The Semigroup of Annuli

In the integrability theorem, we demonstrated that every projective representation of integrates to a projective representation of , and so does . This result initially appears paradoxical, as one might expect a real Lie algebra to integrate to a real Lie group, with its complexification integrating to the complexification of that group. However, this expectation is not met here, as lacks a complexification. Instead, an “almost” complexification of it is a semigroup of annuli.

Cannot be Complexified

Theorem

does not admit a complexification.

Proof We can see this by contradiction. If there were a complex Lie group serving as the complexification, then

The Semigroup of Annuli

Semigroup of Annuli

The semigroup of annuli, denoted , consists of equivalence classes of triples where is an annulus-like surface, and , are orientation preserving diffeomorphic parametrisations of the outer and inner boundaries of , respectively.
Two annuli and are considered equivalent if there exists a biholomorphic such that the following diagram commutes:

We call equivalence classes of annuli annulus classes. The semigroup operation is defined on annulus classes by welding corresponding annulus-like surfaces along their boundaries. Concretely, given two annulus classes and , their product is obtained by identifying the outer boundary of with the inner boundary of via the parametrisations and . The resulting annulus is , and we shall simply denote it by .
We denote the union of and as .

Remark

What we really care about here is not the plain annuli, but the rigged annuli that including the parametrisation information of the boundaries (riggings). If you forget the boundary parametrisations and look only at conformal isomorphism classes of annuli, then every annulus is conformally equivalent to a round annulus , and its conformal type is determined by a single real number .

So far, we have not proved that this welding operation is well defined: namely, that the welded surface carries a complex structure and that the operation is independent of the choice of representative. Clearly, the welding of two annuli remains an annulus topologically, we need to verify that the resulting annulus can be given a complex structure. In fact, since this is a local problem, it suffices to only consider the welding of two disks under some attaching map . In other words, we shall verify that the welding of two disks is conformally equivalent to the Riemann sphere. However, such an equivalence mapping is not unique in general, so we need to fix a normalisation condition to reduce the freedom. Suppose the two disks are and , with containing . Let be the welding map, then the unique conformal maps and such that and satisfies the normalisation condition are called the solution to the -welding problem. The normalisation condition can also be written out explicitly using Laurent series: We show that every -welding problem has a unique solution in conformal welding problem. We first prove the consecutive welding lemma, and then conclude in theorem by using the fact that any topological group is generated by a neighbourhood of the identity. This is, in fact, a topological gluing with extra conformality.

Proposition

is an complex Fréchet manifold, inheriting the complex structure from .

Proof It suffices to show that the inclusion has an open image.

Involution on

There is an involution that turns an annulus inside out, so that the incoming and outgoing circles are exchanged, but their parametrizations remain the same

The Standard Representative

Once we checked that is a well-defined semigroup, for any , we can pick a standard representative, i.e., a round annulus such that and .

Alternative Realizations of

Proposition

Every annulus can be uniquely (up to some normalization) identified as a tuple , where is a Riemann surface conformally equivalent to the Riemann sphere, and and are holomorphic embeddings of the unit disk and .

Proof This is a direct result of the conformal welding. For each annulus , we can weld an upper disk to its inner boundary and a lower disk to its outer boundary, so that the resulting Riemann surface is conformally equivalent to the Riemann sphere. The holomorphic embeddings and are given by the -welding and -welding respectively.

For more alternative realizations of , see

References

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