A Combinatorial Study on Quiver Varieties

Abstract

Abstract. 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.

Fivebranes and 3-manifold homology

“…show that these are not characters of integrable representations of su(p)N , but rather products of characters of su(N )1 integrable representations [43] (see also [44] and references therein):…”

n-th parafermion $$ {\mathcal{W}}_N $$ characters from U(N) instanton counting on ℂ2/ℤn

“…is described by N n free fermions [30][31][32][33][34] (see also [20] for an elegant string theory interpretation by intersecting D4 and D6-branes). Actually, the generating function (2.4) of coloured Young diagrams for general N and n is obtained by [49,50]…”

On ’t Hooft defects, monopole bubbling and supersymmetric quantum mechanics

“…As in the case of the ADHM construction of instanton moduli space on C 2 , the Uhlenbeck compactification of the moduli space M KN is singular. The smooth resolution in this case is the moduli space of Z n -equivariant torsion free sheaves on CP 2 with fixed framing at the line at infinity [5,35,37]. The resolved space M KN (ζ i R ) can again be described as a hyperkähler quotient after introducing stability/FI parameters which deform the real moment map as follows:…”