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Low-functioning pinhead...
October 30, 2005
Boo!
Pumkpin carving, chéz nous, involves a certain amount of thought and planning. The results are often striking, even when the real world disappoints.
But then there are the times when events intervene, derailing the best-laid plans, …
October 28, 2005
Patience is a Virtue
Gawd, how I’ve been waiting for this.
Waiting as Randy “Duke” Cunningham sailed off into the sunset. Waiting, as Jack Abramoff and David Safavian were indicted and the latter pled guilty in that ever-widening scandal. Waiting, as Michael Brown resigned in disgrace — another little bit of Bush cronyism, gone badly awry. Waiting, as Bill Frist is investigated by the SEC for insider trading.
Finally, Tom Delay was indicted for money laundering and funnelling illegal corporate contributions to Texas State political campaigns (yes, pretty minor stuff, on the scale of his machinations but, then, they nabbed Al Capone for tax-evasion).
And now … this.
Scooter Libby indicted on 5 counts of Obstruction of Justice and Perjury, in connection with the Valerie Plame investigation. The Prosecutor has made it clear that this is but the first round. His investigation of Karl Rove (and, perhaps others) continues…
October 26, 2005
CDO and Pure Spinors
One of the drawback of having other things to do besides blogging is the long list of half-completed posts sitting on my computer. One of them was adiscussion of the work of Witten and Kapustin on chiral differential operators. The basic idea is to recast the nontrivial aspects of 2D (supersymmetric) nonlinear -models. One works with an open covering of the target space. On each patch, one can replace the nontrivial -model action by a free theory, with a first-order action
On patch overlaps, one has nontrivial transition functions for the algebra of observables. All of the nontrivial aspects (the -function, anomalies, etc.) of the -model are encoded in how the local observables patch together. The usual global observables (whatever they are) are obtained as global sections of the sheaf of local observables.
I have written some observations on Berkovits’s pure-spinor approach to the superstring. In that approach, the (bosonic) ghosts, , (which contribute to the total central charge) are a Majorana-Weyl spinor of (, if you’re not too particular about reality conditions) satisfying the constraint The ghosts, , and the anti-ghosts, , have a free-field action. But, because of the constraints, rather than really being free fields, we have a nontrivial -model.
Berkovits and Nekrasov realize the space of (Euclidean) pure spinors as a cone over . Since the action is already of the “free”, first-order form, (1) one can adopt the same CDO technique to study it. Urs has posted a nice summary of a talk by Nikita on the subject.
Of course, even adopting this CDO approach, there’s a big difference with the examples studied by Kapustin and Witten. Here, the cone is not a smooth manifold. I’m not sure how that impacts the analysis. But, evidently, there’s some progress in understanding the composite operators which are the analogues of the -antighosts, in the bosonic string, as Čech cohomology classes. This, as you’ll recall, was the point on which my understanding of the pure-spinor approach fell apart.
October 21, 2005
Shih on OSV
David Shih visited with us this week, which gives me an excuse to talk about the checks he and collaborators have been able to carry out on the Ooguri-Strominger-Vafa conjecture. I’ve written about the work of Dabholkar et al, which focusses on “small” BPS blackholes which, in their examples, have heterotic duals, the Dabholhar-Harvey states (whose degeneracies are, therefore, exactly-known). In the end, that wasn’t a very satisfactory test, because — even in the large-charge limit — the volumes of some cycles on turned out to be small. There wasn’t an a-priori reason to expect agreement with OSV.
Shih, Strominger and Yin computed the degeneracy of 1/4 BPS blackholes in string theory (IIA compactified on