Symbolic calculus for Toeplitz operators with half-form (Q873763)

The author considers semi-classical analysis with phase space a compact Kähler manifold endowed with a prequantum line bundle. Let \(M\) be a compact Kähler manifold of complex dimension \(n\). The square root of its canonical bundle \(\Lambda^{n, 0} T^*M\) is called an half-form bundle. If \( \omega \in \Omega^2(M, \mathbb{R})\) is the foundamental form of \(M\) and if \(L \to M\) is a prequantization bundle (i.e., a Hermitian line bundle with a connection \(\nabla^L\) of curvature \(\frac{1}{i}\omega\)), then, for \(K \to M\) a Hermitian line bundle, the quantum space \(\mathcal{H}_k (k \in \mathbb{N})\) is the space of holomorphic sections of \(L^k \otimes K\). If \(M\) carries a half-form bundle \((\delta, \varphi)\), \(\delta \to M\) inherits a Hermitian scalar product. Let \(L_1\) be a holomorphic line bundle such that \(K = L_1 \otimes \delta\); then \(\mathcal{H}_k\) is finite-dimensional and its dimension can be computed by an integral depending on \(\omega\) and \(\omega_1\) where \(\frac{1}{i}\omega_1\) is the curvature of the Chern connection of \(L_1\). If \(M\) does not carry a half-form bundle, the same formula is valid with \(\omega_1 = \omega_k - \frac{\omega_c}{2}\) where \(\frac{1}{i} \omega_i\) is the curvature of the Chern connection of \(\Lambda^{0, n}T^*M\). The Toeplitz operations are defined as follows: let \(\prod_k\) be the orthogonal projection of \(L^2(M, L^k \otimes K)\) onto \(\mathcal{H}_k\). Then a Toeplitz operator is a sequence \(\{T_k : \mathcal{H}_k \to \mathcal{H}_k\}\) with \(T_k = \prod_k f (\cdot, k) + R_k\), where \(f(\cdot, t)\) is a sequence of \(C^\infty(M)\) functions having an asymptotic expansion \(f_0 + \hslash f_1 + \ldots\) (in the \(C^\infty\) topology) and the norms of \(R_k\) are \(O(k^{-\infty})\). The set \(T\) of Toeplitz operators is a semi-classical algebra associated to \((M,\omega)\). The author introduces the normalized symbol of a Toeplitz operator. The Weyl symbol \(\sigma_{\text{norm}}\) [cf. \textit{L. Charles}, Commun. Partial Differ. Equations 28, No. 9--10, 1527--1566 (2003; Zbl 1038.53086)] in case \(K\) is the trivial line bundle is equal to the normalized symbol moduls \(O(\hslash^2)\). The map \(\tau_{\text{norm}}:T \to C^{\infty}[[h]]\) enjoys the same properties as the map \(\sigma_{\text{conv}}\). The \(\ast_{\text{norm}}\) is a normalized star product. The definition of \(\sigma_{\text{norm}}\) somehow agrees with the usual procedure to quantize observables on geometric quantization. If \(M\) is two-dimensional, and \((T_k)\) a self adjoint Toeplitz operator its to \(f_0+\hslash f_1+\dots\) normalized symbol is real valued. The Bohr-Sommerfeld condition gives the spectrum of \(T_k\) on every open interval \(I\) of regular values of \(f_0\) in the semi-classical limit. The Bohr-Sommerfeld conditions are given in terms of \(a^i \in C^{\infty}(I)\) (the principal action), of \(a^i_1\) (the subprincipal action) and of an index \(\varepsilon^i\) from the half-form bundle. Denote with \(\sum^i (k)\) the set of \(\lambda \in Z\) satisfying the B-S condition, \(\sum(k)=\bigcup_i \sum^i (k)\). The multiplicity of \(\lambda \in \sum(k)\) is the number of \(\sum^i (k)\) which contains \(\lambda\). The points of \(\sum(k)\) approximate (in a precise sense given by Theorem~2.1.) the eigenvalues of \(T\). Under more restrictive assumptions a similar result holds also in higher dimensions. It is known that the trace of a Toeplitz operator with normalized symbol admits an asymptotic expansion. From the fact that the quantization by Toeplitz operators is locally equivalent with Weyl quantization, it follows that the trace \(f(h) \to (2\pi h)^{-n} \int_{M} f(h)\,d(h) \frac{\mu^n}{n!}\) is the same as the canonical trace (for the existence of an unique canonical class see \textit{B. Fedosov} [Deformation quantization and index theory, Berlin: Akademie Verlag (1996; Zbl 0867.58061)]). In particular, \(\text{tr}f=(2\pi h)^{-n} \int_M \frac{f(\omega + h\omega_1)^n}{n!}+O(h^{-n+2})\) (\(M\) need not to be 2-dimensional). Using results of Fedosov [loc.\,cit.], one can deduce the Riemann-Roch-Hirzebruch theorem by use of the above formula. If \(i : \Gamma \to M\) is a closed Lagrange embedding, let \(V\) be an open set of \(M\) such that \(U_{\Gamma}=i^{-1}(U)\) is contractible. Consider the formal series \(\sum_{l=0}^{\infty} \hslash^l g_l \in C^{\infty}(U_{\Gamma}, V^{-1}k)[[h]]\) and let \(V\) be open in \(M\), \(\bar{V} \subset U\). The sequence \(\Psi_k \in \mathcal{H}_k\) is a Lagrangian section over \(V\) associated to \((\Gamma, t_{\Gamma})\) with symbol \(\sum \hslash^l g_l\) if \(\Psi_k=\left( \frac{k}{2\pi} \right)^{n/4} F^{k/n} \widetilde g (x,k) + O(k^{-\infty})\) where \(F\) is a section of \(L\) such that \(i^{*} F=t_{\Gamma}\), \(\overline\partial F\equiv 0\), \(t_{\Gamma}\) is a flat unitary section of \(i^{*} L \to U_{\Gamma}\) (whose existence results from the fact that the curvature of \(i^* L\) vanishes), \(\widetilde g (1,k)\) is a sequence in \(C^{\infty}(U,k)\) with asymptotic expansion \(\sum {\bar k}^l \widetilde{g_l}\) such that \(i^* \widetilde{g_l}=g_l\), \(\bar{\partial} \widetilde{g_l}\equiv 0\) modulo a section which vanishes of infinite order along \(i(\Gamma)\). The author proves that for every series \(\sum \hslash^l g_l \in C^{\infty}(U_{\gamma}, i^* k) [[k]]\) there exists a Lagrangian section over \(V\) associated with \((\Gamma, t_{\Gamma})\) with symbol \(\sum \hslash^l g_l\), unique modulo a section which is \(O(k^{-\infty})\) over \(V\). The author also shows that if \(T_k\) is a Toeplitz operator with principal symbol \(f_0\), then \(T_k \Psi_k\) is a Lagrangian section over \(V\) associated with \((\Gamma, t_{\Gamma})\) with symbol \((i^* f_0)g_0 +O(\hslash)\), and obtains estimates of the norm of \(\Psi_k\). This results allow the author to prove the B-S conditions for \(n\) self-adjoint commuting Toeplitz operators \(T^1, \dots, T^n\). In an Appendix the author gives a result from which the existence of a Lagrangian section with an arbitrary symbol follows. There is also a comparison with the cotangent case: for this, the author introduces some kind of Maslov bundle (in the case of a symplectic vector space, one obtains the usual Maslov bundle).