Supersymmetric gauge theory

A supersymmetric gauge theory combines vector superfields with charged matter superfields so that gauge invariance and supersymmetry are both manifest.

Minimal ten-dimensional supersymmetric gauge theory contains a gauge field and an adjoint Majorana-Weyl spinor. They have eight propagating bosonic and eight propagating fermionic states per gauge generator and sixteen real supercharges. A toroidal zero-mode dimensional reduction to four dimensions gives one gauge field, six real adjoint scalar fields, and four adjoint Weyl spinors: a four-dimensional N=4 super Yang-Mills theory. The internal field strength $F_{mn}=-ig[X_m,X_n]$ gives the nonnegative scalar potential $V=\frac{g^2}{4}{\sum }_{m,n}\Vert [X_m,X_n]{\Vert }^2$, where the norm uses a positive invariant inner product on the Lie algebra. Its minima are commuting tuples of adjoint scalar fields.

In four-dimensional $\mathcal{N}=2$ supersymmetric gauge theory, Seiberg–Witten theory determines exact low-energy quantities from the geometry of the vacuum parameter space. For rank one, the central charge in supersymmetry is proportional to $n_{e}a+n_m{a}_D$, with integral electric and magnetic charges and $a_D$ the dual scalar period. The BPS bound in supersymmetry fixes the mass of a short state by this charge combination. Its vanishing identifies loci where a magnetic monopole or dyon can become massless, even when the original electric description is strongly coupled.

At a vacuum obeying F-flatness and D-flatness, charged scalar field expectation values can break an internal gauge symmetry while preserving supersymmetry. For each broken generator, a massless vector multiplet combines with a chiral superfield into a massive N=1 vector multiplet. The vector absorbs one real Goldstone boson; a real scalar remains, and the gaugino mixes with a higgsino. All four bosonic and four fermionic on-shell states have a common mass.

A four-dimensional $\mathcal{N}=4$ supersymmetric gauge theory has one supersymmetric vector multiplet containing a gauge boson, four Weyl spinors, and six real scalar fields in the adjoint group representation. Its massless helicity multiplicities are $1,4,6,4,1$ from helicity $1$ to $-1$.

The scalar potential is a positive sum of squared commutators of the six adjoint scalar fields. It vanishes precisely when they commute. For a compact gauge group, simultaneous conjugation puts a commuting tuple in a Cartan subalgebra. Its arbitrary $6r$ real coordinates are classical flat directions of a scalar potential, with residual identification by the Weyl group. For $SU(N)$ the simultaneous eigenvalue vectors also obey the tracelessness constraint. Generic values break the gauge group to its maximal torus.

An R-symmetry acts nontrivially on the supersymmetry generators. In four-dimensional N=1 supersymmetry, the superspace coordinate $\theta$ has R-charge one, so a superpotential must have R-charge two.

R-charge is the charge under a continuous R-symmetry. In four-dimensional N=1 supersymmetry, the fermion in a chiral multiplet has R-charge one less than its scalar, and every superpotential term has R-charge two.

A scalar chiral primary operator at a four-dimensional N=1 superconformal fixed point saturates a shortening bound and obeys $\Delta =3R/2$.

In two-dimensional N=(2,2) supersymmetry, vector R-symmetry rotates ${\theta }^{+}$ and ${\theta }^{-}$ with the same phase. A chiral superspace measure has vector R-charge minus two.

In two-dimensional N=(2,2) supersymmetry, axial R-symmetry rotates ${\theta }^{+}$ and ${\theta }^{-}$ with opposite phases. A twisted chiral superspace measure has axial R-charge minus two.

A vacuum of a globally supersymmetric gauge theory preserves supersymmetry exactly when every auxiliary F-field and D-field can vanish simultaneously. Its vacuum energy is then zero because the scalar potential is a sum of nonnegative squares.

F-flatness is the system $\partial W/\partial {\varphi }_i=0$ obtained by setting every chiral-multiplet auxiliary field to zero.

D-flatness sets every vector-multiplet auxiliary field to zero. Quotienting its solution set by the compact gauge group is equivalent, under standard stability conditions, to quotienting the F-flat variety by the complexified gauge group.

The vacuum moduli space is the space of simultaneous F-flat and D-flat configurations modulo gauge transformations. Gauge-invariant chiral operators often provide holomorphic coordinates, subject to algebraic constraints.

For canonical chiral superfields, nonzero $\lambda$, a gauged $U(1)$ with charges $(0,+1,-1)$, and no Fayet–Iliopoulos term, F-flatness requires the charged product to vanish and D-flatness requires equal charged magnitudes. Together they force both charged scalar values to zero, leaving the neutral scalar arbitrary. The vacuum moduli space of a supersymmetric gauge theory has zero energy, unbroken gauge symmetry and unbroken supersymmetry. At $\lambda =0$, equal nonzero charged magnitudes instead permit a gauge-Higgsed supersymmetric branch. A neutral modulus alone does not imply internal gauge breaking.

A Higgs branch is a component on which charged scalar expectation values break some or all of the gauge group. It is often obtained as a Kähler or hyperkähler quotient.

A Coulomb branch is a component on which scalar expectation values in vector or neutral multiplets leave an Abelian gauge sector unbroken. Charged fields can become massless at special points where additional branches meet it.

Supersymmetric quantum electrodynamics is an Abelian supersymmetric gauge theory with charged chiral multiplets. Its scalar potential combines F-term constraints with the U(1) D-term and any Fayet–Iliopoulos term.

Supersymmetric quantum chromodynamics is an N=1 supersymmetric $SU(N_c)$ gauge theory with $N_f$ fundamental and $N_f$ antifundamental chiral multiplets.

The supersymmetric conformal window is the range of flavor numbers for which an asymptotically free supersymmetric gauge theory flows to an interacting infrared fixed point. Its lower edge is often detected when a gauge-invariant chiral operator reaches the scalar unitarity bound.

The chiral ring is generated by gauge-invariant chiral operators modulo relations that vanish by F-term equations. In SQCD its basic generators are meson and baryon operators.

An SQCD meson is the gauge-invariant chiral bilinear $M^i{}_j={\tilde{Q}}_j{Q}^i$. It transforms under both flavor groups and parametrizes mesonic directions of the vacuum moduli space.

An SQCD baryon contracts $N_c$ fundamental fields with the gauge epsilon tensor; an antibaryon similarly contracts antifundamentals. Such operators exist when enough flavors are available and obey relations with the mesons.

The holomorphic strong-coupling scale packages the gauge coupling and theta angle into ${\Lambda }^{b_0}$. Treating it as a spurion under anomalous chiral symmetries strongly constrains nonperturbative superpotentials and moduli-space deformations.

For SQCD with $N_f<N_c$, strong dynamics generates the Affleck–Dine–Seiberg superpotential. For $SU(2)$ with one flavor it is proportional to ${\Lambda }^5/M$ and produces a runaway to infinite meson expectation value.

When $N_f=N_c$, SQCD has no generated superpotential but its classical constraint is shifted by the strong-coupling scale. For $SU(2)$ with two flavors, one convention gives $\det M-B\tilde{B}={\Lambda }^4$.

A theory s-confines when its infrared physics is described smoothly everywhere on moduli space by gauge-invariant composites with a local superpotential and no remaining gauge group. SQCD with $N_f=N_c+1$ is the standard example.

Seiberg duality states that two different N=1 supersymmetric gauge theories flow to the same infrared quantum field theory. Gauge-invariant operators, global symmetries, 't Hooft anomalies and deformations match even though the ultraviolet gauge groups and elementary fields differ.

With the index convention $I(\mathbf{N})=1$ for an $SU(N)$ fundamental, an N=1 supersymmetric gauge theory has $b_0=\frac{3}{2}I(\mathrm{adj})-\frac{1}{2}{\sum }_{i}I(R_i)$. Positive $b_0$ gives asymptotic freedom, while negative $b_0$ makes the gauge interaction infrared free near the Gaussian fixed point.