The cohomology of framed moduli spaces and the coordinate ring of torus fixed points of quotient singularities

TL;DR

This paper investigates the relationship between the cohomology of framed moduli spaces and the coordinate rings of fixed points in quotient singularities, providing evidence for the Hikita conjecture in specific cases.

Contribution

It demonstrates an isomorphism of graded vector spaces between cohomology of framed moduli spaces and coordinate rings of fixed points in certain quotient singularities, advancing understanding of the Hikita conjecture.

Findings

Cohomology of framed moduli spaces over the projective plane is computed.

Coordinate rings of fixed points in specific quotient singularities are analyzed.

An isomorphism as graded vector spaces is established between these two structures.

Abstract

If two conical symplectic resolutions $X\to X_0$ and $X^!\to X_0^!$ are symplectic dual, the cohomology ring $H^*(X)$ and the coordinate ring of $\mathbb{C}^*$-fixed points in $X_0^!$ are expected to be isomorphic as graded algebras. This statement is called Hikita conjecture and it is known that the conjecture holds for some cases. In this paper, we deal with the cohomology of framed moduli spaces over the projective plane and the coordinate ring of $\mathbb{C}^*$- fixed points of $\mathbb{C}^{2n}/((\mathbb{Z}/r\mathbb{Z})\wr S_n) $ and show that these are isomorphic as graded vector spaces.