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• CommentRowNumber1.
• CommentAuthorLuigi
• CommentTimeAug 28th 2020

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeAug 28th 2020

Thanks for the pointer.

It seems to have gone largely unrecognized that the cohesive $\infty$-topos-theoretic discussion of the String 2-group is in section 5.1.4 of dcct (p. 583).

• CommentRowNumber3.
• CommentAuthorDavid_Corfield
• CommentTimeAug 28th 2020

Reading that sounds as though the $A$ of the extension could be anything, but it’s limited to being homotopy equivalent to $B U(1)$, right?

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeAug 28th 2020
• (edited Aug 28th 2020)

Yes, on p. 29-30:

[ our definition ] is a generalisation as well as a weakening of the following approach to smooth string group extensions (see, for instance, [FRS16]):

• CommentRowNumber5.
• CommentAuthorDavid_Corfield
• CommentTimeAug 28th 2020

with $A$ not necessarily chosen to be $\mathbf{B}U(1)$ but only of the same homotopy type, …

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeAug 28th 2020

By the way, I think this perspective that the String 2-group, even in the smooth case, “was defined” to be a 3-connected cover is misled:

By it’s very name, the String 2-group is meant to be that $G$ such that $G$-structure encodes cancellation of the Green-Schwarz anomaly.

That for the global GS anomaly (i.e. disregarding differential structure) this is given by a 3-connected cover of the homotopy type pf $Spin$ is a noteworthy phenomenon, but not the definition.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeSep 2nd 2020

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeSep 2nd 2020

• CommentRowNumber9.
• CommentAuthorDavid_Corfield
• CommentTimeFeb 4th 2022

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