Editing Supermultiplet (section)

== Extended supersymmetry (N > 1) ==
This section records some commonly used irreducible supermultiplets in extended supersymmetry in the <math>d = 4</math> case. These are constructed by a [[highest-weight representation]] construction in the sense that there is a vacuum vector annihilated by the supercharges <math>Q^A, A = 1, \cdots, \mathcal{N}</math>. The irreps have dimension <math>2^\mathcal{N}</math>. For supermultiplets representing massless particles, on physical grounds the maximum allowed <math>\mathcal{N}</math> is <math>\mathcal{N} = 8</math>, while for [[renormalization|renormalizability]], the maximum allowed <math>\mathcal{N}</math> is <math>\mathcal{N} = 4</math>.<ref name="kqs">{{cite arXiv |last1=Krippendorf |first1=Sven |last2=Quevedo |first2=Fernando |last3=Schlotterer |first3=Oliver |title=Cambridge Lectures on Supersymmetry and Extra Dimensions |date=5 November 2010|class=hep-th |eprint=1011.1491 }}</ref>

=== N = 2 ===
The <math>\mathcal{N} = 2</math> '''vector''' or '''chiral multiplet''' <math>\Psi</math> contains a [[gauge field]] <math>A_\mu</math>, two [[Weyl fermion]]s <math>\lambda, \psi</math>, and a scalar <math>\phi</math> (which also transform in the [[adjoint representation]] of a [[gauge group]]). These can also be organised into a pair of <math>\mathcal{N} = 1</math> multiplets, an <math>\mathcal{N} = 1</math> vector multiplet <math>W = (A_\mu, \lambda)</math> and chiral multiplet <math>\Phi = (\phi, \psi)</math>. Such a multiplet can be used to define [[Seiberg–Witten theory]] concisely.

The <math>\mathcal{N} = 2</math> '''hypermultiplet''' or '''scalar multiplet''' consists of two Weyl fermions and two complex scalars, or two <math>\mathcal{N} = 1</math> chiral multiplets.

=== N = 4 ===
The <math>\mathcal{N} = 4</math> '''vector multiplet''' contains one gauge field, four Weyl fermions, six scalars, and [[CPT symmetry|CPT]] conjugates. This appears in [[N = 4 supersymmetric Yang–Mills theory]].