Affine Toda system of $\mathbf{A}$ and $\mathbf{C}^t$ type: compactness and affine Weyl group

Abstract: The local mass is a fundamental quantized information that characterizes the blow-up solution to the Toda system and has a profound relationship with its underlying algebraic structure. In \cite{Lin-Yang-Zhong-2020}, it was observed that the associated Weyl group can be employed to represent this information for the $\mathbf{A}_n$, $\mathbf{B}_n$, $\mathbf{C}_n$ and $\mathbf{G}_2$ type Toda systems. The relationship between the local mass of blow-up solution and the corresponding affine Weyl group is further explored for some affine $\mathbf{B}$ type Toda systems in \cite{Cui-Wei-Yang-Zhang-2022}, where the possible local masses are explicitly expressed in terms of $8$ types. The current work presents a comprehensive study of the general affine $\mathbf{A}$ and $\mathbf{C}^t$ type Toda systems with arbitrary rank. At each stage of the blow-up process (via scaling), we can employ certain elements (known as "set chains") in the corresponding affine Weyl group to measure the variation of local mass. Consequently, we obtain the a priori estimate of the affine $\mathbf{A}$ and $\mathbf{C}^t$ type Toda systems with arbitrary number of singularities.