Four-Dimensional Supersymmetry

Four-dimensional supersymmetry has its own spinor notation and its own elementary multiplets. This volume develops them directly from ordinary relativistic field theory. Its shared preparation is the meaning of a supercharge and Grassmann calculus; the three-dimensional and curved-space courses are separate routes.

The aim is to turn an algebra into a model whose field content, action, and vacua you can check. A later exact result is much easier to interpret once those ingredients are explicit.

Begin with Weyl spinors

Weyl spinors and the N=1 supersymmetry algebra defines dotted and undotted indices, the sigma matrices, and the algebra in a declared Lorentzian convention. Work through the matrix identity used in closure before relying on compressed index notation.

The label $\mathcal N=1$ means four real Poincaré supercharges in four dimensions. Counting the two complex entries of a Weyl charge and then counting its adjoint as another independent charge would double the answer incorrectly. Euclidean continuation changes the reality relations used in component calculations; the lesson explains that step separately.

Your first checkpoint is to verify a sigma-matrix product, evaluate the supercharge anticommutator on a momentum eigenstate, and explain the real-supercharge count.

Build fields into multiplets

Four-dimensional chiral and vector multiplets organizes a scalar and Weyl fermion into a chiral multiplet, and a gauge field and gaugino into a vector multiplet. A gaugino is the spinor partner of the gauge field. Auxiliary fields permit the transformations to close before imposing matter equations; they are not additional propagating particle states. The multiplet comparison tracks the same fields through different supersymmetry descriptions and keeps dimension, gauge representation, and auxiliary fields distinct.

The free action provides a concrete test: vary its kinetic and auxiliary terms and identify the total derivative. Matching field counts is useful, but it does not fix all the signs in that cancellation. Afterward, superspace packages those component expressions and makes interactions easier to construct.

From interactions to vacua

Superspace, interactions, and supersymmetric vacua expands the superpotential and gauge-matter integrals into their component terms. Derive the auxiliary equations, substitute them into the scalar potential, and solve the Wess–Zumino and charged-pair examples. A supersymmetric vacuum requires the relevant fermion variations to vanish. A stationary point of the potential is a weaker condition, and a stability calculation is another distinct test.

Does this vacuum preserve supersymmetry? finds the actual minimum of the charged model after an FI deformation, checks its physical fluctuations, and contrasts it with a runaway. This separates an unstable stationary point from a stable state before any quantum lifetime is discussed.

An O’Raifeartaigh model and its Goldstino proves a classical minimum bound, identifies the massless fermion associated with broken supersymmetry and computes the one-loop mass that lifts its scalar partner. A finite positive integral makes the local quantum stability test explicit.

Extended algebras, R-symmetry, and BPS representations diagonalizes the positive supercharge anticommutator, derives the particle mass bound, and counts the states that survive at saturation. Keep the fixed-charge representation distinct from its antiparticle completion. N=2 vectors and hypermultiplets then constructs the interacting field packages and derives a charged hypermultiplet’s mass. These preparations support the independent Seiberg–Witten and $\mathcal N=4$ routes without requiring the curved three-manifold construction.

Choose the next physical question

For four-dimensional N=1 quantum dynamics, begin by removing a heavy chiral field and comparing holomorphic and canonically normalized couplings. This route needs the N=1 action lesson, without the extended-algebra branch. Seiberg–Witten theory begins with the charged threshold of a pure N=2 theory and its remaining low-energy fields. For $\mathcal N=4$ Yang–Mills, begin by specifying the interacting adjoint fields and the supersymmetry made manifest by the chosen notation.

The four-dimensional new minimal rigid background constructs an explicit sphere-circle example, translating the spinor conventions and imposing global spin and source holonomies. Twisted dimensional reduction follows the metric, sources and multiplet fields into three dimensions, checking a nontrivial circle fibration. It has extra geometric prerequisites and is an optional return, not an entrance requirement for the theory paths.

Use the learning-path browser to select the preparation for your destination. At each step, identify whether you have established an algebra, an invariant action, a vacuum, or a quantum observable; one does not stand in for the others.