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Diffeomorphism-invariant generalized measures on the space of connections modulo gauge transformations

John C. Baez

TL;DR

The paper develops a rigorous framework for diffeomorphism-invariant generalized measures on the space of connections modulo gauge transformations by recasting measures as invariant states on holonomy C*-algebras. It reduces the infinite-dimensional problem to finite-dimensional data through lattice-like graphs embedded in the base manifold and characterizes invariant states via covariant, uniformly bounded measures on Hom(pi1(phi),G). Concrete constructions include the flat state and the Ashtekar-Lewandowski state, with a general method to build many more from vertex-type data. The work clarifies the relationship between diffeomorphism-invariant gauge theories and knot theory, and outlines generalizations toward Chern-Simons-type measures, quantum groups, and tube regularizations for framing-aware invariants.

Abstract

The notion of a measure on the space of connections modulo gauge transformations that is invariant under diffeomorphisms of the base manifold is important in a variety of contexts in mathematical physics and topology. At the formal level, an example of such a measure is given by the Chern-Simons path integral. Certain measures of this sort also play the role of states in quantum gravity in Ashtekar's formalism. These measures define link invariants, or more generally multiloop invariants; as noted by Witten, the Chern-Simons path integral gives rise to the Jones polynomial, while in quantum gravity this observation is the basis of the loop representation due to Rovelli and Smolin. Here we review recent work on making these ideas mathematically rigorous, and give a rigorous construction of diffeomorphism-invariant measures on the space of connections modulo gauge transformations generalizing the recent work of Ashtekar and Lewandowski. This construction proceeds by doing lattice gauge theory on graphs analytically embedded in the base manifold.

Diffeomorphism-invariant generalized measures on the space of connections modulo gauge transformations

TL;DR

The paper develops a rigorous framework for diffeomorphism-invariant generalized measures on the space of connections modulo gauge transformations by recasting measures as invariant states on holonomy C*-algebras. It reduces the infinite-dimensional problem to finite-dimensional data through lattice-like graphs embedded in the base manifold and characterizes invariant states via covariant, uniformly bounded measures on Hom(pi1(phi),G). Concrete constructions include the flat state and the Ashtekar-Lewandowski state, with a general method to build many more from vertex-type data. The work clarifies the relationship between diffeomorphism-invariant gauge theories and knot theory, and outlines generalizations toward Chern-Simons-type measures, quantum groups, and tube regularizations for framing-aware invariants.

Abstract

The notion of a measure on the space of connections modulo gauge transformations that is invariant under diffeomorphisms of the base manifold is important in a variety of contexts in mathematical physics and topology. At the formal level, an example of such a measure is given by the Chern-Simons path integral. Certain measures of this sort also play the role of states in quantum gravity in Ashtekar's formalism. These measures define link invariants, or more generally multiloop invariants; as noted by Witten, the Chern-Simons path integral gives rise to the Jones polynomial, while in quantum gravity this observation is the basis of the loop representation due to Rovelli and Smolin. Here we review recent work on making these ideas mathematically rigorous, and give a rigorous construction of diffeomorphism-invariant measures on the space of connections modulo gauge transformations generalizing the recent work of Ashtekar and Lewandowski. This construction proceeds by doing lattice gauge theory on graphs analytically embedded in the base manifold.

Paper Structure

This paper contains 6 sections, 7 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Generalized Measures
  3. The Analytic Holonomy C*-algebra
  4. Reduction to the Finite-Dimensional Case
  5. Constructing Diffeomorphism-invariant States
  6. Generalizations and Conclusions

Key Result

Proposition 1

. Suppose that $\mu \in {\bf H}^\ast_{inv}$. Then there is an ambient isotopy invariant of multiloops ${\cal L}_\mu$ given by for any multiloop $\gamma = \{\gamma_i\}$. Moreover, the map $\mu \mapsto {\cal L}_{\mu}$ is one-to-one.

Theorems & Definitions (7)

  • Proposition 1
  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Proposition 2
  • Theorem 2
  • Corollary 1