Musings

A-Maximization

I haven’t talked about the $a$-maximization proposal of Intriligator and Wecht, nor the interesting followup papers (I, II) by Kutasov and collaborators. But the recent paper by Csaki et al reminded me.

We know know that there is a wealth of interacting 4D $N=1$ superconformal field theories arising as the strongly-coupled fixed point of supersymmetric gauge theories with various matter content. We can’t say much about the physics of such theories, but one thing we ought to be able to calculate is the spectrum of chiral primaries in the theory, superconformal primary fields, $\mathcal{O}$, which saturate the bound

where $R$ is the charge under the $U(1)_R\in SU(2,2|1)$ superconformal symmetry. The difficult part is simply identifying which $U(1)_R$ symmetry of the microscopic theory becomes the R-charge of the superconformal algebra in the IR. In general, there can be a number of nonanomalous global $U(1)$ symmetries, and the desired R-charge is some linear combination

of a valid $U(1)$ R-charge, and the other global $U(1)$ symmetries of the theory. In general, there might be a further complication that the IR fixed point might have additional, “accidental” $U(1)$ symmetries. For instance, if some chiral field $X$ becomes free, and decouples from the rest of the SCFT (more generally, if the IR SCFT breaks up into decoupled sectors), then there is an accidental $U(1)_X$ symmetry, and the “true” R-charge of the SCFT may contain some admixture of $Q_X$.

In a conformal field theory, the $\beta$-function vanishes, and the trace anomaly in a curved background is given by $$\tensor{T}{_^\mu_\mu} = \frac{1}{120 (4\pi)^2} (c W^2 -\frac{a}{4} e)$$ where $W$ is the Weyl tensor,

and $e$ is the Euler density,

The trace-anomaly coefficients, $a,c$, are given by 't Hooft anomaly matching

Cardy conjectured that $a$ decreases along RG flows, $a_{\text{IR}}\lt a_{\text{UV}}$, and is non-negative in unitary four dimensional conformal field theories.

What Intriligator and Wecht showed was that the correct choice of $R$ could be determined by maximizing $a$,

Heuristically, this “explains” why $a_{\text{IR}}\lt a_{\text{UV}}$. A relevant perturbation typically breaks some of the global symmetries and so $a_{\text{IR}}$ is obtained by maximizing only within a subspace of the original parameter space in which one maximized $a_{\text{UV}}$. In any case, $a$-maximization allows one to determine $R$, and hence the spectrum of conformal weights of the chiral primaries.

Csaki et al study $SU(N)$ gauge theory with a 2-index antisymmetric tensor, $F$ fundamentals, and $N+F-4$ anti-fundamentals, as a function of $x=N/F$. Starting in the large-$N,F$ limit, the theory has a Banks-Zaks fixed point near $x\sim .5$. As one increases $x$, the theory remains in a nonabelian Coulomb (SCFT) phase. At some critical value of $x$, the meson $M=\overline{Q}Q$ becomes free and decouples. At a yet-higher value of $x$, $H=\overline{Q} A \overline{Q}$ become free and decouples. When $H$ decouples, the electric description ceases to be effective. For $F\geq 5$, one can use a series of Seiberg dualities to rewrite the theory as an $SU(F-3)\times Sp(2F-8)$ magnetic gauge theory with a superpotential. The $Sp(2F-8)$ is IR-free, whereas the $SU(F-3)$ is in a nonabelian Coulomb phase.

Quite an intricate story, really. And a real testament to how much progress we’ve made in understanding SUSY gauge theories in the past decade.