Moment polyptychs and the equivariant quantisation of hypertoric varieties

Abstract

In this thesis, we develop a method to investigate the geometric quantisation of a hypertoric variety from an equivariant viewpoint, in analogy with the equivariant Verlinde formula for Higgs bundles. We do this by first using the residual circle action on a hypertoric variety to construct its symplectic cut that results in a compact cut space, which is needed for the localisation formulae to be well-defined and for the quantisation to be finite-dimensional. The hyperplane arrangement corresponding to the hypertoric variety is also affected by the symplectic cut, and to describe its effect we introduce the notion of a moment polyptych that is associated to the cut space. Also, we see that the prerequisite isotropy data that is needed for the localisation formulae can be read off from the combinatorial features of the moment polyptych. The equivariant Kawasaki-Riemann-Roch formula is then applied to the pre-quantum line bundle over each cut space, producing a formula for the equivariant character for the torus action on the quantisation of the cut space. Finally, using the quantisation of each cut space, we derive a formula expressing the dimension of each circle weight subspace of the quantisation of the hypertoric variety.