The curve of compactified 6<i>D</i>gauge theories and integrable systems

Braden, H. W.; Hollowood, Timothy J.

doi:10.1088/1126-6708/2003/12/023

Cited by 27 publications

(24 citation statements)

References 35 publications

(59 reference statements)

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“…Finally, it would be interesting to extend analysis of this paper to the curves of the doubleelliptic family, the simplest examples of which were considered in [25,[36][37][38]. Compared to the bi-elliptic curves in the simplest double-elliptic case we have dell : H(z,z) = α(z)cn β(z)z…”

Section: Discussionmentioning

confidence: 99%

WDVV Equations for 6d Seiberg–Witten Theory and Bi-Elliptic Curves

et al. 2007

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Section: Discussionmentioning

confidence: 99%

WDVV Equations for 6d Seiberg–Witten Theory and Bi-Elliptic Curves

et al. 2007

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“…There exists a correspondence between the spectral curves of a family of many-body integrable systems (Calogero-Moser-Sutherland [69][70][71], Ruijsenaars-Schneider [72], and Double-elliptic [73][74][75][76]) and Seiberg-Witten curves of supersymmetric gauge theories with adjoint matter in four, five and six dimensions (see [74] and references therein). These are the examples that we have focussed on in this paper.…”

Section: Integrable Systemsmentioning

confidence: 99%

Defects and quantum Seiberg-Witten geometry

Bullimore

Kim

Koroteev

2015

J. High Energ. Phys.

121

219

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Abstract:We study the Nekrasov partition function of the five dimensional U(N ) gauge theory with maximal supersymmetry on R 4 ×S 1 in the presence of codimension two defects. The codimension two defects can be described either as monodromy defects, or by coupling to a certain class of three dimensional quiver gauge theories on R 2 × S 1 . We explain how these computations are connected with both classical and quantum integrable systems. We check, as an expansion in the instanton number, that the aforementioned partition functions are eigenfunctions of an elliptic integrable many-body system, which quantizes the Seiberg-Witten geometry of the five-dimensional gauge theory.

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“…Recent discussions around of the integrability in Dijkgraaf-Vafa and Seiberg-Witten theory provide us with the "physical" insights to support the relation between the Beauville-Mukai and Double Elliptic IS ( see [6], [5]). …”

Section: Discussion and Future Problemsmentioning

confidence: 99%

Integrable systems associated with elliptic algebras

Odesskii

Rubtsov

2008

Quantum Groups

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Abstract. We construct some new Integrable Systems (IS) both classical and quantum associated with elliptic algebras. Our constructions are partly based on the algebraic integrability mechanism given by the existence of commuting families in skew fields and partly -on the internal properties of the elliptic algebras and their representations. We give some examples to make an evidence how these IS are related to previously studied.Introduction. This paper is an attempt to establish a direct connection between two close subjects of modern Mathematical Physics -the theory of Integrable Systems (IS) and the Elliptic Algebras. The aim of this connection is two-fold: we clarify some our recent results in the both domains and fill the natural gap proving that some large class of the Elliptic Algebras carries in fact the structure of an IS.We will start with a short account in the subject of the story and will describe briefly a type of the IS's under cosideration.In [12], B.Enriquez and second author had proposed a construction of commuting families of elements in skew fields. They explained how to use this construction in Poisson fraction fields to give an another proof of the integrability of Beauville -Mukai integrable systems associated with a K3 surface S ([1]).The Beauville-Mukai systems had appeared as the Lagrangian fibrations which have the formis the Hilbert scheme of g points of S, equipped with a symplectic structure introduced in [19], and L is a line bundle on S. Later, the authors of [8] explained that these systems are natural deformations of the "separated" (in the sense of [13]) versions of Hitchin's integrable systems, more precisely, of their description in terms of spectral curves (already present in [15]). Beauville-Mukai systems can be generalized to surfaces with a Poisson structure (see [3]). When S is the canonical cone Cone(C) of an algebraic curve C these systems coincide with the separated version of Hitchin's systems. The paper [12] shows how the commuting families construction provides a quantization of these systems on the canonical cone.A quantization of Hitchin's system was proposed in [2]. It seems interesting to construct quantization of Beauville-Mukai systems and to compare it with Beilinson -Drinfeld quantization. We had conjectured the correspondence between the results of [2] and a quantization of fraction fields in [9]. A part of this program

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The curve of compactified 6Dgauge theories and integrable systems

Cited by 27 publications

References 35 publications

WDVV Equations for 6d Seiberg–Witten Theory and Bi-Elliptic Curves

WDVV Equations for 6d Seiberg–Witten Theory and Bi-Elliptic Curves

Defects and quantum Seiberg-Witten geometry

Integrable systems associated with elliptic algebras

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