Abstract
We present in the context of supersymmetric gauge theories an extension of the Weyl integration formula, first discovered by Robert Wendt [1], 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 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.
Highlights
Has received much less attention from this perspective, four-dimensional N = 2 theories of this type have been studied recently [9]
We present in the context of supersymmetric gauge theories an extension of the Weyl integration formula, first discovered by Robert Wendt [1], which applies to a class of non-connected Lie groups
We outline a general procedure first presented in [1] that allows to count gauge invariant operators, through integration over the gauge group of the theory using an extension of the Weyl integration formula, that applies to a class of non-connected gauge Lie groups called the principal extensions
Summary
We first review the well-known proof of the Weyl integration formula for a connected Lie group. This serves as a reminder and as a warm-up for the subsection. Where we consider the non-connected case in the framework of principal extensions, which will be defined there. We show how this fairly abstract construction can be applied to the well-known example of O(k) groups, and how it can be visualized via brane constructions in string theory. The readers who are more interested in physical considerations can skip all of this section, taking for granted the formula (2.19) that is used in subsequent developments
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