The algebra of densities can be seen to have origins dating back to the 19th century where densities were used to find invariants of the modular group. Since then they have continued to be a source of projective invariants and cocycles related with the projective group, most notably the Schwarzian derivative. One of the first times that the algebra of densities appears in the literature in a similar guise to the way we shall introduce it, is in the work of T.Y. Thomas. He showed that a projective connection on a manifold allows one to determine a canonical affine connection on the total space of a certain bundle which is now known as Thomas' bundle. More recently they have appeared, with the definition we shall use, by H. Khudaverdian and Th. Voronov when studying second order operators generating certain brackets. Of prime importance in this situation is the case of Gerstenhaber algebras and in particular the Batalin-Vilkovisky operator on the odd cotangent bundle. They have also been used by V.Y. Ovsienko and his group in the area of equivariant quantization which is a topic we shall come across in the text. Densities also regularly appear in physics. For example the correct interpretation of a wavefunction is a half-density on a manifold, and this explains their transformation properties under the Galilean group.These results motivate a study into the geometric structure of the algebra of densities as an object in their own right. We shall see that by considering them as a whole algebra many classical results have a clear geometrical picture. Moreover one finds that there are a wealth of areas within this algebra still to explore. We focused on two fundamental classes of objects, differential operators and Poisson structures. The results we find lead to interesting formula for certain equivariantly defined differential operators which can be applied to gain a wide class of cocycles similar to the Schwarzian derivative. We also find very intimate links with Batalin-Vilkovisky geometry and the methods we use show that it may be useful to consider the full algebra of densities when entering into this arena.
|Date of Award||1 Aug 2014|
- The University of Manchester
|Supervisor||Theodore Voronov (Supervisor) & Hovhannes Khudaverdyan (Supervisor)|