Nieh-Yan term in the mathematics literature?

Question

Let me establish my notation by stating the Cartan structure equations for the frame bundle of a Riemann 4-manifold. The curvature two-form is defined in terms of a ${\rm SO}(4)$ frame bundle connection two-form ${\omega^a}_b$
$$ {\bf R^a}_b = d{\omega^a}_b+ {\omega^a}_c\wedge {\omega^c}_b, $$ and the torsion two-form is given by $$ d {\bf e}^{*a}+ {\omega^a}_b\wedge {\bf e}^{*b}= {\bf T}^a, $$
where ${\bf e}^{*a} \equiv e^{*a}_\mu dx^\mu$ is the one=form dual to the orthonormal frame field ${\bf e}_a$ (As my frame field is orthonormal there is no need to make a distinction between upstairs and downstairs Roman indices on ${\bf R}_{ab}$ and ${\bf T}^a$, but I include a $*$ on the dual frame field so as to distingish the tangent vector field ${\bf e}_a$ from ${\bf e}^{*a}$.)

Now to my question: There are a number of areas in physics where the Nieh-Yan 4-form [Nieh, Yan, JMP 23 (1982) 373] $$ {\bf N}\equiv {\bf T}^a\wedge {\bf T}_a - {\bf e}^{*a}\wedge {\bf e}^{*b}\wedge {\bf R}_{ab} = d({\bf e}^{*a} \wedge {\bf T}_a) $$ plays a central role.

I have seen conflicting statements as to whether the three-form ${\bf e}^{*a} \wedge {\bf T}_a$ is a globally defined quantity, and hence whether ${\bf N}$ is a globally exact form. (The point being that ${\bf e}_a$ and ${\bf e}^{*a}$ are only globally defined when the manifold is paralellizable)

Also there are conflicting statements about the role of ${\bf N}$ in computing the index density for the Dirac opertor in spaces with torsion. (See for example: arXiv:hep-th/9702025. There are a number of statements in the paper that make me uneasy.)

What is true here?

I cannot find any reference to the Nieh-Yan form on Mathematics Stack exchange. Is it used in mathematics at all?

Answer 1

I've worked in differential geometry for 40 years or so (mostly complex differential geometry, admittedly) and have never seen this before. Granted, torsion does not get its due in Riemannian geometry (because the canonical metric connection, of course, is torsion-free).

But, yes, the $3$-form is globally defined. (In spirit, the fact that $\mathbf N$ is exact is similar to way that the Chern-Simons invariant arises.) If you change frame field by $h$, then $\tilde{\mathbf e}{}^{*a} = \sum h^a_b\mathbf e^{*b}$ and $\tilde{\mathbf T}{}^a = \sum h^a_c\mathbf T^c$, so $$\sum\tilde{\mathbf e}{}^{*a}\wedge\tilde{\mathbf T}{}^a = \sum h^a_b h^a_c\mathbf e^{*b}\wedge\mathbf T^c = \sum \mathbf e^{*b}\wedge \mathbf T^b,$$ since $(h^a_b)$ is an orthogonal matrix. (I would ordinarily, when working with orthonormal frames, write everything with lower indices, but ... .)