Quivers from Higgs bundles over P1 and quiver gauge theory

Abstract

In this paper, we associate a Higgs bundle over P1 with a Higgs quiver. We need to consider the Higgs representation of Higgs quivers, where the difference of Higgs representation from the usual linear representation is putting an OP1-module at each vertex in the quiver, and then changing the Higgs field on the underlying vector bundle is equivalent to changing the Higgs representations. We can also treat the space of Higgs fields as a space of the representations of the so-called Higgs structure quiver with the action of a reductive group. We produce some new quivers from the Higgs structure quiver and its representations, such as double quivers and Bogomol’nyi–Prasad–Sommerfield quivers related to quiver gauge theory, and study some topology of moduli space of corresponding stable representations, encoded in the structure of a Higgs bundle. We also introduce some algebraic structures on some vector spaces constructed from the Higgs structure quiver as the generalizations of classical Hall algebra and cohomological Hall algebra via bipartite quivers; the complexities of these algebraic structures are reflected in the nonassociativity of multiplications arising from preservation of some (Z2)r-graded structure.