S-matrices and bi-linear sum rules of conserved charges in affine Toda field theories

TL;DR

This paper uncovers new bi-linear sum rules for conserved charges in affine Toda field theories, derived from exact $S$-matrices, applicable to both simply laced and non-simply laced cases, with explicit examples provided.

Contribution

It introduces a novel class of bi-linear sum rules for conserved quantities in ATFTs based on their exact $S$-matrices, expanding understanding of their algebraic structure.

Findings

Sum rules exist when $S$-matrices have a multiplicative identity.

Results apply to both simply laced and non-simply laced ATFTs.

Explicit examples demonstrate the sum rules in specific models.

Abstract

The exact quantum $S$-matrices and conserved charges are known for affine Toda field theories(ATFTs). In this note we report on a new type of bi-linear sum rules of conserved quantities derived from these exact $S$ matrices. They exist when there is a multiplicative identity among $S$-matrices of a particular ATFT. Our results are valid for simply laced as well as non-simply laced ATFTs. We also present a few explicit examples.