Geometrising the Micro-Cosmos on a Supermanifold

Abstract

For more than half a century, covariant and differential geometric methods have been playing a central role in the development of Quantum Field Theory (QFT). After a brief historic overview of the major scientific achievements using these methods, we will focus on the covariant and differential geometric formalism originally proposed by Vilkovisky and DeWitt (VDW). We discuss recent developmentsmade in addressing the uniqueness of the path-integral measure of the VDW effective action, and so address the problem of quantum frame dependence in cosmologically relevant scalar-tensor theories beyond the classical approximation. Particular attention will be drawn to a long-standing problem concerning the obstacles that the VDW formalism was facing from its original conception in describing generic QFTs that include fermions. We show how in addition to bosons the VDW effective action can be extended to supermanifolds to include fermions. The so-extended formulationappears to be very promising for a complete geometrisation of realistic theories of micro-cosmos, such as the Standard Model and its gravitational sector.