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Yes, but it’s subtle:
First, typically it is fairly immediate that $n$-bundles on $X$ transgress to $(n-1)$-bundles on $L X$.
(This is completely formal for $n$-bundles with structure $n$-group a discrete $\mathbf{B}^{n-1} A$ – as in pp. 5 here – but with some massaging the transgression process typically works more generally.)
But what is more subtle is to characterize the $(n-1)$-bundles on loop spaces $L X$ that appear as transgressions of $n$-bundles this way: These carry some extra structure and it is only with this extra structure that they will be equivalent to their “de-transgressions” on $X$.
The historically most studied case is that of String 2-bundles: As you know, before these were even defined, Witten had essentially thought of them as 1-bundles on loop space. Then in What is an elliptic object? and in The spinor bundle on loop space (pdf) Stolz et al. had sketched out how these “Spin” 1-bundles on loop space $L X$ ought to be equivalent to String 2-bundles on $X$.
It was only much later, namely just recently, that these ideas have been substantiated and proven by Konrad Waldorf et al. First, Konrad pinpointed the extra structure on the transgression of a String 2-bundle to a 1-bundle on loop space, he calls it “fusion structure”. The required equivalence was then essentially proven, I think, in arXiv:2206.09797, see at stringor bundle.
It is hence no less than 36 years after Witten’s “The Index Of The Dirac Operator In Loop Space” that this transgression/de-transgression relation between String 2-bundles and Spinor 1-bundles on loop space has been really understood.
What does remain far less understood (as far as I am aware) is the closer relation of any of this to iterated algebraic K-theory and hence to the redshift conjecture. I think all we really have in hands is:
the relation between String-structure and elliptic cohomology via the string orientation of tmf,
the argument that BDR 2-vector bundles are classified by $K(ku)$ and as such constitute a “form of” elliptic cohomology (for low values of “form”, I suppose).
While suggestive, the concrete relation between these two items remains rather vague, as far as I am aware.
In conclusion, while the redshift conjecture is somewhat reminiscent of what happens in the now well-understood case of String 2-bundles transgressing to 1-bundles on loop space, substantial details on the relation remain scarce.
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