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Low-functioning pinhead...
November 26, 2002
Kick it up a Notch
As I mentioned previously, Paul Krugman is worried about the rise of the New American Plutocracy. If you wait long enough, the Plutocrats have kids, which poses some interesting inter-generational questions about upward mobility in our society.
There’s a lively (as always) discussion over at Brad DeLong’s blog, along with a reprint of Krugman’s op-ed piece.
And Then There Were Three
The latest field-theoretic proof, by Cachazo, Douglas, Seiberg and Witten, of the Dijkgraaf-Vafa conjectures appeared last week. It follows Dijgraaf, Grisaru, Lam, Vafa and Zanon and Ferrari (which I discussed previously and which is, strangely, uncited by Cachazo et al).
There are many points of similarity between these papers — the Konishi anomaly, for instance, figures prominently in any of these derivations — but, as they say, the devil is in the details. After all, the “naive” field-theoretic argument for the result (as sketched in DV’s original paper) is surely wrong. The argument basically says: since we are computing a holomorphic quantity, we can localize the functional integral onto the constant modes of the chiral fields, i.e. we end up with a matrix integral to do.
This argument is surely wrong because
- it is dimension-independent, whereas the result we are interested in is peculiar to 4 dimensions, as it relies on confinement, chiral symmetry breaking, the Konishi anomaly, etc.
- it does not explain why only the planar diagrams contribute (at finite N!).
So a real proof must grapple with the subtleties that a more naive argument leaves out. One of the refreshing things about the present paper is that some of the simplest questions one might ask are finally given a satisfying answer.
Consider an N=1 supersymmetric U(N) gauge theory with a chiral multiplet in the adjoint and a superpotential W(Φ). Generically, the gauge symmetry is broken by the expectation-value of Φ to Π U(Ni), with Σ Ni=N. The chiral fields of the “low-energy” effective action are the gluino condensates, Si= Tr λiλi, of the unbroken U(Ni) and wiα, the gauge field strengths for the U(1) center of U(Ni). But this description is not invariant under the original U(N) gauge symmetry, so it’s not obvious how to relate these fields to the underlying microscopic description.
Cachazo et al find the correct formula in terms of the microscopic fields, and show that these operators obey some Ward identities which follow from a generalized version of the Konishi anomaly (generalizing rescalings, Φ → εΦ, to arbitrary holomorphic reparametrizations, Φ → f(Φ)).
These Ward identities look like nothing else but the Virasoro constraints of the Matrix Model, and, of course, this determines the form of the solution.
The method generalizes quite readily to theories with a “diagonal” U(1) gauge symmetry, under which all the fields are neutral (here, because we only have matter in the adjoint). Such theories have a “hidden”, nonlinearly realized N=2 supersymmetry, under which the photino transforms inhomogeneously. Under this symmetry, the low energy chiral fields transform as
δSi=εα wiα
δwiα=Niεα
which, in turn, is a powerful constraint on the form of the low energy effective action.
November 24, 2002
Remote Root Exploit in Samba
A remotely exploitable buffer overflow has been found in Samba. The version shipping with Jaguar, Samba-2.2.3, is vulnerable. You should either install the latest version or turn off Windows FileSharing (if you had it enabled) until Apple comes out with a Security Update.
Judging by their turn-around time with BIND, that should be in about two weeks.
Hu’s on First
According to Jim Sherman, Abbot and Costello are in charge of the White House. You read it here first.
Plus Ça Change …
We’re leaving a performance of La Traviata, and the psychologist turns to me and says,
“Sorry, that opera was too much like my work.”
“ Really?” I ask, “Do you have a lot of clients dying of tuberculosis?”
“No,” she responds, “AIDS, Hepatitis-C, and cancer.”
“Too much drama, too much anguish, and way too much whining.” she continues.
“But they whine so much more melodically in the opera.” is all I can respond.