Operadic approach to wall-crossing

TL;DR

This paper introduces an operadic framework to understand wall-crossing phenomena, simplifying existing formulas and deriving new ones in algebraic geometry and physics.

Contribution

It proposes a novel operadic approach to encode and analyze wall-crossing transformation rules, providing streamlined proofs and new formulas.

Findings

Simplified proofs of existing wall-crossing formulas

New formulas related to attractor invariants

Operadic framework effectively encodes complex transformations

Abstract

Wall-crossing phenomena are ubiquitous in many problems of algebraic geometry and theoretical physics. Various ways to encode the relevant information and the need to track the changes under the variation of parameters lead to rather complicated transformation rules and non-trivial combinatorial problems. In this paper we propose a framework, reminiscent of collections and plethysms in the theory of operads, that conceptualizes those transformation rules. As an application we obtain new streamlined proofs of some existing wall-crossing formulas as well as some new formulas related to attractor invariants.