Igor G. Korepanov

Functional Tetrahedron Equation

We describe a method for constructing classical integrable models in a (2+1)-dimensional discrete spacetime based on the functional tetrahedron equation, an equation that makes the symmetries of a model obvious in a local form. We construct a very general “block-matrix model,” find its algebraic-geometric solutions, and study its various particular cases. We also present a remarkably simple quantization scheme for one of those cases.

Distinguishing Three-Dimensional Lens Spaces L(7,1) and L(7,2) by Means of Classical Pentagon Equation

Abstract 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 → 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.

Invariants of PL Manifolds from Metrized Simplicial Complexes. Three-Dimensional Case

Abstract An invariant of three-dimensional orientable manifolds is built on the base of a solution of pentagon equation expressed in terms of metric characteristics of Euclidean tetrahedra.

Euclidean 4-Simplices and Invariants of Four-Dimensional Manifolds: II. An Algebraic Complex and Moves 2↔4

We present sequences of linear maps of vector spaces with fixed bases. Each term of a sequence is a linear space of differentials of metric values ascribed to the elements of a simplicial complex determining a triangulation of a manifold. If a sequence is an acyclic complex, then we can construct a manifold invariant using its torsion. We demonstrate this first for three-dimensional manifolds and then construct the part of this program for four-dimensional manifolds pertaining to moves 2↔4.

Euclidean 4-simplices and invariants of four-dimensional manifolds: III. Moves 1 ↔ 5 and related structures

We conclude the construction of the algebraic complex, consisting of spaces of differentials of Euclidean metric values, for four-dimensional piecewise-linear manifolds. Assuming that the complex is acyclic, we investigate how its torsion changes under rebuildings of the manifold triangulation. We first write formulas for moves 3 → 3 and 2 ↔ 4 based on the results in our two previous works and then study moves 1 ↔ 5 in detail. Based on this, we obtain the formula for a four-dimensional manifold invariant. As an example, we present a detailed calculation of our invariant for the sphere S4; in particular, the complex does turn out to be acyclic.

Euclidean 4-simplices and invariants of four-dimensional manifolds: I. Moves 3 → 3

We construct invariants of four-dimensional piecewise linear manifolds, represented as simplicial complexes, with respect to moves that transform a cluster of three 4-simplices having a common two-dimensional face to a different cluster of the same type and having the same boundary. Our construction is based on using Euclidean geometric quantities.

SL(2)-Solution of the Pentagon Equation and Invariants of Three-Dimensional Manifolds

We construct an algebraic complex corresponding to a triangulation of a three-manifold starting with a classical solution of the pentagon equation, constructed earlier by the author and Martyushev and related to the flat geometry, which is invariant under the group SL(2). If this complex is acyclic (which is confirmed by examples), we can use it to construct an invariant of the manifold.

A CLASSICAL SOLUTION OF THE PENTAGON EQUATION RELATED TO THE GROUP SL(2)

A new solution of the pentagon equation related to the flat geometry is obtained; it is invariant under the action of the group SL(2).

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 a triangulated boundary

Geometric torsions are torsions of acyclic complexes of vector spaces consisting of differentials of geometric quantities assigned to the elements of a manifold triangulation. We use geometric torsions to construct invariants for a three-dimensional manifold with a triangulated boundary. These invariants can be naturally combined into a vector, and a change of the boundary triangulation corresponds to a linear transformation of this vector. Moreover, when two manifolds are glued at their common boundary, these vectors undergo scalar multiplication, i.e., they satisfy Atiyah’s axioms of a topological quantum field theory.