N = 4 supersymmetric Yang–Mills theory

N = 4 supersymmetric Yang–Mills (SYM) theory is a relativistic conformally invariant Lagrangian gauge theory describing fermions interacting via gauge field exchanges. In D=4 spacetime dimensions, N=4 is the maximal number of supersymmetries or supersymmetry charges.

It is a toy theory based on Yang–Mills theory that does not model the real world, but is useful because it can act as a proving ground for approaches for attacking problems in more complex theories. It describes a universe containing boson fields and fermion fields which are related by four supersymmetries (this means that transforming bosonic and fermionic fields in a certain way leaves the theory invariant). It is one of the simplest (in the sense that it has no free parameters except for the gauge group) and one of the few ultraviolet finite quantum field theories in 4 dimensions. It can be thought of as the most symmetric field theory that does not involve gravity.

Like all supersymmetric field theories, it may equivalently be formulated as a superfield theory on an extended superspace in which the spacetime variables are augmented by a number of anticommuting Grassmann variables which, for the case N=4, consist of 4 Dirac spinors, making a total of 16 independent anticommuting generators for the extended ring of superfunctions. The field equations are equivalent to the geometric condition that the supercurvature 2-form vanish identically on all super null lines. This is also known as the super-ambitwistor correspondence.

A similar super-ambitwistor characterization holds for D=10, N=1 dimensional super Yang–Mills theory, and the lower dimensional cases D=6, N=2 and D=4, N=4 may be derived from this via dimensional reduction.