Integrability and Seiberg-Witten theory curves and periods

Itoyama, H.; Морозов, А.

doi:10.1016/0550-3213(96)00358-6

Cited by 136 publications

(105 citation statements)

References 47 publications

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“…The only way to actually obey this constraint is to make the number of D4 branes suspended between each two NS5 branes the same, which gives the U(N ) M quiver. The integrable system corresponding to the 4d U(N ) theory with adjoint matter is N -particle elliptic Calogero model [68][69][70], and for necklace quivers it is a certain multipoint generalization of Calogero system [32]. The mass of the adjoint multiplet corresponds to the Calogero coupling constant and the gauge theory coupling constant is encoded in the elliptic parameter of the system.…”

Section: Brane Picturesmentioning

confidence: 99%

Spectral duality in elliptic systems, six-dimensional gauge theories and topological strings

Миронов¹,

Морозов²,

Zenkevich³

2016

J. High Energ. Phys.

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We consider Dotsenko-Fateev matrix models associated with compactified Calabi-Yau threefolds. They can be constructed with the help of explicit expressions for refined topological vertex, i.e. are directly related to the corresponding topological string amplitudes. We describe a correspondence between these amplitudes, elliptic and affine type Selberg integrals and gauge theories in five and six dimensions with various matter content. We show that the theories of this type are connected by spectral dualities, which can be also seen at the level of elliptic Seiberg-Witten integrable systems. The most interesting are the spectral duality between the XYZ spin chain and the Ruijsenaars system, which is further lifted to self-duality of the double elliptic system.

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Section: Brane Picturesmentioning

confidence: 99%

Spectral duality in elliptic systems, six-dimensional gauge theories and topological strings

Миронов¹,

Морозов²,

Zenkevich³

2016

J. High Energ. Phys.

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“…This data allows one to define the notion of prepotential [13,17] related to some integrable system [10].…”

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confidence: 99%

Matrix models vs. Seiberg–Witten/Whitham theories

2003

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We discuss the relation between matrix models and the Seiberg-Witten type (SW) theories, recently proposed by Dijkgraaf and Vafa. In particular, we prove that the partition function of the Hermitean onematrix model in the planar (large N ) limit coincides with the prepotential of the corresponding SW theory. This partition function is the logarithm of a Whitham τ -function. The corresponding Whitham hierarchy is explicitly constructed. The double-point problem is solved. 1.It is well known that partition functions of matrix models are τ -functions of integrable hierarchies of the Toda type [1]. In the specific double scaling limit, these τ -functions become τ -functions of various reduction of the KP hierarchy [2]. If one makes the simplest, large-N (planar) limit, the partition function becomes the τ -function of the dispersionless Toda hierarchy, which in turn becomes the τ -function of the dispersionless (reductions of) KP hierarchy [3] after performing the continuum limit (which basically means working nearby a singularity of the partition function). All these dispersionless hierarchies are just Whitham equations over trivial solutions to integrable (Toda, KP) hierarchies.When solving matrix models, most attention was paid to one-cut solutions where the limiting eigenvalue distribution spans one interval on the real axis [4]. The results on multi-cut solutions [5] were few [6,7,8]. Recently, Dijkgraaf and Vafa proposed [9] the new insight on the multi-cut large-N limit of matrix models. Namely, they associated this limit with a Riemann surface and some related SW system. Its prepotential, which we prove here to be the logarithm of the large-N partition function, is typically associated with the logarithm of some Whitham τ -functions [10,11]. This hints that the matrix matrix model in the large N limit of multi-cut type describes the Whitham system over a non-trivial, finite-gap solution to integrable (Toda, KP) hierarchy. In particular, this solution passes to a finite-gap solution of (reductions of) the KP hierarchy in the continuum limit.In this paper, we restrict ourselves with the simplest example of the Hermitean one-matrix model. We show that coefficients of the potential of the model gives rise to Whitham flows and manifestly construct this Whitham system. In fact, the authors of [9] associated the N = 1 SUSY gauge theory studied in [12] with the SW system related to the multi-cut planar limit of matrix models. From the point of view of N = 1 SUSY theory, these coefficients must be identified with couplings in the tree superpotential, while the SW moduli are associated with v.e.v.'s of the gluino condensates. This gives an interpretation of the results of [12] in the Whitham hierarchy terms. *

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“…From the viewpoint of integrable systems, Γ corresponds to the spectral curve Γ CM of the elliptic Calogero-Moser model [4,61,62] known to have the same geometrical description [63] (generalization to the case of more than two NS5 branes leads to the spin Calogero model, see [17]). To avoid uncertainties in the notation, from now on, we denote the curve Γ of the 4d theory under consideration by Γ CM .…”

Section: Jhep11(2017)023mentioning

confidence: 99%

“…Thus, the prepotential F CM at m = 0 is 10) which is associated with the classical part of the prepotential. To obtain the pure gauge limit of the N = 2 theory, one should bring the value of m and ı τ to infinity in a consistent way (double scaling limit) [61,63,64,66]: 11) JHEP11 (2017)023 so that the resulting cutoff Λ is finite. From the M theory point of view, this limit of infinite mass in the four-dimensional theory is accompanied with the decompactification of the x 6 direction.…”

Section: Jhep11(2017)023mentioning

confidence: 99%

Modular properties of 6d (DELL) systems

Aminov¹,

Миронов

Морозов

2017

J. High Energ. Phys.

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If super-Yang-Mills theory possesses the exact conformal invariance, there is an additional modular invariance under the change of the complex bare charge τ = θ 2π + 4πı g 2 −→ − 1 τ . The low-energy Seiberg-Witten prepotential F (a), however, is not explicitly invariant, because the flat moduli also change a −→ a D = ∂F /∂a. In result, the prepotential is not a modular form and depends also on the anomalous Eisenstein series E 2 . This dependence is usually described by the universal MNW modular anomaly equation. We demonstrate that, in the 6d SU(N ) theory with two independent modular parameters τ andτ , the modular anomaly equation changes, because the modular transform of τ is accompanied by an (N -dependent!) shift ofτ and vice versa. This is a new peculiarity of double-elliptic systems, which deserves further investigation.

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Integrability and Seiberg-Witten theory curves and periods

Abstract: Interpretation of exact results on the low-energy limit of 4d N = 2 SUSY YM in the language of 1d integrability theory is reviewed. The case of elliptic Calogero system, associated with the flow between N = 4 and N = 2 SUSY in 4d, is considered in some detail.

Cited by 136 publications

References 47 publications

Spectral duality in elliptic systems, six-dimensional gauge theories and topological strings

Spectral duality in elliptic systems, six-dimensional gauge theories and topological strings

Matrix models vs. Seiberg–Witten/Whitham theories

Modular properties of 6d (DELL) systems

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