Let be symplectic with a Hamiltonian -action whose moment map is . Consider now the product equipped with the product symplectic structure and the -action
whose moment map is
The symplectic cut is the symplectic quotient of at level .
::: remark Remark
The set decomposes into the disjoint union , where
On , each orbit of the -action contains a unique choice of , where is in fact real and positive. Thus . On the other hand, is just the usual symplectic quotient .
So the symplectic cut can be viewed as , with the -action on the boundary being factored out.
::: example Example