  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.
