We describe the quantization of 2-plectic manifolds as they arise in the
context of the quantum geometry of M-branes and non-geometric flux
compactifications of closed string theory. We review the groupoid approach to
quantizing Poisson manifolds in detail, and then extend it to the loop spaces
of 2-plectic manifolds, which are naturally symplectic manifolds. In
particular, we discuss the groupoid quantization of the loop spaces of R^3, T^3
and S^3, and derive some interesting implications which match physical
expectations from string theory and M-theory.Comment: 71 pages, v2: references adde
