Saw varieties

Abstract

We introduce a new class of quiver varieties, called saw (quiver) varieties. These can be seen as a generalization of the three existing theories: the quiver gauge theory of monopoles, Kac's representation theory of Dynkin types, and Donaldson's theory on the based maps of P1. E.g., in our terminology, the handsaw (quiver) varieties and chainsaw varieties are finite and affine A type saw varieties respectively with the maximal symmetry breaking. These varieties are the monopole spaces and the caloron spaces in 3d gauge theory, respectively. We expand the theory over to the DE type quiver gauge theory.The saw varieties are constructed via a Hamiltonian formalism, so that the moment maps defining these varieties naturally arise. The main assertion of the paper is that the moment map μ defining a saw variety is a flat morphism for arbitrary dimension vector, if and only if the underlying graph is of finite ADE types. Furthermore under the assumption of maximal symmetry breaking or all the higher frame dimension vectors, the affine ADE types also attain the flatness property of μ. We give an algebro-geometric description of the momentum-zero scheme μ−1(0) and the saw variety defined as the Hamiltonian reduction.Various partition functions given rise to by the above mentioned theories are the byproducts of the geometric structure of the corresponding saw varieties.