A new kind of quantum field theory of (n-1)-dimensional defects in 2n dimensions

Abstract

I describe a project to open a new territory of quantum field theory where the fields live not on a space-time manifold but on certain complete metric spaces of (n-1)-dimensional objects (defects) in a 2n-dimensional space-time M. These metric spaces are "quasi Riemann surfaces"; they are formally analogous to Riemann surfaces. Every construction of a 2d conformal field theory is to give an analogous construction of a cft on the quasi Riemann surfaces, and thereby a cft on M. The global symmetry group of the 2d cft becomes a local gauge symmetry. Ordinary local quantum fields in space-time are constructed by restricting to small objects. The project is based on writing the free n-form in 2n dimensions as the 2d gaussian model on the quasi Riemann surfaces. This note is a summary of the main points of arXiv:1605.03279.