Elliptic Calogero-Moser System Research Articles

New results from glueball superpotentials and matrix models: the Leigh-Strassler deformation

Using the result of a matrix model computation of the exact glueball superpotential, we investigate the relevant mass perturbations of the Leigh-Strassler marginal ``q'' deformation of N=4 supersymmetric gauge theory. We recall a conjecture for the elliptic superpotential that describes the theory compactified on a circle and identify this superpotential as one of the Hamiltonians of the elliptic Ruijsenaars-Schneider integrable system. In the limit that the Leigh-Strassler deformation is turned off, the integrable system reduces to the elliptic Calogero-Moser system which describes the N=1^* theory. Based on these results, we identify the Coulomb branch of the partially mass-deformed Leigh-Strassler theory as the spectral curve of the Ruijsenaars-Schneider system. We also show how the Leigh-Strassler deformation may be obtained by suitably modifying Witten's M theory brane construction of N=2 theories.

Hitchin Systems ? Symplectic Hecke Correspondence and Two-Dimensional Version

The aim of this paper is two-fold. First, we define symplectic maps between Hitchin systems related to holomorphic bundles of different degrees. We call these maps the Symplectic Hecke Correspondence (SHC) of the corresponding Higgs bundles. They are constructed by means of the modification of the underlying holomorphic bundles. SHC allows to construct Bäcklund transformations in the Hitchin systems defined over Riemann curves with marked points. We apply the general scheme to the elliptic Calogero-Moser (CM) system and construct SHC to an integrable SL(N,ℂ) Euler-Arnold top (the elliptic SL(N,ℂ)-rotator). Next, we propose a generalization of the Hitchin approach to 2d integrable theories related to the Higgs bundles of infinite rank. The main example is an integrable two-dimensional version of the two-body elliptic CM system. The previous construction allows us to define SHC between the two-dimensional elliptic CM system and the Landau-Lifshitz equation.

Geometric construction of elliptic integrable systems and N=1* superpotentials

We show how the elliptic Calogero-Moser integrable systems arise from a symplectic quotient construction, generalising the construction for A_{N-1} by Gorsky and Nekrasov to other algebras. This clarifies the role of (twisted) affine Kac-Moody algebras in elliptic Calogero-Moser systems and allows for a natural geometric construction of Lax operators for these systems. We elaborate on the connection of the associated Hamiltonians to superpotentials for N=1* deformations of N=4 supersymmetric gauge theory, and argue how non-perturbative physics generates the elliptic superpotentials. We also discuss the relevance of these systems and the associated quotient construction to open problems in string theory. In an appendix, we use the theory of orbit algebras to show the systematics behind the folding procedures for these integrable models.

Elliptic Families of Solutions of the Kadomtsev-Petviashvili Equation and the Field Elliptic Calogero-Moser System ∗

We present a Lax pair for the field elliptic Calogero-Moser system and establish a connection between this system and the Kadomtsev-Petviashvili equation.Namely, we consider elliptic families of solutions of the KP equation such that their poles satisfy a constraint of being balanced.We show that the dynamics of these poles is described by a reduction of the field elliptic CM system. We construct a wide class of solutions to the field elliptic CM system by showing that any N -fold branched cover of an elliptic curve gives rise to an elliptic family of solutions of the KP equation with balanced poles. The main goal of this paper is to establish a connection between the field analog of the elliptic Calogero-Moser system (CM) introduced in (10) and the Kadomtsev-Petviashvili equation (KP). This connection is a next step along the line that goes back to the paper (1), where the dynamics of the poles of elliptic (rational or trigonometric) solutions of the Korteweg-de Vries equation (KdV) was described in terms of commuting flows of the elliptic (rational or trigonometric) CM system. It was shown in the earlier paper (7) of one of the authors that the constrained correspondence between a theory of the elliptic CM system and a theory of the elliptic solutions of the KdV equation becomes an isomorphism for the case of the KP equation.It turns out that a function u(x, y, t) that is an elliptic function with respect to the variable x satisfies the KP equation if and only if it has the form

Calogero-Moser Systems and Hitchin Systems

We exhibit the elliptic Calogero-Moser system as a Hitchin system of G-principal Higgs pairs. The group G, though naturally associated to any root system, is not semi-simple. We then interpret the Lax pairs with spectral parameter of [dP1] and [BSC1] in terms of equivariant embeddings of the Hitchin system of G into that of GL(N).

Vector Bundles and Lax Equations on Algebraic Curves

The Hamiltonian theory of zero-curvature equations with spectral parameter on an arbitrary compact Riemann surface is constructed. It is shown that the equations can be seen as commuting flows of an infinite-dimensional field generalization of the Hitchin system. The field analog of the elliptic Calogero-Moser system is proposed. An explicit parameterization of Hitchin system based on the Tyurin parameters for stable holomorphic vector bundles on algebraic curves is obtained.

Varieties of elliptic solitons

We set out a method for giving explicit algebraic coordinates on varieties of elliptic solitons, which consists of finding the spectral curve by elimination and solving the Jacobi inversion problem by the use of Kleinian functions and their identities. As an example we solve a five-particle elliptic Calogero-Moser system, whose spectral curve turns out to be completely reducible over three equianharmonic elliptic curves.

CalogerosMoser Systems and Super YangsMills with Adjoint Matter

We review the construction of Lax pairs with spectral parameter for twisted and utwisted elliptic Calogero-Moser systems defined by a general simple Lie algebra G, and the corresponding solution of N = 2 SUSY G Yang-Mills theories with a hypermultplet in the adjoint representation of G.

Calogero–Moser Systems and Super Yang–Mills with Adjoint Matter

We review the construction of Lax pairs with spectral parameter for twisted and untwisted elliptic Calogero-Moser systems defined by a general simple Lie algebra , and the corresponding solution of SUSY Yang-Mills theories with a hypermultiplet in the adjoint representation of .

MORE EVIDENCE FOR THE WDVV EQUATIONS IN ${\mathcal N} = 2$ SUSY YANG–MILLS THEORIES

We consider 4D and 5D [Formula: see text] supersymmetric theories and demonstrate that in general their Seiberg–Witten prepotentials satisfy the Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations. General proof for the Yang–Mills models (with matter in the first fundamental representation) makes use of the hyperelliptic curves and underlying integrable systems. A wide class of examples is discussed; it contains few understandable exceptions. In particular, in the perturbative regime of 5D theories, in addition to naive field theory expectations some extra terms appear, as happens in heterotic string models. We consider also the example of the Yang–Mills theory with matter hypermultiplet in the adjoint representation (related to the elliptic Calogero–Moser system) when the standard WDVV equations do not hold.

Elliptic Calogero–Moser systems and isomonodromic deformations

We show that various models of the elliptic Calogero–Moser systems are accompanied with an isomonodromic system on a torus. The isomonodromic partner is a nonautonomous Hamiltonian system defined by the same Hamiltonian. The role of the time variable is played by the modulus of the base torus. A suitably chosen Lax pair (with an elliptic spectral parameter) of the elliptic Calogero–Moser system turns out to give a Lax representation of the nonautonomous system as well. This Lax representation ensures that the nonautonomous system describes isomonodromic deformations of a linear ordinary differential equation on the torus on which the spectral parameter of the Lax pair is defined. A particularly interesting example is the “extended twisted BCl model” recently introduced along with some other models by Bordner and Sasaki, who remarked that this system is equivalent to Inozemtsev’s generalized elliptic Calogero–Moser system. We use the “root-type” Lax pair developed by Bordner et al. to formulate the associated isomonodromic system on the torus.

Seiberg-Witten Theory and Calogero-Moser Systems*

We present a brief account of a series of recent results on twisted and untwisted elliptic Calogero-Moser systems, and on their fundamental role in the Seiberg-Witten solution of gauge theories with one massive hypermultiplet in the adjoint representation of an arbitrary gauge algebra G.

On Bäcklund transformations for many-body systems

Using the n-particle periodic Toda lattice and the relativistic generalization due to Ruijsenaars of the elliptic Calogero-Moser system as examples, we revise the basic properties of the Baecklund transformations (BT's) from the Hamiltonian point of view. The analogy between BT and Baxter's quantum Q-operator pointed out by Pasquier and Gaudin is exploited to produce a conjugated variable mu for the parameter lambda of the BT B_lambda such that mu belongs to the spectrum of the Lax operator L(lambda). As a consequence, the generating function of the composition of n BT's gives rise also to another canonical transformation separating variables for the model. For the Toda lattice the dual BT parametrized by mu is introduced.

Order parameters, free fermions, and conservation laws for Calogero–Moser systems

The classical order parameters for the $\N=2$ supersymmetric SU(N) gauge theory with matter in the adjoint representation are exhibited explicitly as conservation laws for the elliptic Calogero-Moser system. Central to the construction are certain elliptic function identities, which arise from considering Feynman diagrams in a theory of free fermions with twisted boundary conditions.

Quantum integrable systems and differential Galois theory

This paper is devoted to a systematic study of quantum completely integrable systems (i.e., complete systems of commuting differential operators) from the point of view of algebraic geometry. We investigate the eigenvalue problem for such systems and the correspondingD-module when the eigenvalues are in generic position. In particular, we show that the differential Galois group of this eigenvalue problem is reductive at generic eigenvalues. This implies that a system is algebraically integrable (i.e., its eigenvalue problem is explicitly solvable in quadratures) if and only if the differential Galois group is commutative for generic eigenvalues. We apply this criterion of algebraic integrability to two examples: finite-zone potentials and the elliptic Calogero-Moser system. In the second example, we obtain a proof of the Chalyh-Veselov conjecture that the Calogero-Moser system with integer parameter is algebraically integrable, using the results of Felder and Varchenko.

Separation of Variables for the Ruijsenaars System

We construct a separation of variables for the classical n-particle Ruijsenaars system (the relativistic analog of the elliptic Calogero-Moser system). The separated coordinates appear as the poles of the properly normalised eigenvector (Baker-Akhiezer function) of the corresponding Lax matrix. Two different normalisations of the BA functions are analysed. The canonicity of the separated variables is verified with the use of the r-matrix technique. The explicit expressions for the generating function of the separating canonical transform are given in the simplest cases n=2 and n=3. Taking the nonrelativistic limit we also construct a separation of variables for the elliptic Calogero-Moser system.

Geometric construction of the classical r-matrix for the elliptic Calogero-Moser system

By applying the Hamiltonian reduction technique we derive a matrix first order differential equation that yields the classical r-matrices of the elliptic (Euler-) Calogero-Moser systems as well as their degenerations.

Generating equation for r-matrices related to dynamical systems of Calogero type

By applying the Hamiltonian reduction technique we derive a matrix first order differential equation that yields the classical r-matrices of the elliptic (Euler-)Calogero-Moser systems as well as their degenerations.

Holomorphic bundles and many-body systems

We show that spin generalization of elliptic Calogero-Moser system, elliptic extension of Gaudin model and their cousins are the degenerations of Hitchin systems. Applications to the constructions of integrals of motion, angle-action variables and quantum systems are discussed. The constructions of classical systems are motivated by Conformal Field Theory, and their quantum counterparts can be thought of as being the degenerations of the critical level Knizhnik-Zamolodchikov-Bernard equations.