Total masses of solutions to general Toda systems with singular sources

TL;DR

The paper establishes explicit total masses for solutions to general Toda systems with singular sources tied to complex simple Lie algebras, showing $\sigma_i(u)=2\langle \omega_i - \kappa\omega_i, w_0\rangle$ where $\kappa$ is the Weyl group’s longest element. It achieves this by a synthesis of integrable-systems techniques (DPW method, Drinfeld–Sokol gauge, $W$-invariants) and Lie-theoretic data (Kostant’s one-dimensional Toda lattice), deriving asymptotics $U_i(z) \sim 2(\gamma^i - \langle \omega_i - \kappa\omega_i, w_0\rangle)\log|z|$ and connecting the local masses to the Weyl group action. The work generalizes prior results to all simple Lie types, provides explicit mass formulas for several families, and strengthens the bridge between nonlinear PDE analysis and integrable-system methods, with potential implications for a priori bounds on mean-field Toda systems and related geometric problems. Overall, the paper demonstrates a deep link between singular Toda dynamics, Weyl-group symmetries, and holomorphic data via DPW/DS machinery, enabling a unified computation of total masses across Lie types.

Abstract

In this article we obtain total masses of solutions to the Toda system associated to a general simple Lie algebra with singular sources at the origin. The determination of such total masses is one of the important steps towards establishing the a priori bound for solutions to the mean field type of Toda system on compact surfaces. The total mass is found to be related to the longest element $κ$ in the Weyl group of the corresponding Lie algebra. This is the foundation to future work relating the local blowup masses (from analysis) with the Weyl group. This work generalizes the previous works in Lin et al. (2012), Ao et al. (2015) and Nie (20160 for Toda systems of types $A, G_2$ and $B, C$. However, a more Lie-theoretic method is needed here for the general case, and the method relies heavily on the DPW method, Drinfeld-Sokolov gauge and the $W$-invariants. The last crucial step for the total masses is obtained by applying the work of Kostant (1979) on the one dimensional Toda lattice.

Theorems & Definitions (28)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Remark 1.4
• Theorem 1.5
• Lemma 2.1
• proof
• Proposition 2.2
• proof
• Remark 2.3