Equivariant Index on Toric Contact Manifolds
Let (M, D, G) be a (2n + 1)-dimensional toric contact manifold of Reeb type with moment cone C, where n > 1. There exists a G-invariant Sasakian 1-form on M such that D = ker α. This structure allows us to define the horizontal Dolbeault operator on M and more generally the horizontal Dolbeault operator twisted by a G-equivariant vector bundle. This operator is G-transversally elliptic, in the sense of [5], so its index is a formal power series. The goal of this thesis is to calculate the index of the twisted horizontal Dolbeault operator explicitly by identifying the multiplicities of each term in the power series. We apply a localisation method to the symbol of the horizontal Dolbeault operator, decomposing it into a finite sum supported on certain closed orbits of the Reeb vector field. Using Lerman’s classification of toric contact manifolds [30] and the local form for the moment map, we obtain a Lefschetz-type formula for the index. Adapting the Lawrence-Varchenko formula, we obtain a polar decomposition of the moment cone C and relate it to the index of the untwisted horizontal Dolbeault operator. This leads to an explicit formula for the index as a sum of lattice points related to the moment cone C.
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