Quantum field theoretic representation of Wilson surfaces. Part I. Higher coadjoint orbit theory

This is the first of a series of two papers devoted to the partition function re- alization of Wilson surfaces in strict higher gauge theory. A higher version of the Kirillov- Kostant-Souriau theory of coadjoint orbits is presented based on the derived geometric framework, which has shown its usefulness in 4-dimensional higher Chern-Simons theory. An original notion of derived coadjoint orbit is put forward. A theory of derived unitary line bundles and Poisson structures on regular derived orbits is constructed. The proper derived counterpart of the Bohr-Sommerfeld quantization condition is then identified. A version of derived prequantization is proposed. The difficulties hindering a full quantiza- tion, shared with other approaches to higher quantization, are pinpointed and a possible way-out is suggested. The theory we elaborate provide the geometric underpinning for the field theoretic constructions of the companion paper.