On the quantization of polygon spaces.

*(English)*Zbl 1206.47095The quantum space associated to the moduli \({\mathcal M}_\ell\) (the spatial \(n\)-sided polygons with edge length \(\ell=\{\ell_1,\dots, \ell_n\}\)) is studied and a family of Toeplitz operators is constructed (Theorem 3.3). Then by using eigenvectors of these operators, asymptotics of \(6j\)-symbols are related to the geometry of the tetrahedron (Theorem 7.1., see also J. Roberts [Geom. Topol. 3, 21–66 (1999; Zbl 0918.22014)]).

A set of action coordinates of \({\mathcal M}_\ell\) is associated to an admissible graph \(\Gamma\), a connected, acyclic and trivalent graph (§2). \(\ell\) is mostly assumed to satisfy Clebsh-Gordan type condition

\[ \ell_j< \frac{|\ell|}{2},\quad \ell_1\pm\ell_2\pm\cdots\pm \ell_n\neq 0,\quad |\ell|\in 2\mathbb{Z},\;|\ell|= \ell_1+\cdots+\ell_n. \] Let \(S^2_\ell\) be the sphere of radius \(\ell\) and centered at the origin. Then

\[ {\mathcal M}_\ell= \{(x_1,\dots, x_n)\in S^2_{\ell_1}\times\cdots\times x_n= 0\}/\text{SU}(2). \] A Kähler structure, a prequantization line bundle and a half form bundle of \({\mathcal M}_\ell\) are defined by using Fubini-Study metric, the tautological line bundle of \(\mathbb{P}^1(\mathbb{C})\). The quantum space of \({\mathcal M}_\ell\), the space of holomorphic sections of prequantization bundle of \({\mathcal M}_\ell\), is shown to be isomorphic to

\[ {\mathcal H}_\ell= (V_{\ell_1}\otimes\cdots\otimes V_{\ell_n})^{\text{SU}(2)}, \] where \(V_m\) is the \((m+1)\)-dimensional irreducible representation of \(\text{SU}(2)\) with spin \({m\over 2}\). Replacing the prequantum bundle by its \(k\)th power and twisting it by a half form bundle, a quantum space isomorphic to \({\mathcal H}_{k\ell-1}\) is obtained (Theorem 3.1). The semiclassical limit is defined as the limit \(k\to\infty\).

Then a family of Toeplitz operators \(H_{k\ell-1}= H_{a,k\ell-1}\) acting on \({\mathcal H}_{k\ell-1}\) is constructed by using the Casimir operator on \(H^0({\mathcal M}_\ell, L^k_\ell\otimes\delta_\ell)\cong{\mathcal H}_{k\ell-1}\) and an initial vertex \(a\). (Theorem 3.2, the proof is given in §8, the last section). Using eigenvectors belonging to the eigenvalue \(E\) of the operator \(H_{k\ell-1}\) obtained to modify these operators, the basis \(\Psi_{E,k}\) of the quantum space \({\mathcal H}_{k\ell-1}\) concentrated on the Lagrangean submanifold of constant action is obtained (§7). The classical \(6j\)-symbols are defined as the coefficients of the change of basis matrix between the basis coming from \(((V_{\ell_1}\otimes V_{\ell_2})\otimes V_{\ell_3})\otimes V_{\ell_4}\) and \((V_{\ell_1}\otimes (V_{\ell_2}\otimes V_{\ell_3}))\otimes V_{\ell_4}\) as follows:

\[ \left\{\begin{matrix} k\ell_1- 1\;k\ell_2- 1\;k\ell-1\\ k\ell_3- 1\;k\ell_4- 1\;k\ell'- 1\end{matrix}\right\}= (-1)^{k(\ell_1+ \ell_2+ \ell_3+ \ell_4)/2} {(\Psi_{E,k}, \Psi_{E',k}')\over k\sqrt{\ell\ell'}}. \] Here, \((\Psi_1, \Psi_2)\) is defined by \(\int_{{\mathcal M}_ell} (\Psi_1(x), \Psi_2(x))_{L^k_x \otimes\delta_x} \mu_M(x)\).

If \(\sqrt{E}\), \(\sqrt{E'}\), \(\ell_1\), \(\ell_2\), \(\ell_3\) and \(\ell_4\) are the edge lengths of a nondegenerate tetrahedron of volume \(V(E,E')\) and

\[ \theta(E,E')= \alpha\sqrt{E}+ \alpha'\sqrt{E'}+ \sum^4_{i=1} \alpha_i\ell_i, \] where \(\alpha,\alpha', \alpha_1,\dots, \alpha_4\) are the exterior dihedral angles, then

\[ (\Psi_{E,k}, \Psi_{E',k})= C_{k,E, E'} \sqrt{\frac{2}{3\pi}} k^{-\frac12} \frac{(EE')^{\frac14}}{V(E,E')^{\frac12}}\cos\Biggl(\frac{k\theta(E, E')}{2}+\frac\pi2\Biggr)+ O(k^{-\frac32}), \] where \(C_{k,E,E'}\) is a complex number of modulus 1 (Theorem 7.1). The author says that the main part of the proof was already understood by Y. U. Taylor and C. T. Woodward [Sel. Math., New Ser. 11, No. 3–4, 539–571 (2005; Zbl 1162.17306)], except for the delicate phase determination.

Existence of \({\mathcal H}_\ell\) for an admissible graph (Proposition 2.1), nontriviality of \({\mathcal H}_\ell\) when \(|\ell|\) is even (Proposition 2.2) and the definition of \(H_{a,\ell}*{\mathcal H}_\ell\to{\mathcal H}_\ell\) are given in §2. Existence of the isomorphism \(V_{k,\ell}:{\mathcal H}_{k\ell-1}\to H^0({\mathcal M}_\ell, L^k_\ell\otimes\delta_\ell)\) is proved applying the quantization commutes with reduction theorem [V. Guillemin and S. Sternberg, Invent. Math. 67, 515–538 (1982; Zbl 0503.58018)] in §3 (Theorem 3.1). The definition of Toeplitz operator and its symbol are also given in this section. A Toeplitz operator

\[ \Biggl(T_k= \prod_k{\mathcal P}_k(f(\cdot, k))+ R_k: H^0(M, L^k\otimes\delta)\to H^0(M, L^k\otimes\delta)\Biggr)_{k= 1,2,\dots}, \] where \({\mathcal P}_k(f)\) is the rescaled Kostant-Souriau operator [L. Charles, Commun. Partial Differ. Equations 28, No. 9–10, 1527–1566 (2003; Zbl 1038.53086)], has the symbol

\[ f(\cdot, k)= f_0+ k^{-1} f_1+ k^{-2} f_2+\cdots \] (\(f_0\) is called the principal symbol and \(f_1\) the subprincipal symbol) and \(\| R_k\|= O(k^{-\infty})\). Hence we can investigate asymptotics of Toeplitz operators.

\(H_{a,\ell}\) is a Toeplitz operator (Theorem 3.2). This is the most important result in this paper. To show that the corresponding operator on \(S^2_{\ell_1}\times\cdots\times S^2_{\ell_n}\) is a Toeplitz operator follows from general results on Toeplitz operators [cf. L. Boutet de Monvel and V. Guillemin, “The spectral theory of Toeplitz operators” (Annals of Mathematics Studies 99; Princeton University Press and University of Tokyo Press) (1981; Zbl 0469.47021); L. Charles, J. Symplectic Geom. 4, No. 2, 171–198 (2006; Zbl 1123.58016)]. To derive Theorem 3.1 from this fact, any invariant Toeplitz operator of a Hamiltonian space \((M,G)\) is shown to descend to a Toeplitz operator of the quotient \(M/G\) in §8 (Corollary 8.1) via a delicate subprincipal estimate (Theorem 8.5).

In §4, generalizing the result of M. Kapovich and J. J. Millson [J. Differ. Geom. 44, No. 3, 479–513 (1996; Zbl 0889.58017)], an action angle coordinate system is associated to any triangle decomposition (Theorem 4.3). The image of the action and associated torus action are also described (Theorem 4.4 and Theorem 4.2). In §5, joint eigenstates of \(H_{a,k\ell-1}\) are described locally and are deduced from Lagrangean sections (Theorem 5.2). Bohr-Sommerfeld conditions (conditions to patch local sections appearing in Theorem 5.2) are also obtained (Proposition 5.1). §6 studies asymptotics of Lagrangean sections. After these preliminaries, the asymptotics of the \(6j\)-symbol are obtained in §7. The last section (§8) treats the symplectic reduction of Toeplitz operators. Then principal and subprincipal symbols of reduced Toeplitz operators are computed adopting the theory of Fourier integral operators (Theorems 8.4 and 8.5). By these results, Corollary 8.1 is derived and Theorem 3.2 is proved.

A set of action coordinates of \({\mathcal M}_\ell\) is associated to an admissible graph \(\Gamma\), a connected, acyclic and trivalent graph (§2). \(\ell\) is mostly assumed to satisfy Clebsh-Gordan type condition

\[ \ell_j< \frac{|\ell|}{2},\quad \ell_1\pm\ell_2\pm\cdots\pm \ell_n\neq 0,\quad |\ell|\in 2\mathbb{Z},\;|\ell|= \ell_1+\cdots+\ell_n. \] Let \(S^2_\ell\) be the sphere of radius \(\ell\) and centered at the origin. Then

\[ {\mathcal M}_\ell= \{(x_1,\dots, x_n)\in S^2_{\ell_1}\times\cdots\times x_n= 0\}/\text{SU}(2). \] A Kähler structure, a prequantization line bundle and a half form bundle of \({\mathcal M}_\ell\) are defined by using Fubini-Study metric, the tautological line bundle of \(\mathbb{P}^1(\mathbb{C})\). The quantum space of \({\mathcal M}_\ell\), the space of holomorphic sections of prequantization bundle of \({\mathcal M}_\ell\), is shown to be isomorphic to

\[ {\mathcal H}_\ell= (V_{\ell_1}\otimes\cdots\otimes V_{\ell_n})^{\text{SU}(2)}, \] where \(V_m\) is the \((m+1)\)-dimensional irreducible representation of \(\text{SU}(2)\) with spin \({m\over 2}\). Replacing the prequantum bundle by its \(k\)th power and twisting it by a half form bundle, a quantum space isomorphic to \({\mathcal H}_{k\ell-1}\) is obtained (Theorem 3.1). The semiclassical limit is defined as the limit \(k\to\infty\).

Then a family of Toeplitz operators \(H_{k\ell-1}= H_{a,k\ell-1}\) acting on \({\mathcal H}_{k\ell-1}\) is constructed by using the Casimir operator on \(H^0({\mathcal M}_\ell, L^k_\ell\otimes\delta_\ell)\cong{\mathcal H}_{k\ell-1}\) and an initial vertex \(a\). (Theorem 3.2, the proof is given in §8, the last section). Using eigenvectors belonging to the eigenvalue \(E\) of the operator \(H_{k\ell-1}\) obtained to modify these operators, the basis \(\Psi_{E,k}\) of the quantum space \({\mathcal H}_{k\ell-1}\) concentrated on the Lagrangean submanifold of constant action is obtained (§7). The classical \(6j\)-symbols are defined as the coefficients of the change of basis matrix between the basis coming from \(((V_{\ell_1}\otimes V_{\ell_2})\otimes V_{\ell_3})\otimes V_{\ell_4}\) and \((V_{\ell_1}\otimes (V_{\ell_2}\otimes V_{\ell_3}))\otimes V_{\ell_4}\) as follows:

\[ \left\{\begin{matrix} k\ell_1- 1\;k\ell_2- 1\;k\ell-1\\ k\ell_3- 1\;k\ell_4- 1\;k\ell'- 1\end{matrix}\right\}= (-1)^{k(\ell_1+ \ell_2+ \ell_3+ \ell_4)/2} {(\Psi_{E,k}, \Psi_{E',k}')\over k\sqrt{\ell\ell'}}. \] Here, \((\Psi_1, \Psi_2)\) is defined by \(\int_{{\mathcal M}_ell} (\Psi_1(x), \Psi_2(x))_{L^k_x \otimes\delta_x} \mu_M(x)\).

If \(\sqrt{E}\), \(\sqrt{E'}\), \(\ell_1\), \(\ell_2\), \(\ell_3\) and \(\ell_4\) are the edge lengths of a nondegenerate tetrahedron of volume \(V(E,E')\) and

\[ \theta(E,E')= \alpha\sqrt{E}+ \alpha'\sqrt{E'}+ \sum^4_{i=1} \alpha_i\ell_i, \] where \(\alpha,\alpha', \alpha_1,\dots, \alpha_4\) are the exterior dihedral angles, then

\[ (\Psi_{E,k}, \Psi_{E',k})= C_{k,E, E'} \sqrt{\frac{2}{3\pi}} k^{-\frac12} \frac{(EE')^{\frac14}}{V(E,E')^{\frac12}}\cos\Biggl(\frac{k\theta(E, E')}{2}+\frac\pi2\Biggr)+ O(k^{-\frac32}), \] where \(C_{k,E,E'}\) is a complex number of modulus 1 (Theorem 7.1). The author says that the main part of the proof was already understood by Y. U. Taylor and C. T. Woodward [Sel. Math., New Ser. 11, No. 3–4, 539–571 (2005; Zbl 1162.17306)], except for the delicate phase determination.

Existence of \({\mathcal H}_\ell\) for an admissible graph (Proposition 2.1), nontriviality of \({\mathcal H}_\ell\) when \(|\ell|\) is even (Proposition 2.2) and the definition of \(H_{a,\ell}*{\mathcal H}_\ell\to{\mathcal H}_\ell\) are given in §2. Existence of the isomorphism \(V_{k,\ell}:{\mathcal H}_{k\ell-1}\to H^0({\mathcal M}_\ell, L^k_\ell\otimes\delta_\ell)\) is proved applying the quantization commutes with reduction theorem [V. Guillemin and S. Sternberg, Invent. Math. 67, 515–538 (1982; Zbl 0503.58018)] in §3 (Theorem 3.1). The definition of Toeplitz operator and its symbol are also given in this section. A Toeplitz operator

\[ \Biggl(T_k= \prod_k{\mathcal P}_k(f(\cdot, k))+ R_k: H^0(M, L^k\otimes\delta)\to H^0(M, L^k\otimes\delta)\Biggr)_{k= 1,2,\dots}, \] where \({\mathcal P}_k(f)\) is the rescaled Kostant-Souriau operator [L. Charles, Commun. Partial Differ. Equations 28, No. 9–10, 1527–1566 (2003; Zbl 1038.53086)], has the symbol

\[ f(\cdot, k)= f_0+ k^{-1} f_1+ k^{-2} f_2+\cdots \] (\(f_0\) is called the principal symbol and \(f_1\) the subprincipal symbol) and \(\| R_k\|= O(k^{-\infty})\). Hence we can investigate asymptotics of Toeplitz operators.

\(H_{a,\ell}\) is a Toeplitz operator (Theorem 3.2). This is the most important result in this paper. To show that the corresponding operator on \(S^2_{\ell_1}\times\cdots\times S^2_{\ell_n}\) is a Toeplitz operator follows from general results on Toeplitz operators [cf. L. Boutet de Monvel and V. Guillemin, “The spectral theory of Toeplitz operators” (Annals of Mathematics Studies 99; Princeton University Press and University of Tokyo Press) (1981; Zbl 0469.47021); L. Charles, J. Symplectic Geom. 4, No. 2, 171–198 (2006; Zbl 1123.58016)]. To derive Theorem 3.1 from this fact, any invariant Toeplitz operator of a Hamiltonian space \((M,G)\) is shown to descend to a Toeplitz operator of the quotient \(M/G\) in §8 (Corollary 8.1) via a delicate subprincipal estimate (Theorem 8.5).

In §4, generalizing the result of M. Kapovich and J. J. Millson [J. Differ. Geom. 44, No. 3, 479–513 (1996; Zbl 0889.58017)], an action angle coordinate system is associated to any triangle decomposition (Theorem 4.3). The image of the action and associated torus action are also described (Theorem 4.4 and Theorem 4.2). In §5, joint eigenstates of \(H_{a,k\ell-1}\) are described locally and are deduced from Lagrangean sections (Theorem 5.2). Bohr-Sommerfeld conditions (conditions to patch local sections appearing in Theorem 5.2) are also obtained (Proposition 5.1). §6 studies asymptotics of Lagrangean sections. After these preliminaries, the asymptotics of the \(6j\)-symbol are obtained in §7. The last section (§8) treats the symplectic reduction of Toeplitz operators. Then principal and subprincipal symbols of reduced Toeplitz operators are computed adopting the theory of Fourier integral operators (Theorems 8.4 and 8.5). By these results, Corollary 8.1 is derived and Theorem 3.2 is proved.

Reviewer: Akira Asada (Takarazuka)

##### MSC:

47L80 | Algebras of specific types of operators (Toeplitz, integral, pseudodifferential, etc.) |

53D50 | Geometric quantization |

53D20 | Momentum maps; symplectic reduction |

53D30 | Symplectic structures of moduli spaces |

81S10 | Geometry and quantization, symplectic methods |

81Q20 | Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory |