Elliptic Integrable Systems Research Articles

Lax equations for relativistic Gaudin models on elliptic curve

We describe the most general GL NM classical elliptic finite-dimensional integrable system, which Lax matrix has n simple poles on elliptic curve. For M = 1 it reproduces the classical inhomogeneous spin chain, for N = 1 it is the Gaudin type (multispin) extension of the spin Ruijsenaars–Schneider model, and for n = 1 the model of M interacting relativistic GL N tops emerges in some particular case. In this way we present a classification for relativistic Gaudin models on GL-bundles over elliptic curve. As a by-product we describe the inhomogeneous Ruijsenaars chain. We show that this model can be considered as a particular case of multispin Ruijsenaars–Schneider model when residues of the Lax matrix are of rank one. An explicit parametrization of the classical spin variables through the canonical variables is obtained for this model. Finally, the most general GL NM model is also described through R-matrices satisfying associative Yang–Baxter equation. This description provides the trigonometric and rational analogues of GL NM models.

3d $\mathcal{N}=4$ Gauge Theories on an Elliptic Curve

This paper studies 3d3d\mathcal{N}=4𝒩=4 supersymmetric gauge theories on an elliptic curve, with the aim to provide a physical realisation of recent constructions in equivariant elliptic cohomology of symplectic resolutions. We first study the Berry connection for supersymmetric ground states in the presence of mass parameters and flat connections for flavour symmetries, which results in a natural construction of the equivariant elliptic cohomology variety of the Higgs branch. We then investigate supersymmetric boundary conditions and show from an analysis of boundary ’t Hooft anomalies that their boundary amplitudes represent equivariant elliptic cohomology classes. We analyse two distinguished classes of boundary conditions known as exceptional Dirichlet and enriched Neumann, which are exchanged under mirror symmetry. We show that the boundary amplitudes of the latter reproduce elliptic stable envelopes introduced by Aganagic-Okounkov, and relate boundary amplitudes of the mirror symmetry interface to the mother function in equivariant elliptic cohomology. Finally, we consider correlation functions of Janus interfaces for varying mass parameters, recovering the chamber R-matrices of elliptic integrable systems.

E-string quantum curve

In this work we study the quantisation of the Seiberg-Witten curve for the E-string theory compactified on a two-torus. We find that the resulting operator expression belongs to the class of elliptic quantum curves. It can be rephrased as an eigenvalue equation with eigenvectors corresponding to co-dimension 2 defect operators and eigenvalues to co-dimension 4 Wilson surfaces wrapping the elliptic curve, respectively. Moreover, the operator we find is a generalised version of the van Diejen operator arising in the study of elliptic integrable systems. Although the microscopic representation of the co-dimension 4 defect only furnishes an SO(16) flavour symmetry in the UV, we find an enhancement in the IR to representations in terms of affine E8 characters. Finally, using the Nekrasov-Shatashvili limit of the E-string BPS partition function, we give a path integral derivation of the quantum curve.

Integrable extensions of classical elliptic integrable systems

In this article we consider two particular examples of general construction proposed in arXiv:2012.15529. We consider the integrable extensions of the classical elliptic Calogero-Moser model of N particles with spin and the integrable Euler-Arnold top related to the group SL(N,C). The extended systems has additional N-1 degrees of freedom and can be described in terms of the Darboux variables.

Integrable extensions of classical elliptic integrable systems

Рассмотрены два частных примера ранее предложенной общей конструкции: расширение интегрируемого волчка Эйлера-Арнольда, связанного с группой $SL(N,\mathbb{C})$, и интегрируемое расширение классической $N$-частичной эллиптической системы Калоджеро-Мозера со спином. Расширенные системы имеют дополнительные $N-1$ степени свободы и могут быть описаны в терминах переменных Дарбу.

Characteristic determinant and Manakov triple for the double elliptic integrable system

Using the intertwining matrix of the IRF-Vertex correspondence we propose a determinant representation for the generating function of the commuting Hamiltonians of the double elliptic integrable system. More precisely, it is a ratio of the normally ordered determinants, which turns into a single determinant in the classical case. With its help we reproduce the recently suggested expression for the eigenvalues of the Hamiltonians for the dual to elliptic Ruijsenaars model. Next, we study the classical counterpart of our construction, which gives expression for the spectral curve and the corresponding L-matrix. This matrix is obtained explicitly as a weighted average of the Ruijsenaars and/or Sklyanin type Lax matrices with the weights as in the theta function series definition. By construction the L-matrix satisfies the Manakov triple representation instead of the Lax equation. Finally, we discuss the factorized structure of the L-matrix.

The quantum DELL system

We propose quantum Hamiltonians of the double elliptic many-body integrable system (DELL) and study its spectrum. These Hamiltonians are certain elliptic functions of coordinates and momenta. Our results provide quantization of the classical DELL system which was previously found in the string theory literature. The eigenfunctions for the N-body model are instanton partition functions of 6d SU(N) gauge theory with adjoint matter compactified on a torus with a codimension two defect. As a byproduct we discover new family of symmetric orthogonal polynomials which provide an elliptic generalization to Macdonald polynomials.

Relativistic Interacting Integrable Elliptic Tops

We propose relativistic generalization of integrable systems describing $M$ interacting elliptic ${\rm gl}(N)$ tops of the Euler-Arnold type. The obtained models are elliptic integrable systems, which reproduce the spin elliptic ${\rm GL}(M)$ Ruijsenaars-Schneider model for $N=1$ case, while in the $M=1$ case they turn into relativistic integrable ${\rm GL}(N)$ elliptic tops. The Lax pairs with spectral parameter on elliptic curve are constructed.

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.

Real forms of elliptic integrable systems

В рамках общей схемы построения вещественных редукций систем Хитчина описаны вещественные формы таких классических эллиптических интегрируемых систем, как эллиптическая система Калоджеро-Мозера и эллиптический волчок Эйлера-Арнольда.

Exact quantization conditions for the elliptic Ruijsenaars-Schneider model

We propose and test exact quantization conditions for the N-particle quantum elliptic Ruijsenaars-Schneider integrable system, as well as its Calogero-Moser limit, based on the conjectural correspondence to the five-dimensional mathcal{N}={1}^{ast } SU(N) gauge theory in the Nekrasov-Shatashvili limit. We discuss two natural sets of quantization conditions, related by the electro-magnetic duality, and the importance of non-perturbative corrections in the Planck constant. We also comment on the eigenfunction problem, by reinterpreting the Separation of Variables approach in gauge theory terms.

Two-Valued Groups, Kummer Varieties, and Integrable Billiards

A natural and important question of study two-valued groups associated with hyperelliptic Jacobians and their relationship with integrable systems is motivated by seminal examples of relationship between algebraic two-valued groups related to elliptic curves and integrable systems such as elliptic billiards and celebrated Kowalevski top. The present paper is devoted to the case of genus 2, to the investigation of algebraic two-valued group structures on Kummer varieties. One of our approaches is based on the theory of $$\sigma $$ -functions. It enables us to study the dependence of parameters of the curves, including rational limits. Following this line, we are introducing a notion of n-groupoid as natural multivalued analogue of the notion of topological groupoid. Our second approach is geometric. It is based on a geometric approach to addition laws on hyperelliptic Jacobians and on a recent notion of billiard algebra. Especially important is connection with integrable billiard systems within confocal quadrics. The third approach is based on the realization of the Kummer variety in the framework of moduli of semi-stable bundles, after Narasimhan and Ramanan. This construction of the two-valued structure is remarkably similar to the historically first example of topological formal two-valued group from 1971, with a significant difference: the resulting bundles in the 1971 case were ”virtual”, while in the present case the resulting bundles are effectively realizable.

(q, t)-KZ equations for quantum toroidal algebra and Nekrasov partition functions on ALE spaces

We describe the general strategy for lifting the Wess-Zumino-Witten model from the level of one-loop Kac-Moody {U}_q{left(widehat{mathfrak{g}}right)}_k to generic quantum toroidal algebras. A nearly exhaustive presentation is given for both {U}_{q,t}left({widehat{widehat{mathfrak{gl}}}}_1right) and {U}_{q,t}left({widehat{widehat{mathfrak{gl}}}}_nright) when the screenings do not exist and thus all the correlators are purely algebraic, i.e. do not include additional hypergeometric type integrations/summations.Generalizing the construction of the intertwiner (refined topological vertex) of the Ding-Iohara-Miki (DIM) algebra, we obtain the intertwining operators of the Fock representations of the quantum toroidal algebra of type An. The correlation functions of these operators satisfy the (q, t)-Knizhnik-Zamolodchikov (KZ) equation, which features the ℛ-matrix. The matching with the Nekrasov function for the instanton counting on the ALE space is worked out explicitly.We also present an important application of the DIM formalism to the study of 6d gauge theories described by the double elliptic integrable systems. We show that the modular and periodicity properties of the gauge theories are neatly explained by the network matrix models providing solutions to the elliptic (q, t)-KZ equations.

Relativistic elliptic matrix tops and finite Fourier transformations

We consider a family of classical elliptic integrable systems including (relativistic) tops and their matrix extensions of different types. These models can be obtained from the “off-shell” Lax pairs, which do not satisfy the Lax equations in general case but become true Lax pairs under various conditions (reductions). At the level of the off-shell Lax matrix, there is a natural symmetry between the spectral parameter z and relativistic parameter [Formula: see text]. It is generated by the finite Fourier transformation, which we describe in detail. The symmetry allows one to consider z and [Formula: see text] on an equal footing. Depending on the type of integrable reduction, any of the parameters can be chosen to be the spectral one. Then another one is the relativistic deformation parameter. As a by-product, we describe the model of N2 interacting GL(M) matrix tops and/or M2 interacting GL(N) matrix tops depending on a choice of the spectral parameter.

Formal Killing fields for minimal Lagrangian surfaces in complex space forms

The differential system for minimal Lagrangian surfaces in a 2 C -dimensional, non-flat, complex space form is an elliptic integrable system defined on the Grassmann bundle of oriented Lagrangian 2-planes. This is a 6-symmetric space associated with the Lie group SL ( 3 , C ) , and the minimal Lagrangian surfaces arise as the primitive maps. Utilizing this property, we derive the inductive differential algebraic formulas for a pair of the formal loop algebra s l ( 3 , C ) [ [ λ ] ] -valued canonical formal Killing fields. For applications, (a) we give a complete classification of the (pseudo) Jacobi fields for the minimal Lagrangian system, (b) we obtain an infinite sequence of conservation laws from the components of the canonical formal Killing fields.

On elliptic algebras and large-n supersymmetric gauge theories

In this note we further develop the duality between supersymmetric gauge theories in various dimensions and elliptic integrable systems such as Ruijsenaars-Schneider model and periodic intermediate long wave hydrodynamics. These models arise in instanton counting problems and are described by certain elliptic algebras. We discuss the correspondence between the two types of models by employing the large-n limit of the dual gauge theory. In particular, we provide non-Abelian generalization of our previous result on the intermediate long wave model.

Spectral duality in elliptic systems, six-dimensional gauge theories and topological strings

We consider Dotsenko-Fateev matrix models associated with compactified Calabi-Yau threefolds. They can be constructed with the help of explicit expressions for refined topological vertex, i.e. are directly related to the corresponding topological string amplitudes. We describe a correspondence between these amplitudes, elliptic and affine type Selberg integrals and gauge theories in five and six dimensions with various matter content. We show that the theories of this type are connected by spectral dualities, which can be also seen at the level of elliptic Seiberg-Witten integrable systems. The most interesting are the spectral duality between the XYZ spin chain and the Ruijsenaars system, which is further lifted to self-duality of the double elliptic system.

On the N = 1 ∗ $$ \mathcal{N}={1}^{\ast } $$ gauge theory on a circle and elliptic integrable systems

We continue our study of the N=1* supersymmetric gauge theory 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 G2. 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) theory.

Defects and quantum Seiberg-Witten geometry

We study the Nekrasov partition function of the five dimensional U(N) gauge theory with maximal supersymmetry on R^4 x 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 x 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.

Duality and modularity in elliptic integrable systems and vacua of N = 1 ∗ $$ \mathcal{N}={1}^{\ast } $$ gauge theories

We study complexified elliptic Calogero-Moser integrable systems. We determine the value of the potential at isolated extrema, as a function of the modular parameter of the torus on which the integrable system lives. We calculate the extrema for low rank B,C,D root systems using a mix of analytical and numerical tools. For so(5) we find convincing evidence that the extrema constitute a vector valued modular form for a congruence subgroup of the modular group. For so(7) and so(8), the extrema split into two sets. One set contains extrema that make up vector valued modular forms for congruence subgroups, and a second set contains extrema that exhibit monodromies around points in the interior of the fundamental domain. The former set can be described analytically, while for the latter, we provide an analytic value for the point of monodromy for so(8), as well as extensive numerical predictions for the Fourier coefficients of the extrema. Our results on the extrema provide a rationale for integrality properties observed in integrable models, and embed these into the theory of vector valued modular forms. Moreover, using the data we gather on the modularity of complexified integrable system extrema, we analyse the massive vacua of mass deformed N=4 supersymmetric Yang-Mills theories with low rank gauge group of type B,C and D. We map out their transformation properties under the infrared electric-magnetic duality group as well as under triality for N=1* with gauge algebra so(8). We find several intriguing properties of the quantum gauge theories.