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Journal ArticleDOI

Integrable systems and supersymmetric gauge theory

15 Jan 1996-Nuclear Physics (North-Holland)-Vol. 459, Iss: 1, pp 97-112
TL;DR: In this article, it was shown that the dynamics of N = 2 Yang-Mills theory is governed by a spectral curve of the periodic Toda lattice for the dual group GV whose affine Dynkin diagram is the dual of that of GV.
About: This article is published in Nuclear Physics.The article was published on 1996-01-15 and is currently open access. It has received 403 citations till now. The article focuses on the topics: Toda lattice & Seiberg–Witten theory.
Citations
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Journal ArticleDOI
TL;DR: In this paper, the Coulomb branch of a large family of four-dimensional N = 2 field theories with zero or negative beta function is studied, and explicit solutions for Coulomb branches are given.

1,236 citations


Cites background from "Integrable systems and supersymmetr..."

  • ...This structure was noticed in special cases [9-11] and deduced from the generalities of low energy supersymmetric effective field theory [12]....

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Posted Content
TL;DR: In this article, the authors considered BPS states in a large class of d=4, N=2 field theories, obtained by reducing six-dimensional (2,0) superconformal field theories on Riemann surfaces, with defect operators inserted at points of the RiemANN surface.
Abstract: We consider BPS states in a large class of d=4, N=2 field theories, obtained by reducing six-dimensional (2,0) superconformal field theories on Riemann surfaces, with defect operators inserted at points of the Riemann surface. Further dimensional reduction on S^1 yields sigma models, whose target spaces are moduli spaces of Higgs bundles on Riemann surfaces with ramification. In the case where the Higgs bundles have rank 2, we construct canonical Darboux coordinate systems on their moduli spaces. These coordinate systems are related to one another by Poisson transformations associated to BPS states, and have well-controlled asymptotic behavior, obtained from the WKB approximation. The existence of these coordinates implies the Kontsevich-Soibelman wall-crossing formula for the BPS spectrum. This construction provides a concrete realization of a general physical explanation of the wall-crossing formula which was proposed in 0807.4723. It also yields a new method for computing the spectrum using the combinatorics of triangulations of the Riemann surface.

896 citations


Cites background from "Integrable systems and supersymmetr..."

  • ...ched by Witten in Section 2.3 of [2]. It has been observed that the Seiberg-Witten solutions of many N= 2 theories can be understood in terms of known complex integrable systems, as discussed e.g. in [38, 39, 40, 41]. Here we have constructed a large general class of theories for which the relevant complex integrable system is a Hitchin system. 19One ecient proof that the BPS equations are the Hitchin equations ...

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Journal ArticleDOI
TL;DR: The Coulomb branch of N = 2 supersymmetric gauge theories in four dimensions is described in general by an integrable Hamiltonian system in the holomorphic sense as mentioned in this paper.

795 citations

Journal ArticleDOI
TL;DR: In this article, Tate's algorithm was used to identify loci in the moduli of F-theory compactifications corresponding to enhanced gauge symmetry and to test the proposed Ftheory/heterotic dualities in six dimensions.

726 citations


Cites result from "Integrable systems and supersymmetr..."

  • ...ed to derive the dictionary for a large number of cases for the Coulomb branch. In fact, it is quite suggestive that in the description of N = 2 Yang-Mills theories with non-simply laced gauge groups [35], groups appear with a correspondence which is very similar to what we have found with the outer automorphisms (3.1) of Dynkin diagrams. In [35] this corresponds to exchanging the long and short roots...

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Journal ArticleDOI
TL;DR: In this paper, a local geometric realization of quantum field theories together with a local application of mirror symmetry is proposed to reduce non-trivial quantum field theory results to much better understood T -dualities of type 11 strings.

726 citations

References
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Book
01 Jan 1983
TL;DR: The invariant bilinear form and the generalized casimir operator are integral representations of Kac-Moody algebras and the weyl group as mentioned in this paper, as well as a classification of generalized cartan matrices.
Abstract: Introduction Notational conventions 1 Basic definitions 2 The invariant bilinear form and the generalized casimir operator 3 Integrable representations of Kac-Moody algebras and the weyl group 4 A classification of generalized cartan matrices 5 Real and imaginary roots 6 Affine algebras: the normalized cartan invariant form, the root system, and the weyl group 7 Affine algebras as central extensions of loop algebras 8 Twisted affine algebras and finite order automorphisms 9 Highest-weight modules over Kac-Moody algebras 10 Integrable highest-weight modules: the character formula 11 Integrable highest-weight modules: the weight system and the unitarizability 12 Integrable highest-weight modules over affine algebras 13 Affine algebras, theta functions, and modular forms 14 The principal and homogeneous vertex operator constructions of the basic representation Index of notations and definitions References Conference proceedings and collections of paper

4,653 citations

01 Jan 1997

4,469 citations

Journal ArticleDOI
TL;DR: In this article, the authors studied the vacuum structure and spectrum of N = 2 supersymmetric gauge theory in four dimensions, with gauge group SU(2), and obtained exact formulas for electron and dyon masses and the metric on the moduli space of vacua.

4,007 citations


"Integrable systems and supersymmetr..." refers background in this paper

  • ...ar X iv :h ep -t h/ 95 Dedicated to the memory of Claude Itzykson, recalling his gift for combining elegant physics and beautiful mathematics...

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Journal ArticleDOI
TL;DR: In this article, the authors studied four dimensional N = 2 supersymmetric gauge theories with matter multiplets and derived the exact metric on the moduli space of quantum vacua and the exact spectrum of the stable massive states.

2,550 citations

Book
31 Oct 1974
TL;DR: In this article, the authors consider a class of Lie algebras in which every subalgebra is a subideal, and they show that it is possible to construct a locally coalescent class of these classes.
Abstract: 1. Basic concepts.- 1. Preliminaries.- 2. Nilpotency and solubility.- 3. Subideals.- 4. Derivations.- 5. Classes and closure operations.- 6. Representations and modules.- 7. Chain conditions.- 8. Series.- 2. Soluble subideals.- 1. The circle product.- 2. The Derived Join Theorems.- 3. Coalescent classes of Lie algebras.- 1. An example.- 2. Coalescence of classes with minimal conditions.- 3. Coalescence of classes with maximal conditions.- 4. The local coalescence of D.- 5. A counterexample.- 4. Locally coalescent classes of Lie algebras.- 1. The algebra of formal power series.- 2. Complete and locally coalescent classes.- 3. Acceptable subalgebras.- 5. The Mal'cev correspondence.- 1. The Campbell-Hausdorff formula.- 2. Complete groups.- 3. The matrix version.- 4. Inversion of the Campbell-Hausdorff formula.- 5. The general version.- 6. Explicit descriptions.- 6. Locally nilpotent radicals.- 1. The Hirsch-Plotkin radical.- 2. Baer, Fitting, and Gruenberg radicals.- 3. Behaviour under derivations.- 4. Baer and Fitting algebras.- 5. The Levi?-Tokarenko theorem.- 7. Lie algebras in which every subalgebra is a subideal.- 1. Nilpotent subideals.- 2. The key lemma and some applications.- 3. Engel conditions.- 4. A counterexample.- 5. Unsin's algebras.- 8. Chain conditions for subideals.- 1. Classes related to Min-si.- 2. The structure of algebras in Min-si.- 3. The case of prime characteristic.- 4. Examples of algebras with Min-si.- 5. Min-si in special classes of algebras.- 6. Max-si in special classes of algebras.- 7. Examples of algebras satisfying Max-si.- 9. Chain conditions on ascendant abelian subalgebras.- 1. Maximal conditions.- 2. Minimal conditions.- 3. Applications.- 10. Existence theorems for abelian subalgebras.- 1. Generalised soluble classes.- 2. Locally finite algebras.- 3. Generalisations of Witt algebras.- 11. Finiteness conditions for soluble Lie algebras.- 1. The maximal condition for ideals.- 2. The double chain condition.- 3. Residual finiteness.- 4. Stuntedness.- 12. Frattini theory.- 1. The Frattini subalgebra.- 2. Soluble algebras: preliminary reductions.- 3. Proof of the main theorem.- 4. Nilpotency criteria.- 5. A splitting theorem.- 13. Neoclassical structure theory.- 1. Classical structure theory.- 2. Local subideals.- 3. Radicals in locally finite algebras.- 4. Semisimplicity.- 5. Levi factors.- 14. Varieties.- 1. Verbal properties.- 2. Invariance properties of verbal ideals.- 3. Ellipticity.- 4. Marginal properties.- 5. Hall varieties.- 15. The finite basis problem.- 1. Nilpotent varieties.- 2. Partially well ordered sets.- 3. Metabelian varieties.- 4. Non-finitely based varieties.- 5. Class 2-by-abelian varieties.- 16. Engel conditions.- 1. The second and third Engel conditions.- 2. A non-locally nilpotent Engel algebra.- 3. Finiteness conditions on Engel algebras.- 4. Left and right Engel elements.- 17. Kostrikin's theorem.- 1. The Burnside problem.- 2. Basic computational results.- 3. The existence of an element of order 2.- 4. Elements which generate abelian ideals.- 5. Algebras generated by elements of order 2.- 6. A weakened form of Kostrikin's theorem.- 7. Sketch proof of Kostrikin's theorem.- 18. Razmyslov's theorem.- 1. The construction.- 2. Proof of non-nilpotence.- Some open questions.- References.- Notation index.

2,499 citations


"Integrable systems and supersymmetr..." refers background in this paper

  • ...Each affine Dynkin diagram [8] defines a periodic Toda lattice [9, 10] via Lax operators...

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  • ...The fact that the twisted Kac-Moody algebra respects a grading under the ‘dual’ Coxeter element [8] guarantees that the ‘instanton’ terms involving μ will have grade [μ] = 2h∨g ....

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