nLab Weyl bundle

If $(M,\omega)$ is a symplectic manifold then the completed symmetric power of the cotangent bundle $W = \hat{S}(T^* M)$, and sometimes also $W_h = \hat{S}(T^* M)[[h]]$ are called the Weyl bundle. (The same term is used for some other, quite different, notions!) In addition to the commutative symmetric algebra structure, there is a noncommutative product due symplectic structure.
If $a,b\in \Gamma_U(W_h)$ are sections of $W_h$ above open $U\subset M$ then their noncommutative Moyal-Weyl product is

$a \ast b = \left. exp \left(\frac{h}{2}\omega_{j l}(x) \frac{\partial}{\partial y}\frac{\partial}{\partial z}\right) a(y) b(z) \right|_{y=z}$

There is also a grading where $deg h = 2$ and $deg w = l$ for $w\in S^l(T^* M)$. So we get a bundle of noncommutative associative algebras.

Fedosov connection? is a connection on $W_h$ (depending on a choice of a cocycle, the Weyl curvature $\Omega\in Z^2(M)[[h]]$). It has the property that the exponential map identifies the smooth functions on $M$ with horizontal sections of $W_h$ for the connection.

Related entries are deformation quantization

See section 2.2 of

and section 6 of

• Simone Gutt, John Rawnsley, Natural star products on symplectic manifolds and quantum moment maps, pdf

Created on July 21, 2015 at 08:55:25. See the history of this page for a list of all contributions to it.