Equivariant Metaplectic-c Prequantization of Symplectic Manifolds with Hamiltonian Torus Actions

TL;DR

This paper establishes a necessary and sufficient condition for a symplectic manifold with a Hamiltonian torus action to admit an equivariant metaplectic-c prequantization, and applies it to identify quantized energy levels.

Contribution

It generalizes the condition for equivariant metaplectic-c prequantization to symplectic manifolds with torus actions and relates it to quantized energy levels.

Findings

Derived a condition evaluated at fixed points of the momentum map.

Extended previous energy quantization conditions to torus actions.

Provided criteria for the existence of equivariant metaplectic-c prequantizations.

Abstract

This paper determines a condition that is necessary and sufficient for a metaplectic-c prequantizable symplectic manifold with an effective Hamiltonian torus action to admit an equivariant metaplectic-c prequantization. The condition is evaluated at a fixed point of the momentum map, and is shifted from the one that is known for equivariant prequantization line bundles. Given a metaplectic-c prequantized symplectic manifold with a Hamiltonian energy function, the author previously proposed a condition under which a regular value of the function should be considered a quantized energy level of the system. This definition naturally generalizes to regular values of the momentum map for a Hamiltonian torus action. We state the generalized definition for such a system, and use an equivariant metaplectic-c prequantization to determine its quantized energy levels.

Equations

$$\exp (\xi )=(e^{2\pi i{a}_1},\dots ,e^{2\pi i{a}_k}).$$

$${\xi }_M(m)={\frac{d}{dt}|}_{t=0}\exp (t\xi )\cdot m,\forall m\in M.$$

$$d{\Phi }^{\xi }={\xi }_M┘\omega ,\forall \xi \in \mathfrak{t}.$$

$${\varphi }_{\xi }^t(m)=\exp (t\xi )\cdot m,\forall m\in M,$$

$${\mathrm{Mp}}^c(V)=\mathrm{Mp}(V){\times }_{{\mathbb{Z}}_2}U(1).$$

$$1\to U(1)\to {\mathrm{Mp}}^c(V)\stackrel{\sigma }{⟶}\mathrm{Sp}(V)\to 1$$

$$1\to \mathrm{Mp}(V)\to {\mathrm{Mp}}^c(V)\stackrel{\eta }{⟶}U(1)\to 1$$

$$C_g=\frac{1}{2}(g-JgJ).$$

$${\tilde{\varphi }}_{\xi }^t(b)={\varphi }_{\xi \ast }^t{|}_m\circ b,\forall m\in M,\forall b\in \mathrm{Sp}(M,\omega {)}_m.$$

$${\tilde{\xi }}_M(b)={\frac{d}{dt}|}_{t=0}{\tilde{\varphi }}_{\xi }^t(b).$$

$$\gamma ({\xi }_P)=-\frac{1}{i\hslash }{\Pi }^{\ast }{\Phi }^{\xi },$$

$$P{|}_U\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$T^k\stackrel{{\kappa }_z}{⟶}U(V)\subset \mathrm{Sp}(V).$$

$$T_{z}M\cong V,T_{(z,I)}\mathrm{Sp}(M,\omega )\cong V\times \mathfrak{sp}(V),T_{(z,I)}P\cong V\times {\mathfrak{mp}}^c(V)=V\times \mathfrak{sp}(V)\oplus \mathfrak{u}(1).$$

$${\tilde{\varphi }}_{\xi }^t(z,I)=(z,{\kappa }_z(\exp (t\xi ))),$$

$${\tilde{\xi }}_M(z,I)={\frac{d}{dt}|}_{t=0}{\tilde{\varphi }}_{\xi }^t(z,I)=(0,{\kappa }_{z\ast }\xi ).$$

$${\xi }_P(z,I)=\left(0,{\kappa }_{z\ast }\xi \oplus -\frac{1}{i\hslash }{\Phi }^{\xi }(z)\right).$$

$${\psi }_{\xi }^t(z,I)=\left(z,\exp \left[t\left({\kappa }_{z\ast }\xi \oplus -\frac{1}{i\hslash }{\Phi }^{\xi }(z)\right)\right]\right).$$

$$\exp \left[t\left({\kappa }_{z\ast }\xi \oplus -\frac{1}{i\hslash }{\Phi }^{\xi }(z)\right)\right]=\exp \left(t{\kappa }_{z\ast }\xi \oplus 0\right)e^{2\pi it{\Phi }^{\xi }(z)/h},$$

$$\mu (t{)}^2\text{mbox}{Det}_{\mathbb{C}}{C}_{{\kappa }_z(\exp (t\xi ))}=1.$$

$$T^k\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\mu (t{)}^2{w}_z(\exp (t\xi ))=1$$

$$({\kappa }_z(\exp (t\xi )),w_z(\exp (t\xi ){)}^{-1/2}).$$

$$({\kappa }_z(\exp (t\xi )),e^{-\pi it{w}_{z\ast }\xi }),$$

$$\left({\kappa }_z(\exp (t\xi )),e^{-\pi it{w}_{z\ast }\xi }{e}^{2\pi it{\Phi }^{\xi }(z)/h}\right).$$

$$\left({\kappa }_z(\exp (\xi )),e^{-\pi i{w}_{z\ast }\xi }{e}^{2\pi i{\Phi }^{\xi }(z)/h}\right)=(I,1).$$

$$-\pi i{w}_{z\ast }\xi +\frac{2\pi i{\Phi }^{\xi }(z)}{h}=2\pi iN$$

$$\frac{1}{h}{\Phi }^{\xi }(z)-\frac{1}{2}{w}_{z\ast }\xi =N.$$

$$\frac{1}{h}\Phi (z)-\frac{1}{2}{w}_{z\ast }\in {\mathbb{Z}}^{k\ast }\subset {\mathfrak{t}}^{\ast }.$$

$${\xi }_P,{\psi }_{\xi }^t$$

Full text

Given a metaplectic-c prequantized symplectic manifold with a Hamiltonian energy function, the author previously proposed a condition under which a regular value of the function should be considered a quantized energy level of the system. This definition naturally generalizes to regular values of the momentum map for a Hamiltonian torus action. We state the generalized definition for such a system, and use an equivariant metaplectic-c prequantization to determine its quantized energy levels.

1 Introduction

Metaplectic-c quantization was introduced by Hess [4] and further developed by Robinson and Rawnsley [6]. It is a generalization of the Kostant-Souriau quantization procedure with half-form correction that applies to a strictly broader class of symplectic manifolds. The starting point for this paper is a symplectic manifold $(M,\omega)$ that admits a metaplectic-c prequantization.

Suppose there is a Hamiltonian $G$ action on $(M,\omega)$ for some Lie group $G$. Broadly speaking, a prequantization bundle for $(M,\omega)$ is called an equivariant prequantization if the $G$ action lifts to the prequantization bundle in a manner that preserves all of its structures. This concept has been applied to prequantization line bundles in the context of a torus action and an arbitrary compact Lie group action [2]. It has also been applied to spin-c structures in the context of a circle action [1] and a torus action [3].

In this paper, we assume that $(M,\omega)$ is metaplectic-c prequantizable and has an effective Hamiltonian torus action with momentum map $\Phi:M\rightarrow\mathfrak{t}^{*}$, where $\mathfrak{t}^{*}$ is the dual of the Lie algebra $\mathfrak{t}$ for the torus. Section 2 contains our conventions for Hamiltonian torus actions, and reviews the definitions of the metaplectic-c group and a metaplectic-c prequantization.

In Section 3, we further assume that the torus action has at least one fixed point $z$. We give the definition of an equivariant metaplectic-c prequantization, and we determine a condition on the value of the momentum map at $z$ that is necessary and sufficient for $(M,\omega)$ to admit an equivariant metaplectic-c prequantization. For an equivariant prequantization line bundle, a comparable result is known [2]: the value $\frac{1}{h}\Phi(z)$ must be in the integer lattice of $\mathfrak{t}^{*}$, where $h=2\pi\hbar$ is Planck’s constant. The condition that we obtain is shifted from this due to the lift of the torus action to the symplectic frame bundle for $(M,\omega)$. The statement of our equivariance condition is in Theorem 3.1.

In an earlier paper [7], we defined a quantized energy level for the metaplectic-c prequantized system $(M,\omega,H)$, where $H\in C^{\infty}(M)$ is viewed as a Hamiltonian energy function on $(M,\omega)$. This definition has a natural generalization to families of Poisson-commuting functions. In particular, in Section 4, we apply it to the components of the momentum map $\Phi$ for the torus action. Given an equivariant metaplectic-c prequantization for the system $(M,\omega,\Phi)$, we show that the regular values $x$ of $\Phi$ that are quantized are exactly those such that $\frac{1}{h}x$ lies in the integer lattice. This is Theorem 4.2. The section concludes by demonstrating that if the torus acts freely on the level set corresponding to a quantized energy level, then the symplectic reduction is metaplectic-c prequantizable.

Lastly, in Section 5, we consider two examples. First, we obtain the quantized energy levels for a harmonic oscillator of arbitrary dimension. The result includes the half-shift predicted by the standard quantum mechanical calculation. Then we consider the complex projective space $\mathbb{CP}^{2}$ with an action of the two-dimensional torus $T^{2}$ that is induced from a linear $T^{2}$ action on $\mathbb{C}^{3}$. We determine the shift in the momentum map required to satisfy the equivariance condition, and find the quantized energy levels. This example is notable because $\mathbb{CP}^{2}$ admits a metaplectic-c prequantization but not a metaplectic structure, meaning that quantization results for this system cannot be duplicated using Kostant-Souriau quantization with the half-form correction.

2 Hamiltonian Torus Actions and Metaplectic-c Prequantization

Section 2.1 sets up our notation and conventions for the torus and the momentum map. In Section 2.2, we summarize the basic elements of metaplectic-c prequantization. Considerably more detail, including proofs, were given by Robinson and Rawnsley [6]. A similar review also appears in [7].

2.1 Hamiltonian torus actions

Let $T^{k}$ be a $k$-dimensional torus with Lie algebra $\mathfrak{t}$. Write $\tau\in T^{k}$ as $(\tau_{1},\ldots,\tau_{k})$ where each $\tau_{j}\in U(1)$. Let $\left\{\xi_{1},\ldots,\xi_{k}\right\}$ be the standard basis for $\mathbb{R}^{k}$, and identify $\mathfrak{t}$ with $\mathbb{R}^{k}$ such that for any $\xi=\sum_{j=1}^{k}a_{j}\xi_{j}\in\mathbb{R}^{k}$,

[TABLE]

Let $(M^{2n},\omega)$ be a connected symplectic manifold, where $n\geq k$. For the remainder of the paper, we assume that $T^{k}$ has an effective Hamiltonian action on $M$ with momentum map $\Phi:M\rightarrow\mathfrak{t}^{*}$. For all $\xi\in\mathfrak{t}$, we define the vector field $\xi_{M}$ on $M$ by

[TABLE]

Our convention for the momentum map is

[TABLE]

For each $\xi\in\mathfrak{t}$, denote the flow of $\xi_{M}$ on $M$ by $\phi_{\xi}^{t}$. Explicitly,

[TABLE]

and $\phi^{t}_{\xi}$ is a symplectomorphism for all $t$. For any $\xi\in\mathfrak{t}$, the action of the element $\tau=\exp(\xi)\in T^{k}$ on $M$ is given by the map $\phi^{1}_{\xi}$. In particular, if $\xi\in\mathbb{Z}^{k}\subset\mathfrak{t}$, then $\phi^{1}_{\xi}$ is the identity map on $M$.

2.2 Metaplectic-c prequantization

Fix a model $2n$-dimensional symplectic vector space $(V,\Omega)$, together with a compatible complex structure $J$ on $V$. The symplectic group for $(V,\Omega)$ is denoted by $\operatorname{Sp}(V)$. The metaplectic group is the connected double cover $\operatorname{Mp}(V)\stackrel{{\scriptstyle\sigma}}{{\longrightarrow}}\operatorname{Sp}(V)$. The metaplectic-c group is defined to be

[TABLE]

We will make use of the following two group homomorphisms on $\operatorname{Mp}^{c}(V)$. The projection map $\sigma$ appears in the short exact sequence

[TABLE]

and restricts to the double covering map on $\operatorname{Mp}(V)$. The determinant map $\eta$ appears in the short exact sequence

[TABLE]

and acts on $\lambda\in U(1)\subset\operatorname{Mp}^{c}(V)$ by $\eta(\lambda)=\lambda^{2}$. The Lie algebra $\mathfrak{mp}^{c}(V)$ can be identified with $\mathfrak{sp}(V)\oplus\mathfrak{u}(1)$ under $\sigma_{*}\oplus\frac{1}{2}\eta_{*}$.

For any $g\in\operatorname{Sp}(V)$, let

[TABLE]

Then $C_{g}$ commutes with $J$, so it is a complex linear map on $V$. It can be shown [6] that $\mbox{Det}_{\mathbb{C}}C_{g}\neq 0$ for all $g\in\operatorname{Sp}(V)$.

We define an embedding of $\operatorname{Mp}^{c}(V)$ into $\operatorname{Sp}(V)\times\mathbb{C}$ such that each $a\in\operatorname{Mp}^{c}(V)$ is mapped to the pair $(g,\mu)\in\operatorname{Sp}(V)\times\mathbb{C}$, where $\sigma(a)=g$ and $\eta(a)=\mu^{2}\mbox{Det}_{\mathbb{C}}C_{g}$. To resolve the ambiguity in the sign of $\mu$, we assume that $I\in\operatorname{Mp}^{c}(V)$ is mapped to $(I,1)$. Following [6], we refer to $(g,\mu)$ as the parameters of $a\in\operatorname{Mp}^{c}(V)$. Note that if $a\in\operatorname{Mp}(V)=\ker\eta$, then the parameters $(g,\mu)$ of $a$ satisfy $\mu^{2}\mbox{Det}_{\mathbb{C}}C_{g}=1$.

The unitary group $U(V)\subset\operatorname{Sp}(V)$ is the maximal compact subgroup of $\operatorname{Sp}(V)$, and consists of exactly those elements of $\operatorname{Sp}(V)$ that commute with the complex structure $J$. For any $g\in U(V)$, $C_{g}=g$ and $\mbox{Det}_{\mathbb{C}}C_{g}=\mbox{Det}_{\mathbb{C}}g\in U(1)$.

We view the symplectic frame bundle $\operatorname{Sp}(M,\omega)\stackrel{{\scriptstyle\rho}}{{\longrightarrow}}(M,\omega)$ as a right principal $\operatorname{Sp}(V)$ bundle over $M$, defined fiberwise such that for all $m\in M$, every $b\in\operatorname{Sp}(M,\omega)_{m}$ is a linear symplectic isomorphism $b:(V,\Omega)\rightarrow(T_{m}M,\omega_{m})$. The group $\operatorname{Sp}(V)$ acts on the fibers of $\operatorname{Sp}(M,\omega)$ by precomposition.

Definition 2.1**.**

Let $(M,\omega)$ be a symplectic manifold with symplectic frame bundle $\operatorname{Sp}(M,\omega)\stackrel{{\scriptstyle\rho}}{{\longrightarrow}}(M,\omega)$. A metaplectic-c prequantization for $(M,\omega)$ is a triple $(P,\Sigma,\gamma)$, where:

• •

$P\stackrel{{\scriptstyle\Pi}}{{\longrightarrow}}M$ is a right principal $\operatorname{Mp}^{c}(V)$ bundle;

• •

the map $P\stackrel{{\scriptstyle\Sigma}}{{\longrightarrow}}\operatorname{Sp}(M,\omega)$ satisfies $\rho\circ\Sigma=\Pi$ and $\Sigma(q\cdot a)=\Sigma(q)\cdot\sigma(a)$ for all $q\in P$ and $a\in\operatorname{Mp}^{c}(V)$;

• •

$\gamma$ is a $\mathfrak{u}(1)$-valued one-form on $P$, invariant under the principal $\operatorname{Mp}^{c}(V)$ action, such that $d\gamma=\frac{1}{i\hbar}\Pi^{*}\omega$, and for all $\alpha\in\mathfrak{mp}^{c}(V)$, if $\alpha$ generates the vector field $\alpha_{P}$ on $P$, then $\gamma(\alpha_{P})=\frac{1}{2}\eta_{*}\alpha$.

If $(P,\Sigma,\gamma)\rightarrow(M,\omega)$ is a metaplectic-c prequantization, then $(P,\gamma)\stackrel{{\scriptstyle\Sigma}}{{\longrightarrow}}\operatorname{Sp}(M,\omega)$ is a principal circle bundle with connection one-form. The circle that acts on the fibers of $P$ is the center $U(1)\subset\operatorname{Mp}^{c}(V)$.

3 Equivariant Metaplectic-c Prequantization

3.1 Initial constructions

From now on, we assume that $(M,\omega)$ is metaplectic-c prequantizable, and we fix a metaplectic-c prequantization $(P,\Sigma,\gamma)$. We have the bundle projection maps $P\stackrel{{\scriptstyle\Pi}}{{\longrightarrow}}M$ and $\operatorname{Sp}(M,\omega)\stackrel{{\scriptstyle\rho}}{{\longrightarrow}}M$.

Recall from Section 2.1 that there is a Hamiltonian $T^{k}$ action on $(M,\omega)$ with momentum map $\Phi$. Each $\xi\in\mathfrak{t}$ generates the vector field $\xi_{M}$ on $M$ with flow $\phi^{t}_{\xi}$. Since $\phi^{t}_{\xi}$ preserves $\omega$, it can be lifted to a flow $\widetilde{\phi}_{\xi}^{t}$ on $\operatorname{Sp}(M,\omega)$, defined by

[TABLE]

The corresponding vector field on $\operatorname{Sp}(M,\omega)$ is

[TABLE]

Let $\tau=\exp(\xi)\in T^{k}$ act on $\operatorname{Sp}(M,\omega)$ by $\widetilde{\phi}_{\xi}^{1}$. It is easily verified that this definition yields a well-defined group action of $T^{k}$ on $\operatorname{Sp}(M,\omega)$ that lifts the $T^{k}$ action on $M$ and commutes with the principal $\operatorname{Sp}(V)$ action.

Suppose there is a lift of the $T^{k}$ action on $\operatorname{Sp}(M,\omega)$ to one on $P$ that preserves $\gamma$. Then this $T^{k}$ action also commutes with the principal $\operatorname{Mp}^{c}(V)$ action. Let $\xi\in\mathfrak{t}$ generate the vector field $\xi_{P}$ on $P$. It is immediate that $L_{\xi_{P}}\gamma=0$ if and only if $d(\gamma(\xi_{P}))=-\frac{1}{i\hbar}d\Pi^{*}\Phi^{\xi}$. If, in particular,

[TABLE]

then $(P,\Sigma,\gamma)$ is called an equivariant metaplectic-c prequantization for $(M,\omega,\Phi)$. An equivalent definition for spin-c structures appears in [1], although it is stated in terms of equivariant cohomology classes. An analogous definition for an equivariant prequantization line bundle appears in [2].

In the remainder of this section, we determine a necessary and sufficient condition for the metaplectic-c prequantized system $(P,\Sigma,\gamma)\rightarrow(M,\omega,\Phi)$ to admit an equivariant metaplectic-c prequantization. For all $\xi\in\mathfrak{t}$, let $\xi_{P}$ be the vector field on $P$ such that $\xi_{P}$ is a lift of $\widetilde{\xi}_{M}$ and $\gamma(\xi_{P})=-\frac{1}{i\hbar}\Pi^{*}\Phi^{\xi}$. Let $\psi^{t}_{\xi}$ be the flow of $\xi_{P}$ on $P$. If there is a $T^{k}$ action on $P$ such that $\xi\in\mathfrak{t}$ generates the vector field $\xi_{P}$, then $\tau=\exp(\xi)\in T^{k}$ must act on $P$ by the map $\psi^{1}_{\xi}$. We will find a condition that ensures that these maps $\psi^{1}_{\xi}$ yield a well-defined $T^{k}$ action on $P$. It suffices to guarantee that for all $\xi\in\mathbb{Z}^{k}\subset\mathfrak{t}$, $\psi^{1}_{\xi}$ is the identity map on $P$.

The following argument is based on Example 6.10 in [2] (pp. 93-94), which establishes a necessary and sufficient condition for $(M,\omega,\Phi)$ to admit an equivariant prequantization line bundle. Our application of their proof to a metaplectic-c prequantization requires some additional steps concerning the symplectic frame bundle.

Assume that the $T^{k}$ action has a fixed point $z\in M$. For example, it is sufficient to assume that $M$ is compact. However, noncompact examples also exist, and the remainder of the argument does not require compactness. In Section 3.2, we will determine a condition on $\Phi(z)$ such that there is a well-defined $T^{k}$ action on the fiber $P_{z}$. Then, in Section 3.3, we will show that this condition guarantees a $T^{k}$ action on all of $P$.

3.2 $T^{k}$ action over a fixed point

Let $z\in M$ be a fixed point of the $T^{k}$ action. Any $\tau=\exp(\xi)\in T^{k}$ acts on $TM$ by the pushforward $\phi^{1}_{\xi*}$, which preserves the symplectic form. In particular, $\tau:T_{z}M\rightarrow T_{z}M$ is a linear symplectic isomorphism.

Let $U$ be a neighborhood of $z$ in $M$ over which $P|_{U}$ admits a local trivialization: $P|_{U}\cong U\times\operatorname{Mp}^{c}(V)$. This induces a local trivialization $\operatorname{Sp}(M,\omega)|_{U}\cong U\times\operatorname{Sp}(V)$, where the map $\Sigma$ is identified with $(\mbox{Id}_{U},\sigma)$.

[TABLE]

It further induces a local trivialization $TM|_{U}\cong U\times V$, under which there is an identification of the symplectic vector space $(T_{z}M,\omega_{z})$ with $(V,\Omega)$. Since $T^{k}$ acts on $(T_{z}M,\omega_{z})$ by linear symplectic isomorphisms, this identification yields a group homomorphism $T^{k}\stackrel{{\scriptstyle\kappa_{z}}}{{\longrightarrow}}\operatorname{Sp}(V)$. By a suitable adjustment of the choice of trivializing section of $P|_{U}$, we can arrange that

[TABLE]

For emphasis: this property is only required to hold over the single point $z$. On the level of tangent spaces, we obtain identifications

[TABLE]

Let $\xi\in\mathfrak{t}$ be arbitrary. It is immediate that $\xi_{M}(z)=0$. The pushforward $\phi^{t}_{\xi*}$, acting on $T_{z}M$, becomes the symplectic group element $\kappa_{z}(\exp(t\xi))\in\operatorname{Sp}(V)$. The lifted flow $\widetilde{\phi}^{t}_{\xi}$ on $\operatorname{Sp}(M,\omega)$ satisfies

[TABLE]

and so

[TABLE]

The vector field $\xi_{P}$ is the lift of $\widetilde{\xi}_{M}$ to $P$ such that $\gamma(\xi_{P})=-\frac{1}{i\hbar}\Pi^{*}\Phi^{\xi}$. At $(z,I)$, we have

[TABLE]

The flow $\psi^{t}_{\xi}$ of $\xi_{P}$ satisfies

[TABLE]

The desired action of $T^{k}$ on $P_{z}$ exists if and only if $\psi^{1}_{\xi}(z,I)=(z,I)$ for all $\xi\in\mathbb{Z}^{k}\subset\mathfrak{t}$.

Since $U(1)$ is the center of $\operatorname{Mp}^{c}(V)$, we can write the $\operatorname{Mp}^{c}(V)$ term in the above expression for $\psi^{t}_{\xi}(z,I)$ as

[TABLE]

where $h=2\pi\hbar$, and where $\exp\left(t\kappa_{z*}\xi\oplus 0\right)\in\operatorname{Mp}(V)\subset\operatorname{Mp}^{c}(V)$ and $e^{2\pi it\Phi^{\xi}(z)/h}\in U(1)\subset\operatorname{Mp}^{c}(V)$.

The parameters of $\exp(t\kappa_{z*}\xi\oplus 0)\in\operatorname{Mp}(V)$ take the form $(\kappa_{z}(\exp(t\xi)),\mu(t))$ where

[TABLE]

Note that $t=0$ corresponds to $I\in\operatorname{Mp}(V)$, so we must have $\mu(0)=1$. Further, since $\kappa_{z}(\exp(t\xi))\in U(V)$, we have $C_{\kappa_{z}(\exp(t\xi))}=\kappa_{z}(\exp(t\xi))$ and $\mbox{Det}_{\mathbb{C}}\kappa_{z}(\exp(t\xi))\in U(1)$. Let $\Delta$ denote the map from $\operatorname{Sp}(V)$ to $\mathbb{C}$ given by $g\mapsto\mbox{Det}_{\mathbb{C}}C_{g}$. Let $\Delta$ also denote the restriction of this map to $U(V)$, where it is given by $g\mapsto\mbox{Det}_{\mathbb{C}}g$, and note that $U(V)\stackrel{{\scriptstyle\Delta}}{{\longrightarrow}}U(1)$ is a group homomorphism.

[TABLE]

Let $w_{z}=\Delta\circ\kappa_{z}$. Then

[TABLE]

and $\mu(0)=0$, which implies that $\mu(t)=w_{z}(\exp(t\xi))^{-1/2}$. Thus the parameters of $\exp(t\kappa_{z*}\xi\oplus 0)$ are

[TABLE]

Identify $\mathfrak{u}(1)$ with $\mathbb{R}$ such that for all $\lambda\in\mathfrak{u}(1)$, $\exp(\lambda)=e^{2\pi i\lambda}\in U(1)$. Then $w_{z}(\exp(t\xi))=e^{2\pi itw_{z*}\xi}\in U(1)$, and so $w_{z}(\exp(t\xi))^{-1/2}=e^{-\pi itw_{z*}\xi}$. Thus the parameters of $\exp(t\kappa_{z*}\xi\oplus 0)$ are

[TABLE]

which implies that the parameters of $\exp\left[t\left(\kappa_{z*}\xi\oplus-\frac{1}{i\hbar}\Phi^{\xi}(z)\right)\right]$ are

[TABLE]

Now assume that $\xi\in\mathbb{Z}^{k}\subset\mathfrak{t}$, and set $t=1$. The condition $\psi^{1}_{\xi}(z,I)=(z,I)$ is satisfied if and only if

[TABLE]

It is clear that $\kappa_{z}(\exp(\xi))=I$, and it remains to ensure that $e^{-\pi iw_{z*}\xi}e^{2\pi i\Phi^{\xi}(z)/h}=1$. This equation holds if and only if

[TABLE]

for some $N\in\mathbb{Z}$, which rearranges to

[TABLE]

Since an equation of this form must hold for all $\xi\in\mathbb{Z}^{k}\subset\mathfrak{t}$, we conclude that the value $\Phi(z)\in\mathfrak{t}^{*}$ must satisfy

[TABLE]

This is similar to, but shifted from, the result in [2] that an equivariant prequantization line bundle exists if and only if $\frac{1}{h}\Phi(z)$ is in the integer lattice $\mathbb{Z}^{k*}\subset\mathfrak{t}^{*}$ (adjusted for differing $\hbar$ conventions).

3.3 $T^{k}$ action on $P$

We continue to follow a modified version of the argument in Example 6.10 of [2]. Fix $\xi\in\mathbb{Z}^{k}\subset\mathfrak{t}$. Then $\psi^{1}_{\xi}$ is a lift of the identity maps $\widetilde{\phi}^{1}_{\xi}$ on $\operatorname{Sp}(M,\omega)$ and $\phi^{1}_{\xi}$ on $M$.

• •

Since $\psi^{1}_{\xi}$ is a lift of the identity map on $M$, there is a map $R_{\xi}:M\rightarrow\operatorname{Mp}^{c}(V)$ such that $\psi^{1}_{\xi}(q)=q\cdot R_{\xi}(\Pi(q))$ $\forall q\in P.$

• •

Since $\psi^{1}_{\xi}$ is a lift of the identity map on $\operatorname{Sp}(M,\omega)$, there is a map $\widetilde{R}_{\xi}:\operatorname{Sp}(M,\omega)\rightarrow U(1)\subset\operatorname{Mp}^{c}(V)$ such $\psi^{1}_{\xi}(q)=q\cdot\widetilde{R}_{\xi}(\Sigma(q))$ $\forall q\in P$.

These observations together imply that the target of the map $R_{\xi}$ is $U(1)$. That is, there is a map $R_{\xi}:M\rightarrow U(1)$ such that $\psi^{1}_{\xi}(q)=q\cdot R_{\xi}(\Pi(q))$ for all $q\in P$.

Assume that the condition on $\Phi$ derived in the previous section has been satisfied over the fixed point $z$. Then $R_{\xi}(z)=1$. It remains to show that $R_{\xi}$ is constant over $M$. Let $u(s)$ be an arbitrary path in $M$, where $s\in[0,1]$. We will show that $R_{\xi}$ is constant over $u(s)$.

Recall that the determinant map $\operatorname{Mp}^{c}(V)\stackrel{{\scriptstyle\eta}}{{\longrightarrow}}U(1)$ acts on $\lambda\in U(1)\subset\operatorname{Mp}^{c}(V)$ by $\eta(\lambda)=\lambda^{2}$. Let $Y\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}M$ be the circle bundle associated to $P\stackrel{{\scriptstyle\Pi}}{{\longrightarrow}}M$ by $\eta$, and let $P\stackrel{{\scriptstyle H}}{{\longrightarrow}}Y$ be the corresponding bundle map. It is easily verified that $\gamma$ is basic with respect to $H$. Let $\gamma^{\eta}$ be the $\mathfrak{u}(1)$-valued one-form on $Y$ such that $H^{*}\gamma^{\eta}=2\gamma$. Then $\gamma^{\eta}$ is a connection one-form on $Y$, and $d\gamma^{\eta}=\frac{2}{i\hbar}\pi^{*}\omega$.

The pushforward $H_{*}\xi_{P}$ is a well-defined vector field on $Y$, which we denote by $\xi_{Y}$. Let the flow of $\xi_{Y}$ on $Y$ be $\widehat{\chi}^{t}_{\xi}$. The various vector fields and their flows are summarized below.

[TABLE]

Since $H_{*}\xi_{P}=\xi_{Y}$, $H$ intertwines the flows $\psi^{t}_{\xi}$ and $\widehat{\chi}^{t}_{\xi}$. In particular,

[TABLE]

That is, the map $\widehat{\chi}^{1}_{\xi}$ acts on $Y$ by

[TABLE]

From the definitions of $\xi_{Y}$ and $\gamma^{\eta}$, it follows that $\gamma^{\eta}(\xi_{Y})=-\frac{2}{i\hbar}\pi^{*}\Phi^{\xi}$. Let $\partial_{\theta}$ be the vertical vector field on $Y$ such that $\gamma^{\eta}(\partial_{\theta})=2\pi i\in\mathfrak{u}(1)$. Then we have

[TABLE]

where $(\xi_{M})_{hor}$ represents the lift of $\xi_{M}$ to $Y$ that is horizontal with respect to $\gamma^{\eta}$.

Let $C=\mathbb{R}/\mathbb{Z}\times[0,1]$, with coordinates $(r,s)$. Define $F:C\rightarrow M$ by

[TABLE]

and let $\omega_{C}=F^{*}\omega$. Construct the pullback of $(Y,\gamma^{\eta})$ to $C$:

[TABLE]

where $\delta=\widehat{F}^{*}\gamma^{\eta}$ and

[TABLE]

The bundle map $\widehat{F}:D\rightarrow Y$ acts by

[TABLE]

By construction, $(D,\delta)$ is a circle bundle with connection one-form over $(C,\omega_{C})$. Let $\partial_{\theta}$ also denote the vertical vector field on $D$ such that $\delta(\partial_{\theta})=2\pi i\in\mathfrak{u}(1)$, and note that $\widehat{F}_{*}\partial_{\theta}=\partial_{\theta}$.

Abbreviate the vector fields $\frac{\partial{}}{\partial{r}}$ and $\frac{\partial{}}{\partial{s}}$ on $C$ by $\partial_{r}$ and $\partial_{s}$, with flows $\psi^{t}_{r}$ and $\psi^{t}_{s}$ respectively. It is immediate from the definition of $F$ that

[TABLE]

Let $\Psi:C\rightarrow\mathbb{R}$ be given by

[TABLE]

We claim that $\omega_{C}=\frac{\partial{\Psi}}{\partial{s}}dr\wedge ds$. This is established by calculating, at arbitrary $c=(r,s)\in C$,

[TABLE]

and

[TABLE]

Let $\zeta_{r}$ be the vector field on $D$ given by

[TABLE]

with flow $\widehat{\psi}^{t}_{r}$. Then for all $(c=(r,s),y)\in D$,

[TABLE]

Thus $\widehat{F}$ intertwines the flows $\widehat{\psi}^{t}_{r}$ of $\zeta_{r}$ and $\widehat{\chi}^{t}_{\xi}$ of $\xi_{Y}$. In particular, at $t=1$,

[TABLE]

which implies that

[TABLE]

That is, $\widehat{\psi}^{1}_{r}$ fixes the base $C$ and rotates each fiber $D_{(c,y)}$ by $R^{2}_{\xi}(\pi(y))$. Note that $\pi(y)=F(c)$, so we have

[TABLE]

Let $\zeta_{s}=(\partial_{s})_{hor}$ on $D$, with flow $\widehat{\psi}^{t}_{s}$. A standard calculation establishes that $[\zeta_{r},\zeta_{s}]=0$. Hence their flows $\widehat{\psi}^{t}_{r}$ and $\widehat{\psi}^{t}_{s}$ commute. In particular, for any $(c=(r,s),y)\in D$,

[TABLE]

having used the fact that $\widehat{\psi}^{t}_{s}$ is the flow of a horizontal vector field on $D$ and is therefore equivariant with respect to the principal $U(1)$ action. We also calculate

[TABLE]

where we note that $\psi^{t}_{s}(c)=(r,s+t)$. We conclude that $R^{2}_{\xi}(\exp(r\xi)\cdot u(s))=R^{2}_{\xi}(\exp(r\xi)\cdot u(s+t))$ for all $r,s,t$. Hence $R^{2}_{\xi}$ is constant over the path $u$. Since $u$ was arbitrary, $R^{2}_{\xi}$ is in fact constant on $M$.

Recall that $R_{\xi}(z)=1$. Thus $R^{2}_{\xi}=1$ everywhere on $M$, and consequently $R_{\xi}=1$ everywhere on $M$, as needed. Hence the $T^{k}$ action is well defined everywhere on $P$. We have now proved the following theorem.

Theorem 3.1**.**

Let $(M,\omega)$ have an effective Hamiltonian $T^{k}$ action with momentum map $\Phi$ and a fixed point $z$. Then $(M,\omega,\Phi)$ admits an equivariant metaplectic-c prequantization if and only if $(M,\omega)$ is metaplectic-c prequantizable and the momentum map $\Phi$ satisfies

[TABLE]

Assume the hypotheses of the theorem, and let $z$ be a fixed point for the $T^{k}$ action on $(M,\omega)$. By adding a constant to the momentum map $\Phi$, it is always possible to satisfy the condition $\frac{1}{h}\Phi(z)-\frac{1}{2}w_{z*}\in\mathbb{Z}^{k^{*}}$. Thus any metaplectic-c prequantization $(P,\Sigma,\gamma)$ for $(M,\omega)$ can be converted to an equivariant metaplectic-c prequantization by a suitable shift of the momentum map.

4 Quantized Energy Levels

In this section, we extend the concept of a quantized energy level to the equivariant metaplectic-c prequantized system $(P,\Sigma,\gamma)\rightarrow(M,\omega,\Phi)$. Section 4.1 reviews the constructions due to Robinson [5] that we use to define a quantized energy level, and concludes with the generalization of our definition from [7] to a regular value of the momentum map $\Phi$. In Section 4.2, we determine the quantized energy levels of the system $(M,\omega,\Phi)$, assuming that $\Phi$ has been shifted so that the metaplectic-c prequantization is equivariant.

4.1 Descending to the quotient

Assume that $(M,\omega)$ is metaplectic-c prequantizable, and let $(P,\Sigma,\gamma)$ be a metaplectic-c prequantization:

[TABLE]

Assume that there is an effective Hamiltonian $T^{k}$ action on $M$ with momentum map $\Phi$ and at least one fixed point. Shift $\Phi$ if necessary so that $(P,\Sigma,\gamma)$ is an equivariant metaplectic-c prequantization for $(M,\omega,\Phi)$.

Let $x\in\mathfrak{t}^{*}$ be a regular value of the momentum map $\Phi$, and let $S=\Phi^{-1}(x)$. Then $S$ is a codimension-$k$ embedded submanifold of $M$. Recall that $\left\{\xi_{1},\ldots,\xi_{k}\right\}$ is the standard basis for $\mathbb{R}^{k}$. For all $s\in S$, the symplectic orthogonal to $T_{s}S$ is $T_{s}S^{\perp}=\mbox{span}\left\{\xi_{1M}(s),\ldots,\xi_{kM}(s)\right\}\subset T_{s}S$, implying that $S$ is a coisotropic submanifold of $M$. In the special case where $k=n$, $S$ is a Lagrangian submanifold.

Within the model symplectic vector space $(V,\Omega)$, let $W\subset V$ be a coisotropic subspace of codimension $k$, with symplectic orthogonal $W^{\perp}\subset W$. Then $W/W^{\perp}$ is a symplectic vector space with a symplectic structure inherited from $V$. Let $\operatorname{Sp}(V;W)\subset\operatorname{Sp}(V)$ be the subgroup that preserves $W$. There is a natural group homomorphism $\operatorname{Sp}(V;W)\stackrel{{\scriptstyle\nu}}{{\longrightarrow}}\operatorname{Sp}(W/W^{\perp})$.

Let $\operatorname{Mp}^{c}(V;W)\subset\operatorname{Mp}^{c}(V)$ be the preimage of $\operatorname{Sp}(V;W)$ under $\sigma$, and let $\operatorname{Mp}^{c}(W/W^{\perp})$ be the metaplectic-c group for $W/W^{\perp}$. Robinson and Rawnsley [6] showed that $\nu$ lifts to a group homomorphism $\widehat{\nu}$ on the level of metaplectic-c groups:

[TABLE]

The lifted map $\widehat{\nu}$ has the property that $\widehat{\nu}_{*}\circ\eta_{*}=\eta_{*}$.

In the following construction, which is due to Robinson [5], the above diagram serves as a model for fiberwise constructions over $(M,\omega)$. The first column corresponds to the original three-level structure,

[TABLE]

For the second column, let $\operatorname{Sp}(M,\omega;S)\subset\operatorname{Sp}(M,\omega)$ be the subbundle, lying only over $S$, such that for all $s\in S$ and all $b\in\operatorname{Sp}(M,\omega;S)_{s}$, $bW=T_{s}S$. Lastly, let $(P^{S},\gamma^{S})$ be the pullback of $(P,\gamma)$ to $\operatorname{Sp}(M,\omega;S)$.

For the third column, treat $W/W^{\perp}$ as a model symplectic vector space for the symplectic vector bundle $TS/TS^{\perp}\rightarrow S$, so that the symplectic frame bundle $\operatorname{Sp}(TS/TS^{\perp})\rightarrow S$ becomes a right principal $\operatorname{Sp}(W/W^{\perp})$ bundle over $S$. Then $\operatorname{Sp}(TS/TS^{\perp})$ is naturally identified with the bundle associated to $\operatorname{Sp}(M,\omega;S)$ by the map $\nu$. Let $P_{S}$ be the bundle associated to $P^{S}$ by the map $\widehat{\nu}$. The properties of $\widehat{\nu}$ guarantee that there is a one-form $\gamma_{S}$ on $P_{S}$ such that $\widehat{\nu}^{*}\gamma_{S}=\gamma^{S}$. Then $(P_{S},\gamma_{S})\rightarrow\operatorname{Sp}(TS/TS^{\perp})$ is a principal circle bundle with connection one-form.

[TABLE]

Let $\xi\in\mathfrak{t}$ be arbitrary. Construct $\xi_{M}$ with flow $\phi^{t}_{\xi}$ on $M$, $\widetilde{\xi}_{M}$ with flow $\widetilde{\phi}^{t}_{\xi}$ on $\operatorname{Sp}(M,\omega)$, and $\xi_{P}$ with flow $\psi^{t}_{\xi}$ on $P$, as described in Section 3.1. Recall in particular that $\xi_{P}$ is the lift of $\widetilde{\xi}_{M}$ to $P$ such that $\gamma(\xi_{P})=-\frac{1}{i\hbar}\Pi^{*}\Phi^{\xi}$. Let $\widehat{\xi}_{M}$ be the lift of $\widetilde{\xi}_{M}$ to $P$ that is horizontal with respect to $\gamma$, and let its flow be $\widehat{\phi}^{t}_{\xi}$. Each of the vector fields $\xi_{M}$, $\widetilde{\xi}_{M}$, $\widehat{\xi}_{M}$ and $\xi_{P}$ restricts to a vector field on the appropriate manifold in the second column, and descends to a vector field on the manifold in the third column. Moreover, since the $T^{k}$ action commutes with all of the principal actions, the $T^{k}$ action on each level preserves the manifold in the second column, and descends to a $T^{k}$ action on the manifold in the third column.

Viewing $(P,\gamma)$ as a circle bundle with connection one-form over $\operatorname{Sp}(M,\omega)$, let $\partial_{\theta}$ be the vertical vector field on $P$ such that $\gamma(\partial_{\theta})=2\pi i\in\mathfrak{u}(1)$. Then $\xi_{P}=\widehat{\xi}_{M}+\frac{1}{h}\Phi^{\xi}\partial_{\theta}$. We also denote by $\partial_{\theta}$ the restriction of this vector field to $P^{S}$, and the induced vertical vector field on $P_{S}$. In each column, we have $\gamma(\partial_{\theta})=\gamma^{S}(\partial_{\theta})=\gamma_{S}(\partial_{\theta})=2\pi i\in\mathfrak{u}(1)$. Note that for all $s\in S$, $\Phi^{\xi}(s)=x\cdot\xi$, since $S$ is the $x$-level set of $\Phi$. Thus, in columns 2 and 3,

[TABLE]

4.2 Generalized quantized energy levels

In our previous paper [7], we considered the system $(M,\omega,H)$, where $H\in C^{\infty}(M)$ is viewed as a Hamiltonian energy function. Let $(P,\Sigma,\gamma)\rightarrow(M,\omega)$ be a metaplectic-c prequantization. Let $E$ be a regular value of $H$, and use the embedded coisotropic submanifold $S=H^{-1}(E)\subset M$ to construct the three columns of three-level structures as described in the previous section. Let $\xi_{H}$ be the Hamiltonian vector field for $H$ on $M$, and let its lift to $\operatorname{Sp}(M,\omega)$ be $\widetilde{\xi}_{H}$. Then $\xi_{H}$ and $\widetilde{\xi}_{H}$ restrict to column 2 and descend to column 3. We define $E$ to be a quantized energy level for the system $(M,\omega,H)$ if $\gamma_{S}$ has trivial holonomy over all closed orbits of $\widetilde{\xi}_{H}$ on $\operatorname{Sp}(TS/TS^{\perp})$.

This definition has a natural generalization to a regular value of a family of $k$ Poisson-commuting functions. Recall that $\left\{\xi_{1},\ldots,\xi_{k}\right\}$ is the standard basis for $\mathfrak{t}=\mathbb{R}^{k}$. In our context, the family of Poisson-commuting functions consists of the $k$ components of the momentum map, $\Phi^{\xi_{1}},\ldots,\Phi^{\xi_{k}}$. The generalized definition is as follows.

Definition 4.1**.**

Let $(P,\Sigma,\gamma)$ be an equivariant metaplectic-c prequantization for $(M,\omega,\Phi)$. Let $x$ be a regular value of $\Phi$, and let $S=\Phi^{-1}(x)$. Let $F$ be the distribution on $\operatorname{Sp}(TS/TS^{\perp})$ spanned by the vector fields $\left\{\widetilde{\xi}_{1M},\right.\left.\ldots,\widetilde{\xi}_{kM}\right\}.$ Then $x$ is a quantized energy level for $(M,\omega,\Phi)$ if $\gamma_{S}$ has trivial holonomy over all of the leaves of $F$.

The connection one-form $\gamma_{S}$ is flat over each leaf of the distribution spanned by $\left\{\widetilde{\xi}_{1M},\ldots,\widetilde{\xi}_{kM}\right\}$. For the regular value $x$ to be a quantized energy level, it suffices to ensure that $\gamma_{S}$ has trivial holonomy over the orbits of each $\widetilde{\xi}_{jM}$. Given any initial point $b\in\operatorname{Sp}(TS/TS^{\perp})$, the integral curve $\widetilde{\phi}^{t}_{\xi_{j}}(b)$ satisfies $\widetilde{\phi}^{1}_{\xi_{j}}(b)=b$. We need to show that the integral curves $\widehat{\phi}^{t}_{\xi_{j}}(q)$ of the horizontal lift $\widehat{\xi}_{jM}$ on $P_{S}$ satisfy $\widehat{\phi}^{1}_{\xi_{j}}(q)=q$, for all $q\in P_{S}$.

As previously noted, the vector fields $\xi_{jP}$ and $\widehat{\xi}_{jM}$ on $P_{S}$ are related by

[TABLE]

Since these two vector fields differ by a constant multiple of $\partial_{\theta}$ everywhere on $P_{S}$, their flows are related by

[TABLE]

We know that $\tau=\exp(\xi)\in T^{k}$ acts on $P_{S}$ by the map $\psi^{1}_{\xi}$ for all $\xi\in\mathfrak{t}$, and this is a well-defined $T^{k}$ action. In particular, $\exp(\xi_{j})=I\in T^{k}$, so $\psi^{1}_{\xi_{j}}$ is the identity map on $P_{S}$. We have

[TABLE]

Thus $\widehat{\phi}^{1}_{\xi_{j}}(q)=q$ if and only if $e^{2\pi i(x\cdot\xi_{j})/h}=1$, which occurs when $x\cdot\xi_{j}=N_{j}h$ for some $N_{j}\in\mathbb{Z}$. This condition is satisfied for all $\xi_{j}$ exactly when $x\in h\mathbb{Z}^{k^{*}}\subset\mathfrak{t}^{*}$. We conclude the following result.

Theorem 4.2**.**

Let $(M,\omega)$ have an effective Hamiltonian $T^{k}$ action with momentum map $\Phi$ and at least one fixed point. Assume that $(M,\omega)$ is metaplectic-c prequantizable, and shift $\Phi$ by a constant if necessary so that $(M,\omega,\Phi)$ admits an equivariant metaplectic-c prequantization. Then the regular value $x\in\mathfrak{t}^{*}$ for $\Phi$ is a quantized energy level for the system if and only if

[TABLE]

An immediate consequence of this theorem arises in the context of symplectic reduction. Assume the hypotheses of Theorem 4.2, and let $x\in\mathfrak{t}^{*}$ be a regular value of $\Phi$. Further assume that $T^{k}$ acts freely on the level set $S=\Phi^{-1}(x)$. Then, by Marsden-Weinstein reduction, the space of orbits $M_{0}=S/T^{k}$ is a manifold, and it acquires a symplectic form $\omega_{0}$. Let $S\stackrel{{\scriptstyle\varrho}}{{\longrightarrow}}M_{0}$ be the quotient map from $S$ to its orbit space. Then $\varrho^{*}\omega_{0}=\omega_{S}$, where $\omega_{S}$ is the pullback of $\omega$ to $S$.

Let $s\in S$ be arbitrary and let $\varrho(s)=m_{0}$. On the level of tangent spaces, we have the short exact sequence

[TABLE]

implying that $\varrho_{*}|_{s}$ induces a linear symplectic isomorphism between $T_{s}S/T_{s}S^{\perp}$ and $T_{m_{0}}M_{0}$. For all $\xi\in\mathfrak{t}$, the pushforward $\phi^{t}_{\xi*}:TM\rightarrow TM$ preserves both $TS$ and $TS^{\perp}$, and so descends to a map on $TS/TS^{\perp}$. If we let $\tau=\exp(\xi)\in T^{k}$ act on $TS/TS^{\perp}$ by the pushforward $\phi^{1}_{\xi*}$, the result is a $T^{k}$ action on $TS/TS^{\perp}$. Using these observations, it is straightforward to verify that the tangent bundle $TM_{0}$ is naturally identified with the quotient $(TS/TS^{\perp})/T^{k}$. From this it follows that the symplectic frame bundle $\operatorname{Sp}(M_{0},\omega_{0})$ is naturally identified with the quotient $\operatorname{Sp}(TS/TS^{\perp})/T^{k}$.

[TABLE]

The following fact was stated by Robinson [5]. Let $(Y,\gamma)\rightarrow Z$ be a principal circle bundle with connection one-form over an arbitrary manifold $Z$. Let $F$ be a fibrating foliation of $Z$ with leaf space $Z_{0}$ and quotient map $Z\stackrel{{\scriptstyle\varrho}}{{\longrightarrow}}Z_{0}$. Let the curvature of $\gamma$ be $\varpi$. If $\gamma$ has trivial holonomy over the leaves of $F$, then $(Y,\gamma)$ descends to a principal circle bundle with connection one-form $(Y_{0},\gamma_{0})\rightarrow Z_{0}$ such that the curvature $\varpi_{0}$ of $\gamma_{0}$ satisfies $\varrho^{*}\varpi_{0}=\varpi$.

In our case, the base manifold is $\operatorname{Sp}(TS/TS^{\perp})$, and the fibrating foliation is $F=\mbox{span}\left\{\widetilde{\xi}_{1M},\ldots,\right.$ $\left.\widetilde{\xi}_{kM}\right\}$. By Definition 4.1, if $x\in\mathfrak{t}^{*}$ is a quantized energy level for $(M,\omega,\Phi)$, then $\gamma_{S}$ has trivial holonomy over the leaves of $F$. This is exactly the condition required for $(P_{S},\gamma_{S})\rightarrow\operatorname{Sp}(TS/TS^{\perp})$ to descend to a circle bundle

[TABLE]

where $P_{0}=P_{S}/T^{k}$ and $\gamma_{0}$ is a connection one-form on $P_{0}$ such that the curvature of $\gamma_{S}$ is the pullback of the curvature of $\gamma_{0}$. It is easily checked that $(P_{0},\Sigma_{0},\gamma_{0})$ is a metaplectic-c prequantization for $(M_{0},\omega_{0})$. In other words, when $x$ is a quantized energy level for $(M,\omega,\Phi)$, the top row of the diagram above can be completed in the obvious manner, and the result is a metaplectic-c prequantization for the symplectic reduction.

By Theorem 4.2, the quantized energy levels of $(M,\omega,\Phi)$ are the regular values of $\Phi$ that lie in $h\mathbb{Z}^{k*}$. We conclude the following.

Theorem 4.3**.**

Let $(M,\omega)$ have an effective Hamiltonian $T^{k}$ action with momentum map $\Phi$ and at least one fixed point. Assume that $(M,\omega)$ is metaplectic-c prequantizable, and let $(P,\Sigma,\gamma)$ be an equivariant metaplectic-c prequantization for $(M,\omega,\Phi)$. If $x\in\mathfrak{t}^{*}$ is a regular value of $\Phi$ that lies in $h\mathbb{Z}^{k*}$, and if $T^{k}$ acts freely on the level set $S=\Phi^{-1}(x)$, then the symplectic reduction $(M_{0},\omega_{0})$ of this level set acquires a metaplectic-c prequantization by taking the quotient of $(P_{S},\gamma_{S})$ by $T^{k}$.

5 Examples

5.1 Harmonic oscillators

Let $M=\mathbb{R}^{2n}=\mathbb{C}^{n}$, with Cartesian coordinates $(q_{1},\ldots,q_{n},p_{1},\ldots,p_{n})$ and complex coordinates $z_{j}=q_{j}+ip_{j}$, $1\leq j\leq n$. Equip $M$ with the symplectic form $\omega=\sum_{j=1}^{n}dq_{j}\wedge dp_{j}=\frac{1}{2i}\sum_{j=1}^{n}d\overline{z}_{j}\wedge dz_{j}$. Since $M$ is contractible, $(M,\omega)$ admits a metaplectic-c prequantization that is unique up to isomorphism.

Let the circle $T^{1}=U(1)$ act on $M$ as follows: given $\tau\in T^{1}$ and $m=(z_{1},\ldots,z_{n})\in M$,

[TABLE]

Identify $\mathfrak{t}$ with $\mathbb{R}$ such that for all $\xi\in\mathfrak{t}$, $\exp(\xi)=e^{2\pi i\xi}\in T^{1}$. The momentum map $\Phi:M\rightarrow\mathfrak{t}^{*}$ is given, up to an additive constant, by

[TABLE]

The fixed point of the $T^{1}$ action is the origin, $z=(0,\ldots,0)$. If we identify $T_{z}M$ with $\mathbb{R}^{2n}=\mathbb{C}^{n}$ in the obvious way, it is immediate that every $\tau\in T^{1}$ acts on $T_{z}M$ as a complex linear isomorphism. Explicitly, for any $\xi\in\mathfrak{t}$, if $\tau=\exp(\xi)$, then $\tau$ acts on $T_{z}M$ by the complex matrix $\mbox{diag}\left(e^{2\pi i\xi},\ldots,e^{2\pi i\xi}\right)$. The group homomorphism $w_{z}:T^{k}\rightarrow U(1)$, as defined in Section 3.2, is

[TABLE]

Therefore

[TABLE]

Over the fixed point $z$, we find that

[TABLE]

If $n$ is even, then $-\frac{n}{2}\in\mathbb{Z}^{*}\subset\mathfrak{t}^{*}$ and the equivariance condition is satisfied, but not if $n$ is odd. Let

[TABLE]

Then $\Phi^{\prime}$ is also a momentum map for the $T^{1}$ action, and $\frac{1}{h}\Phi^{\prime}(z)-\frac{1}{2}w_{*}\in\mathbb{Z}^{*}\subset\mathfrak{t}^{*}$, for all $n$.

The quantized energy levels of the system $(M,\omega,\Phi^{\prime})$ are the regular values of $\Phi^{\prime}$ that are in $h\mathbb{Z}^{*}\subset\mathfrak{t}^{*}$. Since

[TABLE]

a regular value is $x\in\mathbb{R}$ such that $x<\frac{hn}{2}$. Thus a quantized energy level takes the form

[TABLE]

where $N\in\mathbb{Z}$ is such that $N>-\frac{n}{2}$.

The Hamiltonian energy function for an $n$-dimensional harmonic oscillator is

[TABLE]

Note that $H=-\frac{1}{2\pi}\Phi^{\prime}+\frac{\hbar n}{2}$. Therefore the quantized energy levels for the system $(M,\omega,H)$ take the form

[TABLE]

where $N\in\mathbb{Z}$ is such that $N>-\frac{n}{2}$.

For comparison, the standard quantum mechanical calculation for the energy levels of the quantized harmonic oscillator yields

[TABLE]

where $N\in\mathbb{Z}$ is such that $N\geq 0$. The two calculations do not agree on the starting point for the energy spectrum (but see below). However, the equivariant metaplectic-c prequantization does yield the $\frac{n}{2}$ shift in the energy levels. By contrast, Kostant-Souriau quantization requires the half-form correction to obtain this shift, which uses a choice of polarization.

In quantum mechanics, the quantized energy levels of this system are obtained by solving the Schrödinger equation, which is linear: an $n$-dimensional harmonic oscillator is equivalent to $n$ independent one-dimensional harmonic oscillators. Consider the system described by the functions $H_{1},\ldots,H_{n}$ where $H_{j}=\frac{1}{2}(q_{j}^{2}+p_{j}^{2})$ for each $j$. By an essentially identical calculation, we find that the quantized energy levels for such a system have the form $(E_{1},\ldots,E_{n})$, where for each $j$, $E_{j}=\hbar\left(N_{j}+\frac{1}{2}\right)$ for some $N_{j}\in\mathbb{Z}$ such that $N_{j}>-\frac{1}{2}$. If we view the quantized energy levels of the $n$-dimensional harmonic oscillator as $E=E_{1}+\ldots+E_{n}$, we obtain

[TABLE]

We now have both the $\frac{n}{2}$ shift and the correct starting point, suggesting that this is the more appropriate mathematical interpretation of the physical system.

5.2 Complex projective space

Consider $\mathbb{C}^{n+1}$ with the usual complex coordinates $z=(z_{0},\ldots,z_{n})$, and complex projective space $\mathbb{CP}^{n}$ with the usual homogeneous coordinates $[z]=[z_{0}:\ldots:z_{n}]$. The two-form

[TABLE]

on $\mathbb{C}^{n+1}$ descends to $\mathbb{CP}^{n}$. Let

[TABLE]

on $\mathbb{CP}^{n}$, where $K>0$ is a positive constant to be determined. Then $\omega_{FS}$ is a Kähler form on $\mathbb{CP}^{n}$: specifically, a scalar multiple of the Fubini-Study form.

Robinson and Rawnsley [6] demonstrated that $(\mathbb{CP}^{n},\omega_{FS})$ admits a metaplectic-c prequantization if and only if $K=\hbar\left(N+\frac{n+1}{2}\right)$ for some $N\in\mathbb{Z}$. Note that $\mathbb{CP}^{n}$ admits metaplectic-c prequantizations for all $n$. This is an improvement over the Kostant-Souriau quantization scheme with half-form correction, because $\mathbb{CP}^{n}$ does not admit a metaplectic structure when $n$ is even.111Some additional detail: if we consider a regular level set of the $n$-dimensional harmonic oscillator from Section 5.1 corresponding to energy $K$, the symplectic reduction at this level is the symplectic manifold $(\mathbb{CP}^{n-1},Ki\varpi_{FS})$. The fact that the reduced system admits a metaplectic-c prequantization when $K=\hbar\left(N+\frac{n}{2}\right)$ can be checked directly using properties of $\mathbb{CP}^{n-1}$, as in [6], or it can be seen immediately by applying Theorem 4.3 to the $n$-dimensional harmonic oscillator.

As a concrete example, we consider $\mathbb{CP}^{2}$, which does not admit a metaplectic structure. For any $K$ of the form $K=\hbar\left(N+\frac{3}{2}\right)$, $N\in\mathbb{Z}$, the symplectic manifold $(\mathbb{CP}^{2},\omega_{FS})$ does not admit a prequantization line bundle either, since $\omega_{FS}$ is not integral. However, it does admit a metaplectic-c prequantization for any such $K$. We choose $K=\frac{3}{2}\hbar$.

Define an action of $T^{2}$ on $\mathbb{C}^{3}$ such that if $\tau=(\tau_{1},\tau_{2})\in T^{2}$, then

[TABLE]

where $k^{j}=(k_{1}^{j},k_{2}^{j})\in\mathbb{Z}^{2}$ for $j=1,2$, and $\left\{k^{1},k^{2}\right\}$ is an integer basis for $\mathbb{Z}^{2}$. This action descends to an effective Hamiltonian action of $T^{2}$ on $(\mathbb{CP}^{2},\omega_{FS})$. A calculation establishes that the momentum map takes the form

[TABLE]

where $(C_{1},C_{2})\in\mathbb{R}^{2*}$ is a constant.

The fixed points of the action are $Z_{0}=[1,0,0]$, $Z_{1}=[0,1,0]$ and $Z_{2}=[0,0,1]$. It suffices to find a value of $(C_{1},C_{2})$ such that the value of $\Phi$ at one of these points satisfies the equivariance condition in Theorem 3.1. For example, at $Z_{0}$, we have

[TABLE]

The equivariance condition at this point is

[TABLE]

where $w_{Z_{0}}:T^{k}\rightarrow U(1)$ is defined in terms of the action of $T^{2}$ on the tangent space $T_{Z_{0}}\mathbb{CP}^{2}$. We can satisfy the equivariance condition by taking $(C_{1},C_{2})=\frac{1}{2}w_{Z_{0}*}$. It remains to calculate $\frac{1}{2}w_{Z_{0}*}$.

On the open set $\left\{[z]\in\mathbb{CP}^{2}:z_{0}\neq 0\right\}$, use local coordinates $(\zeta_{1},\zeta_{2})=\left(\frac{z_{1}}{z_{0}},\frac{z_{2}}{z_{0}}\right)$ for $\mathbb{CP}^{2}$. Let $\xi=\sum_{j=1}^{2}a_{j}\xi_{j}\in\mathfrak{t}$ be arbitrary, and let $\tau=\exp(\xi)=(e^{2\pi ia_{1}},e^{2\pi ia_{2}})$. Then $\tau$ acts on the point $(\zeta_{1},\zeta_{2})$ by

[TABLE]

Identify $T_{Z_{0}}\mathbb{CP}^{2}$ with $\mathbb{C}^{2}$ in the natural way. Then the complex matrix corresponding to the action of $\tau$ on $T_{Z_{0}}\mathbb{CP}^{2}$ is

[TABLE]

Therefore

[TABLE]

which implies that

[TABLE]

Hence we set $(C_{1},C_{2})=\frac{1}{2}\left(k_{1}^{1}+k_{1}^{2},k_{2}^{1}+k_{2}^{2}\right)$.

We can verify that this choice of $(C_{1},C_{2})$ also satisfies the equivariance condition over $Z_{1}$. At $Z_{1}$, we have

[TABLE]

We need to calculate $\frac{1}{2}w_{Z_{1}*}$, where $w_{Z_{1}}$ is defined in terms of the $T^{2}$ action on the tangent space $T_{Z_{1}}\mathbb{CP}^{2}$. On the open set $\left\{[z]\in\mathbb{CP}^{2}:z_{1}\neq 0\right\}$, use local coordinates $(\zeta_{0},\zeta_{2})=\left(\frac{z_{0}}{z_{1}},\frac{z_{2}}{z_{1}}\right)$. Then $\tau=(e^{2\pi ia_{1}},e^{2\pi ia_{2}})$ acts by

[TABLE]

An identical calculation to the one performed at $Z_{0}$ yields

[TABLE]

Now,

[TABLE]

as needed. One can similarly check $Z_{2}$.

The image of the momentum map in $\mathbb{R}^{2*}$ (scaled by $\frac{1}{h}$, for simplicity) is the triangle with vertices

[TABLE]

The quantized energy levels correspond to the integer lattice points lying strictly in the interior of the triangle.

As a particularly simple example, consider $k^{1}=(1,0)$ and $k^{2}=(0,1)$. The vertices of the image of the momentum map are

[TABLE]

There is exactly one integer lattice point in the interior of the triangle, namely $(0,0)$. More generally, if we let $K=\hbar\left(N+\frac{3}{2}\right)$ for an arbitrary $N\in\mathbb{Z}$, $N\geq 0$, the vertices are

[TABLE]

and the number of quantized energy levels is $\frac{(N+2)(N+1)}{2}={{N+2}\choose{N}}$. If $N=-1$, the system is metaplectic-c prequantizable and the vertices take the form given above, but there are no quantized energy levels.

The symplectic reduction of the three-dimensional harmonic oscillator at the quantized energy level $K=\hbar\left(N+\frac{3}{2}\right)$ is exactly the symplectic manifold $(\mathbb{CP}^{2},Ki\varpi_{FS})$. We recognize the value ${{N+2}\choose{N}}$ from the quantum mechanical calculation as the multiplicity of the $N$th quantized energy level for $N\geq 0$. This calculation does not yield a quantized energy level corresponding to $N=-1$, and indeed this regular value has multiplicity zero by the above interpretation.

This last example and that in Section 5.1 are different facets of the same system. We will treat the relationships between them in greater generality in a subsequent paper concerning equivariant metaplectic-c prequantizations and quantized energy levels for toric manifolds (in preparation).