On representation spaces in geometric quantization
Let \(F\) be a strongly admissible polarization of a symplectic manifold \((X,\omega)\). If \(D\) is the involutive distribution such that \(F\cap \bar F = D^{\mathbb C}\), it is shown that the integral manifolds of \(D\) are isomorphic to spaces \(R^{d-k}\times T^k\), \(0\le d = \dim D\). Let \(X_k\) be the union of all integral manifolds of \(D\) isomorphic to \(R^{d-k}\times T^k\). Let \((L,V)\) be the complex line bundle with connection, whose curvature is \(-h^{-1}\omega\), and let \(\sqrt{\wedge^nF}\) be the bundle of \(\frac12\)-forms associated to the metalinear bundle of \(F\). This bundle has a canonical partial covariant derivative in the direction of the polarization \(F\). The Bohr-Sommerfeld set \(S\) is the union of all integral manifolds \(\wedge\) of \(D\) such that the holonomy group of the connection on \(L\otimes \sqrt{\wedge^nF}\) is trivial. For each \(k\in \{(0,\ldots,d\}\) a density valued scalar product is introduced on a certain space of sections of \(L\otimes \sqrt{\wedge^nF}\) with supports in \(S_k = S\cap X_k\). \(H^k\) denotes the Hilbert space of square integrable sections, and \(H = d_{k=0}^d H^k\). It is shown that if \(F\) is real, then \(H \ne \{0\}\) if and only if the Bohr-Sommerfeld set \(S\) is nonempty.
Bohr-Sommerfeld set of the representation
scientific article; zbMATH DE number 3615565
scientific article