localization

This refers to supersymmetric localization, the extension of Atiyah-Bott localization formula. It deals with the infinite dimensional (usually over space of fields) path integral in differential geometry/SQFT.
eg. cohomological field theory

When the integral is actually independent of parameters (eg. by Stokes theorem; Witten index), we can put the parameters to such a limit that only a few critical points make leading contributions.
eg. SQM (Witten index) and Atiyah-Singer index theorem (attention: Witten index localizes on constant configurations; only after adding the potential (Morse theory) will we have vacua localized at minima, but the new index counts unpaired zero modes of instantons and has nothing to do with the index theorem)

symplectic form-symplectic integral (finite-dim?)
In Chern-Simons theory, the vortex line is coincident with Wilson line, and reduces to the Chern-character by non-Abelian localization. Besides, the partition function can be expressed by the equivariant cohomology over the moduli space of flat connections due to non-Abelian localization.