrix Models and 2d/4d Correspondence

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