Musings

You Can’t Always Get What You Want

Well, my iBook returned safely¹, with a brand new logic board and brand new hard drive. Recovering the data from the old drive proved to be a more harrowing experience than I could have imagined. CarbonCopyCloner and the underlying commandline utility, ditto, seem to react very badly to certain types of disk errors: freezing up when they encounter some damaged files, silently skipping over others. I think I’ve got everything back in place now, but it took several days of effort (and a good semi-recent backup).

Anyway, Luboš points to a recent note by Vafa, in which Cumrun makes the obvious point that “most” low-energy effective field theories coupled to (super)gravity do not have UV completions. Despite making apparent sense at low energies, they are, in fact, inconsistent. As I’ve been trying to emphasize in recent posts, this is bound to be true, whether you think the short distance behaviour is governed by a UV fixed point or by String Theory.

It’s already true in 10 dimensions. If we demand the cancellation of gauge and gravitational anomalies (an IR phenomenon), there are precisely 4 consistent low-energy theories coupled to 10-dimensional $N=1$ supergravity

1. $SO(32)$
2. $E_8\times E_8$
3. $E_8\times U(1)^{248}$
4. $U(1)^{496}$

But only the first two have UV completions as String Theories. If it’s true in 10 dimensions, it’s surely true after compactification as well, and Cumrun brings up a bunch of examples of generic classes of low-energy theories which do not seem to arise as compactifications of String Theory.

This is interesting. But, ultimately, what will be more interesting is to show not gross features distinguishing “bad” low-energy effective field theories from the “good” (i.e., embeddable in String Theory) ones, but fine features. For instance, nothing about the consistency of low-energy physics (or any anthropic reasoning) requires 3 generations. We could, just as well, get by with 2 or 1. But those 4-d effective field theories don’t seem to be realizable in String Theory².

Despite the enthusiastic belief that ‘anything’ is realizable somewhere on the Landscape, I’m gonna wager that is far, far, from true. And, in figuring out what’s actually possible, we will learn much.

Update (10/6/2005): A 1-Generation Model

In the comments, Volker Braun mentions that it might be possible to construct a 1-generation model as a variant of a heterotic compactification recently considered by him and collaborations at Penn. At least, at the level of crudity that I was making the conjecture, that turns out to be the case.

The Calabi Yau manifold, $X$, that they consider has fundamental group $\mathbb{Z}_3\times\mathbb{Z}_3$, and its universal cover, $\tilde{X}$ is a fiber product of two $dP_9$s. That is, $\tilde{X}$ is elliptically-fibered over a base which is, itself, an elliptically-fibered del Pezzo surface ($\mathbb{P}^2$ blown up at 9 points). In their paper from last October, they work out the geometry of $\tilde{X}$, and show that, with the right sort of fibration structure (three $I_1$ singularities and three $I_3$ singularities), one can find a family of $\tilde{X}$s with a freely-acting $\mathbb{Z}_3\times\mathbb{Z}_3$. The hodge numbers of $\tilde{X}$ are $h^{1,1}(\tilde{X})=h^{2,1}(\tilde{X})=19$ and so $X$ has $h^{1,1}(X)=h^{2,1}(X)=3$.

Equivariant bundles on $\tilde{X}$ descend to bundles on $X$, and in their followup paper from this May, they describe how to construct suitable equivariant bundles on $\tilde{X}$ to build a 3-generation $SO(10)$ model on $X$ (together with suitable doublet-triplet splitting, etc.). To get the rank-4 bundle, $V$, to embed in the visible $E_8$, they follow a 2-step procedure. First construct a rank-2 bundle, $W$ on the base as an extension $$0 \to L^{-1} \to W \to L\otimes \mathcal{I} \to 0$$ and then, after pulling back to $\tilde{X}$, construct another extension $$0\to \mathcal{L}\oplus\mathcal{L} \to V \to \mathcal{L}^{-1}\otimes \pi^*W\to 0$$ Here $L$ is a certain line bundle on the base, $\mathcal{L}$ is a line bundle on $\tilde{X}$, and $\mathcal{I}$ is an ideal sheaf of a set of points, $\{p_i\}$. I’ve brutally suppressed the equivariant structures on all the bundles involved.

Anyway, to get a 3-generation model, they choose $L= \mathcal{O}(2f)$, where $f$ is the divisor class of a generic fiber on $dP_9$ and $\{p_i\}$ is a generic set of 9 points permuted by the $\mathbb{Z}_3\times\mathbb{Z}_3$ action. To get a 1-generation model, Volker says, choose $L= \mathcal{O}(f)$ and $\{p_i\}$ a set of 3 points, the locations on the base of the $I_3$ fibers of $\tilde{X}\to B$. (Note that $\mathbb{Z}_3\times\mathbb{Z}_3$ does not act freely on the base. So, rather trickily, we obtain a rank-4 bundle, $V$, whose index is 1/3 that of the previous case, which yielded 3 generations.)

So, it seems, there is already a counterexample to my conjecture about 1-generation models. But it did give me an excuse to mention this very beautiful construction by the Penn group, so it’s not a total loss.