Interfaces and Quantum Algebras, II: Cigar Partition Function

The supersymmetric cigar (half-)index or cigar partition function of 3d $\mathcal{N}=2$ gauge theories contains a wealth of information. Physically, it captures the spectrum of BPS states, the non-perturbative corrections to various partition functions, the effective twisted superpotential and the data of supersymmetric vacua. Mathematically, it defines the K-theoretic Vertex counting vortices/quasimaps, and connects to quantum K-theory, as well as elliptic cohomology and stable envelopes. We explore these topics from the physics standpoint, systematically developing the foundations and explaining various mathematical properties using the quantum field theory machinery.