Volume 192, Nº 2 (2017)

We start with a Riemann–Hilbert problem (RHP) related toBD.I-type symmetric spaces SO(2r + 1)/S(O(2r − 2s+1) ⊗ O(2s)), s ≥ 1. We consider two RHPs: the first is formulated on the real axis R in the complex-λ plane; the second, on R ⊗ iR. The first RHP for s = 1 allows solving the Kulish–Sklyanin (KS) model; the second RHP is related to a new type of KS model. We consider an important example of nontrivial deep reductions of the KS model and show its effect on the scattering matrix. In particular, we obtain new two-component nonlinear Schrödinger equations. Finally, using the Wronski relations, we show that the inverse scattering method for KS models can be understood as generalized Fourier transforms. We thus find a way to characterize all the fundamental properties of KS models including the hierarchy of equations and the hierarchy of their Hamiltonian structures.

We show that the reflection coefficients in the quantum theory of the Liouville model calculated in the bootstrap and Hamiltonian approaches differ from each other by a phase factor and simply yield different normalizations of vertex operators.

We discuss the correspondence between models solved by the Bethe ansatz and classical integrable systems of the Calogero type. We illustrate the correspondence by the simplest example of the inhomogeneous asymmetric six-vertex model parameterized by trigonometric (hyperbolic) functions.

We propose a quantization of the Kadomtsev–Petviashvili equation on a cylinder equivalent to an infinite system of nonrelativistic one-dimensional bosons with the masses m = 1, 2,.... The Hamiltonian is Galilei-invariant and includes the split and merge terms\(\Psi _{{m_1}}^\dag \Psi _{{m_2}}^\dag {\Psi _{{m_1} + {m_2}}}\)and\(\Psi _{{m_1} + {m_2}}^\dag {\Psi _{{m_1}}}{\Psi _{{m_2}}}\)for all combinations of particles with masses m₁, m₂, and m₁ + m₂for a special choice of coupling constants. We construct the Bethe eigenfunctions for the model and verify the consistency of the coordinate Bethe ansatz and hence the quantum integrability of the model up to the mass M=8 sector.

Let g′ ⊂ g be a pair of Lie algebras of either symplectic or orthogonal infinitesimal endomorphisms of the complex vector spaces C^N−2 ⊂ C^Nand U_q(g′) ⊂ U_q(g) be a pair of quantum groups with a triangular decompositionU_q(g) = U_q(g_-)U_q(g₊)U_q(h). LetZ_q(g, g′) be the corresponding step algebra. We assume that its generators are rational trigonometric functions h ∗ → U_q(g_±). We describe their regularization such that the resulting generators do not vanish for any choice of the weight.

We describe a new generalized Wick theorem for interacting fields in two-dimensional conformal field theory and briefly discuss its relation to the Borcherds identity and its derivation by an analytic method. We give examples of calculating operator product expansions using the generalized Wick theorem including fermionic fields.