It consists of an arbitrary number of chiral and vector supermultiplets whose possible interactions are strongly constrained by supersymmetry, with the theory primarily fixed by three functions: the Kähler potential, the superpotential, and the gauge kinetic matrix.
The particle content of this theory must belong to representations of the super-Poincaré algebra, known as supermultiplets.
Supersymmetry imposes stringent conditions on the way that the supermultiplets can be combined in the theory.
When such mixing occurs, the gauge group must also be consistent with the structure of the chiral sector.
into right-handed Weyl spinors, so the geometry of the scalar manifold must reflect the fermion spacetime chirality by admitting an appropriate decomposition into complex coordinates.
The chirality properties inherited from supersymmetry imply that any closed loop around the scalar manifold has to maintain the splitting between
These manifolds have the useful property that their metric can be expressed in terms of a function known as a Kähler potential
More specifically, they leave the scalar metric as well as the complex structure unchanged.
is that there exists a set of real holomorphic functions known as Killing prepotentials
For abelian subalgebras of the gauge algebra, the Fayet–Iliopoulos terms remain unfixed since these have vanishing structure constants.
Meanwhile, the left-handed and right-handed Weyl fermion projections of the Majorana spinors are denoted by
The third line is the generalized supersymmetric theta-like term for the gauge multiplet, with this being a total derivative when the imaginary part of the gauge kinetic function is a constant, in which case it does not contribute to the equations of motion.
It is these terms that determine the masses of the fermions since in a particular vacuum state with scalar fields expanded around some value
The last line includes the scalar potential where the first term is called the F-term and the second is known as the D-term.
At the quantum level, supersymmetry is broken if the supercharges do not annihilate the vacuum
[6] Since the Hamiltonian can be written in terms of these supercharges, this implies that unbroken supersymmetry corresponds to vanishing vacuum energy, while broken supersymmetry necessarily requires positive vacuum energy.
Due to the superpotential nonrenormalization theorem, which states that the superpotential does not receive corrections at any level of quantum perturbation theory, the above condition holds at all orders of quantum perturbation theory.
Only non-perturbative quantum corrections can modify the condition for supersymmetry breaking.
Spontaneous symmetry breaking of global supersymmetry necessarily leads to the presence of a massless Nambu–Goldstone fermion, referred to as a goldstino
One important set of quantities are the supertraces of powers of the mass matrices
can be expressed as[2]: 297 showing that this vanishes in the case of a Ricci-flat scalar manifold, unless spontaneous symmetry breaking occurs through non-vanishing D-terms.
If instead there are no chiral multiplets, then the theory with a Euclidean gauge kinetic matrix
Super quantum chromodynamics is meanwhile acquired using a Euclidean scalar metric, together with an arbitrary number of chiral multiplets behaving as matter and a single gauge multiplet.
supersymmetry models with particular choices of multiplets, potentials, and kinetic terms.
supergravity and necessarily include additional structures that must be added to the theory.
In particular, 4D N = 1 supergravity has a matter content similar with the case of global supersymmetry except with the addition of a single gravity supermultiplet, consisting of a graviton and a gravitino.
The resulting action requires a number of modifications to account for the coupling to gravity, although structurally shares many similarities with the case of global supersymmetry.
The global supersymmetry model can be directly acquired from its supergravity generalization through the decoupling limit whereby the Planck mass is taken to infinity
[11] This is the minimal extension of the Standard Model that is consistent with phenomenology and includes supersymmetry that is broken at some high scale.
[12] In this approach, Minkowski spacetime is extended to an eight-dimensional supermanifold which additionally has four Grassmann coordinates.