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. It is a special case of 4D N = 1 global supersymmetry. 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 N = 1 {\displaystyle {\mathcal {N}}=1} supersymmetric Yang–Mills theory, which is the theory discussed in this article. The N = 2 {\displaystyle {\mathcal {N}}=2} supersymmetric Yang–Mills theory was studied by Seiberg and Witten in Seiberg–Witten theory. All three theories are based in d = 4 {\displaystyle d=4} 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 G {\displaystyle G} to the theory. The gauge group has associated Lie algebra g {\displaystyle {\mathfrak {g}}} . The field content then consists of
a g {\displaystyle {\mathfrak {g}}} -valued gauge field A μ {\displaystyle A_{\mu }}
a g {\displaystyle {\mathfrak {g}}} -valued Majorana spinor field Ψ {\displaystyle \Psi } (an adjoint-valued spinor), known as the 'gaugino' a g {\displaystyle {\mathfrak {g}}} -valued auxiliary scalar field D {\displaystyle D} . For gauge-invariance, the gauge field A μ {\displaystyle A_{\mu }} is necessarily massless. This means its superpartner Ψ {\displaystyle \Psi } is also massless if supersymmetry is to hold. Therefore Ψ {\displaystyle \Psi } can be written in terms of two Weyl spinors which are conjugate to one another: Ψ = ( λ , λ ¯ ) {\displaystyle \Psi =(\lambda ,{\bar {\lambda }})} , and the theory can be formulated in terms of the Weyl spinor field λ {\displaystyle \lambda } instead of Ψ {\displaystyle \Psi } .
Supersymmetric pure electromagnetic theory When G = U ( 1 ) {\displaystyle G=U(1)} , 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 μ {\displaystyle A_{\mu }} , a Majorana spinor Ψ {\displaystyle \Psi } and a auxiliary real scalar field D {\displaystyle D} . The field strength tensor is defined as usual as F μ ν := ∂ μ A ν − ∂ ν A μ {\displaystyle F_{\mu \nu }:=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }} . The Lagrangian written down by Wess and Zumino is then
L = − 1 4 F μ ν F μ ν − i 2 Ψ ¯ γ μ ∂ μ Ψ + 1 2 D 2 . {\displaystyle {\mathcal {L}}=-{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }-{\frac {i}{2}}{\bar {\Psi }}\gamma ^{\mu }\partial _{\mu }\Psi +{\frac {1}{2}}D^{2}.}
… excerpt ends here. Continue reading the full article.
