Elliptic Calogero-Moser System Research Articles

A focusing–defocusing intermediate nonlinear Schrödinger system

We introduce and study a system of coupled nonlocal nonlinear Schrödinger equations that interpolates between the mixed, focusing–defocusing Manakov system on one hand and a limiting case of the intermediate nonlinear Schrödinger equation on the other. We show that this new system, which we call the intermediate mixed Manakov (IMM) system, admits multi-soliton solutions governed by a complexification of the hyperbolic Calogero–Moser (CM) system. Furthermore, we introduce a spatially periodic version of the IMM system, for which our main result is a class of exact solutions governed by a complexified elliptic CM system.

Novikov-Veselov Symmetries of the Two-Dimensional O(N) Sigma Model

We show that Novikov-Veselov hierarchy provides a complete family of commuting symmetries of two-dimensional O(N) sigma model. In the first part of the paper we use these symmetries to prove that the Fermi spectral curve for the double-periodic sigma model is algebraic. Thus, our previous construction of the complexified harmonic maps in the case of irreducible Fermi curves is complete. In the second part of the paper we generalize our construction to the case of reducible Fermi curves and show that it gives the conformal harmonic maps to even-dimensional spheres. Remarkably, the solutions are parameterized by spectral curves of turning points of the elliptic Calogero-Moser system.

Generalizations of parabolic Higgs bundles, real structures, and integrability

We introduce a notion of quasi-antisymmetric Higgs $G$-bundles over curves with marked points. They are endowed with additional structures, which replace the parabolic structures at marked points in the parabolic Higgs bundles. The latter means that the coadjoint orbits are attached to the marked points. The moduli spaces of parabolic Higgs bundles are the phase spaces of complex completely integrable systems. In our case the coadjoint orbits are replaced by the cotangent bundles over some special symmetric spaces in such a way that the moduli space of the modified Higgs bundles are still phase spaces of complex completely integrable systems. We show that the moduli space of the parabolic Higgs bundles is the symplectic quotient of the moduli space of the quasi-antisymmetric Higgs bundle with respect to the action of product of Cartan subgroups. Also, by changing the symmetric spaces we introduce quasi-compact and quasi-normal Higgs bundles. Then the fixed point sets of real involutions acting on their moduli spaces are the phase spaces of real completely integrable systems. Several examples are given including integrable extensions of the ${\rm SL}(2)$ Euler-Arnold top, two-body elliptic Calogero-Moser system and the rational ${\rm SL}(2)$ Gaudin system together with its real reductions.

Inozemtsev system as Seiberg-Witten integrable system

In this work we establish that the Inozemtsev system is the Seiberg-Witten integrable system encoding the Coulomb branch physics of 4d mathcal{N} = 2 USp(2N) gauge theory with four fundamental and (for N ≥ 2) one antisymmetric tensor hypermultiplets. We describe the transformation from the spectral curves and canonical one-forms of the Inozemtsev system in the N = 1 and N = 2 cases to the Seiberg-Witten curves and differentials explicitly, along with the explicit matching of the modulus of the elliptic curve of spectral parameters to the gauge coupling of the field theory, and of the couplings of the Inozemtsev system to the field theory mass parameters. This result is a particular instance of a more general correspondence between crystallographic elliptic Calogero-Moser systems with Seiberg-Witten integrable systems, which will be explored in future work.

Quantum integrable systems from supergroup gauge theories

In this note, we establish several interesting connections between the super- group gauge theories and the super integrable systems, i.e. gauge theories with supergroups as their gauge groups and integrable systems defined on superalgebras. In particular, we construct the super-characteristic polynomials of super-Toda lattice and elliptic double Calogero-Moser system by considering certain orbifolded instanton partition functions of their corresponding supergroup gauge theories. We also derive an exotic generalization of \U0001d530\U0001d529(2) XXX spin chain arising from the instanton partition function of SQCD with super- gauge group, and study its Bethe ansatz equation.

Quantum elliptic Calogero-Moser systems from gauge origami

We systematically study the interesting relations between the quantum elliptic Calogero-Moser system (eCM) and its generalization, and their corresponding supersymmetric gauge theories. In particular, we construct the suitable characteristic polynomial for the eCM system by considering certain orbifolded instanton partition function of the corresponding gauge theory. This is equivalent to the introduction of certain co-dimension two defects. We next generalize our construction to the folded instanton partition function obtained through the so-called “gauge origami” construction and precisely obtain the corresponding characteristic polynomial for the doubled version, named the elliptic double Calogero-Moser (edCM) system.

Real Forms of Elliptic Integrable Systems

We describe the real forms of classical elliptic integrable systems such as the elliptic Calogero-Moser system and the elliptic Euler-Arnold top in the framework of a general scheme for constructing real reductions for a Hitchin system.

Chiral observables and S-duality in N $$ \mathcal{N} $$ = 2⋆ U(N ) gauge theories

We study N = 2* theories with gauge group U(N) and use equivariant localization to calculate the quantum expectation values of the simplest chiral ring elements. These are expressed as an expansion in the mass of the adjoint hypermultiplet, with coefficients given by quasi-modular forms of the S-duality group. Under the action of this group, we construct combinations of chiral ring elements that transform as modular forms of definite weight. As an independent check, we confirm these results by comparing the spectral curves of the associated Hitchin system and the elliptic Calogero-Moser system. We also propose an exact and compact expression for the 1-instanton contribution to the expectation value of the chiral ring elements.

On Algebraically Integrable Differential Operators on an Elliptic Curve

We study differential operators on an elliptic curve of order higher than 2 which are algebraically integrable (i.e., finite gap). We discuss classification of such operators of order 3 with one pole, discovering exotic operators on special elliptic curves defined over ${\mathbb Q}$ which do not deform to generic elliptic curves. We also study algebraically integrable operators of higher order with several poles and with symmetries, and (conjecturally) relate them to crystallographic elliptic Calogero-Moser systems (which is a generalization of the results of Airault, McKean, and Moser).

Classical conformal blocks from TBA for the elliptic Calogero-Moser system

The so-called Poghossian identities connecting the toric and spherical blocks, the AGT relation on the torus and the Nekrasov-Shatashvili formula for the elliptic Calogero-Moser Yang's (eCMY) functional are used to derive certain expressions for the classical 4-point block on the sphere. The main motivation for this line of research is the longstanding open problem of uniformization of the 4-punctured Riemann sphere, where the 4-point classical block plays a crucial role. It is found that the obtained representation for certain 4-point classical blocks implies the relation between the accessory parameter of the Fuchsian uniformization of the 4-punctured sphere and the eCMY functional. Additionally, a relation between the 4-point classical block and the $N_f=4$, ${\sf SU(2)}$ twisted superpotential is found and further used to re-derive the instanton sector of the Seiberg-Witten prepotential of the $N_f=4$, ${\sf SU(2)}$ supersymmetric gauge theory from the classical block.

Reduction of the elliptic top

We propose a relation between the elliptic top and Toda systems and obtain a new class of integrable systems in a specific limit of the elliptic top. The relation is based on the Inozemtsev limit and a symplectic map from the elliptic Calogero–Moser system to the elliptic top. In the case when N = 2 we use an explicit form of a symplectic map from the phase space of the elliptic Calogero–Moser system to the phase space of the elliptic top and show that the limiting tops are equivalent to the Toda chains. In the case when N > 2 we generalize the above procedure using only the limiting behavior of Lax matrices. In a specific limit we also obtain a more general class of systems and prove the integrability in the Liouville sense of a certain subclass of these systems. This class is described by a classical r-matrix obtained from an elliptic r-matrix.

On elliptic Calogero–Moser systems for complex crystallographic reflection groups

To every irreducible finite crystallographic reflection group (i.e., an irreducible finite reflection group G acting faithfully on an abelian variety X), we attach a family of classical and quantum integrable systems on X (with meromorphic coefficients). These families are parametrized by G-invariant functions of pairs ( T , s ) , where T is a hypertorus in X (of codimension 1), and s ∈ G is a reflection acting trivially on T. If G is a real reflection group, these families reduce to the known generalizations of elliptic Calogero–Moser systems, but in the non-real case they appear to be new. We give two constructions of the integrals of these systems – an explicit construction as limits of classical Calogero–Moser Hamiltonians of elliptic Dunkl operators as the dynamical parameter goes to 0 (implementing an idea of V. Buchstaber, G. Felder and A. Veselov (1994) [BFV]), and a geometric construction as global sections of sheaves of elliptic Cherednik algebras for the critical value of the twisting parameter. We also prove algebraic integrability of these systems for values of parameters satisfying certain integrality conditions.

Towards the proof of complete integrability of quantum elliptic many-body systems with spin degrees of freedom

We consider the problem of finding integrals of motion for quantum elliptic Calogero-Moser systems with arbitrary number of particles extended by introducing spinexchange interaction. By direct calculation, after making certain ansatz, we found first two integrals — quite probably, lowest nontrivial members of the whole commutative ring. This result might be considered as the first step in constructing this ring of the operators which commute with the Hamiltonian of the model.

Classical integrable systems and gauge field theories

In this review we consider the Hitchin integrable systems and their relations with the self-duality equations and twisted super-symmetric Yang-Mills theory in four dimensions. We define the Symplectic Hecke correspondence between different integrable systems. As an example we consider the 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.

On the solutions of 3-particle spin elliptic Calogero–Moser systems

We describe a class of the singular solutions to the multicomponent analogs of the Lamé equation, arising as equations of motion of the elliptic Calogero–Moser systems of particles carrying spin 1/2. At special value of the coupling constant we propose the ansatz which allows one to get meromorphic solutions with two arbitrary parameters. They are quantized upon the requirement of the regularity of the wave function on the hyperplanes at which particles meet and imposing periodic boundary conditions. We find also the extra integrals of motion for three-particle systems which commute with the Hamiltonian for arbitrary values of the coupling constant.

Lectures on Classical Integrable Systems and Gauge Field Theories

In these lectures we consider Hitchin integrable systems and their relations with the self-duality equations and twisted super-symmetric Yang-Mills theory in four dimension. We define the Symplectic Hecke correspondence between different integrable systems. As an example we 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.

Gauge Theory Approach Towards an Explicit Solution of the (Classical) Elliptic Calogero-Moser System

We discuss the relation of the trigonometric Calogero-Moser (CM) system to Yang-Mills gauge theories and its generalization to the elliptic case. This yields a linearization of the time evolution of the elliptic CM system and suggests two promising strategies for finding a fully explicit solution of this model. We also present a large class of integrable spin-particle systems generalizing the elliptic CM system.

The curve of compactified 6Dgauge theories and integrable systems

We analyze the Seiberg-Witten curve of the six-dimensional N=(1,1) gauge theory compactified on a torus to four dimensions. The effective theory in four dimensions is a deformation of the N=2* theory. The curve is naturally holomorphically embedding in a slanted four-torus--actually an abelian surface--a set-up that is natural in Witten's M-theory construction of N=2 theories. We then show that the curve can be interpreted as the spectral curve of an integrable system which generalizes the N-body elliptic Calogero-Moser and Ruijsenaars-Schneider systems in that both the positions and momenta take values in compact spaces. It turns out that the resulting system is not simply doubly elliptic, rather the positions and momenta, as two-vectors, take values in the ambient abelian surface. We analyze the two-body system in some detail. The system we uncover provides a concrete realization of a Beauville-Mukai system based on an abelian surface rather than a K3 surface.

Critical points of glueball superpotentials and equilibria of integrable systems

We compare the matrix model and integrable system approaches to calculating the exact vacuum structure of general N=1 deformations of either the basic N=2 theory or its generalization with a massive adjoint hypermultiplet, the N=2* theory. We show that there is a one-to-one correspondence between arbitrary critical points of the Dijkgraaf-Vafa glueball superpotential and equilibrium configurations of the associated integrable system. The latter being either the periodic Toda chain, for N=2, or the elliptic Calogero-Moser system, for N=2*. We show in both cases that the glueball superpotential at the crtical point equals the associated Hamiltonian. Our discussion includes an analysis of the vacuum structure of the N=1* theory with an arbitrary tree-level superpotential for one of the adjoint chiral fields.

Classicalr-matrices and the Feigin–Odesskii algebra via Hamiltonian and Poisson reductions

We present a formula for a classical $r$-matrix of an integrable system obtained by Hamiltonian reduction of some free field theories using pure gauge symmetries. The framework of the reduction is restricted only by the assumption that the respective gauge transformations are Lie group ones. Our formula is in terms of Dirac brackets, and some new observations on these brackets are made. We apply our method to derive a classical $r$-matrix for the elliptic Calogero-Moser system with spin starting from the Higgs bundle over an elliptic curve with marked points. In the paper we also derive a classical Feigin-Odesskii algebra by a Poisson reduction of some modification of the Higgs bundle over an elliptic curve. This allows us to include integrable lattice models in a Hitchin type construction.