How to determine the number of supercharges (N) for any dimension (D)?

[Osiris]

e.g.
For D=11, the M-theory, we require N=8, ...
For D=10, we require N=16, or 32...
...


So how to determine the number of supercharges, the $$\theta$$s, for arbitrary dimension, e.g. D=1, 2, ...

 

[Physics Monkey]

Hi Osiris,

Do you mean, how do you determine the maximum number of supercharges? If so, the intuitive answer comes from restricting the spin of particles in the supermultiplet. Generically, the supercharges raise or lower the spin of the particles in a supermultiplet. The maximum number of supercharges is determined by the largest supersymmetry algebra that doesn't raise the spin above 1 for theories without gravity or above 2 for theories with gravity. Note also that the counting can be bit funny because the way you organize the supercharges depends on the spinor structure of the dimension you're in. This sort of reasoning can also be used to understand the presence of a maximum dimension for the supersymmetry algebra.