Moment maps

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 26 March 2014

Notes and References

This is a transcript of Work2007/Bites2007/mmtmpbite10.10.06.tex where the picture of the commutative diagrams is a bit nicer.

What is De Rham cohomology?

Let $A$ be a commutative algebra. The de Rham cohomology of the complex $$\begin{array}{ccccccccc}\cdots & ⟶& {\Omega }^{i-1}\left(A\right)& \stackrel{d}{⟶}& {\Omega }^i\left(A\right)& \stackrel{d}{⟶}& {\Omega }^{i+1}\left(A\right)& ⟶& \cdots \end{array}$$ where the $p\text{-differential}$ forms of $A$ is $${\Omega }^p\left(A\right)={\Lambda }^p\left({\Omega }^1\left(A\right)\right),{\Omega }^1\left(A\right)=I/I^2,I=\text{ker}(A\otimes A\to A),$$ and $d$ is the unique antiderivation of degree 1 which extends $$\begin{array}{cccc}d:& A& ⟶& {\Omega }^1\left(A\right)\\ & x& ⟼& x\otimes 1-1\otimes x\end{array}\text{and satisfies}{d}^2=0\text{.}$$

Example. If $A=𝔽[x_1,\dots ,x_n]$ then $$\begin{array}{ccc}{\Omega }^1\left(A\right)& =& A\text{-span}\{d{x}_1,\dots ,d{x}_n\}\text{and}\\ {\Omega }^p\left(A\right)& =& A\text{-span}\{d{x}_{i_1}\wedge \cdots \wedge d{x}_{i_p}\hspace{0.17em}|\hspace{0.17em}1\le i_1<i_2<\cdots <i_p\le n\},\end{array}$$ with $$df=\sum _{i=1}^n\frac{\partial f}{\partial \xi }d\xi \text{and}d\left(f(d{x}_{i_1}\wedge \cdots \wedge d{x}_{i_p})\right)=df\wedge d{x}_{i_1}\wedge \cdots \wedge d{x}_{i_p},$$ for $f\in 𝔽[x_1,\dots ,x_n]\text{.}$

Manifolds. Let $X$ be a manifold. The algebra of differential forms on $X$ is $${\Omega }^{•}=\left\{\text{sections of}\hspace{0.17em}{\Lambda }^p{\left(T\left(X\right)\right)}^{*}\right\},$$ with $${\Omega }^0\left(X\right)=𝒪\left(X\right)\text{and}{\Omega }^1\left(X\right)=\left\{\text{vector fields on}\hspace{0.17em}X\right\}\text{.}$$

Connections

Let $M$ be an $A\text{-module.}$ A connection on $M$ is an $𝔽\text{-linear}$ map $$\nabla :M\to M{\otimes }_A{\Omega }^1\left(A\right)\text{such that}\nabla \left(fm\right)=f\nabla \left(m\right)+m\otimes df,$$ for $f\in A,$ $m\in M\text{.}$ There is a unique extension of $\nabla$ to $$\begin{array}{cc}\begin{array}{ccccccccc}\cdots & ⟶& M{\otimes }_A{\Omega }^{i-1}\left(A\right)& \stackrel{\nabla }{⟶}& M{\otimes }_A{\Omega }^i\left(A\right)& \stackrel{\nabla }{⟶}& M{\otimes }_A{\Omega }^{i+1}\left(A\right)& ⟶& \cdots \end{array}& \text{(1.1)}\end{array}$$ such that $$\nabla \left(x\omega \right)=\left(\nabla x\right)\omega +{(-1)}^{\text{deg}\left(x\right)}xd\omega ,\text{for}\hspace{0.17em}x\in M{\otimes }_A{\Omega }^p\left(A\right),\hspace{0.17em}\omega \in {\Omega }^{\ell }\left(A\right)\text{.}$$ The curvature of $\nabla$ is $$R:M⟶M{\otimes }_A{\Omega }^2\left(A\right)\text{given by}R={\nabla }^1\circ {\nabla }^0,$$ and $\nabla$ is flat if $R=0\text{.}$

If $\nabla$ is flat connection on $M$ then then (??) is a complex and the de Rham cohomology of $(M,\nabla )$ is the homology of (??).

What is a moment map?

A symplectic manifold is a manifold $M$ with a $2\text{-form}$ $$\omega \in {\Omega }^2\left(M\right)\text{such that}d\omega =0\text{.}$$ The form $\omega$ induces a map $$\begin{array}{ccc}\left\{\text{vector fields}\right\}& ⟶& {\Omega }^1\left(M\right)\\ \xi & ⟼& i_{\xi }\omega =\omega (\xi ,·)\end{array}$$ A symplectic vector field is a vector field $\xi$ such that $$d\left(i_{\xi }\omega \right)=0\text{.}$$ There is an exact sequence $$\begin{array}{ccccccccc}0& ⟶& 𝒪\left(M\right)& ⟶& \left\{\text{symplectic vector fields}\right\}& ⟶& H^1(M,ℝ)& ⟶& 0\\ & & & & \xi & ⟼& \omega (\xi ,·)\\ & & f& ⟼& {\xi }_f& ⟼& -df\end{array}$$ where ${\xi }_f$ is the vector field defined by $\omega ({\xi }_f,·)=-df\text{.}$

Let $(M,\omega )$ be a symplectic manifold. Let $G$ be a Lie group acting on $M$ such that $$\omega (gx,gy)=\omega (x,y),\text{for}\hspace{0.17em}g\in G,\hspace{0.17em}m\in M,\hspace{0.17em}x,y\in T_m\left(M\right)\text{.}$$ The action of $G$ induces a map $$\begin{array}{ccc}𝔤& ⟶& \left\{\text{symplectic vector fields}\right\}\\ x& ⟼& \frac{d}{dt}\left(e^{tx}m\right)\end{array}$$ $$\begin{array}{ccccc}& & & & 𝔤\\ & & & & \downarrow \\ 0& ⟶& 𝒪\left(M\right)& ⟶& \left\{symplectic\; vector\; fields\right\}& ⟶& H^1(M,ℝ)& ⟶& 0\end{array}$$ A Hamiltonian is a Lie algebra homomorphism $$\begin{array}{cccc}H:& 𝔤& ⟶& 𝒪\left(M\right)\\ & x& ⟼& H_x\end{array}\text{such that}\begin{array}{ccc}& & 𝔤\\ & \stackrel{H}{\swarrow }& \downarrow \\ 𝒪\left(M\right)& ⟶& \left\{symplectic\; vector\; fields\right\}\end{array}$$ commutes. The moment map is $$\begin{array}{cccc}\mu :& M& ⟶& {𝔤}^{*}\\ & m& ⟼& {\mu }_m\end{array}\text{given by}{\mu }_m\left(x\right)=H_x\left(m\right)\text{.}$$

Favourite example. Let $V={ℂ}^n={ℝ}^{2n}\text{.}$ Then $$\omega =d{z}_1\wedge d{\bar{z}}_1+\cdots +d{z}_n\wedge d{\bar{z}}_n,$$ is a symplectic form on $V$ (coming from a Hermitian inner product on $V\text{).}$ Then $U_n$ acts on $V$ and preserves $\omega$ (because it preserves the inner product).

Favourite example. Let $P$ be a parabolic subgroup of $G\text{.}$ Then $T^{*}(G/P)$ is a symplectic manifold with $$\omega =d\lambda ,\text{where}\hspace{0.17em}\lambda =\text{???}$$ and Hamiltonian $H:𝔤\to 𝒪\left(T^{*}(G/P)\right)$ given by $$\text{???????????}$$ Then $$\begin{array}{ccc}{T}^{*}(G/P)& \stackrel{\sim }{⟶}& G{\times }_P{𝔭}^{\perp }\\ \text{???}& ⟼& \text{???}\end{array}\text{and}\text{diagram}$$

References

[GRa0405333] S. Griffeth and A. Ram, Affine Hecke algebras and the Schubert calculus, European J. Combinatorics 25 (2004) 1263-1283, MR2095481, arXiv:0405333.

page history