# BPS Hall Algebra of Scattering Hall States

## TL;DR

This paper compares two different algebraic constructions of BPS states in supersymmetric theories, demonstrating their equivalence through Harvey-Moore's S-matrix approach and Kontsevich-Soibelman's cohomological Hall algebra framework.

## Contribution

It establishes the equivalence between two seemingly different algebraic definitions of BPS state algebras, bridging physics and mathematics.

## Key findings

- The two algebra constructions are mathematically equivalent.
- The S-matrix based algebra matches the cohomological Hall algebra structure.
- Provides a unified understanding of BPS state algebras in supersymmetric theories.

## Abstract

Starting with a very pedestrian point of view we compare two different at the first glance definitions for an algebra associated to BPS states in supersymmetric fields theories. One proposed by Harvey and Moore exploits $S$-matrices of BPS states as structure constants of a new algebra. Another one proposed by Kontsevich and Soibelman gives a construction according to the structure of cohomological Hall algebras. We show these two constructions give equivalent algebras.