Hamiltonian Group Actions and Equivariant Cohomology

A symplectic manifold is a smooth manifold \( M\) equipped with a non-degenerate closed 2-form \(\omega\).  Symplectic manifolds generalize phase spaces, i.e. position and momentum coordinates.  A smooth function \(H:M \rightarrow \mathbb{R}\) is called a Hamiltonian, and the flow of its associated Hamiltonian vector field \(X_H\), defined by the equation \(dH(\cdot) = \omega(X_H,\cdot)\), is called a Hamiltonian flow. Locally, the Hamiltonian flow is a solution curve to Hamilton's differential equations from classical mechanics.

 

When a Lie group \(G\) acts on a symplectic manifold \((M,\omega)\), it is natural to consider what sort of compatibility there might be between the group action and the symplectic form.  The most obvious compatibility condition is to consider  symplectic actions,  i.e. assume \( g^\ast \omega = \omega \) for all \(g \in G\). A stronger condition is to require a symplectic group action to be Hamiltonian. This means that there is a morphism of Lie algebras \(\tilde{\mu}: \mathfrak{g} \rightarrow C^\infty(M)\) which satisfies a certain compatibility condition with the Hamiltonian vector fields. The momentum mapping associated with a Hamiltonian action of a Lie group \(G\) on a symplectic manifold \((M,\omega)\) is a map \(\mu:M \rightarrow \mathfrak{g}^\ast = \text{Hom}(\mathfrak{g},\mathbb{R})\) such  that           \( \mu(x)(X) = \tilde{\mu}_X(x)\).

 

Many of the results in the book under review concern properties of the momentum mapping associated with a Hamiltonian group action. The main result in the first half of the book is the Convexity Theorem of Atiyah and Guillemin-Sternberg, which says (in part) that the image of the momentum mapping for a Hamiltonian torus action on a compact connected symplectic manifold is the convex hull of the images of the fixed points of the action. In this case, the image \(\Delta = \mu(M) \subset \mathfrak{g}^\ast\) of the momentum mapping is called the associated moment polytope.

 

When the Hamiltonian action of the torus on a compact connected symplectic manifold \((M,\omega)\) is effective and the dimension of the torus \(\mathbb{T}\) is half that of \(M\), the symplectic manifold \((M,\omega)\) is called a toric manifold. In this case, the image of the momentum manifold is a primitive polytope, and a result due to Delzant says that this primitive polytope completely determines the Hamiltonian \(\mathbb{T}\)-space. The authors fully explain this Delzant correspondence in their book, but they only prove half of the correspondence.

 

The later chapters in the book discuss other theorems concerning torus actions on manifolds, including a localization formula for equivariant homology due to Berline-Vergne and Atiyah-Bott, and the Duistermatt-Heckman Theorem, which describes how symplectic quotients of Hamiltonian torus actions vary in a neighborhood of a regular value of the momentum map. The results surveyed in the last two chapters of the book are mostly independent of the earlier results in the book. One chapter contains a brief introduction to geometric quantization and another contains a brief survey of representations of fundamental groups of 2-manifolds.

 

The word ``brief'' is a good descriptor for this book. It's part of the Springer Briefs in Mathematics series which describes itself as, ``concise summaries of cutting-edge research and practical applications''. This book fits that description and contains lots of nice examples. It might be useful to people looking for a summary of the main theorems surrounding momentum mappings of Hamiltonian torus actions. However, for a detailed approach to this topic, the reader should consider Michèle Audin's excellent book Torus Actions on Symplectic Manifolds.

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David Hurtubise is a Professor of Mathematics at Penn State Altoona.  His web page is at http://www.personal.psu.edu/dxh40/.