  We consider $(2, 0)$ theory on a space-time of the form $R \times T^5$, where
the first factor denotes time, and the second factor is a flat spatial
five-torus. In addition to their energy, quantum states are characterized by
their spatial momentum, 't Hooft flux, and $Sp (4)$ $R$-symmetry
representation. The momentum obeys a shifted quantization law determined by the
't Hooft flux. By supersymmetry, the energy is bounded from below by the
magnitude of the momentum. This bound is saturated by BPS states, that are
annihilated by half of the supercharges. The spectrum of such states is
invariant under smooth deformations of the theory, and can thus be studied by
exploiting the interpretation of $(2, 0)$ theory as an ultra-violet completion
of maximally supersymmetric Yang-Mills theory on $R \times T^4$. Our main
example is the $A$-series of $(2,0)$ theories, where such methods allow us to
study the spectrum of BPS states for many values of the momentum and the 't
Hooft flux. In particular, we can describe the $R$-symmetry transformation
properties of these states by determining the image of their $Sp (4)$
representation in a certain quotient of the $Sp (4)$ representation ring.
