A note on quivers with symmetries

TL;DR

This paper links bases of certain modules of non-symmetric Kac-Moody algebras to invariant Lagrangian subvarieties of quiver-associated varieties, with implications for string duality.

Contribution

It establishes a natural identification between module bases and invariant Lagrangian subvarieties in the context of quivers with symmetries, extending geometric representation theory.

Findings

Bases correspond to invariant Lagrangian subvarieties

Identification extends to affine and finite Kac-Moody algebras

Connections to string duality are discussed

Abstract

We show that the bases of irreducible integrable highest weight module of a non-symmetric Kac-Moody algebra, which is associated to a quiver with a nontrivial admissible automorphism, can be naturally identified with a set of certain invariant Langrangian irreducible subvarieties of certain varieties associated with the quiver defined by Nakajima. In the case of non-symmetric affine or finite Kac-Moody algebras, the bases can be naturally identified with a set of certain invariant Langrangian irreducible subvarieties of a particular deformation of singularities of the moduli space of instantons over A-L-E spaces. The motivation of this paper comes from string/string duality and the paper is ended with questions and speculations related to string/string duality.