227 research outputs found
A matrix solution to pentagon equation with anticommuting variables
We construct a solution to pentagon equation with anticommuting variables
living on two-dimensional faces of tetrahedra. In this solution, matrix
coordinates are ascribed to tetrahedron vertices. As matrix multiplication is
noncommutative, this provides a "more quantum" topological field theory than in
our previous works
A Euclidean Geometric Invariant of Framed (Un)Knots in Manifolds
We present an invariant of a three-dimensional manifold with a framed knot in
it based on the Reidemeister torsion of an acyclic complex of Euclidean
geometric origin. To show its nontriviality, we calculate the invariant for
some framed (un)knots in lens spaces. Our invariant is related to a
finite-dimensional fermionic topological quantum field theory
Geometric torsions and invariants of manifolds with triangulated boundary
Geometric torsions are torsions of acyclic complexes of vector spaces which
consist of differentials of geometric quantities assigned to the elements of a
manifold triangulation. We use geometric torsions to construct invariants for a
manifold with a triangulated boundary. These invariants can be naturally united
in a vector, and a change of the boundary triangulation corresponds to a linear
transformation of this vector. Moreover, when two manifolds are glued by their
common boundary, these vectors undergo scalar multiplication, i.e., they work
according to M. Atiyah's axioms for a topological quantum field theory.Comment: 18 pages, 4 figure
Quantum 2+1 evolution model
A quantum evolution model in 2+1 discrete space - time, connected with 3D
fundamental map R, is investigated. Map R is derived as a map providing a zero
curvature of a two dimensional lattice system called "the current system". In a
special case of the local Weyl algebra for dynamical variables the map appears
to be canonical one and it corresponds to known operator-valued R-matrix. The
current system is a kind of the linear problem for 2+1 evolution model. A
generating function for the integrals of motion for the evolution is derived
with a help of the current system. The subject of the paper is rather new, and
so the perspectives of further investigations are widely discussed.Comment: LaTeX, 37page
Geometric torsions and an Atiyah-style topological field theory
The construction of invariants of three-dimensional manifolds with a
triangulated boundary, proposed earlier by the author for the case when the
boundary consists of not more than one connected component, is generalized to
any number of components. These invariants are based on the torsion of acyclic
complexes of geometric origin. The relevant tool for studying our invariants
turns out to be F.A. Berezin's calculus of anti-commuting variables; in
particular, they are used in the formulation of the main theorem of the paper,
concerning the composition of invariants under a gluing of manifolds. We show
that the theory obeys a natural modification of M. Atiyah's axioms for
anti-commuting variables.Comment: 15 pages, English translation (with minor corrections) of the Russian
version. The latter is avaible here as v
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
Form-factors in the Baxter-Bazhanov-Stroganov model I: Norms and matrix elements
We continue our investigation of the Z_N-Baxter-Bazhanov-Stroganov model
using the method of separation of variables [nlin/0603028]. In this paper we
calculate the norms and matrix elements of a local Z_N-spin operator between
eigenvectors of the auxiliary problem. For the norm the multiple sums over the
intermediate states are performed explicitly. In the case N=2 we solve the
Baxter equation and obtain form-factors of the spin operator of the periodic
Ising model on a finite lattice.Comment: 24 page
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