Geometry of field configuration spaces and quantum mass gap

Given a classical bosonic field theory, the kinetic term of its action induces a Riemannian metric on the configuration space. Recent works by Moncrief, Mondal, et al. propose that a 'weighted' Ricci curvature from this metric controls the mass gap of the corresponding quantum field theory. Such curvature, known as Bakry-Emery, is equal to the Ricci tensor plus the Hessian of a functional that is responsible for the normalization of the ground state. This result, although elegant, is heuristic for two reasons: First, it assumes the existence of a rigorous quantum field theory, which is a tall order in itself. Secondly, the curvatures are UV divergent; thus, one must adopt a renormalization scheme to extract the physical mass gap, and this process is yet to be understood. In this presentation, I will give an overview of such arguments with a focus on pure Yang-Mills theory in 3D and 4D. Further, I will discuss ongoing work on how to extend the results to fermionic theories.