projects. However, currently this page is under construction. In
the present form this is a collection of few raw ideas. The
principle of this page in its current form can be formulated as
“One fool man in one minute can ask as many questions that
thousand wise men in thousand years could hardly answer”.
area of multidimensional exactly integrable systems and
presumably the projects for very insolent PhD students (or for
post-docs) The key criterion is: a methodology is not known.
- [Level A+] Extraordinary challenging problems. This is a
long-standing problem to find a proper form of algebraic equations
defining spectra of integrable field theories in three-dimensional
space-time suitable in the truly three-dimensional thermodynamic
limit. This is the open problem for the auxiliary
transfer-matrices, layer-to-layer transfer matrices and for the
evolution operators as well. This subject covers
- The cyclic representation for the Weyl algebra models (see the next paragraph for some progress and more recent development)
- The Fock space representation of q-oscillator model (see also the next paragraph)
- The modular representation — the most complicated one, even in 2D only a few exact results are known.
- [Level A] Challenging problems. This paragraph contains
some known development of the previous one.
There are known spectral equations describing a two-dimensional
lattice Bose gas. The latter model can be seen as a model
describing a non-relativistic Hamiltonian of a real Bose gas in
tow-plus-one dimension of space-time. The spectral equations are
much more complicated than e.g. nested Bethe Ansatz equations. The
real problems are:
- First, to study them numerically and
- To obtain a limiting form of them (i.e. an integral form of equations
in the limit when the number of degrees of freedom goes to
Spin lattices. We know the spectral equations for what
is called the spin lattice (a multidimensional extension of spin
chains) and, moreover, we know the limiting form of these
equations. However, only a structure of the ground state is known.
As well, the model is formulated on the level of auxiliary
transfer matrix (a quantum curve). Some recent results concern a
factorization quantum transfer matrix (much more complicated
object than the quantum curve) and evolution operator. The further
- To find a good numerical and analytical
method to solve the spectral equations, and
- To find a theta-functions analogue of polynomial factorization in the
thermodynamic limit, and
- To find finally a proper form of spectral equations for evolution operator and find a structure of elementary field-theoretical excitations.
- [Level A+] To make all mentioned in paragraph 1 for fermionic
and mixed representations (subject of integrable field theories).
- [Level A-] To classify three-dimensional boundary conditions
(in my draft calculations I’ve seen a lot of unusual scenarios)
- [Level A+] To solve some q-oscillator statistical mechanics for
non-unitary representations (completely positively defined
Boltzmann weights but non-linear boundary conditions — standard
machinery does not work)
- [Level A-] To understand a fixed boundary conditions and
presumably B, C and D series of quantum groups entirely in the
algebraic terms. To prove, at least, that the fixed boundary
conditions indeed give the representation theory for B, C and D.
- [Level A]. There exist BKP (B-type Kadomcev-Petviashvili
hierarchy) solitons. I believe, the existence of solitons
guarantees the existence of quantization. Discrete BKP hierarchy
is very well known, but its quantization scheme is a total
- [Level A-] We know a couple of solutions to quantum 4-simplex
equation. These solutions for a simple square four-dimensional
lattice do not produce an interesting series of three-dimensional
models (the result is a disjoint set of independent Hirota-type
three-dimensional models). A good question for topological
students is: what will happen if we will topologically modify the
simple square lattice?
- [Level A-] Solutions to 4-simplex equation mentioned before
are related to a class of three-dimensional models with Weyl
algebra of observables. There must be their analogue for
q-oscillator algebra of observables. Find them.
- High dimensional simplex equations and high dimensional
- Applications to knot and link invariants. In general, the
tetrahedral scheme provides a definition of Hamiltonian for every
knot and link — does it make any sense?
- 3D models are holographic image of 4-dimensional what?
- Solution of the Tetrahedron equation in R and W forms can be
used to produce an infinite seria of solutions to the Yang-Baxter
equation in R and W forms. However, a solution of the Tetrahedron
equation in S form produce a series of solutions to YBE in a mixed
R and W form which are not yet classified in terms of quantum
groups. Are there non-trivial examples?
- In the framework of quantum curve for q-oscillator model a
very unusual object appeared: a non-local transfer matrix. It has
no interpretation in terms of quantum groups (as well as the
lattice Bose gas mentioned above). What is this?
- Search for solutions of the Tetrahedron equation with
non-local algebra of observables. Such theories a priori will have
no quantum group interpretation. Presumably, a solution of p.1 of “curiuos problems” lays