Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 307 (Supersymmetry)

Treating derivatives with respect to the Grassmann variables as graded derivatives and using $\{{\partial }_{\alpha },{\theta }^{\beta }\}={\delta }_{\alpha }{}^{\beta }$ givesThe terms in $\{{\mathcal{D}}_{\alpha },{\mathcal{D}}_{\beta }\}$ and $\{{\overline{\mathcal{D}}}_{\dot{\alpha }},{\overline{\mathcal{D}}}_{\dot{\beta }}\}$ cancel pairwise, so both vanish.

A chiral superfield obeys ${\overline{\mathcal{D}}}_{\dot{\alpha }}\Phi =0$, while an antichiral superfield obeys ${\mathcal{D}}_{\alpha }{\Phi }^{†}=0$. Since ${\overline{\mathcal{D}}}_{\dot{\alpha }}{y}^{\mu }=0$ for $y^{\mu }=x^{\mu }+i\theta {\sigma }^{\mu }\bar{\theta }$, the general chiral expansion isIn the original coordinates this iswith signs following the conventions of the question.

A holomorphic function $W(\Phi )$ is chiral. Its highest component transforms into a spacetime divergence, so the F-term actionis supersymmetric even though it integrates over only chiral half of superspace.

The non-renormalization theorem says that perturbative loop corrections are full-superspace D-terms and cannot generate a new local superpotential. Holomorphy and spurion symmetries therefore preserveThe Kähler potential is renormalized, however. If its kinetic term is $Z{\Phi }^{†}\Phi$, canonical normalization ${\Phi }_c=Z^{1/2}\Phi$ givesup to scheme and scale conventions. Thus superpotential parameters are holomorphic invariants while physical masses and couplings still run through wave-function renormalization.

Under the Abelian supergauge transformation $V\mapsto V+i(\Omega -{\Omega }^{†})$, the real and imaginary components of $\omega$, together with $\psi$ and $F$, shift $C,\chi$ and $M$. They can be chosen to setleaving Wess-Zumino gaugeThe remaining imaginary scalar gauge parameter acts as the ordinary transformation $A_{\mu }\mapsto A_{\mu }+{\partial }_{\mu }\alpha$.

The chiral field-strength superfieldis chiral and gauge invariant in the Abelian theory. Henceis supersymmetric. Its components are the Maxwell kinetic term, the gaugino kinetic term, and the auxiliary term $D^2/(2e^2)$.

For the component check, vary $F_{\mu \nu }$ using $\delta A_{\mu }=ϵ{\sigma }_{\mu }\bar{\lambda }+\lambda {\sigma }_{\mu }\overline{ϵ}$ and vary the gaugino term using $\delta \lambda =F_{\rho \sigma }{\sigma }^{\rho \sigma }ϵ$ and its conjugate. After integration by parts, the terms proportional to ${\partial }_{\mu }{F}^{\mu \nu }$ cancel between the two variations. The remaining terms reduce, by the stated sigma-matrix identity, to a contraction of ${\partial }_{[\mu }{F}_{\nu \rho ]}$, which vanishes by the Bianchi identity. Thus $\delta \mathcal{L}$ is a total divergence and the action is invariant. The auxiliary field would make this supersymmetry close off shell; setting $D=0$ gives the displayed on-shell transformations.

A gauge anomaly breaks a redundancy needed to remove unphysical states and makes a quantum gauge theory inconsistent unless it cancels. A chiral anomaly is instead the quantum violation of a classical global axial current; the theory remains consistent, but processes such as ${\pi }^0\to \gamma \gamma$ and instanton-induced charge violation become possible. A 't Hooft anomaly is an obstruction to gauging a global symmetry. It is compatible with a consistent theory but is invariant under renormalization-group flow, so any infrared phase must reproduce it through massless fields, spontaneous symmetry breaking, or topological degrees of freedom.

Normalize the cubic anomaly coefficient of the $SU(N)$ fundamental to $+1$. The two-index antisymmetric representation has anomaly coefficient $N-4$, while an antifundamental contributes $-1$. Cancellation of the $SU(N{)}^3$ gauge anomaly requires

Classically the $p$ identical fields ${\psi }^i$ have $U(p)=SU(p)\times U(1{)}_{\psi }$, and $\lambda$ has an independent $U(1{)}_{\lambda }$. One linear combination has an $SU(N{)}^{2}U(1)$ anomaly. For the orthogonal combination chooseIndeed,The continuous quantum global symmetry is thereforeup to discrete identifications. The anomalous orthogonal axial rotation does not survive as a continuous quantum symmetry.

Only the $\psi$ fields transform under flavor $SU(p)$; their gauge-color index gives $N$ copies of its fundamental. With $p=N-4$, the ultraviolet 't Hooft anomalies areThe last two expressions simplify torespectively.

The composite ${\chi }^{ij}$ is in the two-index symmetric representation of $SU(p)$ and hasFor that representation,Its anomalies are consequentlyin the same order as part iii. Every anomaly matches, so confinement to the proposed massless composite is consistent with 't Hooft anomaly matching.

The fermion in a chiral multiplet has R-charge $R[\Phi ]-1$, while the gluino has charge one. Cancellation of the $Sp(N_c{)}^{2}U(1{)}_R$ anomaly givesUsing $I(\mathrm{adj})=2(N_c+1)$ and $I(\square )=1$ yields

The one-loop coefficient isThus asymptotic freedom is lost atand the theory is infrared free for larger $N_f$.

In a four-dimensional supersymmetric conformal field theory, a chiral primary operator in four-dimensional N=1 supersymmetry obeys $\Delta =3R/2$. The meson has R-charge $R[M]=2R[\Phi ]$, henceAt $N_f=3(N_c+1)$ this gives

A free scalar has scaling dimension one. Setting the meson dimension to one givesThe expected interacting supersymmetric conformal window is therefore

The magnetic gauge group is $Sp({\tilde{N}}_c)$ with ${\tilde{N}}_c=N_f-N_c-2$. Repeating the gauge-anomaly cancellation gives

The magnetic superpotential $W\sim qMq$ must have R-charge two, soThereforeexactly matching the R-charge and dimension of the electric meson ${\Phi }^i{\Phi }^j$.

For the magnetic theory,It ceases to be asymptotically free whenThis is precisely the lower edge of the electric conformal window. Inside the window both descriptions flow to the same interacting infrared fixed point; below it the magnetic variables provide a weakly coupled infrared description.

The electric quarks give $2N_c$ copies of the $SU(2N_f)$ fundamental, soIn the magnetic theory the $2{\tilde{N}}_c$ color components of $q$ transform as flavor antifundamentals and contribute $-2{\tilde{N}}_c$. The singlet $M^{ij}$ is the two-index antisymmetric representation of $SU(2N_f)$, whose cubic anomaly coefficient is $2N_f-4$. HenceThis is a direct anomaly check of Seiberg duality.