Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
I have created a minimum at global family (a suitable family of groups in the sense of global equivariant homotopy theory).
Hm, the set of finite subgroups of $SO(3)$ or of $SU(2)$. Is that a global family? I.e. is it closed under quotient groups by normal subgroups?
Edited
See http://projecteuclid.org/euclid.atmp/1408561553 (arXiv: http://arxiv.org/abs/1007.4761) for a table of the normal subgroups of the ADE groups of $SU(2)$ and their quotients (table 5).
It appears to be “almost”, since the SO(3) ADE subgroups $D_{2n},T,O,I$, appear, not their double covers. But maybe the $SO(3)$ ADE subgroups do form such a family.
Ah, thanks, excellent. I am aware of this article, of course, but I forgot that it has this classification.
Hm, right, so it looks like there is a chance the that the finite $SO(3)$-subgroups do form a global family…
How exactly does the quotient-by-normal-subgroup thing work: is it as subgroups of the given ambient Lie group? Or as abstract groups?
Good point. With the definition of global family as given, it must be as abstract groups. Which of course is bad in the given situation.
Let me see if I can gather together here the relevant data.
There’s this list of containment relations for finite subgroups of $SO(3)$.
No issues here, as these are closed under quotients already. All subgroups are cyclic.
This contains as a degenerate case the cyclic group of order 2.
See this M.SE answer for a list of the normal subgroups.
But also Proposition 5 in this PlanetMath article:
Quotient groups of dihedral groups are dihedral, and subgroups of dihedral groups are dihedral or cyclic.
Additionally, the only dihedral normal subgroups, when they exist, have index 2, and there are exactly two of them. See theorem 3.8 here. There is of course a cyclic subgroup of index 2, and all subgroups of this (again cyclic) are also normal subgroups.
So the AD-type groups together are closed under quotients.
$T \simeq A_4$. This has one non-trivial quotient, which is $T/V_4 \simeq \mathbb{Z}/3$, for $V_4$ the Klein 4-group (which is dihedral).
$O \simeq S_4$. This has $O/T \simeq \mathbb{Z}/2$, as usual. Also, from here the only other nontrivial quotient is $O/V_4 \simeq S_3$, which is again a dihedral group.
$I\simeq A_5$, which is simple hence no nontrivial quotients.
Hence the finite ADE subgroups are closed under quotients, and so form a global family. EDIT: let’s call this family $ADE_{glob}$.
I guess if you’re willing to admit all closed subgroups of $SO(3)$, then I think you might also get a global family. Here’s a list of them, and we only have to add $SO(2)$ and $O(2)$, which of course have cyclic subgroups (get nothing new there), and dihedral subgroups in the case of $O(2)$, a quotient by which I believe should just be isomorphic to $SO(2)$.
Also, if you’re willing to admit the finite subgroups of both $SO(3)$ and $SU(2)$, these form a global family too.
So I guess with Schwede’s Remark 5.11 you could cook up mixed model structures using a mixture of these global families.
However - $ADE_{glob}$ is certainly not a multiplicative global family. For that you’d have to complete under finite products, but hmm, not sure that’s a simple operation: there might now be extra normal subgroups… (as is pointed out just before Proposition 5.12: something in the family $\mathcal{F}\times \mathcal{F}$ is of the form $(K\times L)/N$ for $K,L\in \mathcal{F}$ and $N$ normal.
My guess is that you’re aiming to establish something like corollary 6.31 for the setup at hand?
Let me put down some thoughts about how close $ADE_{glob}$ is to being multiplicative. Or rather, how well can we describe $ADE_{glob}\times ADE_{glob}$? (using Schwede’s notation)
Subgroups of products $G\times H$ are all of the form $K\times_L M$, for $K\lt G$ and $M \lt H$. Here we can take the cospan $K\to L \leftarrow M$ to consist of quotient maps. If such a subgroup $N = K\times_L M$ is normal, then we must have $K$ normal in $G$ and $M$ normal in $H$ (this is only necessary, not sufficient, though). Hence every group in sight is in $ADE_{glob}$ apart from the pullback and the resulting quotient. So to get our hands on the $K\times_L M$ we can consider for now the special case that $K=G$, $M=H$. ADDED: there’s a small twist, in that we can in fact add in an outer automorphism of $L$, so get the cospan $K \to L \stackrel{\sim}{\to} L \leftarrow M$. This is discussed and then all possible finite subgroups of $SU(2)\times SU(2)$ are given in section 6.3 of Half-BPS M2-brane orbifolds, which gives an upper bound on the task of looking at these (I wish I knew how to use GAP right about now!)
Some trivial facts
Now for the dihedral groups things are a little more interesting. We can consider cospans of quotients of the form $O \to S_3 \leftarrow D_{2n}$ or $O \to \mathbb{Z}/2 \leftarrow D_{2n}$ for products involving the E-series, and cospans of the form $\mathbb{Z}/2k \to \mathbb{Z}/2 \leftarrow D_{2n}$ for the A-series. There are restrictions on the possible $n$ that work here: an even/odd distinction (the order-2 quotient only exists for even $n$) and a divisibility criterion to get the $S_3$ quotient.
(Fibre products of dihedral groups seems to be a little complicated)
But it seems we can (in principle) just do a case-by-case analysis and see what we get.
ADDED2: In the notation above, normal subgroups $N = K\times_L M$ of $G\times H$ have to contain the product $[K,G]\times[M,H] \lt G \times H$. This trims things down a bit.
References just so I can read them more carefully later:
I had thought you were wondering about this question back here.
I’ve added all finite groups as a global family. Symmetric spectra model global homotopy theory of finite groups talks about this case.
I should point out that, given the points in the discussion David C reminds us of, we can’t exclude cyclic groups here, since $T/V_4 = C_3$. (I think I’ll switch from using $\mathbb{Z}/n$ for this thread from now on)
OK, I had a very tentative conjecture that perhaps $ADE_{glob}\times ADE_{glob}$ was given solely by products of things in $ADE_{glob}$, when a priori it consists of things that are subgroups of quotients of the form $(G\times H)/N$, in particular all subgroups of $G\times H$ for arbitrary $G,H\in ADE_{glob}$ (I missed the ’subgroups of’ when reading the definition yesterday).
Hence $ADE_{glob}\times ADE_{glob}$ contains for instance the (normal) subgroup $T\times_{C_3} T \lt (T \times T)/\{1\}$, and I don’t know this can be written as a product of ADE groups.
I do wonder whether it is possible to use slightly weaker assumptions on the global family to still get a symmetric monoidal model structure, something like laxly multiplicative – not asking there is an equality , but some other relation.
Does every global family give rise to a form of global cohesion?
That “universal compact Lie group” idea of Schwede’s here looks interesting.
@David #12
I think it might. Schwede proves that one gets the right sort of model category I think, but I haven’t checked the adjoints are in his book.
I added some more global families from the different notes by Schwede ’Global stable homotopy theory’. It mentions
all finite p-groups; all finite p’-groups
Would the latter be groups whose element orders are not divisible by p?
Thanks for all this!
I was on vacation and need some time to catch up now.
Welcome back, Urs! Hope you had a good restful vacation. :-)
Yes, welcome back :-) I didn’t put this in the nLab page yet as I wasn’t sure where we would go next.
Thanks. Yes, I had a good restful vacation, indeed, with family. For the first time in over two years, I feel that I have caught enough sleep.
Sorry for still not getting back to the above discussion. I wanted to figure out some more consistency checks with the physics story that I am trying to formalize, in order to know more precisely which mathematical structure precisely I need to consider. But I feel a little stuck.
So, to recall, the issue is that we proved that rationally the coefficients for M2/M5-brane charges are the quaternionic Hopf fibration,
$\mathcal{F}\times\mathcal[ \array{ S^7 \to S^4 \to \mathbf{B} SU(2) \to \mathbf{B}^3 U(1) }$regarded as an object in the slice topos
$\left[ \array{ S^4 \\ \downarrow \\ \mathbf{B}^3 U(1) } \right] \;\;\; \in \mathbf{H}_{/\mathbf{B}^3 U(1)}$classifying a twisted cohomology theory.
The question is what this should be away from the rational approximation. In particular, the physics story says that non-rationally we should see special effects associated with the cohomology of $G_{ADE}$-orbifolds, where $G_{ADE}$ is a finite subgroup of $SU(2)$. This is very suggestive, since the quaternionic Hopf fibration is of course controled by $SU(2)$, but I still need to get a better grip on what exactly ought to be going on.
Maybe the strongest hint is not from the D6-brane charges that I had been focusing on (M-theory lift of gauge enhancement on D6-branes) but from the M2-brane charges that are implicit in
To see this, first recall the direct way to see the M-brane charges have coefficients in $S^4$:
It’s all a generalization of the classical case (Dirac charge quantization), where a magnetically charged point mass in 4d has a “near horizon geometry” of the form $AdS_2 \times S^2$, homotopy equivalent to $S^2$, and the magnetic charge of this spacetime, which is felt by an electron propagating near the magnetic point charge, is measured by maps $AdS_2 \times S^2 \simeq S^2 \to \mathbf{B} U(1)$.
Similarly, the near horizon geometry of an M2-brane is $AdS_4 \times S^7$ and the charge of this which is felt by an M5-brane propagating near the M2-brane pole is measured rationally by maps $AdS_4 \times S^7 \simeq S^7 \to S^4$. Dually, the near horizon geometry of an M5-brane is $AdS_7 \times S^4$ and the charge of this as seen by an M2-brane propagating nearby is measure by maps $AdS_7 \times S^4 \simeq S^4 \to S^4$.
This is one way to see that $S^4$ is, rationally, just the right coefficient space to measure these M2- and M5-brane charges, since it has non-torsion homotopy groups $\simeq \mathbb{Z}$ precisely in degrees 7 and 4.
But now this is supposed to be the story only for “abelian” M-branes. There are also supposed to be non-abelian variants. One formalization of what this means is given in the above article. There it is shown that the “more than 1/2 BPS configurations” of near horizon geometries of M2-branes as above are more generally given precisely by choosing a finite subgroup $G_{ADE} \subset SU(2)$ and replacing $AdS_4 \times S^7$ by $AdS_4 \times S^7/G_{ADE}$, where $G_{ADE}$ acts on $S^7$ via the canonical action of $SU(2)$ that gives the quaternionic Hopf fibration.
(For $\leq 1/2$ BPS configurations we still have a quotient of $S^7$ by some $G_{ADE}$, but it is no longer via the canonical action of $SU(2)$. I’d like to ignore this more general case for the moment.)
Also, the above article explains (section 8.3) that for the M5-brane then all $\gt 1/2$ BPS configurations are the round $S^4$ as before.
So this means that the statement is essentially that the rational charges of $\gt 1/2$ BPS M-branes are encoded not by the plain quaternionic Hopf fibration
$\array{ S^7 \to S^7 / SU(2) \simeq S^4 \to \mathbf{B} SU(2) }$but by a finite quotient version
$\array{ S^7 \to S^7 / G_{ADE} \to \mathbf{B}G_{ADE} } \,.$Somehow. Here is my idea for how to conceptualize this:
(continued in next comment)
In fact from the higher super-Cartan geometry picture we want the rational part of the spacetime charge $X \to \mathbf{B}^3 U(1)_{conn}$ to be fixed, so that the only remaining freedom here is addition of flat 2-gerbes. $X \to \flat \mathbf{B}^3 U(1)$. Now, since the map $S^4 \to \mathbf{B}^3 U(1)$ canonically factors, in the present context, through $\mathbf{B}SU(2)$, it is somehow natural (but this needs to be better understood) to require that such flat maps factor through $\flat \mathbf{B}SU(2)$, which in turn is naturally achieved by factoring through $\mathbf{B}G_{ADE}$.
So possibly the role of the spacetimes $AdS_4 \times S^7 / G_{ADE}$, which are supposed to be the elemenary carriers of the charge groups that our would-be cohomology theory is to measure, is that they fit not just into diagrams
$\array{ AdS_4 \times S^7/G_{ADE} && \longrightarrow && S^4 \\ & \searrow &\swArrow& \swarrow \\ && \mathbf{B}^3 U(1) }$being morphisms in $\mathbf{H}_{/\mathbf{B}^3 U(1)}$
but even in diagrams of the form
$\array{ AdS_4 \times S^7/G_{ADE} & \to & S^7 /SU(2) & \simeq & S^4 \\ \downarrow && & \swarrow \\ \mathbf{B} G_{ADE} &\to & \mathbf{B} SU(2) \\ && \downarrow \\ && \mathbf{B}^3 U(1) }$which lift these to morphisms in $\mathbf{H}_{/\mathbf{B}G_{ADE}}$ through the canonical functor $\mathbf{H}_{/\flat \mathbf{B}^3 U(1)} \longrightarrow \mathbf{H}_{/\mathbf{B}^3 U(1)}$.
So to amplify: this is essentially guaranteed to model the relevant physics story, but the question is how to say “XYZ cohomology theory” such that this is forced upon us from just unwinding what cocycles in XYZ-cohomology theory are.
As a mathematical side remark, notice that the above scenario has the pleasant side effect that trivializing the torsion part of the M2-brane charge $X \to \mathbf{B}G_{ADE} \to \mathbf{B}^3 U(1)$ (say over a Horava-Witten boundary to find the heterotic string) yields precisely Epa-Ganter’s Platonic 2-groups (Epa-Ganter 16).
So somehow all the ingredients are there. But I am still not sure how exactly to organize them all into one neat single concept of equivariant cohomology theory.
So maybe what I am saying is simply that the part of the cohomology which corresponds to the flat C-field component is, rationally, twisted cohomology with coefficients in
$\left[ \array{ S^7 / \flat SU(2) \\ \downarrow \\ \flat \mathbf{B} SU(2) } \right] \;\; \in \mathbf{H}_{/\flat \mathbf{B}SU(2)} \longrightarrow \in \mathbf{H}_{/\mathbf{B}^3 U(1)} \,,$with coefficients of the homotopy pullback of $S^4 \to \mathbf{B} SU(2)$ along the inclusion of the discrete part $\flat \mathbf{B} SU(2) \to \mathbf{B} SU(2)$
$\array{ S^7/ \flat SU(2) &\longrightarrow& S^7 / SU(2) \simeq S^4 \\ \downarrow &(pb)& \downarrow \\ \flat \mathbf{B}SU(2) &\longrightarrow& \mathbf{B}SU(2) } \,.$Now $S^7 / \flat SU(2)$ is not a manifold or orbifold, as $\flat SU(2)$ is not finite, and so if we demand that spacetime is a manifold (or orbifold), then any cocycle on spacetime with these coefficients will have to factor through a further restriction to a finite quotient
$\array{ S^7 / G_{ADE} &\longrightarrow& S^7/ \flat SU(2) &\longrightarrow& S^7 / SU(2) \simeq S^4 \\ \downarrow &(pb)& \downarrow &(pb)& \downarrow \\ \mathbf{B}G_{ADE} &\longrightarrow& \flat \mathbf{B}SU(2) &\longrightarrow& \mathbf{B}SU(2) } \,.$This is looking good to me. If this is the right picture,it implies that the bosonic flat part of the C-field on a spacetime with ADE-singularities is necessarily induced by the flat $SU(2)$-connection that is associated with the ADE-singularity, i.e. that it is necessarily the 3-cocycle classifying the corresponding Platonic 2-group.
I would like to check if this statement has any counterparts in existing literature. Unfortunately, it seems that discussion of this point is missed in
There in section 4.1 the flat C-fields on $G_2$-manifolds locally modeled on cones $X$ over $Y = S^3 \times S^3$ are discussed, but precisely the case of interest, namely where instead one has a cone on the orbifold $Y_{G_{ADE}} = S^3 / G_{ADE} \times S^3$ (introduced at the end of section 2.5) is not discussed (my $G_{ADE}$ is denoted $\Gamma$ in that article). In section 5.4 the discussion gets back to $Y_{G_{ADE}}$, but now it is $H_3(Y_{G_{ADE}}, \mathbb{Z})$ instead of $H^3(Y_{G_{ADE}}, U(1)) = \pi_0 \mathbf{H}( Y_{G_{ADE}}, \flat \mathbf{B}^3 U(1) )$ which is discussed. Too bad.
[edit: I have forwarded this question to PhysicsOverflow here ]
Idle thought: if global families are defined in terms of a class of compact Lie groups, and if p-compact groups share many properties with compact Lie goups, is there a global equivariance in the p-compact case?
I guess that would depend on whether the reasons compactness is wanted in the ordinary case, as outlined by Charles Rezk here, continue in the p-compact case.
1 to 24 of 24