Nenad Manojlovic

Creation operators and Bethe vectors of the osp(1|2) Gaudin model

A Gaudin model based on the orthosymplectic Lie superalgebra osp(1|2) is studied. The eigenvectors of the osp(1|2) invariant Gaudin Hamiltonians are constructed by algebraic Bethe ansatz. Corresponding creation operators are defined by a recurrence relation. Furthermore, explicit solution to this recurrence relation is found. The action of the creation operators on the lowest spin vector yields Bethe vectors of the model. The relation between the Bethe vectors and solutions to the Knizhnik–Zamolodchikov equation of the corresponding super-conformal field theory is established.

Quantum symmetry algebras of spin systems related to Temperley-Lieb R-matrices

A reducible representation of the Temperley–Lieb algebra is constructed on the tensor product of n-dimensional spaces. One obtains as a centralizer of this action a quantum algebra (a quasitriangular Hopf algebra) Uq with a representation ring equivalent to the representation ring of the sl2 Lie algebra. This algebra Uq is the symmetry algebra of the corresponding open spin chain.

Trigonometric osp(1|2) Gaudin model

The problems connected with Gaudin models are reviewed by analyzing model related to the trigonometric osp(1|2) classical r-matrix. The eigenvectors of the trigonometric osp(1|2) Gaudin Hamiltonians are found using explicitly constructed creation operators. The commutation relations between the creation operators and the generators of the trigonometric loop superalgebra are calculated. The coordinate representation of the Bethe states is presented. The relation between the Bethe vectors and solutions to the Knizhnik–Zamolodchikov equation yields the norm of the eigenvectors. The generalized Knizhnik–Zamolodchikov system is discussed both in the rational and in the trigonometric case.

Schlesinger transformations for elliptic isomonodromic deformations

Schlesinger transformations are discrete monodromy-preserving symmetry transformations of the classical Schlesinger system. Generalizing well-known results from the Riemann sphere we construct these transformations for isomonodromic deformations on genus one Riemann surfaces. Their action on the system’s tau-function is computed and we obtain an explicit expression for the ratio of the old and the transformed tau-function.

TRIGONOMETRIC sℓ(2) GAUDIN MODEL WITH BOUNDARY TERMS

We review the derivation of the Gaudin model with integrable boundaries. Starting from the non-symmetric R-matrix of the inhomogeneous spin-½ XXZ chain and generic solutions of the reflection equation and the dual reflection equation, the corresponding Gaudin Hamiltonians with boundary terms are calculated. An alternative derivation based on the so-called classical reflection equation is discussed.

Jordanian deformation of the open XXX spin chain

We find the general solution of the reflection equation associated with the Jordanian deformation of the SL(2)-invariant Yang R-matrix. A special scaling limit of the XXZ model with general boundary conditions leads to the same K-matrix. Following the Sklyanin formalism, we derive the Hamiltonian with the boundary terms in explicit form. We also discuss the structure of the spectrum of the deformed XXX model and its dependence on the boundary conditions.

G 2-Calogero-Moser Lax operators from reduction

Abstract We construct a Lax operator for the G 2-Calogero-Moser model by means of a double reduction procedure. In the first reduction step we reduce the A 6-model to a B 3-model with the help of an embedding of the B 3-root system into the A 6-root system together with the specification of certain coupling constants. The G 2-Lax operator is obtained thereafter by means of an additional reduction by exploiting the embedding of the G 2-system into the B 3-system. The degree of algebraically independent and non-vanishing charges is found to be equal to the degrees of the corresponding Lie algebra.

Asymptotic behaviour of cylindrical waves interacting with spinning strings

We consider a family of cylindrical spacetimes endowed with angular momentum that are solutions to the vacuum Einstein equations outside the symmetry axis. This family was recently obtained by performing a complete gauge fixing adapted to cylindrical symmetry. In this paper, we find boundary conditions that ensure that the metric arising from this gauge fixing is well defined and that the resulting reduced system has a consistent Hamiltonian dynamics. These boundary conditions must be imposed both on the symmetry axis and in the region far from the axis at spacelike infinity. Employing such conditions, we determine the asymptotic behaviour of the metric close to and far from the axis. In each of these regions, the approximate metric describes a conical geometry with a time dislocation. In particular, around the symmetry axis the effect of the singularity consists in inducing a constant deficit angle and a timelike helical structure. Based on these results and on the fact that the degrees of freedom in our family of metrics coincide with those of cylindrical vacuum gravity, we argue that the analysed set of spacetimes represent cylindrical gravitational waves surrounding a spinning cosmic string. For any of these spacetimes, a prediction of our analysis is that the wave content increases the deficit angle at spatial infinity with respect to that detected around the axis.

Infinite dimensional algebras and quantum integrable systems

Gaudin Model and Opers.- Integrable Models with Unstable Particles.- Quantum Reduction in the Twisted Case.- Representation Theory and Quantum Integrability.- Connecting Lattice and Relativistic Models via Conformal Field Theory.- Elliptic Spectral Parameter and Infinite-Dimensional Grassmann Variety.- Trigonometric Degeneration and Orbifold Wess-Zumino-Witten Model. II.- Weil-Petersson Geometry of the Universal Teichmuller Space.- Duality for Knizhinik-Zamolodchikov and Dynamical Equations, and Hypergeometric Integrals.

Algebraic Bethe Ansatz for deformed Gaudin model

The Gaudin model based on the sl2-invariant r-matrix with an extra Jordanian term depending on the spectral parameters is considered. The appropriate creation operators defining the Bethe states of the system are constructed through a recurrence relation. The commutation relations between the generating function t(λ) of the integrals of motion and the creation operators are calculated and therefore the algebraic Bethe ansatz is fully implemented. The energy spectrum as well as the corresponding Bethe equations of the system coincide with the ones of the sl2-invariant Gaudin model. As opposed to the sl2-invariant case, the operator t(λ) and the Gaudin Hamiltonians are not Hermitian. Finally, the inner products and norms of the Bethe states are studied.