Musings

Raoul Bott was a huge influence on my life. I learned Differential Geometry from him, in a reading course at Dunster House, where he was House Master, and I was an undergraduate. I and a couple of other Dunster House undergraduates met with him once a week, while working through the notes to his graduate course. Of course, there was that one, somewhat intimidating, week, when Sir Michael Atiyah showed up in his stead…

Raoul started out studying Engineering at McGill and, I think, that “nuts 'n bolts” approach to things mathematical never left him (to be fair, though, Marc Grisaru, who knew him from those days, reports that, “Even then, we knew he was a Mathematician.”). He was one of the people responsible for the close working relationship between Physics and Mathematics at Harvard, and was known to admonish his more skeptical colleagues, “Don’t argue with the physicists; they know how to compute!”

He was a warm and approachable House Master, greeting everyone at the Master’s Teas (back in the days when it was still legal to serve sherry to the undergraduates) with a big bear hug. He and Phyllis also had continuous stream of interesting visitors staying with them at Dunster: from the aforementioned Sir Michael to trumpeter Red Rodney to poet Allan Ginsburg.

But I think what impressed my fellow (non-Physics/Math) Dunsterites most was that, on Martha’s Vineyard, where he and Phyllis summered, Raoul was known as “King of the Nude Beach” — a claim-to-fame few other Harvard House Masters could match.

There’s more from Luboš and Sean and the Harvard Math Department.

There’s an interesting paper by Diaconescu, Florea, Kachru and Svrček on gauge-mediated supersymmetry-breaking in String Theory. In perturbative heterotic string theory and, I think, in heterotic M-theory as well, supersymmetry-breaking is generally expected to occur in a hidden sector, associated to the second $E_8$, and is communicated only indirectly to the visible sector via gravity-mediation. Gauge mediation requires a messenger field charged under both the visible and the hidden gauge group. There are such fields in the heterotic string but their masses are string-scale in the weakly-coupled heterotic string and super-Planckian in heterotic M-theory. So gravity-mediation typically dominates¹.

Gauge-mediated SUSY-breaking is attractive, because it solves the “flavour problem” by making the SUSY-breaking squark masses flavour-independent. That’s because the messenger(s) couple directly only to the gauge sector of the Standard Model. Gaugino masses are a 1-loop effect. Squark and slepton masses are a 2-loop effect, and are flavour-independent. Gravity-mediation (despite the name) has no such flavour-universality (a-priori).

Diaconescu et al find some new classes of heterotic models in which gauge-mediated SUSY-breaking dominates. In the dual F-theory picture (these models all have F-theory duals), SUSY is broken on a stack of D3-branes probing a singularity corresponding to a shrunken del Pezzo. The corresponding quiver gauge theories are known to be supersymmetric to all orders in perturbation theory, but to dynamically break supersymmetry at the nonperturbative level (see also Berenstein et al and Bertolini et al).

If this stack of D-branes is located sufficiently close to the stack on which the Standard Model arises, then the ground states of the open strings between the two stacks are the desired messengers for gauge-mediated SUSY-breaking.

Most of the discussion takes place in the context of the elliptically-fibered Calabi-Yaus explored by the Penn Group (for some recent papers getting ever-closer to the precise MSSM field content and couplings, see Bouchard and Donagi and Braun, He, Ovrut, and Pantev). So there’s a good chance that one can actually build a fully-realistic compactification along these lines. (Foes of F-theory flux vacua are permitted, at this point, to go berserk.)

So much for “top-down,” what about “bottom-up”? What will we be able to learn about SUSY at the LHC? I’ve mentioned before the paper by Arkani-Hamed, Kane, Thaler and Wang, which has finally appeared. In it, they discuss the “inverse problem” of deducing a model in the MSSM parameter space from the LHC data. To study the question, they simulate a huge number of MSSM models, look at the experimental signatures they produce. When the sleptons are somewhat heavy (so that they don’t appear in the decay-chain of neutralinos which result in opposite-sign dilepton events), there’s a relatively large² degeneracy (~10-100 models), corresponding to different orderings of the slepton masses. If the sleptons are lighter, then the degeneracies are smaller, and they present a rough taxonomy of some of the degenerate models.

Flippers

The spectrum of masses of the Electroweak 'inos is fixed, but the identities of which ino has which mass are exchanged.

Sliders

The Electroweak 'ino masses are moved up or down, keeping the mass splittings fixed.

Squeezers

The information of some of the Electroweak 'inos is hidden because the mass splittings are small enough that the leptons in the decay products are too soft to be seen.

With the limited available signatures, the LHC will be unable to distinguish between these radically models.

But, before losing hope, one of the key messages to take away from their paper is that, if you have some reason to choose between the degenerate models (either a theoretical prejudice, or some experimental signature that they did not consider), then the LHC data can determine the parameters of that single model to very high accuracy.

While Trackback spam is a source of continuing fascination hereabouts, Comment spam is but a fading memory. Yes, we occasionally get the odd piece of hand-entered Comment spam from India or Thailand or the Former Soviet Bloc, but Comment Spambots pretty much pass us by. Which is a shame, really, because I’d like to keep abreast of developments in that field.

So you can imagine my delight in finding that this blog had been visited by a new (to me, at least) Comment Spambot the other day.

In the space of 14 minutes, it

• made 2257 requests
• from 91 distinct IP addresses (all, as far as I can tell, zombie PCs)
• of which, 467 were requests for my comment script
• among which were 151 (unsuccessful, of course) attempts to POST a comment
• which resulted in 48 new IP addresses automagically added to my IP-banlist

How do I know all these details? Because the Spambot issues a malformed HTTP REQUEST header. (Fortunately, Apache is liberal in what it accepts, and equanimously records the malformed header to the logs.) I guess the Spambot author found the HTTP 1.1 Specification too difficult to understand.

Perhaps some public-spirited person, like Sam, could put together a Spambot Validation Service, in the interest of improving the overall quality of the Web.

Kieran Healy has written up an excellent list of rationales for supporting the President’s Executive Order authorizing domestic spying without Judicial oversight. I’d been trying to draft such a list myself, but Kieran’s is wittier than I could have come up with.

But the thing that I find most puzzling about the current fuss over renewing the Patriot Act has not been much discussed. It’s apparent that, under the Yoo Doctrine, the President can, by Executive Order, authorize all of the activities covered by the Patriot Act, with or without Congressional approval.

So who cares whether the Patriot Act is renewed?

I assume it’s just a matter of principle. One can’t let the namby-pamby Democratic “enemies of freedom” in Congress give aid and comfort to the terrorists by denying the President their approval for his activities.

After considerable frustration with Apache 2.2.0, I was about to throw in the towel and downgrade to 2.0.55. On a lark, I decided to download the latest Development Version and try that instead.

Upgrading from version 2.0.55 to 2.2.0 of the world’s dominant WebServer was a bit more of a hassle than I thought it was going to be. So, herewith, some notes.

Leigh, Minic and Yelnikov have a very interesting announcement of new results on 2+1 D Yang Mills Theory. Using a formalism pioneered by Karabali and Nair, they compute the glueball spectrum analytically at large-$N$. The result is expressed in terms of zeroes of the Bessel function.

I’ve been reading the recent Beasley-Witten paper on instanton effects in string theory.

As you know, instantons can have dramatic effects on the vacuum structure of supersymmetric gauge theories. They can induce a superpotential that lifts degeneracies that are present to all orders in perturbation theory. More subtly, as found by Seiberg, in the case of $SU(N_c)$ QCD">SQCD, with $N_f=N_c$ flavours, instanton effects can change the topology of the vacuum manifold. The singular affine variety,

$$\det(M)- B\tilde B = 0$$ (where $M$ is an $N_c\times N_c$ complex matrix) is deformed to

$$\det(M)- B\tilde B = \Lambda^{2 N_c}$$

In their previous paper, Beasley and Witten argued that this effect can be understood as the generation of an exotic sort of “superpotential” of the form

$$W = \omega_{\overline{\imath}\overline{\jmath}}(\Phi,\overline{\Phi}) D_{\dot{\alpha}}\Phi^{\overline{\imath}}D^{\dot{\alpha}}\Phi^{\overline{\jmath}}$$ where $\Phi^{\overline{\imath}}$ are anti-chiral superfields, parametrizing the moduli space, $M$, and $$\omega_{\overline{\imath}\overline{\jmath}} = \frac{1}{2} \left(g_{\overline{\imath}k} \tensor{\omega}{_\overline{\imath}_^k} + g_{\overline{\jmath}k} \tensor{\omega}{_\overline{\imath}_^k}\right)$$ where $g_{\overline{\imath}j}$ is the Kähler form on $M$ and $\tensor{\omega}{_\overline{\imath}_^j}\in H^1(M, T_M)$ is the deformation of complex structure of $M$ that deforms (1) into (2)¹.

In this particular case, this fancy formalism is somewhat superfluous. The constraint (2) can simply be imposed by introducing an additional chiral superfield, $S$, and a garden-variety superpotential

$$W= S(\det(M)- B\tilde B - \Lambda^{2 N_c})$$ For $\Lambda\neq0$ (and, even for $\Lambda=0$, away from the origin in field space), $S$ and some particular combination of the other fields are massive, and can be integrated out. When $\Lambda=0$, all the fields are massless at the origin, and so integrating out $S$ leads to the singular Lagrangian, of the form (3), constructed in detail by Beasley and Witten for $SU(2)$.

In String Theory, alas, you can’t willy-nilly integrate-in fields. So there are situations where, presumably, you need to represent the instanton-induced deformation of the classical moduli space by an exotic superpotential of the form (3), instead of the more transparent (4).

Beasley and Witten lay out a couple of instances where they argue that’s the case, and show how you can calculate a superpotential of the form (3), induced by worldsheet instantons in heterotic string theory.

What’s quite interesting is that some of their heterotic computations are closely related to the higher-genus worldsheet instanton computations in the topological A-model. These compute what are, by now, quite well-known corrections to the $N=2$ supergravity action that one obtains from a Type-IIA compactification on a Calabi-Yau. Beasley and Witten compute the analogous corrections to the $N=1$ theory arising from a heterotic compactification on the same Calabi-Yau (they’re F-terms involving higher powers of the supersymmetric gauge field strength squared, $W_\alpha W^\alpha$).

I’ve been a Time-Warner Roadrunner cable modem subscriber for many years now and, generally, the service has been decent, if somewhat expensive. Recently, though, it’s gone seriously downhill.

I generally have a couple of terminal sessions open with golem. With alarming frequency, nowadays, when I click on the terminal window and start typing, nothing echoes back. Sure enough, I click over to my web browser and try opening a new web page; the internet is unreachable. After a couple of minutes of pounding my fist and cursing at the computer, the network comes back. Web pages open again, the terminal sessions start echoing characters again and life returns to normal. For a while. Later in the evening, the same thing will happen again … and again.

Some have suggested that I call the Service department, and have them swap out my “bad” cable modem for a new one. But I don’t quite see why a modem that functioned just fine for years should suddenly turn “bad.” It’s not as if it’s broken down; 95% of the time, the connection is working fine.

On a hunch, I decided to do a little experiment and run a process which pings golem at regular intervals (one 64 byte ICMP packet every 10 seconds). And the problem … seems to have gone away. My best guess is that Time Warner’s router keeps flushing its ARP table and “inactive” connections (if you haven’t opened a web page, or typed something in the terminal in the past N minutes) get dropped. By keeping the connection perpetually active, I avoid being dropped.

A somewhat hackish solution, but it beats complaining about the crappy service.

Luboš blogs about Witten’s Loeb Lecturestalks at Harvard on Geometric Langlands, and reminds me of one of the most beautiful (and, at least in the Mathematics community, unexploited) aspects of the subject: the connection between topological twistings of $N=4$ SYM and manifolds of special holonomy.

So I thought I’d post a little review…

Remember this map?

Turns out that Justice Department staffers unanimously agreed that it violated the Voting Rights Act. But they were overruled by senior political appointees, and their 73 page memo was buried, never presented to the 3-judge panel which ultimately approved the Texas Redistricting plan.

Mark Posner, a longtime Justice Department lawyer who now teaches law at American University, said it was “highly unusual” for political appointees to overrule a unanimous finding such as the one in the Texas case.

“In this kind of situation, where everybody agrees at least on the staff level … that is a very, very strong case,” Posner said. “The fact that everybody agreed that there were reductions in minority voting strength, and that they were significant, raises a lot of questions as to why it was” approved, he said.

Maybe questions for you, Mr. Posner. But, in this Administration, it would only have been surprising were this plan to have been rejected. In fact, I don’t know why we haven’t seen more fake news stories to the effect that minority voters in Texas are actually happier under the new plan.