Invariance of polarization induced by symplectomorphisms (Q6613466)

A famous body of results in geometric quantization are the invariance of polarization results, where it can be shown that two naturally arising polarizations induce the same quantization. Two standard examples are the Gelfand-Zeitlin system [\textit{V. Guillemin} and \textit{S. Sternberg}, J. Funct. Anal. 52, 106--128 (1983; Zbl 0522.58021), Theorem 6.1] and the moduli space of flat \(SU(2)\) connections [\textit{L. C. Jeffrey} and \textit{J. Weitsman}, Commun. Math. Phys. 150, No. 3, 593--630 (1992; Zbl 0787.53068), Theorem 8.3].\N\NThis paper shows a new kind of invariance theorem arising from the action of a symplectomorphism on the polarizations. The author obtains the following two theorems, where \underline{\(\mathbb{C}\)}\(_{M}^{\times}\) denotes the sheaf of locally constant, nonvanishing complex functions on \(M\).\N\NTheorem. Let \((M,\omega)\) be a symplectic manifold, \(P\) a real polarization, \(\Phi:M\rightarrow M\) a symplectomorphism, and \((L,\nabla)\rightarrow M\) a prequantum line bundle. If \(H^{1}(M,\underline{\mathbb{C}}_{M}^{\times})=0\), then \(\Phi\) induces an isomorphism between the sheaf quantizations\N\[\NQ_{shf}(M.P,\nabla)\rightarrow Q_{shf}(M.\Phi^{\ast }P,\nabla)\N\]\Nwhere \(Q_{shf}(M.P,\nabla)\) is defined by\N\[\NQ_{shf}(M.P,\nabla)=\bigoplus\limits_{n}H^{n}(M,S_{(P,\nabla)})\N\]\Nwith the Čech cohomology groups \(H^{n}(M,S_{(P,\nabla )})\) associated to the sheaf.\N\NTheorem. Let \((M,\omega)\) be a symplectic manifold, \(P\) a real polarization, \(\Phi:M\rightarrow M\) a symplectomorphism, and \((L,\nabla)\rightarrow M\) a prequantum line bundle. If \(H^{1}(M,\underline{\mathbb{C}}_{M}^{\times})=0\), then \(\Phi\) induces an isomorphism between the Bohr-Sommerfeld quantizations.\N\NFor the entire collection see [Zbl 1540.57001].