Spaces of fields and the factorizable Feigin-Frenkel center

We prove a factorizable version of the Feigin-Frenkel theorem. Given a simple finite dimensional Lie algebra $\mathfrak{g}$ and a smooth curve $C$ over $\mathbb{C}$, we canonically identify the (a fortiori) factorization algebra of the center of the enveloping algebra of the affine Kac-Moody algebra at critical level $\hat{\mathfrak{g}}_{C,k_c}$ with the factorization algebra of functions on the space of $\check{\mathfrak{g}}$-Opers over the pointed disk $\Fun(\Op_{\check{\mathfrak{g}}}(D^*))$, where $\gogl$ is the Langlands dual Lie algebra. In order to do so we develop an extension of the usual concept of field in vertex algebra theory which is well adapted to deal with factorization properties and makes it possible to upgrade vertex algebra statements to their global and factorizable versions.