On the 5d instanton index as a Hilbert series

TL;DR

This paper explores the relationship between the 5d superconformal instanton index and the Hilbert series of instanton moduli spaces, proposing a method to compute the index via Hilbert series and validating it with known results.

Contribution

It introduces a novel identification of fugacities linking the instanton index to the Hilbert series, enabling exact computations for pure gauge theories.

Findings

Agreement with existing literature results

Exact index computation for pure U(1) gauge theory

Validation of the proposed fugacity identification

Abstract

The superconformal index for N=2 5d theories contains a non-perturbative part arising from 5d instantonic operators which coincides with the Nekrasov instanton partition function. In this note, for pure gauge theories, we elaborate on the relation between such instanton index and the Hilbert series of the instanton moduli space. We propose a non-trivial identification of fugacities allowing the computation of the instanton index through the Hilbert series. We show the agreement of our proposal with existing results in the literature, as well as use it to compute the exact index for a pure U(1) gauge theory.