Analysis of blowup masses in constant-coefficient Toda systems: W-invariants and Weyl orbits

We classify the local blowup masses of constant-coefficient Toda systems for all simple Lie algebras with one prescribed singular source, bounded component masses, and bounded boundary oscillations. We encode the source strengths and the blowup masses by two elements of the Cartan subalgebra: the grading element $p_μ$ and the mass element $q_σ$. We show that $p_μ-q_σ=wp_μ$ for some Weyl group element $w\in W$. The proof uses the Drinfeld-Sokolov gauge and the resulting holomorphic $W$-invariants for Toda systems. Asymptotic analysis near the source and on a shrinking circle enclosing the total local mass gives two evaluations of the leading Laurent coefficients of the $W$-invariants, which agree by holomorphicity. This agreement enables us to conclude the above Weyl-orbit result. In particular, the quadratic constraint is the local Pohozaev identity. Only with the full set of $W$-invariants can we strengthen earlier orbit-plus-integer restrictions to an exact Weyl-orbit characterization in the constant-coefficient setting. Together with the author's explicit families in [Nie], this determines all attainable local mass vectors.