Quantisation on hyper-Kähler spaces


Geometric quantisation has proven an effective approach to many problems in mathematical physics. Many examples have been shown of theories whose classical solutions form geometric spaces with rich and interesting structures, which may then be used for quantisation. Sometimes, however, there is just too much structure, and it can become difficult to pick on to use. This is the case, for example, for hyper-Kähler manifolds, which come with infinite families of symplectic forms.

In a recent work with J.E. Andersen and G. Rembado, we proposed a new paradigm for quantisation of hyper-Kähler spaces assuming sufficient symmetry, which opens the way to exploration in many different directions. There are many examples of spaces to which this quantisation can be applied, including several from mathematical physics, and there are many famous results in “ordinary” quantisation that should be tested for this new version, notably the famous statement that quantisation commutes with reduction. In this talk I will give a panoramic of known results and possible research directions, including ongoing work with M. Mayrand.