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September 29, 2005

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=1N=1 supergravity

  1. SO(32)SO(32)
  2. E 8×E 8E_8\times E_8
  3. E 8×U(1) 248E_8\times U(1)^{248}
  4. U(1) 496U(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, XX, that they consider has fundamental group 3× 3\mathbb{Z}_3\times\mathbb{Z}_3, and its universal cover, X˜\tilde{X} is a fiber product of two dP 9dP_9s. That is, X˜\tilde{X} is elliptically-fibered over a base which is, itself, an elliptically-fibered del Pezzo surface ( 2\mathbb{P}^2 blown up at 9 points). In their paper from last October, they work out the geometry of X˜\tilde{X}, and show that, with the right sort of fibration structure (three I 1I_1 singularities and three I 3I_3 singularities), one can find a family of X˜\tilde{X}s with a freely-acting 3× 3\mathbb{Z}_3\times\mathbb{Z}_3. The hodge numbers of X˜\tilde{X} are h 1,1(X˜)=h 2,1(X˜)=19h^{1,1}(\tilde{X})=h^{2,1}(\tilde{X})=19 and so XX has h 1,1(X)=h 2,1(X)=3h^{1,1}(X)=h^{2,1}(X)=3.

Equivariant bundles on X˜\tilde{X} descend to bundles on XX, and in their followup paper from this May, they describe how to construct suitable equivariant bundles on X˜\tilde{X} to build a 3-generation SO(10)SO(10) model on XX (together with suitable doublet-triplet splitting, etc.). To get the rank-4 bundle, VV, to embed in the visible E 8E_8, they follow a 2-step procedure. First construct a rank-2 bundle, WW on the base as an extension 0L 1WL0 0 \to L^{-1} \to W \to L\otimes \mathcal{I} \to 0 and then, after pulling back to X˜\tilde{X}, construct another extension 0V 1π *W0 0\to \mathcal{L}\oplus\mathcal{L} \to V \to \mathcal{L}^{-1}\otimes \pi^*W\to 0 Here LL is a certain line bundle on the base, \mathcal{L} is a line bundle on X˜\tilde{X}, and \mathcal{I} is an ideal sheaf of a set of points, {p i}\{p_i\}. I’ve brutally suppressed the equivariant structures on all the bundles involved.

Anyway, to get a 3-generation model, they choose L=𝒪(2f)L= \mathcal{O}(2f), where ff is the divisor class of a generic fiber on dP 9dP_9 and {p i}\{p_i\} is a generic set of 9 points permuted by the 3× 3\mathbb{Z}_3\times\mathbb{Z}_3 action. To get a 1-generation model, Volker says, choose L=𝒪(f)L= \mathcal{O}(f) and {p i}\{p_i\} a set of 3 points, the locations on the base of the I 3I_3 fibers of X˜B\tilde{X}\to B. (Note that 3× 3\mathbb{Z}_3\times\mathbb{Z}_3 does not act freely on the base. So, rather trickily, we obtain a rank-4 bundle, VV, 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.


1 Due to the wisdom of someone at Apple, it was sent, not to Houston, but to Memphis for repair.

2 If, in fact, we can show that they aren’t, then we have an answer to I. I. Rabi’s famous question.

Posted by distler at September 29, 2005 10:19 AM

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8 Comments & 0 Trackbacks

Re: You Can’t Always Get What You Want

Ah, but is N=1 supergravity the only candidate for quantum field theories in 10D with a UV completion?

Are you saying that it’s hard to come up with 1 or 2 generations of Standard Model fermions from string theory?

Posted by: Jason on September 29, 2005 3:57 PM | Permalink | Reply to this

Re: You Can’t Always Get What You Want

Ah, but is N=1 supergravity the only candidate for quantum field theories in 10D with a UV completion?

No, there’s also (pure) N=2 supergravity (which comes in two types).

[Just to be clear, we are discussion low-energy theories including gravity. In low enough dimensions (10 is not low enough), there are plenty of non-gravitational theories which are UV-complete.]

Are you saying that it’s hard to come up with 1 or 2 generations of Standard Model fermions from string theory?

Yes.

Posted by: Jacques Distler on September 29, 2005 4:22 PM | Permalink | PGP Sig | Reply to this

Re: You Can’t Always Get What You Want

Are you saying that it’s hard to come up with 1 or 2 generations of Standard Model fermions from string theory?

Yes.

Is that because the GUT breaking chain

E 8E 6×SU(3) F 3E_8 \supset \frac{E_6 \times SU(3)_F}{\mathbb{Z}_3}

contains the “family symmetry group” SU(3) FSU(3)_F and that one of the E 6×SU(3) F 3\frac{E_6 \times SU(3)_F}{\mathbb{Z}_3} representations coming from the adjoint representation of E 8E_8 is (27,3)(27,3)? Reps like (27,3)(27,3) are allowed but not reps like (27,1)(27,1).

Or is there some more fundamental reason? The previous “explanation” seems to be very model dependant.

Posted by: Jason on September 29, 2005 5:31 PM | Permalink | Reply to this

Flavours

No, I wasn’t thinking about embedding the flavour group in E 8E_8, or anything fancy (but model-dependent) like that.

I was just making an empirical observation about known string vacua and the evident difficulty of constructing one with 1 (or even 2) generations of chiral matter.

Not a proof, but a conjecture. It’s not obvious why such a theory should not be embeddable in String Theory, any more than Cumrun’s example of pure N=2N=2 gauge theory coupled to supergravity.

But, of course, it’s precisely because it’s not obvious that one learns something deep if you manage to understand why it’s true …

Posted by: Jacques Distler on September 30, 2005 1:33 AM | Permalink | PGP Sig | Reply to this

Re: Flavours

I am pretty sure that we could construct models with 1 or 2 generations (and 0 anti-generations) in the framework of our Het[E8] non-standard embedding models. Of course I have not worked out all the details, but I do not see any problem.

Is the scarcity of such vacua your conjecture, or are there any references for that?

Posted by: Volker Braun on September 30, 2005 6:58 PM | Permalink | Reply to this

Re: Flavours

I am pretty sure that we could construct models with 1 or 2 generations (and 0 anti-generations) in the framework of our Het[E 8]\text{Het}[E_8] non-standard embedding models.

Really? That would be interesting, if true.

Is the scarcity of such vacua your conjecture, or are there any references for that?

No reference. Just folklore.

Posted by: Jacques Distler on September 30, 2005 8:32 PM | Permalink | PGP Sig | Reply to this

Re: You Can’t Always Get What You Want

http://www.macosxhints.com/article.php?story=20050720092514388 “and character” query=gnu+dd

This macosxhints article describes how to recover data from a bad drive using unix tools that are very robust in the face of a bad drive.
It may help you if you are still at it, or help you next time. The idea is to use a gnu app called ddrescue (which works fine on HFS+, because it’s below the file system).

WTF??? Your insane comment system won’t allow people to post URLs, and complains by throwing up vast streams of XML minutiae that no-one in their right mind is going to bother to read.

Posted by: Maynard Handley on September 29, 2005 5:01 PM | Permalink | Reply to this

Recovery

Thanks for the link, Maynard. Sorry you had difficulty entering it (you need to replace “&” by “&”).

ddrescue sounds a lot more user-friendly than using dd for block-level disk recovery. I did that once, years ago, on the first generation golem. ‘Twas not a pleasurable experience.

My downfall, in this case, was that I really didn’t think the disk was bad when I sent the machine in for repair. All indications were of a bad controller chip. Apple’s DiskUtility.app still says the disk checks out OK, though it seems pretty clear that there are some bad blocks (or something) screwing up the reading of data from it. Once I got the true measure of the problem, it was not hard to deal with.

There were a dozen files where bad blocks made the process of recovering the data freeze the machine, and another 20 or so files which were mysteriously skipped over. All of these were easily recovered from backups.

Posted by: Jacques Distler on September 30, 2005 2:04 AM | Permalink | PGP Sig | Reply to this

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