Topology, Algebraic Geometry, and Dynamics Seminar (TADS): Quantum Field Theory and Derived Differential Geometry (Part I)

Speaker:  David Carchedi, GMU

Title:  Quantum Field Theory and Derived Differential Geometry (Part I)

Abstract:  In a quantum theory, physicists express expectation values of observable properties formally as an integral over an infinite dimensional space of fields. In actuality, these values are computed as a sum of contributions of certain graphs called Feynman diagrams. When computing with the Feynman diagrams of non-abelian gauge theories, like those involved in quantum chromodynamics, physicists discovered that they had to add auxiliary “ghost fields” in order to extract sensible answers. Similarly, while quantizing supergravity, physicists introduced “antifields” to partner with every field. In this talk, we will show how these extra fields pop up naturally from considerations in derived geometry.

Derived differential geometry is an enhancement of the theory of manifolds developed to handle non-transverse intersections. In its essence, it is a hybrid of differential geometry, homological algebra, and algebraic geometry. We will derive an explicit model for derived manifolds starting from a simple universal property. We will then explain how to construct from a classical field theory an infinite dimensional moduli space of derived solutions to the PDEs describing its dynamics, and how ghost fields and anti-fields occur naturally in its tangent complex. We will work this out explicitly for Chern-Simons and Yang-Mills.

Time:  Friday, September 30, 2022, 1:30pm – 2:30pm

Place:  Exploratory Hall, room 4208