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I. G. Korepanov

Publications and source records attributed to I. G. Korepanov.

At least 19 recordsLinked to original sources
arXiv2016

"Nonconstant cohomology" of Hietarinta's two-color solutions to four-simplex equation

"Nonconstant cohomologies" are introduced for solutions of set-theoretical four-simplex equation (FSE). While usual cohomologies lead to solutions of constant quantum FSE, our "nonconstant cohomologies" lead to solutions of nonconstant quantum FSE. Computer calculations are presented showing that large spaces of such cohomologies exist for all Hietarinta's two-color linear solutions to set-theoretical FSE. After taking a partial trace of the corresponding quantum operators, combined with one additional trick, this leads to solutions of tetrahedron equation, including those with non-negative matrix elements, and not reducible to a permutation, even with cocycle multipliers.

math-ph↗
arXiv2016

Three-dimensionalizing the eight-vertex model

A simple ansatz is proposed for two-color R-matrix satisfying the tetrahedron equation. It generalizes, on one hand, a particular case of the eight-vertex model to three dimensions, and on another hand - Hietarinta's permutation-type operators to their linear combinations. Each separate R-matrix depends on one parameter, and the tetrahedron equation holds provided the quadruple of parameters belongs to an algebraic set containing five irreducible two-dimensional components.

math-ph↗
arXiv2015

Cohomology of the tetrahedral complex and quasi-invariants of 2-knots

This paper explores a particular statistical model on 6-valent graphs with special properties which turns out to be invariant with respect to certain Roseman moves if the graph is the singular point graph of a diagram of a 2-knot. The approach uses the technic of the tetrahedral complex cohomology. We emphasize that this model considered on regular 3d-lattices appears to be integrable. We also set out some ideas about the possible connection of this construction with the area of topological quantum field theories in dimension 4.

math-ph↗
arXiv2009

The tetrahedral analog of Veneziano amplitude

In solv-int/9812016 it was shown that the Veneziano amplitude in string theory comes naturally from one of the simplest solutions of the functional pentagon equation (FPE). More generally, FPE is intimately connected with the duality condition for scattering processes. Here I find the amplitude that comes the same way from a solution of the functional tetrahedron equation, with the duality replaced by the local Yang - Baxter equation.

solv-int↗
arXiv2009

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.

math-ph↗
arXiv2008

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.

math.GT↗
arXiv2004

Euclidean tetrahedra and knot invariants

We construct knot invariants on the basis of ascribing Euclidean geometric values to a triangulation of sphere S^3 where the knot lies. The main new feature of this construction compared to the author's earlier papers on manifold invariants is that now nonzero "deficit angles" (in the terminology of Regge calculus) can also be handled. Moreover, the knot goes exactly along those edges of triangulations that have nonzero deficit angles.

math.GT↗
arXiv2003

SL(2)-solution of the pentagon equation and invariants of three-dimensional manifolds

Building on a classical solution to the pentagon equation, constructed earlier by the author and E.V. Martyushev and related to the flat geometry invariant under the group SL(2), we construct an algebraic complex corresponding to a triangulation of a three-manifold. In case if this complex is acyclic (which is confirmed by examples), we use it for constructing a manifold invariant .

math.AT↗
arXiv2003

Exact solution for a matrix dynamical system with usual and Hadamard inverses

Let A be an n*n matrix with entries a_ij in the field C. Consider the following two involutive operations on such matrices: the matrix inversion I: A -> A^-1 and the element-by-element (or Hadamard) inversion J: a_ij -> a_ij^-1. We study the algebraic dynamical system generated by iterations of the product JI. In the case n=3, we give the full explicit solution for this system in terms of the initial matrix A. In the case n=4, we provide an explicit ansatz in terms of theta-functions which is full in the sense that it works for a Zariski open set of initial matrices. This ansatz also generalizes for higher n where it gives partial solutions.

nlin.SI↗
arXiv2002

Distinguishing three-dimensional lens spaces L(7,1) and L(7,2) by means of classical pentagon equation

We construct new topological invariants of three-dimensional manifolds which can, in particular, distinguish homotopy equivalent lens spaces L(7,1) and L(7,2). The invariants are built on the base of a classical (not quantum) solution of pentagon equation, i.e.algebraic relation corresponding to a ``2 tetrahedra to 3 tetrahedra'' local re-building of a manifold triangulation. This solution, found earlier by one of the authors, is expressed in terms of metric characteristics of Euclidean tetrahedra.

math.GT↗
arXiv2000

The method of vacuum vectors in the theory of Yang - Baxter equation

In modern terminology, this is the first published paper where the solutions of Yang - Baxter equation "at roots of unity" were analyzed and shown to be related to algebraic curves of genus >1. They are also known now to be connected with the "chiral Potts model". The paper's abstract as written in 1986 reads: "Vacuum vectors of an L-operator form a holomorphic bundle over the vacuum curve of that operator. These notions, as well as the theory of commutation relations of the 6-vertex model, are used in this work for constructing solutions of the Yang - Baxter equation that do not possess a spectral parameter of traditional type".

nlin.SI↗
arXiv2000

A formula with hypervolumes of six 4-simplices and two discrete curvatures

One of the generalizations of the pentagon equation to higher dimensions is the so-called "six-term equation". Geometrically, it corresponds to one of the "Alexander moves", that is elementary rebuildings of simplicial complexes, namely, replacing a "cluster" of three 4-simplices by another "cluster", also of three 4-simplices and with the same boundary. We present a formula containing the euclidean volumes of the simplices in the first cluster in its l.h.s., and those in the second cluster - in its r.h.s. The formula also involves "discrete curvatures" appearing when we slightly deform the euclidean space.

nlin.SI↗
arXiv2000

A formula with volumes of five tetrahedra and discrete curvature

Given five points in a three-dimensional euclidean space, one can consider five tetrahedra, using those points as vertices. We present a pentagon-like formula containing the product of three volumes of those tetrahedra in its l.h.s. and the product of the two remaining tetrahedron volumes in its r.h.s., as well as the derivative of the "discrete curvature" which arises when we slightly deform our euclidean space.

nlin.SI↗
arXiv1999

Multidimensional analogs of geometric s<-->t duality

The usual propetry of s<-->t duality for scattering amplitudes, e.g. for Veneziano amplitude, is deeply connected with the 2-dimensional geometry. In particular, a simple geometric construction of such amplitudes was proposed in a joint work by this author and S.Saito (solv-int/9812016). Here we propose analogs of one of those amplitudes associated with multidimensional euclidean spaces, paying most attention to the 3-dimensional case. Our results can be regarded as a variant of "Regge calculus" intimately connected with ideas of the theory of integrable models.

solv-int↗
arXiv1998

Finite-dimensional analogs of string s <-> t duality and pentagon equation

We put forward one of the forms of functional pentagon equation (FPE), known from the theory of integrable models, as an algebraic explanation to the phenomenon known in physics as s<->t duality. We present two simple geometrical examples of FPE solutions, one of them yielding in a particular case the well-known Veneziano expression for 4-particle amplitude. Finally, we interpret our solutions of FPE in terms of relations in Lie groups.

solv-int↗