Cohomology and representation theory of infinite dimensional Lie groups

This is a mathematics proposal, belonging to the field of (differential) geometry and Lie theory. It concerns the cohomology and representation theory of infinite dimensional Lie groups, with special focus on gauge groups. The aim is twofold. First, we aim to develop the structure theory and cohomology of Generalised Lie Algebra Sheaves (GLAS). Technically, GLAS are defined as (certain) subsheaves of the sheaf of sections of a Lie algebroid. This is a class of infinite dimensional Lie algebras that is specially adapted to geometric situations, and wide enough to encompass many examples from geometry and physics, including gauge algebras. We expect to discover a rich structure theory, and a wide range of tools for calculating Lie algebra cohomology. Second, we aim to develop the theory of projective unitary representations for infinite dimensional Lie groups in general, and for gauge groups in particular. The above mentioned GLAS cohomology should yield methods to calculate the second Lie algebra cohomology, a vital and nontrivial ingredient of projective representation theory. In quantum theory, projective unitary representations determine the relation between symmetries and conserved quantities. Hence, this project is strongly related to (Mathematical) Physics. One of the overall techniques we develop is the theory of Reflection Positivity, which originated in QFT and (Quantum) Statistical Physics, but which we aim to apply in representation theory. Applying these (and other) techniques to the gauge groups, we aim to prove classification theorems for a wide class of projective unitary representations.