# Non-Connected Gauge Groups and the Plethystic Program

## TL;DR

This paper extends the Weyl integration formula to non-connected gauge groups in supersymmetric theories, enabling systematic counting of gauge-invariant operators and computing Hilbert series for instanton moduli spaces, validated through mirror symmetry.

## Contribution

It introduces a novel extension of the Weyl integration formula for non-connected Lie groups, facilitating new calculations in gauge theory and instanton moduli spaces.

## Key findings

- Extended Weyl integration formula for non-connected groups
- Computed Hilbert series for instanton moduli spaces
- Validated results with Coulomb branch and mirror symmetry

## Abstract

We present in the context of supersymmetric gauge theories an extension of the Weyl integration formula, first discovered by Robert Wendt, which applies to a class of non-connected Lie groups. This allows to count in a systematic way gauge-invariant chiral operators for these non-connected gauge groups. Applying this technique to $\mathrm{O}(n)$, we obtain, via the ADHM construction, the Hilbert series for certain instanton moduli spaces. We validate our general method and check our results via a Coulomb branch computation, using three-dimensional mirror symmetry.