Vertex Operator Algebra - Heisenberg Lie Algebra Example

Heisenberg Lie Algebra Example

The Heisenberg Lie algebra is defined by the commutation relations:

One representation is to define the operators b, in terms of the dummy variables x as:

and b₀ = 0.

This can be made into a vertex algebra by the definition:

where :..: denotes normal ordering (i.e. moving all derivatives in x to the right). Thus Y(1,z) = Id.

T is defined by the conditions T.1 = 0 and

Setting

it follows that

and hence:

The vertex operators may also be written as a functional of a multivariable function f as:

if we understand that each term in the expansion of f is normal ordered.