Witt Algebra and Virasoro Algebra

Diffeomorphisms of the Circle

Before introducing the diffeomorphism group of the circle, let us first see why it is important in 2d conformal field theory. Suppose we have a Minkowski spacetime of two dimension with coordinates , where is the time coordinate and is the space coordinate. We can introduce the light-cone coordinates so that the metric tensor becomes A conformal transformation is a transformation that preserves angles but not necessarily lengths. In two dimensions, conformal transformations can be characterized by holomorphic functions. Specifically, any transformation of the form where and are holomorphic functions, is a conformal transformation.

Diffeomorphism Group of the Circle

Denote the group of diffeomorphisms of the circle as , its subgroup of orientation preserving diffeomorphisms as .

Proposition

has two path-connected components corresponding to orientation preserving and orientation reversing diffeomorphisms. That is, with the -action given by conjugation with some fixed orientation-reversing diffeomorphism . In other words, there is a short exact sequence of Lie groups

Proof Idea Any diffeomorphism is either orientation-preserving or orientation-reversing, and this property is locally constant: the sign of the derivative gives a continuous map with discrete target, so it is constant on connected components. Thus any path in starting at the identity (orientation-preserving) stays in , and no orientation-reversing diffeomorphism can lie in the same connected component

The most important property of is that it is a simple group, and it can be generated by the exponential maps.

Lemma

is algebraically simple.

Proposition

is an infinite dimensional Lie group whose Lie algebra is identified with the real topological vector space of smooth vector fields on with the usual topology.

Proof See Milnor, Remarks on Infinite Dimensional Lie Groups.

Now let us see properties of the orientation-preserving diffeomorphisms:

Proposition

Suppose is the subgroup of that consists of all orientation-preserving diffeomorphisms fixing . Then , where the action satisfies for all and .

Vector Fields on a Circle

Definition

Define as the Lie algebra of complex vector fields on a circle. That is equipped with the usual Lie bracket where are derivatives with respect to . For each , we can define a norm on by

Proposition

For all and , there holds

Witt Algebra

The Witt algebra is a subalgebra of with elements having finite Fourier series: A convenient basis for is given by the vector fields which satisfy the commutation relations

Canonical 2-cocycle on

On , there exists a unique 2-cocycle on up to scalar multiple and coboundary, such that is zero on . Concretely, is given by

Virasoro Algebra

The Virasoro algebra is the unique nontrivial central extension of the Witt algebra, often denoted as or . That is, for some in the center of , equipped with the Lie bracket A useful and standard way to present the Virasoro algebra is to decompose it into three parts: where , and .

Induced Representations of

For any highest -weight representation for , we can define an induced representation of for each , by where is a standard basis of such that

References

Goodman and Wallach, Projective unitary positive-energy representations of