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The Geometry of Loop Spaces II: Characteristic Classes

Yoshiaki Maeda, Steven Rosenberg, Fabián Torres-Ardila

TL;DR

The paper develops a loop-space Chern-Weil/Chern-Simons framework using the Wodzicki residue to define Wodzicki-Chern-Simons (WCS) classes on the loop space $LM$ of an odd-dimensional manifold. It proves that residue characteristic forms vanish on mapping spaces and provides explicit, computable local expressions for the WCS forms, including a regularized, $s$-independent version and a vanishing result for $CS^W_3(g)$. The main application shows that, for circle bundles $\overline M_p$ over integral Kähler surfaces with Sasakian structure, the loop-space invariant $\int_{[a^L]} CS^W_5$ is nonzero for large $|p|$, implying $|\pi_1(\mathrm{Diff}(\overline M_p))|=\infty$; this yields infinite families of 5-manifolds with nontrivial diffeomorphism-group topology. The results are illustrated with several examples (e.g., $T^4$, $\mathbb{C}P^2$, $S^2\times S^2$, K3, and $S^2\times S^3$ via Sasaki-Einstein metrics), highlighting how WCS invariants inform the diffeomorphism-group structure in dimension five.

Abstract

Using the Wodzicki residue, we build Wodzicki-Chern-Simons (WCS) classes in $H^{2k-1}(LM)$ associated to the residue Chern character on the loop space $LM$ of a Riemannian manifold $M^{2k-1}$. These WCS classes are associated to the $L^2$ connection and the Sobolev $s=1$ connections on $LM.$ The WCS classes detect several families of 5-manifolds whose isometry group has infinite fundamental group. These manifolds are the total spaces of the circle bundles associated to a multiple $pω, |p|\gg 0$, of the Kähler form $ω$ over an integral Kähler surface.

The Geometry of Loop Spaces II: Characteristic Classes

TL;DR

The paper develops a loop-space Chern-Weil/Chern-Simons framework using the Wodzicki residue to define Wodzicki-Chern-Simons (WCS) classes on the loop space of an odd-dimensional manifold. It proves that residue characteristic forms vanish on mapping spaces and provides explicit, computable local expressions for the WCS forms, including a regularized, -independent version and a vanishing result for . The main application shows that, for circle bundles over integral Kähler surfaces with Sasakian structure, the loop-space invariant is nonzero for large , implying ; this yields infinite families of 5-manifolds with nontrivial diffeomorphism-group topology. The results are illustrated with several examples (e.g., , , , K3, and via Sasaki-Einstein metrics), highlighting how WCS invariants inform the diffeomorphism-group structure in dimension five.

Abstract

Using the Wodzicki residue, we build Wodzicki-Chern-Simons (WCS) classes in associated to the residue Chern character on the loop space of a Riemannian manifold . These WCS classes are associated to the connection and the Sobolev connections on The WCS classes detect several families of 5-manifolds whose isometry group has infinite fundamental group. These manifolds are the total spaces of the circle bundles associated to a multiple , of the Kähler form over an integral Kähler surface.

Paper Structure

This paper contains 11 sections, 23 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Chern-Simons Classes on Loop Spaces
  3. Chern-Weil and Chern-Simons Theory for Finite Dimensional Bundles
  4. Chern-Weil and Chern-Simons Theory for $\Psi{\rm DO}_0^*$-Bundles
  5. Local calculations
  6. WCS Classes and Diffeomorphism Groups of Sasakian $5$-Manifolds
  7. Circle actions
  8. Sasakian structures over Kähler surfaces and diffeomorphism groups
  9. Examples
  10. Proof of Lemma \ref{['lemapp']}
  11. Topologies on $LM$

Key Result

Theorem 2.1

Let $\nabla$ be a connection on $F\longrightarrow M$ with curvature $\Omega\in \Lambda^2(M,Q\times_\rho {\mathfrak g})$. For $P\in I^k(G)$, $P(\Omega)$ is a closed $2k$-form on $M$, and so determines a de Rham cohomology class $[P(\Omega)]\in H^{2k}(M).$ The Chern-Weil map is a well-defined algebra homomorphism, and in particular is independent of the choice of connection on $F$.

Theorems & Definitions (60)

  • Remark 2.1
  • Theorem 2.1: The Chern-Weil Homomorphism
  • Remark 2.2
  • Definition 2.1
  • Remark 2.3
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • ...and 50 more