On Differential Structure for Projective Limits of Manifolds

TL;DR

This paper compares two differential calculi on projective limits of manifolds, establishing their equivalence in certain cases and exploring applications in String Theory and gauge theories.

Contribution

It analyzes and compares Ashtekar-Lewandowski and Froehlicher-Kriegl calculi, proving their equivalence for product manifolds and discussing examples in physics.

Findings

Boman theorem proved for product manifolds

Equivalence of the two calculi in specific cases

Applications in String Theory and loop quantum gravity

Abstract

We investigate the differential calculus defined by Ashtekar and Lewandowski on projective limits of manifolds by means of cylindrical smooth functions and compare it with the C^infty calculus proposed by Froehlicher and Kriegl in more general context. For products of connected manifolds, a Boman theorem is proved, showing the equivalence of the two calculi in this particular case. Several examples of projective limits of manifolds are discussed, arising in String Theory and in loop quantization of Gauge Theories.