CAPPED VERTEX FUNCTIONS FOR HILBERT SCHEMES OF POINTS AND CYCLIC QUIVER VARIETIES

Capped vertex functions are central objects within the intersection of Mathematical Physics and $K$-theoretic enumerative geometry. Originally appearing as the partition functions from the Higgs branch of certain quantum field theories, these objects have been recently explored as enumerative objects via counts of quasimaps to the Higgs branch. In the geometric sense these objects are Laurent series with coefficients in the non-localized equivariant $K$-theory of the moduli space of these quasimaps, and thus are computationally difficult to work with. The works of Okounkov and Okounkov-Smirnov have shown that the computation of capped vertex functions arises from computing two objects in localized $K$-theory: the bare vertex function and the capping operator. The bare vertex function is a generalization of $q$-hypergeometric functions, and like the capped vertex function are an enumerative count of quasimaps. The capping operator is a tensor in two variables that originates in geometric representation theory as either the solution of the qKZ equation, or as the solution of the dynamical equation. Okounkov-Smirnov, and recent work of Zhu in the affine ADE case have shown that the representation theoretic formula for the capping operator has its starting point given by the so-called quantum torodial algebra. Despite these well known localized objects, the computation of the capped vertex function remains complex, and there is an open conjecture which states that this is the Laurent expansion of a rational function. The purpose of this thesis is give an explicit formula for the capped vertex function for the Nakajima quiver varieties $\text{Hilb}^{n}(\mathbb{C}^{2})$ and $\text{Hilb}^{n}([\mathbb{C}^{2}/\mathbb{Z}_{l}])$. In both varieties the formula can be recast as a purely combinatorial statement involving Macdonald, or Wreath Macdonald polynomials respectively. The corollary of these results are a confirmation of the conjecture mentioned above.