Tetrahedron Equation and Quantum R Matrices for Spin Representations of B^{(1)}_n, D^{(1)}_n and D^{(2)}_{n+1}

It is known that a solution of the tetrahedron equation generates infinitely many solutions of the Yang-Baxter equation via suitable reductions. In this paper this scheme is applied to an oscillator solution of the tetrahedron equation involving bosons and fermions by using special 3d boundary conditions. The resulting solutions of the Yang-Baxter equation are identified with the quantum R matrices for the spin representations of B^{(1)}_n, D^{(1)}_n and D^{(2)}_{n+1}.Comment: 17 pages, 7 figures, minor misprint correcte

Spectral equations for the modular oscillator

Motivated by applications for non-perturbative topological strings in toric Calabi--Yau manifolds, we discuss the spectral problem for a pair of commuting modular conjugate (in the sense of Faddeev) Harper type operators, corresponding to a special case of the quantized mirror curve of local $\mathbb{P}^1\times\mathbb{P}^1$ and complex values of Planck's constant. We illustrate our analytical results by numerical calculations.Comment: 23 pages, 9 figures, references added and interpretation of the numerical results of Section 6 correcte

Geometry of quadrilateral nets: second Hamiltonian form

Discrete Darboux-Manakov-Zakharov systems possess two distinct Hamiltonian forms. In the framework of discrete-differential geometry one Hamiltonian form appears in a geometry of circular net. In this paper a geometry of second form is identified.Comment: 6 page

Zamolodchikov's Tetrahedron Equation and Hidden Structure of Quantum Groups

The tetrahedron equation is a three-dimensional generalization of the Yang-Baxter equation. Its solutions define integrable three-dimensional lattice models of statistical mechanics and quantum field theory. Their integrability is not related to the size of the lattice, therefore the same solution of the tetrahedron equation defines different integrable models for different finite periodic cubic lattices. Obviously, any such three-dimensional model can be viewed as a two-dimensional integrable model on a square lattice, where the additional third dimension is treated as an internal degree of freedom. Therefore every solution of the tetrahedron equation provides an infinite sequence of integrable 2d models differing by the size of this "hidden third dimension". In this paper we construct a new solution of the tetrahedron equation, which provides in this way the two-dimensional solvable models related to finite-dimensional highest weight representations for all quantum affine algebra $U_q(\hat{sl}(n))$, where the rank $n$ coincides with the size of the hidden dimension. These models are related with an anisotropic deformation of the $sl(n)$-invariant Heisenberg magnets. They were extensively studied for a long time, but the hidden 3d structure was hitherto unknown. Our results lead to a remarkable exact "rank-size" duality relation for the nested Bethe Ansatz solution for these models. Note also, that the above solution of the tetrahedron equation arises in the quantization of the "resonant three-wave scattering" model, which is a well-known integrable classical system in 2+1 dimensions.Comment: v2: references adde

Tetrahedron Equation and Quantum RR Matrices for modular double of Uq(Dn+1(2)),Uq(A2n(2))U_q(D^{(2)}_{n+1}), U_q(A^{(2)}_{2n}) and Uq(Cn(1))U_q(C^{(1)}_{n})

We introduce a homomorphism from the quantum affine algebras $U_q(D^{(2)}_{n+1}), U_q(A^{(2)}_{2n}), U_q(C^{(1)}_{n})$ to the $n$-fold tensor product of the $q$-oscillator algebra ${\mathcal A}_q$. Their action commute with the solutions of the Yang-Baxter equation obtained by reducing the solutions of the tetrahedron equation associated with the modular and the Fock representations of ${\mathcal A}_q$. In the former case, the commutativity is enhanced to the modular double of these quantum affine algebras.Comment: 11 pages, minor correction

Functional Bethe Ansatz for a sinh⁡\sinh-Gordon model with real qq

Recently, Bazhanov and Sergeev have described an Ising-type integrable model which can be identified as a $\sinh$-Gordon-type model with an infinite number of states but with a real parameter $q$. This model is the subject of Sklyanin's Functional Bethe Ansatz. We develop in this paper the whole technique of the FBA which includes: 1. Construction of eigenstates of an off-diagonal element of a monodromy matrix. Most important ingredients of these eigenstates are the Clebsh-Gordan coefficients of the corresponding representation. 2. Separately, we discuss the Clebsh-Gordan coefficients, as well as the Wigner's 6j symbols, in details. The later are rather well known in the theory of $3D$ indices. Thus, the Sklyanin basis of the quantum separation of variables is constructed. The matrix elements of an eigenstate of the auxiliary transfer matrix in this basis are products of functions satisfying the Baxter equation. Such functions are called usually the $Q$-operators. We investigate the Baxter equation and $Q$-operators from two points of view. 3. In the model considered the most convenient Bethe-type variables are the zeros of a Wronskian of two well defined particular solutions of the Baxter equation. This approach works perfectly in the thermodynamic limit. We calculate the distribution of these roots in the thermodynamic limit, and so we reproduce in this way the partition function of the model. 4. The real parameter $q$, which is the standard quantum group parameter, plays the role of the absolute temperature in the model considered. Expansion with respect to $q$ (tropical expansion) gives an alternative way to establish the structure of the eigenstates. In this way we classify the elementary excitations over the ground state.Comment: References update

On Faddeev's Equation

Faddeev' equations are a set-theoretical and an operator forms of the star-triangle equation. Known solutions of the quantum star-triangle equation, related to the Faddeev equations, are based on various forms of the modular double of the Weyl algebra including its cyclic representation. We show in this paper that Fadeev's equation also leads to a solution of the quantum star-triangle equation even in the case of a simple Weyl algebra with $|q|<1$. This paper can be seen as an addendum to the recent paper "V. Bazhanov and S. Sergeev, A distant descendant of the six-vertex model, arXiv:2310.08427".Comment: References update