How are the classifying space of $E_8$ and $K(\mathbb{Z},4)$ related?

Question

I recently heard the following fact :

Up to the $15$th skeleton, the classifying space $BE_8$ and $K(\mathbb{Z},4)$ are homotopy equivalent?

I have two questions on this :

(1) Is there any easy way to see this? Of course, knowing the first fourteen homotopy groups of $E_8$ is enough but then the question is how does one compute them?

(2) Is there any feasible explanation that suggests that $4$th cohomology classes (possibly related to gerbes), i.e., elements in $H^4(X;\mathbb{Z})$, arise from physical considerations and if $X$ is of dimension $14$ or less then we're classifying $E_8$-bundles on $X$, thereby suggesting that $E_8$ arises out of physical considerations?

The last question is a little vague but any pointers would be great!

Answer 1

Given a simple Lie group $G$, you can check how far $G$ is from beeing a $K(\mathbb Z,3)$ by looking at the place where the affine vertex gets glued onto the Dynkin diagram, and measuring the length of that tail. For $E_8$, it's the longest, and so $E_8$ is the best possible approximation to a $K(\mathbb Z,3)$.

$$\bullet - \bullet - \stackrel{\stackrel{\displaystyle\bullet}|}{\bullet} - \underbrace{\bullet - \bullet - \bullet - \bullet}_{\text{long tail}} - \circ$$

This is done by labelling the cells of the affine Grassmannian $\Omega G$ by data from the dynlin Diagram, and checking how far you need to go for $\Omega G$ to start looking different than $\mathbb C \mathbb P^\infty$.

---

the affine Grassmannian $\Omega G$ is a very nice space: it's a complex (ind-)variety, and it is stratified by finite dimensional cells. In particular, it has a natural CW-decomposition. Each cells of $\Omega G$ is isomorphic to $\mathbb C^n$, and is in particular of even (real) dimension.

Moreover, $\Omega G$ is a coadjoint orbit of the infinite dimensional Lie group $S^1\ltimes \widetilde {LG}$. Here, the tilde refers to the universal central extension of the loop group $LG$, and $S^1$ acts by reparametrizing the loops.

The inclusion $\Omega G\to Lie(S^1\ltimes \widetilde {LG} )^* $ can be composed with the projection $$Lie(S^1\ltimes \widetilde {LG})^* \twoheadrightarrow (\mathfrak t_{S^1\ltimes \widetilde {LG}})^* \cong \mathfrak t^* \oplus \mathbb R \oplus \mathbb R$$ (here $\mathfrak t$ denotes the Lie algebra of the maximal torus $T$ of $G$). It turns out that the composite lands in a translated copy of $\mathfrak t^* \oplus \mathbb R$, and so one gets map $$ \mu:\Omega G \to \mathfrak t^* \oplus \mathbb R $$ called the moment map (for the $T_{S^1\ltimes LG}$ action).

What is important, is that the space $t^* \oplus \mathbb R$ has a natural basis that is indexed by the vertices of the extended Dynkin diagram: those are the simple coroots. I will denote each cell by the moment map image in $\mathfrak t^* \oplus \mathbb R$ of its center point (in the basis of simple coroots).

Now, let me specialize to the case $G=E_8$. Here we go:

0-dimensional cell: $\qquad\begin{matrix} 0 - 0 - \stackrel{\stackrel{\displaystyle 0}|}{0} - 0 - 0 - 0 - 0 - 0 \end{matrix}$

2-dimensional cell: $\qquad\begin{matrix} 0 - 0 - \stackrel{\stackrel{\displaystyle 0}|}{0} - 0 - 0 - 0 - 0 - 1 \end{matrix}$

4-dimensional cell: $\qquad\begin{matrix} 0 - 0 - \stackrel{\stackrel{\displaystyle 0}|}{0} - 0 - 0 - 0 - 1 - 1 \end{matrix}$

6-dimensional cell: $\qquad\begin{matrix} 0 - 0 - \stackrel{\stackrel{\displaystyle 0}|}{0} - 0 - 0 - 1 - 1 - 1 \end{matrix}$

8-dimensional cell: $\qquad\begin{matrix} 0 - 0 - \stackrel{\stackrel{\displaystyle 0}|}{0} - 0 - 1 - 1 - 1 - 1 \end{matrix}$

10-dimensional cell: $\qquad\begin{matrix} 0 - 0 - \stackrel{\stackrel{\displaystyle 0}|}{0} - 1 - 1 - 1 - 1 - 1 \end{matrix}$

12-dimensional cell: $\qquad\begin{matrix} 0 - 0 - \stackrel{\stackrel{\displaystyle 0}|}{1} - 1 - 1 - 1 - 1 - 1 \end{matrix}$

14-dimensional cell: $\qquad\begin{matrix} 0 - 0 - \stackrel{\stackrel{\displaystyle 1}|}{1} - 1 - 1 - 1 - 1 - 1 \end{matrix}$

other 14-dimensional cell: $\qquad\begin{matrix} 0 - 1 - \stackrel{\stackrel{\displaystyle 0}|}{1} - 1 - 1 - 1 - 1 - 1 \end{matrix}$

As you can see, $ H^* (\Omega G) = H^* (\mathbb C \mathbb P^\infty ) $ for $*\le 13$. Even better: the varieties $\Omega G$ and $\mathbb C \mathbb P^\infty$ are isomorphic in complex dimensions $\le 6$ [added later: I take back that claim. I don't konw how to prove that the varieties $\Omega G$ and $\mathbb C \mathbb P^\infty$ are isomorphic in complex dimensions $\le 6$ (it might still be true)]. In particular, the CW-complexes $\Omega G$ and $\mathbb C \mathbb P^\infty$ are isomorhpic in dimensions $\le 13$. Taking classifying spaces, we get that the CW-complexes $G$ and $K(\mathbb Z,3)$ are isomorphic in dimensions $\le 14$.

Answer 2

This answer is only a bit of a sketch for (2), because it's been a while since I thought/heard about this. And I interpret 'physical considerations' loosely, since this all comes from string theory, whose status as a physical theory is open to debate (at the very least, nothing below comes from experiments!).

There is, in M-theory, a locally-defined 3-form 'higher connection', whose 'curvature' is a 4-form. This is related to the existence of a bundle 2-gerbe/circle 3-bundle whose characteristic class is this 4-form. Really all that people see of this is the low-energy limit which is 11-dimensional supergravity. Since spacetime $X^{11}$ as conceived in M-theory is 11-dimensional, any classifying map $X^{11} \to K(\mathbb{Z},4)$ for one of these higher structures wouldn't be able to distinguish $K(\mathbb{Z},4)$ from $BE_8$. Thus the bundle 2-gerbe might secretly be an $E_8$-bundle.

Alternatively, in regular string theory one has the Kalb-Ramond field and its $H$-flux, which is 3-form, and is commonly understood to be the curvature 3-form associated to a bundle gerbe/circle 2-bundle. However exactly the same argument as before means that perhaps what is going on is that there is an $\Omega E_8$-bundle instead of a bundle gerbe (this is the point of view espoused to me by Jarah Evslin a few years back). Due to the heterotic string theory $E_8\times E_8$, and various models with compactifications, it is not wholly unreasonable to expect $\Omega E_8$ to turn up at some point.

It should be noted that I don't think that these ideas are widely accepted, though.

Edit: I add that the only source I have for this is various talks at least half a decade ago. There is substantial work by others that I was previously unaware of, and I'm sure I have no idea about who did what.

As far as (1) goes, which version of $E_8$ are you after? Assuming it is a compact real Lie group, it is 2-connected and we have $\pi_3(E_8) = \mathbb{Z}$. After that I'm not sure how the homotopy groups are calculated. Morse theory?

Answer 3

In addition to the work mentioned in David's very useful answer I would suggest that you take a look at http://arXiv.org/pdf/hep-th/0312069 by Diaconescu, Moore and Freed. They give a mathematically precise definition of the M-theory 3-form in terms of the Chern-Simons term of $E_8$ gauge theory and apply the formalism to study M-theory on manifolds with boundary. My understanding is that the formalism "works" in the sense of giving mathematically well defined answers which agree with various physical constraints, but whether the $E_8$ gauge field is fundamental or not remains elusive.

Answer 4

re 1): André's answer is superb, but just for the record, original references determining the homotopy groups of E8 are here: http://ncatlab.org/nlab/show/E8#HomotopyGroupsReferences

re 2):

One way to think about the phenomenon $B E_8 \simeq_{15} B^3 U(1) \simeq K(\mathbb{Z},4)$ from the point of view of string theory is to compare it to

a) the equivalence $B \mathrm{PU}(\mathcal{H}) \simeq B^2 U(1) \simeq K(\mathbb{Z},3)$ that controls Freed-Witten anomaly cancellation over D-branes

b) generalized complex geometry and exceptional generalized geometry that controls various other geometric structures in string theory.

In all these cases, one is looking at geometry which arises from reduction of structure groups along maps $G \to K$ of groups with the property that they are weak homotopy equivalences.

This is true for the inclusions of maximal compact subgroups that control generalized complex and exceptional generalized geometry, hence the U-duality symmetry of supergravity theories in various dimensions.

Notice that these inclusions are far from being equivalences as morphisms of Lie groups. But they are equivalences of the underlying topological spaces.

This situation now has a good analog in higher smooth geometry, which "explains" the role of $E_8$.

Namely, there is a smooth 3-group $\mathbf{B}^2 U(1)$ (a smooth group 2-stack) and the universal degree 4-class on $B E_8$ has a smooth refinement to a morphism of smooth 3-groups (group 2-stacks)

$$ \Omega \mathbf{a} : E_8 \to \mathbf{B}^2 U(1) . $$

There is a higher analog of the notion of "reduction of structure groups" along such higher maps, and this controls the geometry of the supergravity C-field. For comparison, there is similarly a morphism of smooth 2-groups (smooth group stacks)

$$ \Omega \mathbf{dd} : \mathrm{PU}(\mathcal{H}) \to \mathbf{B}U(1) $$

and its induced "generalized geometry" by "reduction of higher structure groups" controls precisely the Chan-Paton bundles on D-branes twisted by the $B$-field.

Both of these morphisms of smoth higher stacks become equivalences of topological spaces under geometric realization (the first on 15-coskeleta, hence over the relevant spacetimes). So we may think of this as saying that:

"The Lie group $E_8$ is a generalized maximal compact subgroup of the smooth 3-group $\mathbf{B}^2 U(1)$. The geometry of the $C$-field is the 'generalized geometry' controled by this 'inclusion'."

For more details on all this, see around section 4.3 of

http://arxiv.org/abs/1201.5277

and the big overview tables in section 4.4.1 of

http://ncatlab.org/schreiber/files/cohesivedocumentv032.pdf

Answer 5

For (1), after searching a bit, I think the original reference in the physics literature is Witten's "Topological Tools in Ten-Dimensional Physics", Int. J. Mod. Phys. A 1, 39 (1986). I think there's a reference to the fact about homotopy groups there, but I haven't read it in years.

Just to expand a bit on Jeff's answer for (2), M-theory contains a three-form with a four form curvature. Horava and Witten shower that one could associate the $E_8 \times E_8$ heterotic string with M-theory on $S^1/\mathbb{Z}_2$. The boundaries each have an $E_8$ gauge theory on them. Soon after, Witten used $E_8$ bundles to determine the quantization of the four form in M-theory in hep-th/9609122. This quantization is, interestingly, shifted from being integral. $E_8$ index theory was used spectacularly to compare the partition functions of IIA and M-theory in the ginormous paper of Diaconsecu, Moore and Witten, hep-th/0005090. As Jeff says, the paper of Diaconescu, Moore and Freed is the most modern way of looking at the subject using (shifted) differential cohomology. One of the conclusions of that paper is we just don't know whether the use of $E_8$ bundles for quantizing the M-theory four-form is real or just a convenient trick. But given the other ways $E_8$ seems to be hanging around M-theory (for example, the split real form of $E_8$ shows up when you compactify M-theory on $T^8$), I'd guess the former.