### E_{8} Quillen Superconnection

#### Posted by Urs Schreiber

A remark on the nature of Quillen superconnections with values in $\mathbb{Z}_2$-graded Lie algebras, such as $e_8$.

**Quick remark on the background for those unfamiliar with Yang-Mills (super) gauge theory**

For decades physicicts have been playing around with symbols, trying to identify the mathematical structure that underlies the mess they read off their particle accelerators.

The biggest general success has been the identification that all forces in nature can be identified with the mathematical structure of a connection on a fiber bundle. Known as “Yang-Mills theory”, this is the heart of the standard model of particle physics today.

Apart from forces, there is matter. It turns out that while forces are described by connections on fiber bundles, these fiber bundles are spin bundles and matter turns out to be encoded by sections of these Spin bundles. The picture thus obtained is called the standard model.

While nice, this insight makes some people think that one should look for mathematical structures further unifying this picture. There is an obvious guess: extend the structure Lie group (called the *gauge Lie group* in this context) of your fiber bundles to a super Lie group. That will make the connection have odd graded components which may possibly be identified with the spin sections we had before. Hence it would allow to identify forces and matter alike with the mathematical structure given by a connection, distinguished just by the super degree of the components of that connection.

And in fact, in supergravity theories it works just like this: the gravitational force itself is a connection with value in the Poincaré Lie algebra $iso(n,m)$, and the “matter” piece in this context, called the gravitino, is simply the odd part obtained when extending this to a super Poincaré connection with values in $s iso(n,m)$.

But that also makes it clear that, unlike the matter observed in the standard model, gravitinos are spinor-valued *1-forms*, not 0-forms.

**What structure?**

In physics people often feel free to manipulate their symbols first, and worry about finding the mathematical interpretation of these manipulations later. One needs to be careful with this, but lots of interesting structure was found this way over the centuries, beginning with Newton’s “fluxions”.

In arXiv:0711.0770 the proposal is – implicitly – not to worry about the above problem – that a superconnection is locally given by a super 1-form – and just formally add to an ordinary connection 1-form an odd 0-form: $\underbrace{A }_{\in \Omega^{(1,even)}(Y,g)} + \underbrace{\Psi }_{\in \Omega^{(0,odd)}(Y,g)} \;\;\;\; (1)$ where $g$ is some $\mathbb{Z}_2$-graded Lie algebra, for instance $e_8$ I, II.

The proposal that this might be the key to making big progress with understanding the standard model of particle physics had created an unprecedented amount of attention and lots of criticism.

Much of that criticism revolved around the interpretation of the formal sum above. While arXiv:0711.0770 offers an interpretation, this was perceived as unviable.

**Quillen superconnections on $\mathbb{Z}_2$-graded bundles**

Since I have never seen it mentioned in any of these discussions, all I want to point out here is that (1), for $g$ a $\mathbb{Z}_2$-graded Lie algebra, can be read as a Quillen superconnection on a (necessarily) $\mathbb{Z}_2$-graded vector bundle $E \to X$ over spacetime $X$, which is associated to a principal $G$-bundle by some representation of $G$ on $\mathbb{Z}_2$-graded vector spaces:

$\mathbf{A} := d + A + \Psi$ $\mathbf{A} : (\Omega^\bullet(X,E))_{even,odd} \to (\Omega^\bullet(X,E))_{odd,even}$ $\forall \omega \in \Omega^1(X) : [\mathbf{A},\omega] = d\omega \,.$

Notice two things:

1) Quillen superconnections are different from other notions of superconnections. In particular, Quillen superconnections do not come from a path-lifting property and are not related to an ordinary notion of parallel transport. For a discussion of Quillen superconnections and also of super parallel transport I can recommend

Florin Dumitrscu,
*Superconnections and Parallel Transport*

(pdf).

2) It is crucial to distinguish here between $\mathbb{Z}_2$-graded Lie algebras and super Lie algebras. As follows:

there are precisely two different symmetric braided monoidal structures on the category of $\mathbb{Z}_2$-graded vector spaces: the trivial one and the one which introduces a sign when two odd vectors are exchanged in a tensor product.

The symmetric monoidal category with the trivial braiding here is $\mathbb{Z}_2 Vect$. The other one is $S Vect$.

A $\mathbb{Z}_2$-graded Lie algebra is a Lie algebra internal to $\mathbb{Z}_2 Vect$. A super Lie algebra is a Lie algebra in $S Vect$.

$\mathbb{Z}_2$-graded Lie algebras such as $e_8$ are not super Lie algebras: while they look alike a lot (especially $e_8$ does look a lot alike a super $so(16)$ Lie algebra) they are different.

**Main point, summarized.**

Whatever the physical viability of the proposal of arXiv:0711.0770, the expression in equation (3.1) on p. 23 is to be interpreted as a Quillen superconnection $\mathbf{A}$ on a $\mathbb{Z}_2$-graded $e_8$ associated vector bundle and (3.2) is the corresponding Quillen curvature $F_{\mathbf{A}} = \mathbf{A}^2 \,.$

So if one wants to examine the possibility of describing particle physics with this approach, the mathematical structure to determine would seem to be something like “Quillen Yang-Mills theory”.

## Re: E8 Quillen Superconnection

Wow, this sounds cool Urs.