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

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.