The Non-Orientable Map Asymptotics Constant $p_g$

Abstract

Using the pfaffian structure of the generating series for locally orientable maps, we show that the generating series satsifies a nonlinear differential equation called the BKP equation. Using this we are able to derive a cubic differential equation which is satisfied by the generating series for locally orientable triangulations. As a result, we prove a conjecture of Garoufalidis and Mari\~no concerning the constant $p_g$ which appears in asymptotic formulas for a variety of rooted maps on non-orientable surfaces. This allows one to determine the asymptotic expansion for $p_g$ up to an unknown Stokes constant.