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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJul 18th 2020

    I am finally splitting this off from G-structure. Have added comments on the disambiguation both here and there

    v1, current

    • CommentRowNumber2.
    • CommentAuthorLuigi
    • CommentTimeMay 5th 2021
    • (edited May 5th 2021)

    I would have a question about something that is really interesting me.

    In generalised geometry we have a generalised notion of GG-structure, which is defined on the generalised tangent bundle ETMT *ME \simeq T M \oplus T^\ast M, rather than on the tangent bundle. Since the generalised tangent bundle canonically corresponds to a cocycle MBO(d,d)M \rightarrow \mathbf{B}O(d,d), these generalised structures are given by a lift of this cocycle via BGBO(d,d)\mathbf{B}G \hookrightarrow \mathbf{B}O(d,d). Examples: G=U(d/2,d/2)G=U(d/2,d/2) and G=O(d)×O(d)G=O(d)\times O(d).

    My question is the following: is it possible to generalise the notion of cobordism of GG-structures to a “generalised cobordism of generalised GG-structures”? Can it give rise to some form of generalised Thom spectrum (or, maybe, a generalised Madsen-Tillmann spectrum)?

    Since these structures are related (in some sense) a more general notion of string compactification, this could have some relevance in the cobordism swampland conjecture.

    Many thanks in advance for any reply!

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMay 5th 2021

    It can certainly be defined. But I am not sure if I remember anyone proving results about it.

    As proof of principle, something closely related is well-studied: cobordism with 2-framing: which is reduction of TXTXT X \oplus T X. I know that Sawin04 (pdf) discusses 2-framed cobordism, but there must be more. Googling for “2-framed cobordism” yields hits.