On the geometric quantization of contact manifolds

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Suppose that ( M , E ) is a compact contact manifold, and that a compact Lie group G acts on M transverse to the contact distribution E . In an earlier paper, we defined a G -transversally elliptic Dirac operator ⁄ D b , constructed using a Hermitian metric h and connection ∇ on the symplectic vector bundle E → M , whose equivariant index is well-defined as a generalized function on G , and gave a formula for its index. By analogy with the geometric quantization of symplectic manifolds, the virtual G -representation Q ( M ) = [ ker ⁄ D b ] − [ ker ⁄ D b ∗ ] can be interpreted as the “quantization” of the contact manifold ( M , E ) ; the character of this representation is then given by the equivariant index of ⁄ D b . By defining contact analogues of the algebra of observables, prequantum line bundle and polarization, we further extend the analogy by giving a contact version of the Kostant–Souriau approach to quantization, and discussing the extent to which this approach is reproduced by the index-theoretic method.