Shigeyuki Fujii | Searcharxiv

arXiv2017

A Combinatorial Study on Quiver Varieties

This is an expository paper which has two parts. In the first part, we study quiver varieties of affine $A$-type from a combinatorial point of view. We present a combinatorial method for obtaining a closed formula for the generating function of Poincaré polynomials of quiver varieties in rank 1 cases. Our main tools are cores and quotients of Young diagrams. In the second part, we give a brief survey of instanton counting in physics, where quiver varieties appear as moduli spaces of instantons, focusing on its combinatorial aspects.

math.AG↗

arXiv2008

Instanton calculus and chiral one-point functions in supersymmetric gauge theories

We compute topological one-point functions of the chiral operator Tr ϕ^k in the maximally confining phase of U(N) supersymmetric gauge theory. These one-point functions are polynomials in the equivariant parameter \hbar and the parameter of instanton expansion q=Λ^{2N} and are of particular interest from gauge/string theory correspondence, since they are related to the Gromov-Witten theory of P^1. Based on a combinatorial identity that gives summation formula over Young diagrams of relevant functions, we find a relation among chiral one-point functions, which recursively determines the \hbar expansion of the generating function of one-point functions. Using a result from the operator formalism of the Gromov-Witten theory, we also present a vacuum expectation value of the loop operator Tr e^{itϕ}.

hep-th↗