Geometric Subsystem Quantisation: Toward a Rigorous Field Theory

We present a comprehensive synthesis of the geometric subsystem quantisation programme for topological solitons, starting from the classification of ODE reductions of nonlinear wave equations and culminating in a precise description of the current boundary: the need for an intrinsic geometric renormalisation to incorporate radiative corrections. The programme provides a rigorous finite-dimensional symplectic reduction and deformation quantisation of collective coordinates, yet it halts at the one-loop meson correction, where standard field-theoretic renormalisation must be imported. We analyse this barrier in depth and outline a systematic path toward a fully geometric, infinite-dimensional formulation in which the renormalisation procedure itself may be understood as a choice of polarisation – a consistent complex structure on the field phase space. The role of determinant line bundles, Quillen metrics, and the Batalin–Vilkovisky formalism is discussed, together with an honest inventory of what has been proved, what is conjectured, and what remains to be developed. This document is intended as a detailed roadmap, free of speculation, that builds the logical chain step by step and explains why each step is taken, while acknowledging the mathematical challenges ahead.