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March 11, 2004

A-Maximization

I haven’t talked about the aa-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=1N=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, 𝒪\mathcal{O}, which saturate the bound

(1)Δ(𝒪32|R(𝒪)| \Delta(\mathcal{O} \geq \textstyle{\frac{3}{2}} |R(\mathcal{O})|

where RR is the charge under the U(1) RSU(2,2|1)U(1)_R\in SU(2,2|1) superconformal symmetry. The difficult part is simply identifying which U(1) RU(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)U(1) symmetries, and the desired R-charge is some linear combination

(2)R=R 0+ ic iQ i R=R_0+\sum_i c_i Q_i

of a valid U(1)U(1) R-charge, and the other global U(1)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)U(1) symmetries. For instance, if some chiral field XX 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) XU(1)_X symmetry, and the “true” R-charge of the SCFT may contain some admixture of Q XQ_X.

In a conformal field theory, the β\beta-function vanishes, and the trace anomaly in a curved background is given by T μ μ =1120(4π) 2(cW 2a4e) \tensor{T}{_^\mu_\mu} = \frac{1}{120 (4\pi)^2} (c W^2 -\frac{a}{4} e) where WW is the Weyl tensor,

(3)W 2=R μνρσR μνρσ2R μνR μν+13R 2 W^2 = R_{\mu\nu\rho\sigma} R^{\mu\nu\rho\sigma}- 2R_{\mu\nu} R^{\mu\nu} + \textstyle{\frac{1}{3}}R^2

and ee is the Euler density,

(4)e=4R μνρσR μνρσ16R μνR μν+4R 2 e= 4R_{\mu\nu\rho\sigma} R^{\mu\nu\rho\sigma}- 16R_{\mu\nu} R^{\mu\nu} + 4R^2

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

(5)a=332(3TrR 3TrR),c=132(9TrR 35TrR) a = \frac{3}{32} (3 Tr R^3 - Tr R),\qquad c = \frac{1}{32} (9 Tr R^3 - 5Tr R)

Cardy conjectured that aa decreases along RG flows, a IR<a UVa_{\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 RR could be determined by maximizing aa,

(6)ac i=0, 2ac ic j<0 \frac{\partial a}{\partial c_i} =0,\qquad \frac{\partial^2 a}{\partial c_i \partial c_j} \lt 0

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

Csaki et al study SU(N)SU(N) gauge theory with a 2-index antisymmetric tensor, FF fundamentals, and N+F4N+F-4 anti-fundamentals, as a function of x=N/Fx=N/F. Starting in the large-N,FN,F limit, the theory has a Banks-Zaks fixed point near x.5x\sim .5. As one increases xx, the theory remains in a nonabelian Coulomb (SCFT) phase. At some critical value of xx, the meson M=Q¯QM=\overline{Q}Q becomes free and decouples. At a yet-higher value of xx, H=Q¯AQ¯H=\overline{Q} A \overline{Q} become free and decouples. When HH decouples, the electric description ceases to be effective. For F5F\geq 5, one can use a series of Seiberg dualities to rewrite the theory as an SU(F3)×Sp(2F8)SU(F-3)\times Sp(2F-8) magnetic gauge theory with a superpotential. The Sp(2F8)Sp(2F-8) is IR-free, whereas the SU(F3)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.

Posted by distler at March 11, 2004 2:58 AM

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