Non-existence of genuine (compact) quantum symmetries of compact, connected smooth manifolds

TL;DR

This paper proves that any faithful smooth action of a compact quantum group on a compact, connected manifold must be classical, confirming the conjecture that such quantum symmetries do not exist beyond classical group actions.

Contribution

It establishes that compact quantum groups cannot act faithfully and smoothly on connected manifolds unless they are classical, resolving a longstanding conjecture.

Findings

Quantum symmetries are always classical for connected manifolds.

The proof employs probabilistic techniques involving Brownian stopping times.

No non-commutative quantum symmetries exist in this setting.

Abstract

Suppose that a compact quantum group ${\mathcal Q}$ acts faithfully on a smooth, compact, connected manifold $M$, i.e. has a $C^{\ast}$ (co)-action $\alpha$ on $C(M)$, such that $\alpha(C^\infty(M)) \subseteq C^\infty(M, {\mathcal Q})$ and the linear span of $\alpha(C^\infty(M))(1 \otimes {\mathcal Q})$ is dense in $C^\infty(M, {\mathcal Q})$ with respect to the Frechet topology. It was conjectured by the author quite a few years ago that ${\mathcal Q}$ must be commutative as a $C^{\ast}$ algebra i.e. ${\mathcal Q} \cong C(G)$ for some compact group $G$ acting smoothly on $M$. The goal of this paper is to prove the truth of this conjecture. A remarkable aspect of the proof is the use of probabilistic techniques involving Brownian stopping time.