March 11, 2003

Some Physics Notes

David Berenstein has a followup to his previous paper discussing the application of Dijkgraaf-Vafa techniques to $\mathcal{N}=1$ supersymmetric gauge theories with moduli spaces of vacua.

Most of the applications to date have been to theories with isolated vacua. Understanding the vacuum structure of such theories is interesting but it is in some sense more challenging to study theories with continuous moduli spaces of vacua. In particular, it’s interesting to understand the quantum modifications to the geometry (indeed, to the topology) of the “classical” moduli space.

David studies deformations of $\mathcal{N}=4$ $U(N)$ SYM by a superpotential of the form

(1)$W= \tr [ X Y Z - q X Z Y + V(X) ]$

where

(2)$q^n=1,\,\,\, V(X) = \textstyle{\frac{a_{n+1}}{n+1}}X^n+ \sum_{k=1}^{n-1} \textstyle{\frac{a_k}{k}} X^k,\,\,\, n\leq N$

($q=1$ and $a_k=0$ is $\mathcal{N}=4$ SYM.)

In the corresponding matrix model, the loop equations degenerate, and David actually turns matters around to use information about the quantum-corrected moduli space of the gauge theory to provide a conjectured solution to the matrix model.

Strominger and Thompson write about “quantum violations” of the Bousso entropy bound. At first blush, this is a rather surprising subject, as the statement, by Hawking and Bekenstein, that a black hole has a temperature and an entropy, is an intrinsically quantum-mechanical one. One would have thought that Bousso’s bound, like Bekenstein’s , already incorporated the effects of quantum mechanics.

Well, in the black hole case, there is the assumption that the black hole has a well-defined mass (and hence a well-defined temperature), which is true only in the semiclassical regime, i.e. for large-enough black holes. Moreover, even semiclassically, one must count the entropy carried by the outgoing Hawking radiation in the “generalized second law of thermodynamics”

(3)$\textstyle{\frac{1}{4}} \Delta A_H + \Delta S_{\text{hr}} \geq \Delta S_M$

where $A_H$ is the area of the event horizon, $S_{\text{hr}}$ is the entropy of the outgoing Hawking radiation and $\Delta S_M$ is the entropy carried by the matter falling into the black hole.

The “classical” Bousso bound (as proven by Flanagan, Marolf and Wald) is the generalization of the above inequality without the second term on the left hand side. Strominger and Thompson make a proposal for this missing term.

Finally, Basu and Sethi have a paper on (0,2) gauged linear $\sigma$-models (a subject dear to my heart). They argue that worldsheet instantons do not generate a (worldsheet) superpotential in such theories. I’ve only ever looked the conformally-invariant case (i.e., where the GL$\sigma$M flows to a (0,2) SCFT in the infrared) where — in all the examples I’ve ever looked at — this is perfectly obvious. They’re more interested in “massive” models where, perhaps, there’s something to prove. Though, I have to say, since the theory is strongly-coupled in the IR, I don’t see the relevance of the dilute instanton gas approximation. Moreover, at least in the (2,2) case, I’m fairly sure there is the analogue of a Veneziano-Yankielowicz superpotential generated. In the (0,2) case, there isn’t enough supersymmetry to protect the result, but I doubt that you get “zero”.

Anyway, they then go on to try to adduce, from the absence of an induced worldsheet superpotential, evidence for the nonperturbative nonrenormalization of the Fayet-Iliopoulos parameter, $r$ (i.e., in the string theory context, for the absence of a spacetime superpotential). Sigh! I tried to warn them …

Posted by distler at March 11, 2003 4:11 PM

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