Genus one correction to Seiberg-Witten prepotential from β-deformed matrix model

Baek, Jong-Hyun

doi:10.1007/jhep04(2013)120

Cited by 6 publications

(9 citation statements)

References 51 publications

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“…Let us for instance consider a rooted cluster integral I(y) and attach a G-link < xy > to y, such that the new root is x. Assume that I(y) satisfies (A.1), and take the kernel relevant to N = 2 SYM in (1.4), 15 one obtains…”

Section: A3 G-linksmentioning

confidence: 99%

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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: A3 G-linksmentioning

confidence: 99%

“…15) where we used (2.23) to replace Y with ρ. This expression should be compared with the collective action associated to the partition function…”

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confidence: 99%

Confinement and Mayer cluster expansions

Bourgine

2014

Int. J. Mod. Phys. A

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“…In particular, in [46] the expression in terms of modular forms of the first few F (n,g) 's, for generic deformation and masses, was inferred from their instanton expansion; these findings suggest a recursion relation among the coefficient of the expansion of the prepotential for large a which is equivalent to the modular anomaly equation (1.9). Finally, to obtain explicit expressions and uncover the modular properties of the deformed theories, one can exploit the AGT relation, as recently done in [48,49], or the deformed matrix models [50]. In this paper we address various issues.…”

Section: Jhep11(2013)123mentioning

confidence: 99%

Modular anomaly equation, heat kernel and S-duality in $ \mathcal{N}=2 $ theories

Billó

Frau

Gallot

et al. 2013

J. High Energ. Phys.

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We investigate ǫ-deformed N = 2 superconformal gauge theories in four dimensions, focusing on the N = 2 * and N f = 4 SU(2) cases. We show how the modular anomaly equation obeyed by the deformed prepotential can be efficiently used to derive its non-perturbative expression starting from the perturbative one. We also show that the modular anomaly equation implies that S-duality is implemented by means of an exact Fourier transform even for arbitrary values of the deformation parameters, and then we argue that it is possible, perturbatively in the deformation, to choose appropriate variables such that it reduces to a Legendre transform.

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“…On the other hand, a single term in (2.2) corresponds to many terms of A and B series, since a tree T l can be rooted from any of its vertices, leading to l terms in A, and has l − 1 links that can be associated to the main link of a B-term. 11 The fact that the formula (2.6) holds has to do with the property that a tree T l has exactly l links for l − 1 vertices. The strategy we follow is to consider the terms in A and B cluster expansions that can be identified with a given cluster T l of the summation (2.2).…”

Section: A Demonstration Of the Tree Level Free Energy Formulamentioning

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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Genus one correction to Seiberg-Witten prepotential from β-deformed matrix model

Cited by 6 publications

References 51 publications

Confinement and Mayer cluster expansions

Confinement and Mayer cluster expansions

Modular anomaly equation, heat kernel and S-duality in $ \mathcal{N}=2 $ theories

Notes on Mayer expansions and matrix models

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