90 research outputs found

### 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

### 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

### 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

### 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

### An integrable 3D lattice model with positive Boltzmann weights

In this paper we construct a three-dimensional (3D) solvable lattice model
with non-negative Boltzmann weights. The spin variables in the model are
assigned to edges of the 3D cubic lattice and run over an infinite number of
discrete states. The Boltzmann weights satisfy the tetrahedron equation, which
is a 3D generalisation of the Yang-Baxter equation. The weights depend on a
free parameter 0<q<1 and three continuous field variables. The layer-to-layer
transfer matrices of the model form a two-parameter commutative family. This is
the first example of a solvable 3D lattice model with non-negative Boltzmann
weights.Comment: HyperTex is disabled due to conflicts with some macro

### Exact solution of the Faddeev-Volkov model

The Faddeev-Volkov model is an Ising-type lattice model with positive
Boltzmann weights where the spin variables take continuous values on the real
line. It serves as a lattice analog of the sinh-Gordon and Liouville models and
intimately connected with the modular double of the quantum group U_q(sl_2).
The free energy of the model is exactly calculated in the thermodynamic limit.
In the quasi-classical limit c->infinity the model describes quantum
fluctuations of discrete conformal transformations connected with the
Thurston's discrete analogue of the Riemann mappings theorem. In the
strongly-coupled limit c->1 the model turns into a discrete version of the D=2
Zamolodchikov's ``fishing-net'' model.Comment: 4 page

### Comment on star-star relations in statistical mechanics and elliptic gamma-function identities

We prove a recently conjectured star-star relation, which plays the role of
an integrability condition for a class of 2D Ising-type models with
multicomponent continuous spin variables. Namely, we reduce this relation to an
identity for elliptic gamma functions, previously obtained by Rains.Comment: 8 pages, 3 figure

### Ground states of Heisenberg evolution operator in discrete three-dimensional space-time and quantum discrete BKP equations

In this paper we consider three-dimensional quantum q-oscillator field theory
without spectral parameters. We construct an essentially big set of eigenstates
of evolution with unity eigenvalue of discrete time evolution operator. All
these eigenstates belong to a subspace of total Hilbert space where an action
of evolution operator can be identified with quantized discrete BKP equations
(synonym Miwa equations). The key ingredients of our construction are specific
eigenstates of a single three-dimensional R-matrix. These eigenstates are
boundary states for hidden three-dimensional structures of U_q(B_n^1) and
U_q(D_n^1)$.Comment: 13 page

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