Hamiltonian Group Action

Definition

First recall that the coadjoint action $\text{ad}^\ast:G\times\mathfrak{g}^\ast\rightarrow\mathfrak{g}^\ast$ of a Lie group $G$ on the dual of its Lie algebra $\mathfrak{g}^\ast$ is defined as $$ \langle \text{ad}^\ast_g \eta, X\rangle = \langle \eta, \text{ad}_g X\rangle, $$ where $\text{ad}:G\times\mathfrak{g} \rightarrow \mathfrak{g}$ is the pointwisely (for points in $G$) defined as the differential $\text{ad}_g = d_e\text{Ad}_g: T_e G \rightarrow T_{e= \text{Ad}_g e} G$ of the adjoint $\text{Ad}_g: h \mapsto g h g^{-1}$.

The $G$-action $\rho:G\times M \rightarrow M$ give rise to a Lie algebra homomorphism $\mathfrak{g}\rightarrow \Gamma(TM)=\text{Vect}(M):\xi\mapsto \xi^M$ where $\xi^M_p = \frac{d}{dt}\rho(\exp(t\xi),p)$ is the fundamental vector field associated to $\xi$. Associated to a presymplectic form $\omega$ that is $G$-invariant, the moment map is a smooth map $$ \mu:M\rightarrow \mathfrak{g}^* $$ that is equivariant w.r.t. to the $G$-action on $M$ and the coadjoint action on $\mathfrak{g}^*$, such that the components $\mu^\xi=\langle\mu,\xi\rangle$ (which are $\mathbb{R}$-valued functions) satisfy the Hamilton’s equation $$ d\mu^\xi = \iota_{\xi^M} \omega $$ for all $\xi\in\mathfrak{g}$. Hence for every $X\in \Gamma(TM)$ we have $X\mu^\xi = \iota_X d\mu^\xi = \omega(\xi^M,X)$.

Definition(Hamiltonian group action) . A smooth Lie group $G$-action on a presymplectic manifold is Hamiltonian if it is a symplectomorphism that admits a moment map.

Results

The area of symplectic geometry which deals with the global properties of Hamiltonian group action is to a large extent a branch of equivariant topology that involves manifolds equipped with some unconventional structure. We present some examples

Duistermaat-Heckman formula / Equivariant cohomology class of degree two

The Duistermaat-Heckman theorem states that the oscillatory integral for the moment map of a torus action on a symplectic manifold is exactly equial to the leading term of its aymptoptic expansion. This provides a formula for the Fourier transform of the push-forward of the Liouville measure by the moment map in terms of the fixed points of the action.

More explicitly, given an action of the circle $\mathbb{T}$ on a compact symplectic manifold $(M,\omega)$ with isolated fixed points forming a set $M^G$, and a moment map $\mu: M \rightarrow \mathfrak{g}^\ast = \mathbb{R}$, the Duistermaat-Heckman formula is $$ \frac{1}{n!}(1/2\pi)^n \int_M e^{\mu}\omega^n = \sum_{p\in M^G} \frac{e^{\mu(p)}}{\prod_j\alpha_{j,p}} $$ where $\alpha_{j,p}$ are the weights of the linearized action of $G$ on $T_p M$ at a fixed point $p$.

It was soon discovered by Berne, Vergne and Atiyah, Bott that the theorem is a particular case of a general localization formula in equivariant cohomology for torus actions. This formula is of a purely topological nature, and equates the integral of an equivariant cohomology class over $M$ and the sum of the integrals of this class over the components of the fixed point set with corrections coming from the actions on the normal bundles to the components. Explicitly, $$ \int_M u = \sum_F \int_F \frac{u|_F}{eu(\mathcal{N}_F)} $$ where $u$ is an equivariant cohomology class on $M$, $F$ are the components of the fixed point set, and $eu(\mathcal{N}_F)$ is the equivariant Euler class of the normal bundle to $F$.

If $u=e^{-(\omega-\mu)}$ the localization theorem becomes the Duistermaat-Heckman formula.

Geometric Quantization / Indices of Dirac operators associated with symplectic structures

When a Dirac operator associated with a symplectic structure is invariant under a compact group action, the index becomes a virtual representation, and the character of the virtual representation can be evaluated as the integral of an equivariant cohomology class (by a generalizeation of the Atiyah-Singer index theorem).

The geometric quantization $\mathcal{Q}(M,\omega)$ is defined as the virtual representation, so the index, of the Dirac operator, hence $$ \mathcal{Q}(M,\omega) = \int_M e^{\omega/2\pi} \text{Td}(M). $$ In the equivariant situation $\text{Td}(M)$ should be replaced by $\text{Td}^G(M)$, and $\omega$ by the equivariant symplectic form $\omega-\mu$ where $\mu$ is the moment map.

Note that in this formula the integrand depends only on the cohomology class of the symplectic structure and the Chern classes of the almost complex structure associated with the symplectic form. Actually this Chern class is just an invariant of the ‘stable comlex structure’ associated with the almost complex structure. A cohomology class, a stable complex structure and an orientation are sufficient to define a Dirac operator with the correct index.

A variant of this, which only depends on the cohomology class and the orientation, is “$\text{Spin}^c$ quantization”.

When the group acting is abelian, the index is determined by the fixed point data. Applying the localization theorem to the integrand above, one can obtain an explicit formula, called the Atiyah-Bott-Lefschetz fixed point theorem.

Guillemin-Sternberg

This asserts that the $G$-invariant part of the quantization of a symplectic $G$-manifold is equal to the quantization of the reduction at the zero level of the moment map. Under certain additional assumptions the theorem are obtained using equivariant $K$-theory.

Non-abelian localization theorem of Jeffrey and Kirwan

This is closely related to the Guillemin-Sternberg. It expresses the integral over the reduced space of an equivariant cohomology class coming from a Hamiltonian $G$-manifold in terms of the fixed points of the action.

Generalization to stacky Hamiltonian action of group stacks on symplectic stack

For Hamiltonian action of an etale Lie group stack on an etale symplectic stack