Here I am going to ask if there is any work on field theories with "weak" metric structure exhibited not by vielbein fields on the tangent bundle, but by the analog kind of "viel+1-bein fields" on the direct sum of the tangent bundle with the trivial line bundle.

A vielbein field configuration on a manifold \(X\) is locally a choice of trivialization of the tangent bundle, and if it is also globally so then it is a choice of global parlallelism, a "framing". Traditionally in physics this receives attention in the corner of "teleparallel gravity", but this is not what I am after here.

In mathematics, notably in generalized cohomology, a major role is played by the weaker concept of "stable framings" which are framings not necessarily of the tangent bundle \(TX\) itself, but of the sum of that with a trivial vector bundle of any rank. In between that very weak concept and the ordinary strong concept of framing (global parallelism) is that of "\((d+1)\)-framing", a choice of trivialization of \(TX \oplus \underline{\mathbb{R}}\) on a \(d\)-dimensional manifold \(X\).

There is some sense in which a \((d+1)\)-framing is analogous to a conformal structure. A conformal structure, too, may be thought of as given by a local vielbein field but relaxing its conditions a bit -- by allowing a common rescaling of all the beins in the vielbein. Here for a \((d+1)\)-framing we don't have rescalings, but we have the extra freedom of moving around one extra "bein" and of rotating all the "beins" in one more dimension than the manifold has.

A more precise way to say how \((d+1)\)-framings play a role analogous to that of conformal structures in conformal field theory is the cobordism hypothesis-theorem, which says among many other things that if we produce a Chern-Simons-like topological field theory of dimension \((d+1)\),then its Wess-Zumino-Witten-like holographic dual theory in dimension \(d\) has a flat bundle of "conformal blocks" on the moduli space of the "\((d+1)\)-structures" on its spacetime \(X\). If the dual CS theory is fully anomaly-free in that it is framed, then this means that the \(d\)-dimensional boundary theory has a flat bundle of "conformal blocks" on the moduli space of its \((d+1)\)-framings. (More in this MO question which, it turns out, has a positive answer.)

Has any field theory of such kind ever surfaced anywhere in the physics literature? Maybe in discussion of generalized WZW-like theories or in discussion of variants of AdS/CFT. Or elsewhere?