Total masses of solutions to general Toda systems with singular sources

TL;DR

This paper calculates the total masses of solutions to general Toda systems with singular sources, linking these masses to Lie algebra Weyl group elements, advancing understanding of Toda systems on compact surfaces.

Contribution

It generalizes previous results to arbitrary simple Lie algebras using Lie-theoretic methods and the DPW approach, establishing a key link between total masses and Weyl group elements.

Findings

Total masses relate to the longest element in the Weyl group.

The method extends to general Lie algebras beyond types A, G2, B, C.

Uses Lie-theoretic techniques and the DPW method for analysis.

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 $\kappa$ 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.