Torsional Topological Invariants (and their relevance for real life)

TL;DR

This paper introduces torsional topological invariants analogous to classical characteristic classes, explores their properties, and discusses their implications for stable configurations and anomalies in spacetimes with torsion.

Contribution

It defines new torsional topological invariants, analyzes their properties, and demonstrates their relevance for stable configurations and anomalies in higher-dimensional spacetimes.

Findings

Existence of torsional invariants similar to Chern/Pontryagin classes.

Explicit examples of stable configurations with nonzero instanton number.

Potential anomalies in chiral theories due to torsion-related invariants.

Abstract

The existence of topological invariants analogous to Chern/Pontryagin classes for a standard $SO(D)$ or $SU(N)$ connection, but constructed out of the torsion tensor, is discussed. These invariants exhibit many of the features of the Chern/Pontryagin invariants: they can be expressed as integrals over the manifold of local densities and take integer values on compact spaces without boundary; their spectrum is determined by the homotopy groups $\pi_{D-1}(SO(D))$ and $\pi_{D-1}(SO(D+1))$. These invariants are not solely determined by the connection bundle but depend also on the bundle of local orthonormal frames on the tangent space of the manifold. It is shown that in spacetimes with nonvanishing torsion there can occur topologically stable configurations associated with the frame bundle which are independent of the curvature. Explicit examples of topologically stable configurations carrying nonvanishing instanton number in four and eight dimensions are given, and they can be conjectured to exist in dimension $4k$. It is also shown that the chiral anomaly in a spacetime with torsion receives a contribution proportional to this instanton number and hence, chiral theories in $4k$-dimensional spacetimes with torsion are potentially anomalous.