Local gauge field theory from the perspective of non-linear PDE geometry

Abstract: Higher structures and derived geometry have become ubiquitous tools when studying the mathematics of quantum field theory. Specifically, shifted Poisson structures and their quantization have found application in quantum field and string theory with derived symplectic geometry providing a powerful reinterpretation of the AKSZ formalism.

In the most `basic' setting, these notions appear when describing the homotopical space of critical points of an action functional. Rather than start with the critical locus, we would like to study the corresponding space of solutions of the equation of motion and the natural geometric structures it possesses. The upshot of this type of an approach is that we can study non-linear PDEs which are not necessarily of Euler-Lagrange form.

In my talk I will describe a functorial approach to non-linear PDEs in the presence of symmetries. We will pay special attention to describing gauge field theories and the derived covariant phase space, equipped with its canonical shifted symplectic form.

mathematical physicsalgebraic geometrydifferential geometrydynamical systemssymplectic geometry

Audience: researchers in the topic

Junior Global Poisson Workshop II