Beyond wild walls there is algebraicity and exponential growth (of BPS indices)

The BPS spectrum of pure SU(3) four-dimensional super Yang-Mills with N=2 supersymmetry (a theory of class S(A)) exhibits a surprising phenomenon: there are regions of the Coulomb branch where the growth of BPS-indices with the charge is exponential. We show this using spectral networks and, independently, using wall-crossing formulae and quiver methods. The technique using spectral networks hints at a general property dubbed "algebraicity": generating series for BPS-indices in theories of class S(A) (a class of N=2 four-dimensional field theories) are secretly algebraic functions over the rational numbers. Kontsevich and Soibelman have an independent understanding of algebraicity using indirect techniques, however, spectral networks give a distinct reason for algebraicity with the advantage of providing explicit algebraic equations obeyed by generating series; along these lines, we provide a novel example of such an algebraic equation, and explore some relationships to Euler characteristics of Kronecker quiver stable moduli. We conclude by proving that exponential asymptotic growth is a corollary of algebraicity, leading to the slogan "there are either finitely many BPS indices or exponentially many" (in theories of class S(A)).