On the 5d instanton index as a Hilbert series

TL;DR

The paper addresses computing the non-perturbative part of the 5d $ ext{N}=2$ superconformal index by relating it to the Hilbert series of the instanton moduli space. A nontrivial fugacity mapping is proposed so that the Nekrasov instanton partition function can be obtained from the Hilbert series, with the identification $ ext{I}_{ ext{inst}}^{(k)} ightleftharpoons ext{HS}_{k}$. Explicit checks for pure $SU(2)$ confirm agreement with known results including symmetry enhancement, and an exact index for a pure $ ext{U}(1)$ theory is derived. The work provides a computational bridge between Hilbert-series techniques and Nekrasov-style indices in 5d and suggests extensions to flavored instantons and holographic duals.

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.

Figures (2)

• Figure 1: Quiver engineering $k$-instantons of $U(2)$ --the same as 1-instanton of $SU(2)\sim USp(2)$--.
• Figure 2: Quiver engineering $k$-instantons of $U(1)$.