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

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

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

Quantum Geometry of 3-Dimensional Lattices and Tetrahedron Equation

We study geometric consistency relations between angles of 3-dimensional (3D) circular quadrilateral lattices -- lattices whose faces are planar quadrilaterals inscribable into a circle. We show that these relations generate canonical transformations of a remarkable "ultra-local" Poisson bracket algebra defined on discrete 2D surfaces consisting of circular quadrilaterals. Quantization of this structure allowed us to obtain new solutions of the tetrahedron equation (the 3D analog of the Yang-Baxter equation) as well as reproduce all those that were previously known. These solutions generate an infinite number of non-trivial solutions of the Yang-Baxter equation and also define integrable 3D models of statistical mechanics and quantum field theory. The latter can be thought of as describing quantum fluctuations of lattice geometry.Comment: Plenary talk at the XVI International Congress on Mathematical Physics, 3-8 August 2009, Prague, Czech Republi