BPS sectors in $\cal{N}=4$ SYM

Question

I am familiar with the idea of a BPS bound as in a lower limit on the mass of supermultiplets given by a certain function of the central charge and when I think of $\cal{N}=4$ SYM I see a complicated lagrangian in my mind.

Somehow the above picture seems insufficient to understand the classification of what are called "BPS sectors" in $\cal{N}=4$ SYM.

I would like to know what is the meaning and the derivation of the following listing for $\cal{N}=4$ SYM ,

• $\frac{1}{2}$ BPS sector consists of multi-trace operators involving a single bosonic operator $Z$

• $\frac{1}{4}$ BPS sector consists of multi-trace operators involving two bosonic operator $Z, Y$

• $\frac{1}{8}$ BPS sector consists of multi-trace operators involving three bosonic operator $Z, Y, X$ and two fermionic operators $\lambda$, $\bar{\lambda}$

(everything above is apparently assumed to be in the the Lie algebra of $U(N)$)

I would also like to know what is the relation between the above classification and thinking in terms of short/semi-short/long multiplets.

I would be happy to know of expository references for the above topics. I haven't been able to locate any on my own.

Answer 1

The BPS conditions in $\mathcal{N}=4$ SYM is just another name for the shortening conditions in the $psu(2,2|4)$ symmetry algebra of the theory. This algebra has 32 odd generators: 16 supercharges $Q_{\alpha a}$ and $\tilde{Q}_{\dot{\alpha}}^a$ and 16 superconformal charges $S^a_\alpha$ and $\tilde{S}_{\dot\alpha a}$. Here the indices $\alpha,\dot\alpha=1,2$ are Weyl spinor indices and $a=1,\dotsc,4$ is an index in the fundamental representation of the $SU(4)$ R-symmetry group. The generators $S$ and $\tilde{S}$ have dimension $-1/2$ and hence annihilates a primary state. The highest weight state of a BPS multiplet is furthermore annihilated by one or more of the supercharges. In particular a $\frac{1}{2}$-BPS state, or chiral primary, is annihilated by half of the charges $Q$ and $\tilde{Q}$. Similarly $\frac{1}{4}$-BPS ($\frac{1}{8}$-BPS) states are annihilated by a quarter (an eighth) of these charges. The exact details of what fields make up the operators in the various subsectors of course depend on the choice of simple roots (or lowering/raising operators) in the algebra, but the notation in the question seems fairly standard.

The details of the symmetry algebra should be covered in any review of $\mathcal{N}=4$ SYM or the AdS/CFT-correspondence. See eg arXiv:1012.3938, which however only discusses the planar theory where only single trace operators are relevant. From an algebra point of view the more general case should however be essentially the same.