[Paper Review] Almost commuting elements in compact Lie groups

Armand Borel, Robert Friedman+1 more|ArXiv.org|Jul 1, 1999

Advanced Algebra and Geometry8 references18 citations

TL;DR

This paper classifies almost commuting pairs and triples of elements in compact, simply connected Lie groups, focusing on their moduli spaces via holonomy representations and Chern-Simons invariants. It proves Witten's 'Clockwise Symmetry Conjecture' by showing the Chern-Simons invariant induces a bijection between components of order $k$ in the moduli space and points of order $k$ in $\mathbb{R}/\mathbb{Z}$, particularly for non-cyclic $\langle C\rangle$. The classification relies on root systems, Weyl groups, and diagram automorphisms of the extended Dynkin diagram.

ABSTRACT

We describe the components of the moduli space of conjugacy classes of commuting pairs and triples of elements in a compact Lie group. This description is in terms of the extended Dynkin diagram of the simply connected cover, together with the coroot integers and the action of the fundamental group. In the case of three commuting elements, we compute Chern-Simons invariants associated to the corresponding flat bundles over the three-torus, and verify a conjecture of Witten which reveals a surprising symmetry involving the Chern-Simons invariants and the dimensions of the components of the moduli space.

Motivation & Objective

• To classify isomorphism classes of flat connections on principal bundles over the two- and three-tori using holonomy representations.
• To characterize almost commuting $N$-tuples in compact, simply connected Lie groups via group-theoretic and geometric invariants.
• To prove Witten's 'Clockwise Symmetry Conjecture' concerning the Chern-Simons invariant on the moduli space of flat $G$-bundles over the three-torus.
• To establish a correspondence between components of the moduli space and the values of the Chern-Simons invariant modulo $\mathbb{Z}$.

Proposed method

• Uses the holonomy representation to translate flat bundle classification into the classification of almost commuting $N$-tuples in the simply connected cover $G$ of a compact semisimple group $K$.
• Applies the extended Dynkin diagram of $G$, the action of $\pi_1(K)$, and coroot integers to characterize the moduli space structure.
• Employs the Weyl group $W(S,G)$ and root systems on fixed subspaces to analyze centralizers and component groups.
• Utilizes group cohomology and component group computations to study $\pi_0(Z(x,y))$ and $\pi_0(H^\sigma)$ for automorphisms $\sigma$.
• Computes the Chern-Simons invariant via flat connections and curvature forms, relating it to the $c$-triple and $C$-triple structures.
• Applies the basic equation in cohomology and the homomorphism $\delta$ to relate component groups to root systems and generalized Cartan matrices.

Experimental results

Research questions

• RQ1 How are almost commuting $N$-tuples in compact, simply connected Lie groups classified up to simultaneous conjugation?
• RQ2 What is the structure of the moduli space of flat $G$-bundles over the two- and three-tori, and how does it relate to the Chern-Simons invariant?
• RQ3 How does the Chern-Simons invariant behave under orientation reversal and covering maps, and what does this imply for component structure?
• RQ4 What is the precise relationship between the component group of the centralizer and the generalized Cartan matrix associated with the root system?
• RQ5 How does the automorphism group of $G$ act on the space of rank-zero $c$-triples and $C$-triples?

Key findings

• The Chern-Simons invariant induces a bijection between components of order $k$ in the moduli space ${\cal T}_G(C)$ and points of order $k$ in $\mathbb{R}/\mathbb{Z}$, for every positive integer $k$ dividing 4.
• For $\langle C\rangle$ cyclic, the Chern-Simons invariant takes values $\pm 1/4 \mod \mathbb{Z}$ on the two components of order 4, with opposite signs on opposite components.
• The $c_0$-triple $\hat{\bf u} = (u,v,w^2)$ lies in the non-trivial component of the moduli space of $c_0$-triples, while $\hat{\bf x} = (x,y,z^2)$ lies in the trivial component.
• The Chern-Simons invariant satisfies $\mathrm{CS}(r^*\Gamma) \equiv -\mathrm{CS}(\Gamma) \mod \mathbb{Z}$ under orientation reversal, confirming symmetry under time-reversal.
• The component group of $H^\sigma$ is computed via the homomorphism $\delta$ and the torus of the restricted root system, linking it to $\mathrm{Tor}((\Lambda/Q^\vee_H)_\sigma)$.
• The proof of Witten's Clockwise Symmetry Conjecture is completed for non-cyclic $\langle C\rangle$ using Corollary 12.4.4 and the component count from Lemma 12.3.1 and Corollaries 12.3.6 and 12.3.9.