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By Howie J.M. (ed.)

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Let (M, ω, Ψ) be a Hamiltonian G-manifold. Let π : P → M be a pre-quantization circle bundle. This means that P is a G-equivariant principal circle bundle with a connection one-form Θ, such that dΘ = −π ∗ ω, and such that the moment map Ψ is given by π ∗ Ψξ = Θ(ξP ) for all ξ ∈ g. ) Then the pullback π ∗ Ψ : P → g∗ is an exact moment map. 13. A linear combination of abstract moment maps is an abstract moment map. 14. If Ψ0 and Ψ1 are exact moment maps, then so is (1−ρ)Ψ0 +ρΨ1 for any smooth function ρ : M → [0, 1].

2) is called exact (cf. 8). A compatible two-form is then given by ω = −dµ. An exact moment map has the property that ΨH vanishes on M H for all H ⊆ G. 27. Many “classical” moment maps, such as the canonical moment map on a cotangent bundle, or the moment map on a pre-quantization circle bundle, are exact. 12. The following two examples exhibit functoriality properties of abstract moment maps with respect to G and M , respectively. 9. Let M be a G-manifold with an abstract moment map Ψ : M → g∗ .

Let M be a manifold with a G-action with isolated fixed points, and let Ψ : M → R be a polarized abstract moment map. 1) (M, Ψ) ∼ (Tp M, Ψ# p ), p∈M G 45 46 4. 1. A proper function on (0,1] where for each fixed point p ∈ M G , the vector space Tp M is equipped with the linear isotropy G-action induced from M , and Ψ# p : Tp M → R is a polarized abstract (0) = Ψ(p). 3. 1) is not specified. However, it is unique up to proper cobordism. Indeed, two such maps are cobordant through the trivial cobordism [0, 1]×Tp M with the polarized abstract moment map # ˜ v) = (1 − t)(Ψ# Ψ(t, p )1 (v) + t(Ψp )2 (v).

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