Title:
SUSY localization for Coulomb branch operators in omega-deformed 3d N=4 gauge theories

Abstract:
Part I: We first review basics of 3d N=4 supersymmetric (SUSY) gauge theories;  Higgs branch, Coulomb branch, and mirror symmetry etc.
Next we review the construction of abelianized Coulomb branch chiral rings and their deformation quantizations
in terms of abelianization and abelian mirror symmetry in arXiv:1503.04817 [hep-th]. The authors of arXiv:1503.04817 [hep-th] conjectured that
the Coulomb branch chiral rings and their deformation quantizations are isomorphic to  sub rings of abelianized ones.

Part II: We study SUSY localization for vacuum expectation values of Coulomb branch operators
(vector multiplet scalars, bare monopole operators, and dressed operators)  in 3d N=4 SUSY gauge theories in the presence of an omega background.
In our SUSY localization formula, non-perturbative effects called monopole bubbling effects play important roles
in order to reproduce properties of Coulomb branch chiral rings and their deformation quantizations.
We evaluate monopole bubbling effects in terms of D-brane pictures and show that
wall-crossing phenomena in monopole bubbling effects correlate with non-commutativity of the deformation quantization.
Part II is based on a joint work arXiv:1910.01802 [hep-th]  with Takuya Okuda.