A Torsional Topological Invariant

TL;DR

This paper discusses a torsional topological invariant in spacetime geometry, highlighting its significance in gauge theories with spin and its recent applications beyond traditional metric-based invariants.

Contribution

It introduces and reviews a torsional topological invariant discovered in 1982, emphasizing its relevance in modern theoretical physics involving torsion and spin.

Findings

Highlights the role of torsion in spacetime topology

Connects the invariant to recent developments in gauge theories

Provides insights into non-metric spacetime properties

Abstract

Curvature and torsion are the two tensors characterizing a general Riemannian spacetime. In Einstein's general theory of gravitation, with torsion postulated to vanish and the affine connection identified to the Christoffel symbol, only the curvature tensor plays the central role. For such a purely metric geometry, two well-known topological invariants, namely the Euler class and the Pontryagin class, are useful in characterizing the topological properties of the spacetime. From a gauge theory point of view, and especially in the presence of spin, torsion naturally comes into play, and the underlying spacetime is no longer purely metric. We describe a torsional topological invariant, discovered in 1982, that has now found increasing usefulness in recent developments.