Weyl bundle

If (M,ω)(M,\omega) is a symplectic manifold then the completed symmetric power of the cotangent bundle W=S^(T *M)W = \hat{S}(T^* M), and sometimes also W h=S^(T *M)[[h]]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,bΓ U(W h)a,b\in \Gamma_U(W_h) are sections of W hW_h above open UMU\subset M then their noncommutative Moyal-Weyl product is

a*b=exp(h2ω jl(x)yz)a(y)b(z)| y=z 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 degh=2deg h = 2 and degw=ldeg w = l for wS l(T *M)w\in S^l(T^* M). So we get a bundle of noncommutative associative algebras.

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