An article that we have written:
Domenico Fiorenza, Chris Rogers, Urs Schreiber
Higher geometric prequantum theory
lecture notes:
Urs Schreiber, Higher geometric prequantum theory and The brane bouquet, talk at Bayrischzell workshop 2013 (pdf slides)
on the higher geometric refinement of prequantum field theory and geometric prequantization formulated generally in cohesive higher geometry.
In the companion article L-∞ algebras of local observables from higher prequantum bundles we discuss the L-∞ algebraic aspects in more detail. See also the discussion at higher Atiyah groupoid.
How higher geometric prequantum theory serves as a foundation for traditional classical field theory, local prequantum field theory and local quantum field theory is further expanded on in
Applications include
We develop the refinement of geometric prequantum theory to higher geometry (higher stacks), where a prequantization is given by a higher principal connection (a higher gerbe with connection). We show how there is canonically a tower of higher “gauge groupoids” assigned to a higher prequantization and establish the higher and integrated Kostant-Souriau extension of higher Hamiltonian symplectomorphisms fairly generally. In the special case of higher differential geometry over smooth manifolds we find the L-∞ algebra extension of Hamiltonian vector fields which is the higher-Poisson bracket of local observables and show that it is equivalent to the construction in n-plectic geometry proposed by Rogers. Finally we indicate examples of applications of higher prequantum theory in the extended geometric quantization of local quantum field theories and specifically in string geometry.
differential cohomology in a cohesive topos
Higher geometric prequantum theory