Supersymmetry (SUSY)

A postulated symmetry between bosonic and fermionic fields in quantum field theories and string theories. In non-technical terms, this means that each bosonic field or particle has a fermionic superpartner and vice versa.

The theory of Supersymmetry has been incorporated in the standard-model (mssm), yang-mills theory (Super-Yang-Mills Theory), and most famously String Theory (superstrings).

While Supersymmetry remains experimentally unconfirmed, one of its greatest achievements is that the MSSM (which also appears in realistic m-theory vacua) predicts a Higgs of mass 125 GeV (which was measured by the LHC recently.), which is contrary to the standard-model, which predicts such a mass to be rather unlikely.

Technical details

There are two types of supersymmetry; worldsheet supersymmetry, and spacetime supersymmetry

Worldsheet supersymmetry

The Ramond-Neveu-Schwarz formalism has explicit worldsheet supersymmetry. Since the RNS Action is given by adding the Polyakov Action to the Dirac action, it is given by:

$${{\mathsf{\mathcal{L}}}_ {RNS}}=\frac{T}{2} h^{\alpha \beta} \left( \partial_\alpha X^\mu \partial_\beta X^\nu +i\hbar c_0 \bar{\psi_\mu} \not{\partial} \psi^\mu \right) g_{\mu\nu}$$

The supersymmetric transformations on the worldsheet can therefore be (almost trivially, by taking variations of this above action) shown to be:

$$\begin{gathered} \delta {X^\mu } \to \bar \epsilon {\psi ^\mu } ; \\ \delta {\psi ^\mu } \to - i \not \partial {X^\mu }\epsilon \\ \end{gathered} $$

Spacetime Supersymmetry

The Green-Schwarz formalism, or the superspace-formalism, are with explicit spacetime supersymmetry. The supersymmetric transformations on spacetime are (which is rather intuitive if you compare this to the RNS Worldsheet supersymmetry transformations) given by:

$$\begin{gathered} \delta {\Theta ^{Aa}} \leftrightarrow {\varepsilon ^{Aa}} ; \\ \delta {X^\mu } \leftrightarrow {{\bar \varepsilon }^A}{\gamma ^\mu }{\Theta ^A} ;\\ \end{gathered} $$