Topological invariants, instantons, and the chiral anomaly on spaces with torsion

In a spacetime with nonvanishing torsion there can occur topologically stable configurations associated with the frame bundle which are independent of the curvature. The relevant topological invariants are integrals of local scalar densities first discussed by Nieh and Yan (NY). In four dimensions, the NY form ${N=(T}^a\wedge T_a-R_{\mathrm{ab}}\wedge e^a\wedge e^b)$ is the only closed four-form invariant under local Lorentz rotations associated with the torsion of the manifold. The integral of $N$ over a compact $D$-dimensional (Euclidean) manifold is shown to be a topological invariant related to the Pontryagin classes of SO$\mathrm{}(D+1\mathrm{})\mathrm{}$ and SO$\mathrm{}\left(D\right)\mathrm{}$. An explicit example of a topologically nontrivial configuration carrying a nonvanishing instanton number proportional to $\int N$ is constructed. The chiral anomaly in a four-dimensional spacetime with torsion is also shown to contain a contribution proportional to $N$, in addition to the usual Pontryagin density related to the spacetime curvature. The violation of chiral symmetry can thus depend on the instanton number of the tangent frame bundle of the manifold. Similar invariants can be constructed in $D>\mathrm{}4\mathrm{}$ dimensions and the existence of the corresponding nontrivial excitations is also discussed.