Algebraicity and Asymptotics: An explosion of BPS indices from algebraic generating series (1606.02693v1)

Abstract: It is an observation of Kontsevich and Soibelman that generating series that produce certain (generalized) Donaldson Thomas invariants are secretly algebraic functions over the rationals. From a physical perspective this observation arises naturally for DT invariants that appear as BPS indices in theories of class S[A]: explicit algebraic equations (that completely determine these series) can be derived using (degenerate) spectral networks. In this paper, we conjecture an algebraic equation associated to DT invariants for the Kronecker 3-quiver with dimension vectors (3n,2n), n>0 in the non-trivial region of its stability parameter space. Using a functional equation due to Reineke, we show algebraicity of generating series for Euler characteristics of stable moduli for the Kronecker m-quiver assuming algebraicity of generating series for DT invariants. In the latter part of the paper we deduce very explicit results on the asymptotics of DT invariants/Euler characteristics under the assumption of algebraicity of their generating series; explicit large n asymptotics are deduced for dimension vectors (3n,2n) for the Kronecker 3-quiver. The algebraic equation is derived using spectral network techniques developed by Gaiotto-Moore-Neitzke, but the main results can be understood without knowledge of spectral networks.