The Large N limits of integrable models (original) (raw)
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Classical integrable systems and gauge field theories
Physics of Particles and Nuclei, 2009
In these lectures I consider the Hitchin integrable systems and their relations with the self-duality equations and the twisted super-symmetric Yang-Mills theory in four dimension follow Hitchin and Kapustin-Witten. I define the Symplectic Hecke correspondence between different integrable systems. As an example I consider Elliptic Calogero-Moser system and integrable Euler-Arnold top on coadjoint orbits of the group GL(N,C) and explain the Symplectic Hecke correspondence for these systems.
String Theory and Integrable Systems
This is mainly a brief review of some key achievements in a "hot" area of theoretical and mathematical physics. The principal aim is to outline the basic structures underlying integrable quantum field theory models with infinite-dimensional symmetry groups which display a radically new type of quantum group symmetries. Certain particular aspects are elaborated upon with some detail: integrable systems of Kadomtsev-Petviashvili type and their reductions appearing in matrix models of strings; Hamiltonian approach to Lie-Poisson symmetries; quantum field theory approach to two-dimensional relativistic integrable models with dynamically broken conformal invariance. All field-theoretic models in question are of primary relevance to diverse branches of physics ranging from nonlinear hydrodynamics to string theory of fundamental particle interactions at ultra-high energies.
hep-th/9310113 STRING THEORY AND INTEGRABLE SYSTEMS1
2016
This is mainly a brief review of some key achievements in a "hot" area of theoretical and mathematical physics. The principal aim is to outline the basic structures underlying integrable quantum field theory models with infinite-dimensional symmetry groups which display a radically new type of quantum group symmetries. Certain particular aspects are elaborated upon with some detail: integrable systems of Kadomtsev-Petviashvili type and their reductions appearing in matrix models of strings; Hamiltonian approach to Lie-Poisson symmetries; quantum field theory approach to two-dimensional relativistic integrable models with dynamically broken conformal invariance. All field-theoretic models in question are of primary relevance to diverse branches of physics ranging from nonlinear hydrodynamics to string theory of fundamental particle interactions at ultra-high energies.
Two-dimensional massive integrable models on a torus
Journal of High Energy Physics
The finite-volume thermodynamics of a massive integrable QFT is described in terms of a grand canonical ensemble of loops immersed in a torus and interacting through scattering factors associated with their intersections. The path integral of the loops is evaluated explicitly after decoupling the pairwise interactions by a Hubbard-Stratonovich transformation. The HS fields are holomorphic fields depending on the rapidity and can be expanded in elementary oscillators. The torus partition function is expressed as certain expectation value in the Fock space of these oscillators. In the limit where one of the periods of the torus becomes asymptotically large, the effective field theory becomes mean field type. The mean field describes the infinite-volume thermodynamics which is solved by the Thermodynamical Bethe Ansatz.
Geometric transitions and integrable systems
Nuclear Physics B, 2006
We consider B-model large N duality for a new class of noncompact Calabi-Yau spaces modeled on the neighborhood of a ruled surface in a Calabi-Yau threefold. The closed string side of the transition is governed at genus zero by an A 1 Hitchin integrable system on a genus g Riemann surface Σ. The open string side is described by a holomorphic Chern-Simons theory which reduces to a generalized matrix model in which the eigenvalues lie on the compact Riemann surface Σ. We show that the large N planar limit of the generalized matrix model is governed by the same A 1 Hitchin system therefore proving genus zero large N duality for this class of transitions.
Gauge theories, Simple Groups and Integrable Systems
In this review we discuss interrelations between classical Hitchin integrable systems, monodromy preserving equations and topological field theories coming from N=4 supersymmetric Yang-Mills theories developed by Gukov, Kapustin and Witten. In particular, we define the systems related to bundles with nontrivial characteristic classes and discuss relations of the characteristic classes with monopole configurations in the Yang-Mills theory. 2 Classical Integrable Systems 2.1 Integrability We consider here the integrable systems of classical mechanics [17]. In this case the notion of complete integrability can be formulated correctly, while in a field theory there are subtleties in its definition. Consider a smooth symplectic manifold R of dim(R) = 2n. It means that there exists a closed non-degenerate two-form ω, and the inverse bivector π (ω a,b π bc = δ c a), such that the space of smooth functions C ∞ (R) becomes a Poisson algebra with respect to the Poisson brackets {F, G} = dF |π|dG = ∂ a F π ab ∂ b G ≡ ∂ a F π ab ∂ b G. (2.1) In terms of the bi-vector π ab the Jacobi identity for the brackets assumes the form π ab ∂ b π cd + π cb ∂ b π da + π db ∂ b π ca = 0 .
Journal of High Energy Physics
We continue our study of the N = 1 * supersymmetric gauge theory on R 2,1 × S 1 and its relation to elliptic integrable systems. Upon compactification on a circle, we show that the semi-classical analysis of the massless and massive vacua depends on the classification of nilpotent orbits, as well as on the conjugacy classes of the component group of their centralizer. We demonstrate that semi-classically massless vacua can be lifted by Wilson lines in unbroken discrete gauge groups. The pseudo-Levi subalgebras that play a classifying role in the nilpotent orbit theory are also key in defining generalized Inozemtsev limits of (twisted) elliptic integrable systems. We illustrate our analysis in the N = 1 * theories with gauge algebras su(3), su(4), so(5) and for the exceptional gauge algebra G 2. We map out modular duality diagrams of the massive and massless vacua. Moreover, we provide an analytic description of the branches of massless vacua in the case of the su(3) and the so(5) theory. The description of these branches in terms of the complexified Wilson lines on the circle invokes the Eichler-Zagier technique for inverting the elliptic Weierstrass function. After fine-tuning the coupling to elliptic points of order three, we identify the Argyres-Douglas singularities of the su(3) N = 1 * theory.
Quantization of Integrable Systems and Four Dimensional Gauge Theories
XVIth International Congress on Mathematical Physics, 2010
We study four dimensional N = 2 supersymmetric gauge theory in the Ωbackground with the two dimensional N = 2 super-Poincare invariance. We explain how this gauge theory provides the quantization of the classical integrable system underlying the moduli space of vacua of the ordinary four dimensional N = 2 theory. The ε-parameter of the Ω-background is identified with the Planck constant, the twisted chiral ring maps to quantum Hamiltonians, the supersymmetric vacua are identified with Bethe states of quantum integrable systems. This four dimensional gauge theory in its low energy description has two dimensional twisted superpotential which becomes the Yang-Yang function of the integrable system. We present the thermodynamic-Bethe-ansatz like formulae for these functions and for the spectra of commuting Hamiltonians following the direct computation in gauge theory. The general construction is illustrated at the examples of the many-body systems, such as the periodic Toda chain, the elliptic Calogero-Moser system, and their relativistic versions, for which we present a complete characterization of the L 2 -spectrum. We very briefly discuss the quantization of Hitchin system.
Supersymmetric Yang-Mills theory and integrable systems
Nuclear Physics B, 1996
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. A natural construction of such systems comes from two-dimensional gauge theory and spectral curves. Starting from this point of view, we propose an integrable system relevant to the N = 2 SU(n) gauge theory with a hypermultiplet in the adjoint representation, and offer much evidence that it is correct. The model has an SL(2, Z) S-duality group (with the central element -1 of SL(2,Z) acting as charge conjugation); SL(2, Z) permutes the Higgs, confining, and oblique confining phases in the expected fashion. We also study more exotic phases.
Integrable tops and non-commutative torus
We consider the hydrodynamics of the ideal fluid on a 2-torus and its Moyal deformations. The both type of equations have the form of the Euler-Arnold tops. The Laplace operator plays the role of the inertia-tensor. It is known that 2-d hydrodynamics is non-integrable. After replacing of the Laplace operator by a distinguish pseudo-differential operator the deformed system becomes integrable. It is an infinite rank Hitchin system over an elliptic curve with transition functions from the group of the non-commutative torus. In the classical limit we obtain an integrable analog of the hydrodynamics on a torus with the inertia-tensor operator∂ 2 instead of the conventional Laplace operator ∂∂. 1 For Manakov's top lim N → ∞ was considered in Ref.[13].