Why the maximum number of supercharges in supergravity must be $Q=32$?

Question

For a supergravity theory not to have particles with spin greater than 2, all books state that $Q\leq 32$, where $Q$ is the number of fermionic supercharges and for a given dimension $D$ it's related to the number of supersymmetries $\mathcal{N}$ through the number of components of the fundamental spinors in that dimension, $C$ as $Q=\mathcal{N} C$.

Naively, I would expect each supercharge $Q$ to raise or lower the spin of a given particle by $1/2$, as it does in the 4 dimensional case, but I suspect this is not the case in different dimensions (though all the books I have read are pretty confusing, just analyzing $D=4$ in detail using the chiral properties in that dimension and then hand-waving their way to higher dimensions).

If each of the $\mathcal{N}$ supersymmetries could be used once and only once to raise the spin of the states, then I would expect the limit to be $\mathcal{N} \leq 8$ instead of a bound on $Q$, but this is not the case.

According to one of my professors, $Q=32$ is just the consequence of wanting $\mathcal{N}=8$ at the most in 4 dimensions, where $C=4$, so that we can compactify the higher dimensional theory and obtain an acceptable model for our world.

However, the $D=11$ supergravity should only include the graviton in that case, right? And $D=10$ theories would only have the graviton and $\mathcal{N}$ gravitinos, which is not true. So what's the right justification for $Q \leq 32$?

Answer 1

You have to first understand the contruction of massless multiplets which is found in the beginning of every introduction to supersymmetry, so I won't repeat it here. Then the argument goes like this:

• Given $Q$ supercharges, half of them will be zero for the massless case, thus leaving you with $Q/2$ non-zero supercharges.

• From the remaining $Q/2$ supercharges we can construct $Q/4$ lowering operators, and $Q/4$ raising operators.

• Every raising/lowering operator changes the helicity $\lambda$ by $\pm 1/2$. So avoiding helicities $\lambda$ greater than $|2|$ requires that

\begin{equation} Q/4\leq 8 \end{equation}

Therefore the maximum number of supercharges is $Q=32$ for any supergravity.