Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 307 / 4

Two theories are Seiberg duals when they are distinct ultraviolet gauge theories that flow to the same infrared quantum field theory. Their gauge-invariant operator spectra, global symmetries, 't Hooft anomalies and responses to deformations agree, even though one description may be strongly coupled where the other is weakly coupled.

Quantum consistency requires cancellation of the $SU(N{)}^3$ gauge anomaly. The two-index symmetric square has cubic anomaly coefficient $A(S)=N+4$, whereas each antifundamental has coefficient $-1$. Therefore

The mixed $SU(N{)}^{2}U(1{)}_R$ anomaly receives $I(\mathrm{adj})=2N$ from the gaugino. A chiral multiplet of R-charge $r$ contains a fermion of charge $r-1$. Withthe symmetric-tensor fermion contributeswhile every antifundamental fermion contributes zero. The total is $2N-2N=0$, so this is a nonanomalous R-symmetry of the quantum theory.

Using the supplied Dynkin indices and $p=N+4$, the one-loop beta function of a supersymmetric gauge theory isThus $b_0>0$ for every nontrivial $SU(N)$ in this family, and the electric theory is asymptotically free.

In the proposed Spin(8) theory, the gaugino contributes $I(\mathrm{adj})=6$ to the mixed gauge-R anomaly. The $p$ fields $q$ have superfield R-charge one, so their fermions contribute zero. The spinor $t$ has fermion R-charge $-5-1=-6$ and index one, giving $-6$. The singlets $M$ and $U$ do not enter the gauge anomaly. Henceand the proposed R-symmetry survives quantization.

Gauge invariance, $SU(p)$ flavor symmetry and R-charge two permitwith the Spin(8) vector and spinor indices contracted by their invariant bilinear forms. Indeed, $R[Mqq]=0+1+1=2$ and $R[Utt]=12-5-5=2$. Powers of the matching scale can be inserted to give fields whichever canonical mass dimensions are chosen.

Only the $p$ Spin(8) vectors $q$ and the one spinor $t$ contribute to the magnetic gauge beta function. The supplied indices giveThe magnetic theory is infrared free when $b_0^{\mathrm{magnetic}}<0$, namelyAt $N=13$ the one-loop coefficient vanishes and higher-order dynamics decides the flow. For $N>13$, Seiberg duality says that the strongly coupled low-energy limit of the asymptotically free electric $SU(N)$ theory is described by the weakly coupled Spin(8) fields and the superpotential from part v.

Choose the flavor orientation in which each electric $\tilde{\Phi }$ is a fundamental of $SU(p)$. Its $N$ color components then giveIn the magnetic theory, the eight Spin(8) components of $q$ transform in the antifundamental and contribute $-8$. The singlet $M$ transforms in the symmetric representation of $SU(p)$, whose cubic anomaly coefficient is $p+4$, while $t$ and $U$ are flavor singlets. Thereforewhere $p=N+4$ was used. The two $SU(p{)}^3$ 't Hooft anomalies match, as required by Seiberg duality.