Space-time Supersymmetry

Hi,

I think this distinction comes in the context of string theory.

In string theory one has a string (duh), and on this string you define oscillator modes. The simplest string theories are just bosonic. You can also add fermionic degrees of freedom to your string in order to be able to describe fermions. However, these are degrees of freedom on your worldsheet. So you need to distinguish between supersymmetry on your worldsheet (turning bosonic degrees of freedom into fermionic ones) and supersymmetry in spacetime! Ofcourse, it turns out that fermionic and bosonic degrees of freedom on the worldsheet can be interpreted as bosons and fermions in spacetime, but this is not trivial! These particles are in a representation of the Poincaré group and it's not trivial that your worldsheet degrees of freedom neatly fit into these Poincare representations.

Now, space-time supersymmetry is the ordinary supersymmetry you encounter when you don't talk about strings. This is not the same as supergravity! Space-time supersymmetry is a global symmetry; the transformation parameters don't depend on coordinates. However, if you gauge this space-time symmetry, you can show that you introduce diffeomorphism invariance in the theory. A gauge theory with diffeomorphism invariance necessarily contains a dynamical metric, a graviton and hence describes gravity: Supergravity! (I believe this has to do with the fact that as soon as you start to quantize a spin-2 gauge theory you need diffeomorphism invariance to avoid negative-norm states, and the other way around can also be shown).

Hope this helps, but I'm not an expert on this, so maybe I say things which are not entirely true :P

 

Typically in QM and in QFT for each symmetry you have a set of generators. Angular momentum operators generate rotations, for example. They are a subset of the Poincare algebra which consists of rotations, boosts and translations (space- and timelike).

These generators are conserved quantities due to the Noether theorem:
Lagrangian with symmetry (*) => conserved charge dQ/dt=0 => charge operator => qm generator of the symmetry (*)

Now you have such conserved charges which act as generators of SUSY. These charges do not transform as Lorentz-scalars but as Lorentz spinors!!! And the commutators of these charges are generators of the well-known Poincare algebra.

 

The notes by Aitchison or Bilal are very nice, I think :)

 

One can find quite some material for instance at
http://web.mit.edu/redingtn/www/netadv/Xsusy.html