Quantomorphisms

Suppose that exists an -bundle over :

with a connection such that the curvature is equal to . Then we can realize the central extension as the group of automorphisms of this bundle.

We have the exact sequence:

which can be thought of as an exact sequence of algebroids over , or just of Lie superalgebras. It involves the Atiyah algebroid whose anchor is . The kernel of is the -dimensional space . A connection is a split:

Suppose that we can find a “symplectic potential” such that . Then we can use it to construct the connection satisfying:

where is the vector field arizing from the action of on . (We can think of as a coordinate in the fiber; it is only defined locally, but is globally well-defined.) Explicitly:

Let us consider the subalgebra consisting of Hamiltonian vector fields. For every even (we will restrict to even vector fields for simplicity) consider the following vector field on :

It is defined to preserve the connection. An explicit calculation shows that the Lie derivative vanishes:

Notice that the vertical component of (with respect to the connection defined in Eq. (1)) is . By construction, the space of vector fields of this form is closed under commutator. We can check it directly, using the formula:

As a Lie algebra this is . It integrates to the group of automorphisms of the fiber bundle which preserve the connection defined in Eq. (1).