See also the previous Seminar on derived critical loci .
But in the practice of physics, much of the work happens before a quantum field theory is even obtained: in the process of quantization of a classical field theory. This process involves a multitude of further higher geometric tools.
A grand overview of the topic complex Classical field theory, Quantization, Quantum field theory;
For a large class of comparatively simple field theories – most of them still very rich – path integral quantization can be made sense of as a kind of higher integral transform given by pull-tensor-push operations in cohomology.
Also string topology can be interpreted this way, at least in parts. See
The action functionals that govern quantum field theory come from Lagrangians on jet bundles. The differential calculus on these, that governs the construction of covariant phase spaces as the solution space to Euler-Lagrange equations of motion is variational calculus .
The jet bundle geometry that supports variational calculus has its natural home in D-geometry, the higher geometry over a infinitesimal path ∞-groupoid/de Rham space : for any bundle, its jet bundle is the direct image along the constant infintisimal path inclusion .
A quasicoherent sheaf over is equivalently a D-module on . Therefore algebraically one characterizes spaces over as D-schemes (BeilinsonDrinfeld. All of variational calculus has an elegant and powerful formalization in this D-geometry . See (Paugam) for a remarkably comprehensive discussion in the context of quantum field theory.
A remarkable attempt at collecting together the latest modern mathematical developments on various aspects of quantum field theory is in
The D-geometric-formulation of variational calculus has been introduced in chapter 2 of