Sobolev Maps into Compact Lie Groups and Curvature

TL;DR

This paper explores the curvature properties of Sobolev groups of maps from Riemannian manifolds into compact Lie groups, revealing Einstein manifold structures in critical cases relevant to quantum field theory.

Contribution

It extends Freed's work by analyzing curvature of Sobolev Lie groups at critical Sobolev exponents using advanced trace techniques, establishing Einstein properties.

Findings

W^{1/2}(S^1,K)/K is an Einstein manifold with PSU(1,1) invariance

W^1(Σ,K)/K is an Einstein manifold with conformal invariance

Curvature formulas are surprisingly simple in these critical Sobolev cases

Abstract

These are notes on seminal work of Freed, and subsequent developments, on the curvature properties of (Sobolev Lie) groups of maps from a Riemannian manifold into a compact Lie group. We are mainly interested in critical cases which are relevant to quantum field theory. For example Freed showed that, in a necessarily qualified sense, the quotient space $W^{1/2}(S^1,K)/K$ is a (positive constant) Einstein `manifold' with respect to the essentially unique PSU(1,1) invariant metric, where $W^{s}$ denotes maps of $L^2$ Sobolev order s. In a similarly qualified sense, and in addition making use of the Dixmier trace/Wodzicki residue, we show that for a Riemann surface Sigma, $W^1(\Sigma,K)/K$ is a (positive constant) Einstein `manifold' with respect to the essentially unique conformally invariant metric. As in the one dimensional case, invariance implies Einstein, but the sign of the Ricci curvature has to be computed. Because of the qualifications involved in these statements, in practice it is necessary to consider curvature for $W^s(\Sigma,K)$ for s above the critical exponent, and limits. The formula we obtain is surprisingly simple.