TITLE: What is an Oper?
QUESTION [14 upvotes]: Given a curve C, and a reductive group G, there is a moduli stack Loc_G(C), the stack of G-local systems.  I keep reading that there's a substack of "opers" but am having trouble locating a definition.  So what's an oper, and how should I think about them?

REPLY [2 votes]: For reviews, one can try the articles of Frenkel and Teschner.
In the physics context, a fairly recent article which uses opers is this one, where Gaiotto and Witten define an oper for $G = \text{SU}(2)$ to be a flat rank two complex bundle $E$ over a Riemann surface $C$, with structure group $SL(2, \mathbb{C})$, together with a holomorphic line sub-bundle $L$ in $E$ such that $L$ is nowhere invariant under parallel transport by the connection $\mathcal{D}_z$ on $E$.
An oper which is endowed with a covariantly constant reduction of its structure group to a Borel subgroup (ie. the group of upper triangular matrices) is called a Miura oper.  This notion is also described in detail in the above article by Frenkel.