Musings

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(N_i), with Σ N_i=N. The chiral fields of the “low-energy” effective action are the gluino condensates, S_i= Tr λ_iλ_i, of the unbroken U(N_i) and w_iα, the gauge field strengths for the U(1) center of U(N_i). 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

δS_i=ε^α w_iα
δw_iα=N_iε_α

which, in turn, is a powerful constraint on the form of the low energy effective action.