We present in the context of supersymmetric gauge theories an extension of the Weyl integration formula, 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. As an example we show how to apply the above technique to theories with O(n) gauge group. This way we can obtain the Hilbert series for certain instanton moduli space. In the second part of the talk we focus on a different kind of four dimensional N=2 QFT, with gauge group given by the principal extension of SU(N). We analyse some of the consequences of this choice. We find that the Coulomb branch is in general non-freely generated and that the global symmetry of the Higgs branch is modified in a non-trivial way.
References:
[1] A. Bourget and A. Pini, "Non-Connected Gauge Groups and the Plethystic Program,'' [arXiv:1706.03781 [hep-th]].
[2] A. Bourget, A. Pini and D. Rodríguez-Gómez, "The Importance of Being Disconnected, A Principal Extension for Serious Groups,'' arXiv:1804.01108 [hep-th].
