  In various subdisciplines of optics and photonics, Mie theory has been
serving as a fundamental language and play indispensable roles widely.
Conventional studies related to Mie scattering largely focus on local
properties such as differential cross sections and angular polarization
distributions. Though spatially integrated features of total cross sections in
terms of both scattering and absorption are routine for investigations, they
are intrinsically dependent on the specific morphologies of both the scattering
bodies and the incident waves, consequently manifesting no sign of global
invariance. Here we propose global Mie scattering theory to explore topological
invariants for characterizations of scatterings by any obstacles of arbitrarily
structured or polarized coherent light. It is revealed that, independent of
distributions and interactions among the scattering bodies of arbitrary
geometric and optical parameters, in the far field inevitably there are
directions where the scatterings are either zero or circularly polarized.
Furthermore, for each such singular direction we can assign a half-integer
index and the index sum of all those directions are bounded to be a global
topological invariant of $2$. The global Mie theory we propose, which is
mathematically simple but conceptually penetrating, can render new perspectives
for light scattering and topological photonics in both linear and nonlinear
regimes, and would potentially shed new light on the scattering of acoustic and
matter waves of various forms.
