Geometric torsions and invariants of manifolds with a triangulated boundary

Abstract

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.

A Euclidean Geometric Invariant of Framed (Un)Knots in Manifolds

“…In doing so, we are guided by the idea of constructing eventually a topological quantum field theory (TQFT) according to some version of Atiyah's axioms [1] where, as is known, the boundary plays a fundamental role. Some fragments of this theory have been already developed in our works; in particular, it was shown in papers [11] and [12] that, indeed, a TQFT is obtained this way. This TQFT is fermionic: the necessary modification of Atiyah's axioms is that the usual composition of tensor quantities, corresponding to the gluing of manifolds, is replaced with Berezin (fermionic) integral in anticommuting variables.…”

Geometric torsions and an Atiyah-style topological field theory

“…Here, we finish constructing a version of invariants of three-dimensional manifolds with triangulated boundaries begun in [1]. Namely, we present a construction of these invariants for a compact connected three-dimensional manifold with an arbitrary number of boundary components, while we restricted ourself to not more than one component in [1].…”

“…Namely, we present a construction of these invariants for a compact connected three-dimensional manifold with an arbitrary number of boundary components, while we restricted ourself to not more than one component in [1].…”

“…In [1], we demonstrated how to construct not one but many algebraic complexes for a manifold with a one-component boundary with a fixed triangulation given on the boundary. Such a complex depends on two arbitrary sets of boundary edges C and D of equal cardinality; we gave examples showing that such a complex is acyclic in sufficiently many cases (and if it is not acyclic, then its torsion is assumed to be zero).…”

A matrix solution of the pentagon equation with anticommuting variables

“…We also note that it is a long way from formula (5) in[5], on which our research on pentagon equations is based, to the TQFT in[6],[7].…”