Learning to count BPS states

### The general idea

Several interesting problems appear when one wants to count states that preserve some fraction of the supersymmetry in quantum field theory or string theory or in M-theory. Dualities relate counting problems that appear to be rather different and unrelated. For instance, the counting of chiral primaries in a superconformal field theory gets mapped to counting different supergravity solutions using the AdS-CFT correspondence.

In most counting problems, it turns out to be simpler to construct generating functions as one does in statistical mechanics. Recall, that in statistical mechanics the canonical partition function may be thought of as a weighted sum over configurations with a fixed energy, $E$, with the weight given by $\exp(-\beta E)$. Similarly, the grand partition function introduces chemical potentials for every species of particles in the system — thus each term in grand partition gives the number of configurations with a fixed energy as well as fixed number of particles. The counting of BPS states proceeds in a similar manner — every BPS state carries a certain number of charges, introduce the analogue of the chemical potential for every independent charge — both electric and magnetic charges are to be included. The Dirac-Schwinger-Zwanziger quantisation makes these charges live on a lattice. One thus schematically writes

(1)
\begin{align} \mathcal{Z}(\mathbf{q})=\sum_{\textrm{lattice}}\ d(\mathbf{n}) \ \mathbf{q}^{\mathbf{n}} \ , \end{align}

where $\mathbf{n}$ runs over some lattice, $d(\mathbf{n})$ is the number of configurations associated with the charge vector $\mathbf{n}$ and $\mathbf{q}$ is the generalised fugacity vector. Now apriori, the fugacity seems like just a mathematical curiosity. However, it usually turns out to be related to some physical parameter in the system like a Kahler modulus or string coupling and so on. The lattice that appears is usually an interesting one — for instance, in $\mathcal{N}=4$ string theory, it may be the Narain lattice and hence is even self-dual and Lorentzian. String and other dualities usually lead to additional properties — it may lead to a reduction in the effective number of fugacities that are really needed. Further, it may turn out that $\mathcal{Z}(\mathbf{q})$ may be a modular function of some group acting on the space of fugacities. Then, the modular property enables one to more or less uniquely fix this function.

### Some examples

Let us list a few examples of such counting problems