Quiver varieties, affine Lie algebras, algebras of BPS states, and semicanonical basis

TL;DR

This paper proposes a conjectural basis construction for affine Lie algebras of type ADE using quiver varieties, linking it to BPS states and exploring related combinatorial questions about root systems and Lusztig's semicanonical basis.

Contribution

It introduces a new conjectural basis for affine Lie algebras connected to quiver varieties and BPS states, expanding the understanding of their combinatorial and geometric structures.

Findings

Conjectural basis indexed by quiver variety components

Connections between BPS states and affine Lie algebra structures

New combinatorial insights into root systems and Lusztig's basis

Abstract

We suggest a (conjectural) construction of a basis in the plus part of the affine Lie algebra of type ADE indexed by irreducible components of certain quiver varieties. This construction is closely related to a string-theoretic construction of a Lie algebra of BPS states. We then study the new combinatorial questions about the (classical) root systems naturally arising from our constructions and Lusztig's semicanonical basis.