Quantum field theories of extended objects

TL;DR

This paper proposes a framework to construct quantum field theories of extended objects in higher dimensions by leveraging 2D conformal field theories, aiming to unify and extend existing QFT techniques.

Contribution

It introduces a novel approach to build higher-dimensional QFTs of extended objects from 2D CFTs, connecting geometric spaces of cycles to quantum fields.

Findings

Initial steps in defining the space of extended objects as metric spaces.

Proposed mapping of higher-dimensional fields to 2D CFT observables.

Lays groundwork for future construction of non-conformal QFTs of extended objects.

Abstract

First steps are taken in a project to construct a general class of conformal and perhaps, eventually, non-conformal quantum field theories of (n-1)-dimensional extended objects in a d=2n dimensional conformal space-time manifold M. The fields live on the spaces E of relative integral (n-1)-cycles in M -- the integral (n-1)-currents of given boundary. Each E is a complete metric space geometrically analogous to a Riemann surface $\Sigma$. For example, if $M=S^d$, $\Sigma = S^2$. The quantum fields on E are to be mapped to observables in a 2d CFT on $\Sigma$. The correlation functions on E are to be given by the 2d correlation functions on $\Sigma$. The goal is to construct a CFT of extended objects in d=2n dimensions for every 2d CFT, and eventually a non-conformal QFT of extended objects for every non-conformal 2d QFT, so that all the technology of 2d QFT can be applied to the construction and analysis of quantum field theories of extended objects. The project depends crucially on settling some mathematical questions about analysis in the spaces E. The project also depends on extending the observables of 2d CFT from the finite sets of points in a Riemann surface to the integral 0-currents.