In theoretical physics, more specifically in quantum field theory and supersymmetry, supersymmetric Yang–Mills, also known as super Yang–Mills and abbreviated to SYM, is a supersymmetric generalization of Yang–Mills theory, which is a gauge theory that plays an important part in the mathematical formulation of forces in particle physics.
Super Yang–Mills was studied by Julius Wess and Bruno Zumino in which they demonstrated the supergauge-invariance of the theory and wrote down its action, alongside the action of the Wess–Zumino model, another early supersymmetric field theory.
The treatment in this article largely follows that of Figueroa-O'Farrill's lectures on supersymmetry and of Tong.
While N = 4 supersymmetric Yang–Mills theory is also a supersymmetric Yang–Mills theory, it has very different properties to supersymmetric Yang–Mills theory, which is the theory discussed in this article. The supersymmetric Yang–Mills theory was studied by Seiberg and Witten in Seiberg–Witten theory. All three theories are based in super Minkowski spaces.
The supersymmetric Yang–Mills action
Preliminary treatment
A first treatment can be done without defining superspace, instead defining the theory in terms of familiar fields in non-supersymmetric quantum field theory.
Spacetime and matter content
The base spacetime is flat spacetime (Minkowski space).
SYM is a gauge theory, and there is an associated gauge group to the theory. The gauge group has associated Lie algebra .
The field content then consists of
a -valued gauge field
a -valued Majorana spinor field (an adjoint-valued spinor), known as the 'gaugino' a -valued auxiliary scalar field .
For gauge-invariance, the gauge field is necessarily massless. This means its superpartner is also massless if supersymmetry is to hold. Therefore can be written in terms of two Weyl spinors which are conjugate to one another: , and the theory can be formulated in terms of the Weyl spinor field instead of .
Supersymmetric pure electromagnetic theory
When , the conceptual difficulties simplify somewhat, and this is in some sense the simplest gauge theory. The field content is simply a (co-)vector field , a Majorana spinor and a auxiliary real scalar field .
The field strength tensor is defined as usual as .
The Lagrangian written down by Wess and Zumino is then
This can be generalized to include a coupling constant , and theta term , where is the dual field strength tensor
and is the alternating tensor or totally antisymmetric tensor. If we also replace the field with the Weyl spinor , then a supersymmetric action can be written as
This can be viewed as a supersymmetric generalization of a pure gauge theory, also known as Maxwell theory or pure electromagnetic theory.
Supersymmetric Yang–Mills theory (preliminary treatment) In full generality, we must define the gluon field strength tensor,
and the covariant derivative of the adjoint Weyl spinor,
To write down the action, an invariant inner product on is needed: the Killing form is such an inner product, and in a typical abuse of notation we write simply as , suggestive of the fact that the invariant inner product arises as the trace in some representation of .
Supersymmetric Yang–Mills then readily generalizes from supersymmetric Maxwell theory. A simple version is
while a more general version is given by
Superspace treatment
Superspace and superfield content
The base superspace is super Minkowski space.
The theory is defined in terms of a single adjoint-valued real superfield , fixed to be in Wess–Zumino gauge.
Supersymmetric Maxwell theory on superspace
The theory is defined in terms of a superfield arising from taking covariant derivatives of : . The supersymmetric action is then written down, with a complex coupling constant , as
where h.c. indicates the Hermitian conjugate of the preceding term.
Supersymmetric Yang–Mills on superspace
For non-abelian gauge theory, instead define
and . Then the action is
Symmetries of the action
Supersymmetry
For the simplified Yang–Mills action on Minkowski space (not on superspace), the supersymmetry transformations are
where .
For the Yang–Mills action on superspace, since is chiral, then so are fields built from . Then integrating over half of superspace, , gives a supersymmetric action.