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1. • CommentRowNumber1.
   • CommentAuthorZhen Huan
   • CommentTimeSep 8th 2023
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   Author: Zhen Huan
   Format: TextI have a maybe naive question. Is there a geometric interpretation of Redshift conjecture? For instance, should a n-vector bundle over a "space" be equivalent to a (n-1)-vector bundle over a suitable loop space of it?

   I have a maybe naive question. Is there a geometric interpretation of Redshift conjecture? For instance, should a n-vector bundle over a "space" be equivalent to a (n-1)-vector bundle over a suitable loop space of it?
2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeSep 8th 2023
   • (edited Sep 8th 2023)
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   Author: Urs
   Format: MarkdownItexYes, 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](https://arxiv.org/pdf/2212.13836.pdf#page=5) -- 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%3F|What is an elliptic object?]]* and in *The spinor bundle on loop space* ([pdf](https://ncatlab.org/nlab/files/Stolz-Teichner-SpinorOnLoopSpace.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 bundle|fusion structure]]". The required equivalence was then essentially proven, I think, in [arXiv:2206.09797](https://arxiv.org/abs/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: 1. the relation between String-structure and elliptic cohomology via the [[string orientation of tmf]], 1. 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.

   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:

   1. the relation between String-structure and elliptic cohomology via the string orientation of tmf,

   2. 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.

3. • CommentRowNumber3.
   • CommentAuthorZhen Huan
   • CommentTimeSep 8th 2023
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   Author: Zhen Huan
   Format: TextThank you very much, Urs.

   Thank you very much, Urs.