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created an entry for caloron correspondence with an Idea-section and references.
The caloron correspondence is (or has been used as) an attempt to mathematically formalize aspects of what in physics is called double dimensional reduction: $p$-branes (or their charge cohomology classes) on the total space of a circle bundle turn into $p$-branes and $(p-1)$-brane on the base space, by a process that involves, but is more refined that, fiber integration over the circle fibers.
A deficiency of the caloron correspodence (as far as I am aware) is that it actually only applies to trivial circle bundles. This is related to the fact that for
$\alpha \;\colon\; X \longrightarrow \mathbf{B}G$a cocycle on the total space $X$ of the circle bundle $X \to Y$, one wants to first form loops $\mathcal{L}\alpha = [S^1, \alpha]$, and then precompose with a canonical map
$Y \longrightarrow \mathcal{L}X$which assigns to each point of $Y$ the $S^1$-fiber over it. This only works for $X = Y \times S^1$ (in which case the map in question is the product/hom-adjunction unit).
But stated this way, we see that this works more generally if instead of free loop spaces $\mathcal{L}X$ we form the stacky homotopy quotients $\mathcal{L}(-) / \!/ S^1$. Then there is a well defined map
$Y \longrightarrow \mathcal{L}X / \!/ S 1$assigning fibers, and hence composition allows us to turn $\alpha \colon X \to \mathbf{B}G$ into
$Y \to \mathcal{L} X / \! / S^1 \overset{\mathcal{L}\alpha / \! / S^1}{\longrightarrow} \mathcal{L} \mathbf{B}G / \! / S^1 \,.$I don’t know in general to which extent this operation yields an exquivalence. But I know that this kind of operation yields the correct equivalence in the archetypical example of double dimensional reduction in physics: that from rational M-brane charges to type IIA F1/D$p$/NS5-brane charges. This is the content of section 3 of Rational sphere valued supercocycles in M-theory (schreiber); it uses the model for the rational free loop space discussed at Sullivan model of free loop space.
For the moment this is just an observation in itself. But it seems suggestive of some interesting further story. In particular the cohomology of $\mathcal{L}(-)/ \! / S^1$ is of course cyclic cohomology, and so it seems to be suggesting that double dimensional reduction is generally to be thought of as a process of passing to cyclic cohomology. That in turn is suggestive of a lot of other things.
A deficiency of the caloron correspodence (as far as I am aware) is that it actually only applies to trivial circle bundles.
It actually applies to any circle fibration, or even other bundles. This is work of Vozzo, Murray and others (Hekmati for instance), eg The general caloron correspondence
True, I should have said this more precisely: there is a caloron correspondence for more general fibrations, but it is no longer in general of the form that prinicpal bundles correspond to loop group principal bundles.
In any case, my point is not so much about the caloron correspondence in itself, but about attempts to use it to explain double dimensional reduction, in the style going back to
There have been various followups on this, but, as far as I am aware, the implication remains inconclusive.
What I mean to highlight is that, in contrast, a simple modification of the caloron correspondence “for products” makes everything work regarding these approaches to reduce M-brane charges to D-brane charges: simply quotient out rotations of loops after forming free loops. Then the original construction first of all generalizes to nontrivial bundles, and it does reproduce the physics story. If, that is, one starts not with a $G$-principal bundle, but with a cocycle in cohomotopy.
So maybe I should have started this discussion here in a thread with a different title.
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