Typo fixed in link.

]]>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

- Domenico Fiorenza, Hisham Sati, Urs Schreiber, Section 3 of
*T-Duality from super Lie n-algebra cocycles for super p-branes (schreiber)*ATMP Volume 22 (2018) Number 5 (arXiv:1611.06536)

and in full homotopy theory in

- Vincent Braunack-Mayer, Hisham Sati, Urs Schreiber, Section 2.2 of
*Gauge enhancement of Super M-Branes via rational parameterized stable homotopy theory*(arXiv:1806.01115)

Exposition is in

- Urs Schreiber, Section 4 of
*Super Lie n-algebra of Super p-branes (schreiber)*, talks at Fields, Strings, and Geometry Seminar, Surrey, 5th-9th Dec. 2016

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

$\left( \hat X \longrightarrow A \right) \;\;\; \leftrightarrow \;\;\; \left( \array{ X && \longrightarrow && [G,A]/G \\ & \searrow && \swarrow \\ && \mathbf{B}G } \right)$ ]]>Expressed in HoTT, I imagine that could look beautifully simple.

]]>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.

]]>added to *double dimensional reduction* a formal definition for double dimensional reduction of cocycles in differential cohomology.

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.

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?

]]>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.

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