Geometric quantization via cotangent models

Abstract

In this article we give a universal model for geometric quantization associated to a real polarization given by an integrable system with non-degenerate singularities. This universal model goes one step further than the cotangent models in [13] by both considering singular orbits and adding to the cotangent models a model for the prequantum line bundle. These singularities are generic in the sense that are given by Morse-type functions and include elliptic, hyperbolic and focus-focus singularities. Examples of systems admitting such singularities are toric, semitoric and almost toric manifolds, as well as physical systems such as the coupling of harmonic oscillators, the spherical pendulum or the reduction of the Euler’s equations of the rigid body on $T^*(SO(3))$ to a sphere. Our geometric quantization formulation coincides with the models given in [11] and [21] away from the singularities and corrects former models for hyperbolic and focus-focus singularities cancelling out the infinite dimensional contributions obtained by former approaches. The geometric quantization models provided here match the classical physical methods for mechanical systems such as the spherical pendulum as presented in [4]. Our cotangent models obey a local-to-global principle and can be glued to determine the geometric quantization of the global systems even if the global symplectic classification of the systems is not known in general.

A The sheaf cohomology of the cylinder

We reproduce here the calculations of the sheaf cohomology of the cylinder. In Section 5 we change a piece of the cohomology and with this appendix we want to recall the whole picture of the construction of the covering and the cohomology groups of flat sections.

We start by calculating the cohomology on a finite subset of the cylinder $M={\mathbb {R}}\times S^1$. Let U be a band of the cylinder, i.e., a subset of M of the form $I\times S^1$, with $I \subset {\mathbb {R}}$ open and bounded. Assume that U contains at most one Bohr-Sommerfeld leaf and construct a cover ${\mathcal {A}}$ of U by three rectangles A, B, C that slightly overlap as in Figure 5.

Fig. 5

A band U of the cylinder $M={\mathbb {R}}\times S^1$ covered by three rectangles A, B, C. The rectangles overlap slightly in the thick joints

We want to identify the Čech 0-cochains and 1-cochains with respect to the cover ${\mathcal {A}}$. A 0-cochain $\alpha$ is an assignment of a flat section over A, B and C to that same subsets. Then, $\alpha$ assigns A to the section $a_A(t)e^{it\theta }$, B to $a_B(t)e^{it\theta }$ and C to $a_C(t)e^{it\theta }$.

The angular coordinate $\theta$ can not be defined on all of $S^1$ so a branch of $\theta$ has to be fixed on each rectangle. We choose the branches so that $\theta _A = \theta _B$ on $A\cap B$, $\theta _B = \theta _C$ on $B\cap C$, and $\theta _C = \theta _A + 2\pi$ on $A\cap C$. The coboundary operator $\delta$ acts on $\alpha$ as

$$\begin{aligned} (\delta \alpha )_{ij} = \eta _j - \eta _i = a_j(t)\, e^{it\theta _j} - a_i(t)\, e^{it\theta _i}, \end{aligned}$$

for $i,j\in \{A,B,C\}$. We impose that, at the three intersections, $\delta \alpha$ is 0, obtaining the following three equations:

$$\begin{aligned} 0&= a_B(t)\, e^{it\theta _B} - a_A(t)\, e^{it\theta _A} \qquad \text {on } A\cap B \end{aligned}$$

(A.1)

$$\begin{aligned} 0&= a_C(t)\, e^{it\theta _C} - a_B(t)\, e^{it\theta _B} \qquad \text {on } B\cap C \end{aligned}$$

(A.2)

$$\begin{aligned} 0&= a_A(t)\, e^{it\theta _A} - a_C(t)\, e^{it\theta _C} \qquad \text {on } C\cap A \end{aligned}$$

(A.3)

Then, $\alpha$ is a cocycle if and only if the following three equations simultaneously:

$$\begin{aligned} a_B(t)&= a_A(t) \qquad \qquad \text {on } A\cap B \end{aligned}$$

(A.4)

$$\begin{aligned} a_C(t)&= a_B(t) \qquad \qquad \text {on } B\cap C \end{aligned}$$

(A.5)

$$\begin{aligned} a_A(t)&= a_C(t)\, e^{2\pi it} \qquad \text {on } C\cap A \end{aligned}$$

(A.6)

which is not possible since $e^{2\pi it}$ can not equal 1 in an entire interval of values of t. We conclude that are no 0-cocycles and that $H^0 = 0$.

Now, a 1-cochain $\beta$ is an assignment of a flat section over $A\cap B$, $B\cap C$ and $C\cap A$ to that same subsets. Then, $\beta$ assigns $A\cap B$ to the section $b_{AB}(t)e^{it\theta }$, $B\cap C$ to $b_{BC}(t)e^{it\theta }$ and $C\cap A$ to $b_{CA}(t)e^{it\theta }$. There only possible triple intersection in the cover ${\mathcal {A}}$ is empty and 2-cochains do not exist, implying that every 1-cochain is a cocycle. Since the 1-cochain is determined essentially by the three smooth functions $b_{ij}(t)$ on I, the space of 1-cocycles is isomorphic to $C^\infty (I)^3$.

A 1-cochain $\beta$ is a coboundary if there exists a 0-cochain $\alpha = \{a_A(t) e^{it\theta _A}, a_B(t) e^{it\theta _B}, a_C(t) e^{it\theta _C}\}$ with $\delta \alpha = \beta$. Or, equivalently, if these three equations are satisfied:

$$\begin{aligned} \beta _{AB}&= \alpha _B - \alpha _A \quad \text {on } A\cap B \end{aligned}$$

(A.7)

$$\begin{aligned} \beta _{BC}&= \alpha _C - \alpha _B \quad \text {on } B\cap C \end{aligned}$$

(A.8)

$$\begin{aligned} \beta _{CA}&= \alpha _A - \alpha _C \quad \text {on } C\cap A \end{aligned}$$

(A.9)

Giving the sections explicitly, these equations transform to:

$$\begin{aligned} b_{AB}(t)e^{it\theta _A}&= a_B(t) e^{it\theta _B} - a_A(t) e^{it\theta _A} \qquad \text {on }A\cap B \end{aligned}$$

(A.10)

$$\begin{aligned} b_{BC}(t)e^{it\theta _B}&= a_C(t) e^{it\theta _C} - a_B(t) e^{it\theta _B} \qquad \text {on }B\cap C \end{aligned}$$

(A.11)

$$\begin{aligned} b_{CA}(t)e^{it\theta _C}&= a_A(t) e^{it\theta _A} - a_C(t) e^{it\theta _C} \qquad \text {on }C\cap A \end{aligned}$$

(A.12)

Notice that, on each ordered intersection of two sets $E\cap F$, we use the $\theta$ coordinate from E. In each equation all the $\theta$ coordinates coincide except in Equation A.12, where they differ by a factor of $2\pi$. Then, we obtain a system of equations in the three unknown functions $a_A$, $a_B$, and $a_C$ which has to be true for each value of t in I and which has, as a matrix of coefficients, the following:

$$\begin{aligned} \begin{pmatrix} -1&{} 1&{} 0\\ 0&{} -1&{} 1\\ e^{-2\pi it}&{} 0&{} -1 \end{pmatrix} \end{aligned}$$

(A.13)

We observe that the matrix has rank 3 (and therefore the system has a unique solution) when $e^{-2\pi it} \ne 1$. Then if $e^{-2\pi it}$ is never 1 on U, every cocycle is a coboundary, and U has the zero cohomology. Otherwise, if $e^{-2\pi it}=1$ somewhere in I, which happens if and only if I contains an integer m, the system only has a solution if the matrix

$$\begin{aligned} \begin{pmatrix} -1&{} 1&{} 0&{} b_{AB}(m)\\ 0&{} -1&{} 1&{} b_{BC}(m)\\ e^{-2\pi it}&{} 0&{} -1&{} b_{CA}(m) \end{pmatrix} \end{aligned}$$

(A.14)

has rank 2, i.e., if $\beta$ satisfies the condition

$$\begin{aligned} b_{AB}(m) + b_{BC}(m) + b_{CA}(m) = 0. \end{aligned}$$

(A.15)

Then, we have the following result:

Proposition A.1

the cohomology group $H^1$ of U is precisely

$$\begin{aligned} H^1 = C^\infty (I)^3 / \{b_{AB}(m) + b_{BC}(m) + b_{CA}(m) = 0\}, \end{aligned}$$

(A.16)

which is isomorphic to ${\mathbb {C}}$.

Observe that for $k>1$ there are no $(k+1)$-fold intersections in the cover ${\mathcal {A}}$. Therefore, all the cohomology groups $H^k_{{\mathcal {A}}}$ are zero for $k>1$.

The condition $e^{2\pi it}=1$ is satisfied exactly at the Bohr-Sommerfeld leaves, so we conclude that if U is a band on the cylinder, the sheaf cohomology of U with respect to the cover ${\mathcal {A}}$ by the three rectangles is trivial if U does not contain a Bohr-Sommerfeld leaf, and it is:

$$\begin{aligned} H^k_{{\mathcal {A}}} (U;{\mathcal {J}}) \cong {\left\{ \begin{array}{ll} {\mathbb {C}}&{}k = 1\\ 0 &{}k\ne 1 \end{array}\right. } \end{aligned}$$

One can see that if another cover ${\mathcal {B}}$ of U is made by k rectangles instead of 3, the cohomology calculated with respect to ${\mathcal {B}}$ is the same as that calculated with respect to ${\mathcal {A}}$.