  In this article we present a pedagogical discussion of some of the
optomechanical properties of a high finesse cavity loaded with ultracold atoms
in laser induced synthetic gauge fields of different types. Essentially, the
subject matter of this article is an amalgam of two sub-fields of atomic
molecular and optical (AMO) physics namely, the cavity optomechanics with
ultracold atoms and ultracold atoms in synthetic gauge field. After providing a
brief introduction to either of these fields we shall show how and what
properties of these trapped ultracold atoms can be studied by looking at the
cavity (optomechanical or transmission) spectrum. In presence of abelian
synthetic gauge field we discuss the cold-atom analogue of Shubnikov de Haas
oscillation and its detection through cavity spectrum. Then, in the presence of
a non-abelian synthetic gauge field (spin-orbit coupling), we see when the
electromagnetic field inside the cavity is quantized, it provides a quantum
optical lattice for the atoms, leading to the formation of different quantum
magnetic phases. We also discuss how these phases can be explored by studying
the cavity transmission spectrum.
