Wall-crossing formula for framed quiver moduli

We investigate the wall-crossing phenomena for moduli of framed quiver representations. These spaces are expected to be highly useful in capturing the representation theoretic essence of special functions in integrable systems. Within this class of moduli spaces, we focus on the type $A$ flag manifold, type $A$ affine Laumon spaces, Nakajima quiver variety, and framed moduli of sheaves on the projective plane and the blow-up as main motivating examples. Specifically, we examine the wall-crossing formulas for integrals of Euler classes over these moduli spaces.