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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
Added the remark that the differential geometry of the double dimensional reduction of the M2-brane- and M5-brane-charges was maybe first clearly written out in:
One can regard Proposition 2.3 to be a statement about the double-dimensional reduction of $Fields$. But there should be a more general statement concerning the reduction of theories. For instance, if $Fields$ carries a prequantum circle n-bundle (e.g. take $Fields =BG_{conn}$ equipped with a characteristic class, say the Chern-Simons 3-bundle to $B^3U(1)$), then one should be able to do double-dimensional reduction not only of the fields but also of the corresponding theory. This I guess will involve some statement similar to Proposition 2.3 involving a more general comma category. Is this sorted out anywhere? I’m particularly asking because there have been some recent papers arguing one can get Symmetry TFT’s from doing dimensional reduction “of” M-theory, so it would be nice to make this precise.
Just to note that the double dimensional reduction by cyclification does not involve truncation:
The space/stack of dd-reduced fields is still equivalent to that of the original fields (the “reduction/oxidation” equivalence).
In that sense there is no necessary cause to change the prequantum $\infty$-bundle.
And if one does want to truncate some of the KK-modes, that would typically amount to restricting to a sub-stack of the dd-reduced fields, for which the appropriate prequantum bundle would just be the restriction (pullack along the inclusion) of the original one.
Is it described anywhere how this process leads to getting WZW from CS on a $\Sigma\times S^1$ worldvolume? That is, how one uses this to go from $\Sigma\times S^1\to BG \to B^3 U(1)_{conn}$ to $\Sigma\to G\to B^2 U(1)_{conn}$ where in the first case one has the CS 2-gerbe and in the second case it is the WZW 1-gerbe?
Yes, this is pp. 40 in arXiv:1304.0236, following arXiv:1207.5449 (pp. 3) and arXiv:0410013.
Thanks. A couple of questions. If you start with an action $G\to \text{Aut}(\nabla_{CS})$ of a higher group $G$ on Chern-Simons, what happens to this action when passing to WZW? And also, in p.41 it is mentioned that the sections of the WZW gerbe are twisted unitary bundles, referencing Example 2.5.4 for this, which does not talk about gerbes with connection. But isn’t the fact that the sections of a line bundle gerbe with connection $\mathcal{G}$ can be expressed as bundle gerbe morphisms from a trivial gerbe $\mathcal{I}_{\omega}$ to $\mathcal{G}$, that is as twisted vector bundles with connection, only valid for torsion Dixmier-Douady classes?
Right, the twisted bundles discussed there are finite-rank, hence the corresponding gerbe must have torsion DD-class.
But this is the case after pullback to D-brane worldvolume submanifolds $Q \hookrightarrow G$ as considered there, if by Freed-Witten anomaly cancellation and assuming $\mathrm{Spin}^c$-structure, the restriction of the class of the gerbe to the brane worldvolume $Q$ is torsion.
On the other hand, in a more sophisticated perspective the quantization is anyways described directly by pushforward in K-theory twisted by that gerbe, as in section 5.2.4 of Joost Nuiten’s thesis (pp. 115).
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