The Gaiotto-Kim Trace Formula and Line Defects

Abstract: We study the insertion of 4d half-BPS line defects into the trace formula proposed by Gaiotto and Kim, which computes the ellipsoid partition function of the 3d topological quantum field theory (TQFT) arising from a 4d $\mathcal{N}=2$ superconformal field theory. This yields a homomorphism to 3d Wilson loops, preserving the fusion algebra of the 4d defects. We also present a systematic algorithm for determining the simple objects of a 3d TQFT in terms of Wilson loops in the UV 3d $\mathcal{N}=2$ abelian Chern-Simons-matter theory. Utilizing supersymmetric localization, we characterize the modular $S$- and $T$-matrices of the TQFT. We apply our algorithm to various 3d TQFTs arising from the trace formula with higher-power monodromy of the $(A_1,G)$ Argyres-Douglas theories.