### GraviGUT

One of the disadvantages of having waded into the Lisi affair is that I keep getting asked about related ill-conceived ideas for “theories of everything.” For the most-part, such ideas don’t have a relentless publicity machine behind them. Nor do they receive what can only be described as a credulous reception in certain corners of the Mathematics community. But still, it is *assumed* that one has an opinion about them.

One such idea is the $Spin(3,11)$ “GraviGUT” of Nesti and Percacci. The bosonic fields consist of a connection for a noncompact $Spin(3,11)$ gauge group, and a 1-form, $\theta$, transforming in the 14-dimensional vector representation. More formally, we assume that $P\to M$ is a $Spin(3,11)$ principal bundle, and $E\to M$ a vector bundle, associated associated to $P$ via the 14-dimensional vector representation. The fields consist of a connection on $P$, and a 1-form, $\theta$, with values in sections of $E$.

At least at the classical level, there’s something superficially attractive about the idea. As Percacci explains, $40=4\times 10$ of the components of $\theta$ can be used to Higgs $Spin(3,11)$ down to $Spin(3,1)\times Spin(10)$. The remaining $16=4\times 4$ components of $\theta$ form the usual vierbein. For that interpretation to hold, he requires that $\theta(x)$, viewed as a map from the tangent space at $x$ to the fiber of $E$, have rank-4. This is the higher-dimensional analogue of demanding that the vierbein be invertible.

There’s a very nice analogy, that he draws, between the resulting theory — in which the 40 broken generators of $Spin(3,11)$ are nonlinearly-realized — and the universal low-energy description of the electroweak theory as a gauged nonlinear $\sigma$-model (with target space $S^3$, where we gauge an $SU(2)\times U(1)$ subgroup of the $SU(2)\times SU(2)$ isometry group).

At this stage, there are a bunch of massive higher-spin bosonic fields, along with the massless fields of gravity (in the Palatini form) coupled to $Spin(10)$ Yang-Mills. There are no scalar bosons, and hence no candidate for fields that Higgs $Spin(10)$. But, hey, let’s not worry about that.

The analogy with the low-energy description of the electroweak theory should both please and trouble you. The gauged nonlinear $\sigma$-model can only be a low-energy description. It necessarily breaks down above some cutoff scale, $\Lambda \leq 4\pi f$, where — in the case of the electroweak theory — $f=249 \text{GeV}$. New degrees of freedom must enter the theory at the scale $\Lambda$. Perhaps a fundamental Higgs, perhaps something more exotic. In any case, the LHC is supposed to tell us what those new degrees of freedom are.

Surely, the same is true of the nonlinearly realized “graviGUT” symmetry? As it is a field theory of massive higher-spin bosons, it must break down above some cutoff scales, where $f$ could be as high as the Planck scale. And, indeed, you are right. The graviGUT cannot be a good Lagrangian description to arbitrarily high energies.

But, in this regard, Percacci argues, we no worse off than we were with ordinary gravity. There, too, the effective Lagrangian description breaks down above the Planck scale.

Here’s where the 21st century version of magic pixie dust comes to the rescue. Perhaps the graviGUT is asymptotically-safe. Instead of simply breaking down at high energies, perhaps the spontaneously-broken $Spin(3,11)$ gauge theory is controlled by a nontrivial UV fixed-point.

Even if that were true (and there’s no reason to believe that it is), surely the indefinite signature of the Cartan form on $so(3,11)$ (a generic problem, if you want play around with noncompact gauge groups, like $Spin(3,11)$) renders the whole theory — even if it is controlled by a nontrivial fixed point — non-unitary. Consider, in particular, the massive 1-forms in the $(2,2;10)$. Unless I am very much mistaken, their kinetic terms have indefinite signature, as a direct consequence of the indefinite signature of the Cartan form on $so(3,11)$.

Despite these obvious objections, Nesti and Percacci are excited, because it’s possible to introduce fermions in the irreducible chiral spinor representation of $Spin(3,11)$. These correspond to chiral fermions (a left-handed $16$ of $Spin(10)$, along with its Hermitian conjugate, a right-handed $\overline{16}$), when decomposed under the unbroken subgroup. To build a kinetic term for the fermions, they need to introduce yet another field, $\phi$, a nowhere-vanishing section of $\wedge^4 E$. Such a field is probably familiar from my previous discussion of the Plebanski action.

I suppose that, after Lisi’s “theory”, writing down a model that admits chiral fermions must feel like success. But that’s setting the bar *awfully low* …

## Re: GraviGUT

The ghost issue that occurs after the effective theory breaks down is sidestepped in the paper in a way that I don’t quite understand. In the conclusions Percacci mentions:

“Here we may add that since they occur near or beyond the transition to a topological phase, the standard tree level analysis is questionable.”

Why and how would transitioning to the symmetric topological phase somehow cure a nonunitary theory (even when analyzed at tree level), especially since it is precisely in that phase where the full signature is manifest? It seems to me that you would require a lot more structure than simple asymptotic safety to ensure miraculous cancellations of that nature? Or am I completely missing something obvious.