Wall crossing, string networks and quantum toroidal algebras

TL;DR

This paper explores the algebraic structure of BPS states and string networks in 4d N=4 supersymmetric Yang-Mills theory, revealing a connection to quantum toroidal algebras and their representations, with implications for wall crossing phenomena.

Contribution

It introduces a novel interpretation of line operator algebras as tensor products of quantum toroidal algebra representations, linking wall crossing to algebraic twists and R-matrices.

Findings

Identifies the algebra of line operators with tensor products of vector representations.

Connects wall crossing operators to Drinfeld twists of the coproduct.

Relates the spectrum generator to the universal R-matrix of the quantum toroidal algebra.

Abstract

We investigate BPS states in 4d N=4 supersymmetric Yang-Mills theory and the corresponding (p, q) string networks in Type IIB string theory. We propose a new interpretation of the algebra of line operators in this theory as a tensor product of vector representations of a quantum toroidal algebra, which determines protected spin characters of all framed BPS states. We identify the SL(2,Z)-noninvariant choice of the coproduct in the quantum toroidal algebra with the choice of supersymmetry subalgebra preserved by the BPS states and interpret wall crossing operators as Drinfeld twists of the coproduct. Kontsevich-Soibelman spectrum generator is then identified with Khoroshkin-Tolstoy universal R-matrix.