The Poisson bracket invariant on surfaces (Q6635457)

The authors study a Poisson bracket invariant on closed symplectic surfaces and obtain its several estimates from below by a universal constant.\N\N\textit{L. Polterovich} [Lett. Math. Phys. 102, No. 3, 245--264 (2012; Zbl 1261.81078)] introduced the so-called Poisson bracket invariant on a symplectic manifold, a quantitative measure of Poisson non-commutativity of functions forming a partition of unity \(\mathcal F\), telling that his invariant relates to operational quantum mechanics, that is, gives a lower bound on the statistical noise of measurements of the collections of quantum observables associated to \(\mathcal F\) via Berezin-Toeplitz quantization.\NTherefore its lower bounds become interesting.\N\NThe paper under review considers the invariant when the symplectic manifolds are closed surfaces equipped with an area form.\NThe main result is:\N\NThere exist constants \(C, C^{\prime}>0\) such that the following holds. Let \((M, \omega)\) be a closed symplectic surface and \(\mathcal{F}\) be a partition of unity subordinate to a finite open cover \(\mathcal{U}=\left\{U_i\right\}_{i \in I}\) by topological discs of area no larger than \(c\). Let \(X=\left\{x_1, \ldots, x_m\right\} \subset M\) (note that \(m \geq 1\) and for \(M=S^2\), \(m \geq 3\)) be points at which \(\mathcal{U}\) localizes, i.e., each disc \(U_i \in \mathcal{U}\) contains no more than one \(x_k \in X\). Then\N\[\Np b(\mathcal{F}) c \geq C \quad \text { and } \quad p b(\mathcal{F}) \operatorname{Area}(M, \omega) \geq C^{\prime} m\N\]\Nwhere\N\[\Np b(\mathcal{F}):=\sup _{0\leq a_j, b_j \leq 1}\left\|\left\{\sum_{i \in I} a_i f_i, \sum_{j \in I} b_j f_j\right\}\right\|.\N\]\NA more precise version is given in Theorem 1.4.17 (Poisson bracket theorem).\N\NThe results confirm Polterovich's conjecture [\textit{L. Polterovich}, Commun. Math. Phys. 327, No. 2, 481--519 (2014; Zbl 1291.81198)] for those open covers \(\mathcal{U}\) on \(M=S^2\).\NA positive answer to Polterovich's conjecture for all closed symplectic surfaces was recently given by Buhovsky-Logunov-Tanny [\textit{L. Buhovsky} et al., Comment. Math. Helv. 95, No. 2, 247--278 (2020; Zbl 1454.53063)]. Concerning these estimates, the author also discusses sharpness of his results.