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I am finally splitting this off from G-structure. Have added comments on the disambiguation both here and there
I would have a question about something that is really interesting me.
In generalised geometry we have a generalised notion of $G$-structure, which is defined on the generalised tangent bundle $E \simeq T M \oplus T^\ast M$, rather than on the tangent bundle. Since the generalised tangent bundle canonically corresponds to a cocycle $M \rightarrow \mathbf{B}O(d,d)$, these generalised structures are given by a lift of this cocycle via $\mathbf{B}G \hookrightarrow \mathbf{B}O(d,d)$. Examples: $G=U(d/2,d/2)$ and $G=O(d)\times O(d)$.
My question is the following: is it possible to generalise the notion of cobordism of $G$-structures to a “generalised cobordism of generalised $G$-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!
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 $T X \oplus T X$. I know that Sawin04 (pdf) discusses 2-framed cobordism, but there must be more. Googling for “2-framed cobordism” yields hits.
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