Example: \((\mathbb{R}^2,dx \wedge dy) = (\mathbb{C},\frac{i}{2}dz \wedge d\bar{z})\)

Non-Example: \((\mathbb{R}^3, dx \wedge dy)\)

For any vector \(v\), \(dx \wedge dy\left(\frac{\partial}{\partial z}, v\right) = 0\), so \(dx \wedge dy\) is degenerate.

Non-Example: \((\mathbb{R}^4, dx_1 \wedge dy_1 + y_1 dx_2 \wedge dy_2)\)

Because of the coefficient \(y_1\), this is not closed (“not divergence-free”).

\((S^2,d\theta\wedge dz)\)

\((S^2,\omega)\), where \(\omega_p(u,v) = (u \times v) \cdot p\)

\((\mathbb{R}^2,\omega)\) where \(\omega(u,v) = \langle i u, v \rangle \)

\((\mathbb{C}^n, \frac{i}{2} \sum dz_k \wedge d\overline{z}_k)\)

\((\mathbb{C}^{m \times n}, \omega)\) with \(\omega(X_1,X_2) = -\operatorname{Im} \operatorname{trace}(X_1^* X_2)\).

\((T^* \mathbb{R}^n,\sum dq_i \wedge dp_i)\)

phase space

position

momentum

If \(M\) is compact, this can be normalized to give a probability measure.

Example. \(H: (S^2, d\theta\wedge dz) \to \mathbb{R}\) given by \(H(\theta,z) = z\).

\(dH = dz = \iota_{\frac{\partial}{\partial \theta}}(d\theta\wedge dz)\), so \(X_H = \frac{\partial}{\partial \theta}\).

\(H\) is constant on orbits of \(X_H\):

\(\mathcal{L}_{X_H}(H) = dH(X_H)=\omega(X_H,X_H) = 0\)

\(S^1=U(1)\) acts on \((S^2,d\theta \wedge dz)\) by

For \(r \in \mathbb{R} \simeq \mathfrak{u}(1)\), \(X_r = r\frac{\partial}{\partial \theta}\).

\(= (a,b,c) \times (x,y,z)\)

For \(V_{(a,b,c)} = \begin{bmatrix} 0 & -c & b \\ c & 0 & -a \\ -b & a & 0 \end{bmatrix} \in \mathfrak{so}(3)\),

\(X_{V_{(a,b,c)}}((x,y,z))\)

\(= (bz-cy)\frac{\partial}{\partial x} + (cx - az) \frac{\partial}{\partial y}  +(ay - bx) \frac{\partial}{\partial z}\)

\(X = \frac{\partial}{\partial \theta}\)

\(\mu(\theta,z) = z\)

\(\iota_X\omega = \iota_{\frac{\partial}{\partial \theta}} d\theta \wedge dz = dz \)

\(X_{V_{(a,b,c)}}(x,y,z) = (a,b,c) \times (x,y,z)\)

\((\iota_{X_{V_{(a,b,c)}}}\omega)_{(x,y,z)} = a dx + b dy + c dz \)

\(\mu(x,y,z)(V_{(a,b,c)}) = (x,y,z)\cdot(a,b,c)\)

Theorem [Atiyah, Guillemin–Sternberg] 

Let \((M^{2n},\omega)\) be a compact connected symplectic manifold with a Hamiltonian \(k\)-torus action with momentum map \(\mu: M \to \mathbb{R}^k\). Then

• the nonempty level sets of \(\mu\) are connected;
• the image of \(\mu\) is convex (called the moment polytope);
• the image of \(\mu\) is the convex hull of the images of fixed points of the action.

Theorem [Duistermaat–Heckman]

If \(k=n\), then the pushforward of Liouville measure to the moment polytope is a constant multiple of Lebesgue measure.

Theorem [with Needham, 2021]

Fix the spectrum of \(FF^\ast\) and fix \(\|f_1\|,\dots,\|f_N\|\). There are three possibilities:

• It is impossible for a frame to have these data
• It is impossible for a frame with these data to have full spark
• A random frame with these data will have full spark with probability 1.

Proof Idea

(A suitable quotient) of the space of such frames is almost toric, and spark-deficient frames lie over the boundary of the moment polytope.

Theorem [Duistermaat–Heckman]

The pushforward of Liouville measure to the moment polytope is a constant multiple of Lebesgue measure.

Example. \(\theta: \mathfrak{sl}_n(\mathbb{R}) \to \mathfrak{sl}_n(\mathbb{R})\) defined by \(\theta(X) = -X^T\). This is the derivative of the involution on \(SL_n(\mathbb{R})\) given by \(U \mapsto (U^{-1})^T\).

\(E_{+1} =: \mathfrak{k}\) consists of skew-symmetric matrices

\(E_{-1} =: \mathfrak{p}\) consists of (traceless) symmetric matrices

\(\mathfrak{sl}_n(\mathbb{R}) = \mathfrak{k} \oplus \mathfrak{p}\) is an orthogonal decomposition — the Cartan decomposition

\([\mathfrak{k},\mathfrak{p}]\subseteq \mathfrak{p}\), so the adjoint action of \(SO(n) = \exp(\mathfrak{k})\) restricts to an action on \(\mathfrak{p}\)

\([\mathfrak{p},\mathfrak{p}]\subseteq\mathfrak{k}\), so any Lie subalgebra of \(\mathfrak{p}\) is abelian.

Theorem. \(G\) non-compact, semisimple, and \(\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}\) a Cartan decomposition with \(K = \exp(\mathfrak{k})\) and \(P = \exp(\mathfrak{p})\), then \(G = KP\) and \(T_{[I]} G/K \simeq \mathfrak{p}\).

The adjoint action of \(K\) on \(\mathfrak{p}\) is often called the isotropy representation.

Example.

\(G = SL_n(\mathbb{C})\), \(K=SU(N)\), \(\mathfrak{k} = \mathfrak{su}(N)\), \(\mathfrak{p} = \mathscr{H}_0(N)\), \(a = \operatorname{diag}(a_1, \dots , a_N)\),

\(\mathcal{O}_a = \{UaU^\ast | U \in SU(N)\}\)

\(P(\mathcal{O}_a) = \operatorname{conv}(S_N \cdot a)\) is essentially Schur–Horn.

Can play the same game with real or quaternionic matrices!

This implies the generalized frame homotopy theorem for complex frames.

\(z+w \boldsymbol{j} \mapsto \begin{pmatrix} z & w \\ -\overline{w} & \overline{z}\end{pmatrix}\)

\(\mathbb{H}^d\) is a right vector space over the skew-field \(\mathbb{H}\).

\(\Psi(\operatorname{Sp}(N)) \subset SU(2N)\) is the fixed point set of the involution

\(U \mapsto \Omega^\ast \overline{U} \Omega\),

where \(\Omega = \begin{pmatrix} 0 & I \\ -I & 0 \end{pmatrix}\).

Corresponding Cartan decomposition of the Lie algebra:

\(\mathfrak{su}(2N) = \mathfrak{k} \oplus \mathfrak{p} \simeq \Psi(\mathfrak{sp}(N)) \oplus \Psi(\mathscr{H}_0(N))\).