  In this work we compare different descriptions of the space of vacua of
certain three dimensional N=4 superconformal field theories, compactified on a
circle and mass-deformed to N=2 in a canonical way. The original N=4 theories
are known to admit two distinct mirror descriptions as linear quiver gauge
theories, and many more descriptions which involve the compactification on a
segment of four-dimensional N=4 super Yang-Mills theory. Each description gives
a distinct presentation of the moduli space of vacua. Our main result is to
establish the precise dictionary between these presentations. We also study the
relationship between this gauge theory problem and integrable systems. The
space of vacua in the linear quiver gauge theory description is related by
Nekrasov-Shatashvili duality to the eigenvalues of quantum integrable spin
chain Hamiltonians. The space of vacua in the four-dimensional gauge theory
description is related to the solution of certain integrable classical
many-body problems. Thus we obtain numerous dualities between these integrable
models.
