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am in the process of adding some notes on how the D=5 super Yang-Mills theory on the worldvolume of the D4-brane is the double dimensional reduction of the 6d (2,0)-superconformal QFT in the M5-brane.
started a stubby double dimensional reduction in this context and added some first further pointers and references to M5-brane, to D=5 super Yang-Mills theory and maybe elsewhere.
But this still needs more details to be satisfactory, clearly.
What happens to the S-duality connected to 6d (2,0)-superconformal QFT when undergoing this reduction? What happens to holographic duals when one is reduced?
This is I think part of the original Witten and Kapustin-Witten story on geometric Langlands. Roughly like this:
the 6d $(2,0)$-superconformal QFT on the worldvolume of the M5-brane in 11-d SuGra has a conformal invariance, specifically Moebius transformations when taken to be a product of a 4d space with a torus
double dimensional reduction makes this the 5d super-Yang-Mills theory on the worldvolume of the D4-brane in 10-d SuGra
further ordinary dimensional reduction of the 5d worldvolume theory to a 4d theory yields 4d Yang-Mills and its topological twists. Now the Montonen-Olive S-duality of that theory is supposed to be the shadow of the original conformal invariance of the (2,0)-theory on the torus which was “dimensionally reduced”.
further compactifying down to d=2 turns this into geometric Langlands duality.
added to double dimensional reduction a formal definition for double dimensional reduction of cocycles in differential cohomology.
Meanwhile we have a much more sophisticated formulation of double dimensional reduction. It’s not reflected in the entry yet. But I am writing an exposition as talk notes here.
Expressed in HoTT, I imagine that could look beautifully simple.
I have moved over (here) at least statement and proof of the abstract $\infty$-topos theoretic formulation of double dimensional reduction, in the following form:
Let $\mathbf{H}$ be any (∞,1)-topos and let $G$ be an ∞-group in $\mathbf{H}$. There is a pair of adjoint ∞-functors of the form
$\mathbf{H} \underoverset {\underset{[G,-]/G}{\longrightarrow}} {\overset{hofib}{\longleftarrow}} {\bot} \mathbf{H}_{/\mathbf{B}G} \,,$where
$[G,-]$ denotes the internal hom in $\mathbf{H}$,
$[G,-]/G$ denotes the homotopy quotient by the conjugation ∞-action for $G$ equipped with its canonical ∞-action by left multiplication and the argument regarded as equipped with its trivial $G$-$\infty$-action, hence for $G = S^1$ this is the cyclic loop space construction.
Hence for
$\hat X \to X$ a $G$ principal ∞-bundle
$A$ a coefficient object, such as for some differential generalized cohomology theory
then there is a natural equivalence
$\underset{ \text{original} \atop \text{fluxes} }{ \underbrace{ \mathbf{H}(\hat X\;,\; A) } } \;\; \underoverset {\underset{oxidation}{\longleftarrow}} {\overset{reduction}{\longrightarrow}} {\simeq} \;\; \underset{ \text{doubly} \atop { \text{dimensionally reduced} \atop \text{fluxes} } }{ \underbrace{ \mathbf{H}(X \;,\; [G,A]/G) } }$given by
$\left( \hat X \longrightarrow A \right) \;\;\; \leftrightarrow \;\;\; \left( \array{ X && \longrightarrow && [G,A]/G \\ & \searrow && \swarrow \\ && \mathbf{B}G } \right)$I have taken the liberty of adding pointers to our formalization of double dimensional reduction:
Formalization of double dimensional reduction is discussed in rational homotolpy theory in
and in full homotopy theory in
Exposition is in
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