We explore aspects of the correspondence between Seifert 3-manifolds and 3d $\mathcal{N}=2$ supersymmetric theories with a distinguished abelian flavour symmetry. We give a prescription for computing the squashed three-sphere partition functions of such 3d $\mathcal{N}=2$ theories constructed from boundary conditions and interfaces in a 4d $\mathcal{N}=2^*$ theory, mirroring the construction of Seifert manifold invariants via Dehn surgery. This is extended to include links in the Seifert manifold by the insertion of supersymmetric Wilson-'t Hooft loops in the 4d $\mathcal{N}=2^*$ theory. In the presence of a mass parameter for the distinguished flavour symmetry, we recover aspects of refined Chern-Simons theory with complex gauge group, and in particular construct an analytic continuation of the $S$-matrix of refined Chern-Simons theory.
Refined Painlevé/gauge theory correspondence and quantum tau functions
International School for Advanced Studies (SISSA)（高级研究国际学院）；National Institute for Nuclear Physics (INFN), Section of Trieste（意大利国家核物理研究所的里雅斯特分部）；Institute for Geometry and Physics (IGAP)（几何与物理研究所）
Refined Vafa-Witten invariants for toric surfaces from supersymmetric localization in 5D gauge theory
Philippe Meyer Institute, Physics Department, École Normale Supérieure (ENS), Université PSL（菲利普·梅耶研究所，物理系，巴黎高等师范学院，巴黎文理研究大学）；Laboratoire de Physique Théorique et Hautes Energies (LPTHE), UMR 7589 CNRS-Sorbonne Université（理论与高能物理实验室，CNRS-索邦大学联合研究单位）；SISSA（的里雅斯特国际高等研究学校）；INFN, Sezione di Trieste（意大利国家核物理研究所，的里雅斯特分部）；Institute for Geometry and Physics, IGAP（几何与物理研究所）
