Introducing Quaternionic Gerbes .

The notion of a quaternionic gerbe is presented as a new way of bundling algebraic structures over a four manifold. The structure groupoid of this fibration is described in some detail. The Euclidean conformal group RSO(4) appears naturally as a (non-commutative) monoidal structure on this groupoid. Using this monoidal structure we indicate the existence of a canonical quaternionic gerbe associated to a conformal structure on a four manifold. It is natural to think that quaternionic algebra and four dimensional geometry should be closely linked. Certainly complex algebra and analysis provide indispensable tools for exploring two dimensional Riemaniann geometry. However, despite many attempts, quaternionic algebra has not been usefully applied to the differential geometry of four manifolds. The most commonly held view is that quaternionic algebra is too rigid to be useful in studying four manifolds. It is generally assumed that the natural setting for quaternionic differential geometry is hyperKähler or hypercomplex. [10] The purpose of this talk/article is to present the notion of a quaternionic gerbe, and to demonstrate that they appear naturally as a quaternionic algebraic structure on four manifolds. This work appears as part of an effort to realize the goal of “doing four dimensional geometry and topology with quaternionic algebra.” Although quaternionic structures are defined [8] for all 4n dimensional manifolds, the basic structures and difficulties are already present in only four dimensions. The notion of “quaternionic curve” has been equated with that of a “self dual conformal” structure.[2] Note that even this class of manifolds is strictly larger than the hyperKähler manifolds. Here we restrict our attention to smooth oriented four manifolds, including hyperKähler and self dual conformal manifolds. It is proposed that a “quaternionic structure” on a four manifold is essentially a Euclidean conformal structure. This compares favourably with the two dimensional case where fixing a complex structure is equivalent to fixing a conformal structure.