We are developing local prequantum field theory, in between classical field theory and quantum field theory, that takes global coherence data into account – such as cancellation of classical anomalies – and does so in a fully local way, lifting Kostant-Souriau pre-quantization from mechanics to local field theory.
The local variational aspect of prequantum field theory is discussed in
Prequantum covariant field theory
( talk slides 1: pdf, talk slides 2: talk slides pdf)
For motivation, exposition and survey see
Given an $n$-dimensional local prequantum field theory, it naturally organizes its evolution in n-fold correspondences in the slice (∞,1)-topos over differential cohomology coefficients. This aspect is discussed in
Urs Schreiber, based on discussion with Domenico Fiorenza
Local prequantum field theory
(pdf)
(now section 3.7.11 (4.2.19) of dcct)
Particularly, this discusses examples of ∞-Chern-Simons theory and its boundary ∞-Wess-Zumino-Witten theory.
The discussion of these prequantized Lagrangian correspondences is inspired on the one hand from observations by Domenico Fiorenza, and Alessandro Valentino about boundary conditions in local prequantum field theory, which meanwhile has become available as
and on the other from observations by Hisham Sati on towers of higher codimension boundary field theories/corner field theories arising in string theory/M-theory:
The conjectures about the behaviour of $n$-fold correspondences have meanwhile been proven in
Discussion of traditional classical field theory in this context is at Classical field theory via Cohesive homotopy types and at
Higher prequantum geometry I: The need for prequantum geometry
Higher prequantum geometry II: The principle of extremal action – Comonadically
Higher prequantum geometry III: The global action functional – Cohomologically
Higher prequantum geometry IV: The covariant phase space – Transgressively
Higher prequantum geometry V: The local observables – Lie theoretically.
The quantization of this local prequantum field theory is to proceed by Motivic quantization of local prequantum field theory, details are in
and in
The higher extensions of diffeomorphism groups involved are discussed at
Another survey of the general perspective is at Synthetic Quantum Field Theory.