Compactifying Hypertoric Manifolds via Symplectic Cutting

Benjamin Brown

The University of Edinburgh

Structure Of This Presentation

  • Delzant Polytopes and Toric Symplectic Manifolds

  • Their Hypertoric Analogues

  • Compactification via Symplectic Cutting

  • Outlook

Delzant Polytopes

A Delzant polytope \(\Delta \subseteq \mathbb{R}^{n}\) is a convex polytope satisfying:

  • (simple); \(n\) edges meet at each vertex;

  • (rational); each edge meeting a vertex \(p\) is of the form \(p + tu_{i}\), \(t \geq 0\), \(u_{i} \in \mathbb{Z}^{n}\);

  • (smooth); for each vertex, respective edge vectors \(u_{1}, \ldots, u_{n}\), can be chosen to form a \(\mathbb{Z}\)-basis for \(\mathbb{Z}^{n}\).

Each \(\Delta \subseteq \mathbb{R}^{n}\) can be written as

\[ \Delta = \bigcap\limits_{i} \{ x \in (\mathbb{R}^{n})^{\ast}\, : \, \langle x ,\, u_{i} \rangle + \lambda_{i} \geq 0 \},\quad \lambda_{i} \in \mathbb{R}, \] where \(u_{i} \in \mathbb{Z}^{n}\) are the inward-pointing normals to the facets of \(\Delta\).

Symplectic Toric Manifolds

Definition: A \(2n\)-dimensional symplectic toric manifold is a compact connected symplectic manifold \((M^{2n},\omega)\) with an effective Hamiltonian action of an \(n\)-torus \(T^{n}\), with corresponding moment map \(\mu : M \rightarrow\operatorname{Lie}(T^{n})^{\ast} \cong (\mathbb{R}^{n})^{\ast}\).

This definition is easier to elaborate upon with an example.

Example

\(T^{n}\) acts on \(\mathbb{C}^{n}\) diagonally:

\[ (t_{1}, \ldots, t_{n}) \cdot (z_{1}, \ldots, z_{n}) = (t_{1}z_{1}, \ldots, t_{n}z_{n}), \]

Moment map for the action is: \(\mu : \mathbb{C}^{n} \longrightarrow\mathbb{R}^{n}\),

\[ \mu(z) = \frac{1}{2} \sum\limits_{k = 1}^{n} |z_{k}|^{2}e_{k} \in \mathbb{R}^{n}. \]

Abuse of notation: Identify \((\mathbb{R}^{n})^{\ast} = \mathbb{R}^{n}\) and omit constants for moment maps.

Toric Varieties

\(T^{n}\) acts diagonally on \(\mathbb{C}^{n}\) preserving the Kähler structure.

Let \(\{ u_{1}, \ldots, u_{n} \}\) be inner-normals to some Delzant \(\Delta\). They generate a Lie algebra \(\mathfrak{n}\) for some sub-torus \(N \subseteq T^{n}\).

\(\implies\) exact sequences, where \(\pi: e_{i} \mapsto u_{i}\):

\[ 0 \longrightarrow\mathfrak{n} \overset{\imath}{\longrightarrow} \mathbb{R}^{n} \overset{\pi}{\longrightarrow} \mathbb{R}^{d} \longrightarrow 0, \]

dualising

\[ 0 \longleftarrow\mathfrak{n}^{\ast} \overset{\imath^{\ast}}{\longleftarrow} \mathbb{R}^{n} \overset{\pi^{\ast}}{\longleftarrow} \mathbb{R}^{d} \longleftarrow 0, \]

or exponentiating

\[ 1 \longrightarrow N \overset{\imath}{\longrightarrow} T^{n} \overset{\pi}{\longrightarrow} T^{d} \longrightarrow 1. \]

\(N\) acts on \(\mathbb{C}^{n}\) via \(\imath\), with moment map \[ \bar{\mu}(z) = (\imath^{\ast} \circ \mu)(z) = \frac{1}{2}\sum\limits_{k = 1}^{n} |z_{k}|^{2} \alpha_{k} \in \mathfrak{n}^{\ast}, \]

with \(\alpha_{k} = \imath^{\ast}(e_{k})\).

If \(0\) is a regular value for \(\bar{\mu}\), then

\[ X = \bar{\mu}^{-1}(0)/N = (\imath^{\ast} \circ \mu)^{-1}(0)/N \]

is a smooth Kähler quotient (assuming \(\{u_{1},\ldots,u_{n}\}\) come from Delzant \(\Delta\)).

Convexity

Residual \(T^{d} = T^{n}/N\) action on \(X\), moment map \(\phi : X \rightarrow(\mathbb{R}^{d})^{\ast}\).

For \(X\) compact, Atiyah-Guillemin-Sternberg theorem \(\implies \operatorname{Im}(\phi)\) is a convex polytope \(\Delta\), and fixed-points of \(T^{d}\) are its vertices.

Example

\[ 0 \longrightarrow\mathfrak{n} \overset{\imath}{\longrightarrow} \mathbb{R}^{3} \overset{\pi}{\longrightarrow} \mathbb{R}^{2} \longrightarrow 0 \]

\[ u_{1} = (1,0),\ u_{2} = (0,1),\ u_{3} = (-1,-1), \] \[ \ker(\pi) = \langle e_{1} + e_{2} + e_{3} \rangle \subset \mathbb{R}^{3} \] \[ \implies \imath(t) = (t,t,t) \implies \imath^{\ast}(x,y,z) = x + y + z. \]

\(T^{3}\) on \(\mathbb{C}^{3}\) moment map: \(\mu(z) = \tfrac{1}{2}\sum_{k = 1}^{3} |z_{k}|^{2}e_{k}\), so \(N\) moment map is:

\[ \bar{\mu}(z) = (\imath^{\ast}\circ \mu)(z) = \tfrac{1}{2}\|z\|^{2} \]

\[ X = \mu^{-1}(c)/N = \{ \|z\|^{2} = 2c \}/N \cong S^{5}/S^{1} \cong \mathbb{C}\mathbb{P}^{2}. \]

\[ X = \mu^{-1}(c)/N \cong \mathbb{C}\mathbb{P}^{2} \]

has residual \(T^{2} = T^{3}/N\) action:

\[ (t_{1}, t_{2}) \cdot [z_{0}: z_{1}: z_{2}] = [z_{0}:t_{1}z_{1}:t_{2}z_{2}]. \]

Moment map

\[ \phi(z) = \frac{1}{2}\bigg( \frac{|z_{1}|^{2}}{\|z\|^{2}},\, \frac{|z_{2}|^{2}}{\|z\|^{2}} \bigg),\quad \text{with } \operatorname{Im}(\phi) = \Delta_{2}. \]

Fixed-points of \(T^{2}\):

\[ \begin{array}{ccc} [1:0:0] & \mapsto & (0,0) \\ [0:1:0] & \mapsto & (1/2,0) \\ [0:0:1] & \mapsto & (0,1/2) \end{array} \]

Hyperkähler Moment Maps

Analogous though now with \(\mathbb{H}^{n}\).

Flat hyperkähler with three complex structures \(J_{1}, J_{2}\), and \(J_{3}\).

Fix \(J_{1}\) so \(\mathbb{H}^{n} \cong T^{\ast}\mathbb{C}^{n}\).

\(T^{n}\)-action on \(\mathbb{C}^{n}\) induces \(T^{n}\)-action on \(T^{\ast}\mathbb{C}^{n}\).

Hyperkähler moment maps

\[ \mu_{\mathbb{R}}(z,w) =\frac{1}{2}\sum_{k = 1}^{n}( |z_{k}|^{2} - |w_{k}|^{2} )e_{k} \in \mathbb{R}^{n}, \]

\[ \mu_{\mathbb{C}}(z,w) = \sum\limits_{k = 1}^{n}(z_{k}w_{k})e_{k} \in \mathbb{C}^{n}. \]

Hypertoric Analogues

Choose \(\{u_{1}, \ldots, u_{n}\}\) to get \(N \overset{\imath}{\hookrightarrow} T^{n}\).

Mutatis mutandi, same construction as before:

\[ \bar{\mu}_{\mathbb{R}}(z,w) := (\imath^{\ast} \circ \mu_{\mathbb{R}})(z,w) = \tfrac{1}{2} \imath^{\ast}\bigg(\sum\limits_{k = 1}^{n}\big( |z_{k}|^{2} - |w_{k}|^{2} \big) e_{k} \bigg), \]

\[ \bar{\mu}_{\mathbb{C}}(z,w) := (\imath_{\mathbb{C}}^{\ast} \circ \mu_{\mathbb{C}})(z,w) = \imath^{\ast}_{\mathbb{C}}\bigg( \sum\limits_{k = 1}^{n}(z_{k}w_{k}) e_{k} \bigg). \]

Hyperkähler analogue \(M\) to the Kähler quotient \(X\) is

\[ M := \big( \bar{\mu}_{\mathbb{R}}^{-1}(\lambda) \cap \bar{\mu}_{\mathbb{C}}^{-1}(0) \big) / N. \]

Hyperplane Arrangements

Residual \(T^{d} = T^{n}/N\)-action on \(M\); has hyperkähler moment maps

\[ \phi_{\mathbb{R}}[z,w] = \tfrac{1}{2} \sum\limits_{k = 1}^{n}( |z_{k}|^{2} - |w_{k}|^{2} - \lambda_{k} )\alpha_{k}, \]

\[ \phi_{\mathbb{C}}[z,w] = \sum\limits_{k = 1}^{n} (z_{k}w_{k}) \alpha_{k}. \]

Image \(\operatorname{Im}(\phi_{\mathbb{R}}) \subseteq \mathbb{R}^{d}\) decomposes into a hyperplane arrangement: for \(y \in \mathbb{R}^{d}\),

\[ F_{i} = \{ y \cdot u_{i} + \lambda_{i} \geq 0 \},\quad G_{i} = \{ y \cdot u_{i} + \lambda_{i} \leq 0 \},\]

\[ H_{i} = F_{i} \cap G_{i}. \]

Example - Hypertoric Analogue for \(\mathbb{C}\mathbb{P}^{2}\)

Extend \(T^{3}\) diagonal action on \(\mathbb{C}^{3}\) to \(T^{\ast}\mathbb{C}^{3}\); now \(N\) acts as \(t \cdot (z,w) = (tz,t^{-1}w)\).

Hyperkähler quotient

\[ M = \Big( \bar{\mu}_{\mathbb{R}}^{-1}(\lambda) \cap \bar{\mu}_{\mathbb{C}}^{-1}(0) \Big) / N \cong T^{\ast}\mathbb{C}\mathbb{P}^{2} \]

has residual \(T^{2}\)-action.

\[ \phi_{\mathbb{R}}[z,w] = \tfrac{1}{2}\sum\limits_{k=1}^{3}(|z_{k}|^{2} - |w_{k}|^{2}) - \lambda_{3}. \]

The \(\{ H_{i} \}\) divide \((\mathbb{R}^{d})^{\ast}\) into a union of closed convex polyhedra

\[ \Delta_{A} = \bigcap\limits_{i \in A} F_{i} \cap \bigcap\limits_{i \not\in A} G_{i}. \]

Set \(\mathcal{E} := \phi_{\mathbb{C}}^{-1}(0) = \{ [z,w] \in M : z_{i}w_{i} = 0,\, \text{for all } i \} \subseteq M\), which further decomposes

\[ \mathcal{E}_{A} := \{ w_{i} = 0 \text{ for all } i \in A, \text{ and } z_{i} = 0 \text{ for all } i \not\in A \}, \] for subsets \(A \subseteq \{1,\ldots,n\}\).

Lemma: If \(w_{i} = 0\) then \(\phi_{\mathbb{R}}[z,w] \in F_{i}\), and if \(z_{i} = 0\), then \(\phi_{\mathbb{R}}[z,w] \in G_{i}\).

The Core and the Extended Core of \(M\)

Call \(\mathcal{E}_{A} = \phi_{\mathbb{C}}^{-1}(0)\) the extended core of \(M\):

Each \(\mathcal{E}_{A} \subseteq M\) is a \(d\)-dimensional Kähler subvariety with effective Hamiltonian \(T^{d}\)-action.

Lemma: \(\phi_{\mathbb{R}}(\mathcal{E}_{A}) \cong \bigcap\limits_{i \in A} F_{i} \cap \bigcap\limits_{i \not\in A} G_{i} =: \Delta_{A}\), and \(\Delta_{A}\) is corresponding Delzant polytope to \(\mathcal{E}_{A}\).

We call \(\mathcal{L} := \bigcup\limits_{\Delta_{A} \text{ bounded}} \mathcal{E}_{A}\), the core of \(M\).

Residual \(S^{1}\)-Action

Additional \(S^{1}\)-action on \(T^{\ast}\mathbb{C}^{n}\):

\[ \tau \cdot (z,w) = (z,\tau w). \]

Descends to \(M\), but only preserves \(J_{1}\) structure, not \(J_{2}\) nor \(J_{3}\).

Does not act on \(M\) as a sub-torus of \(T^{d}\), but does when restricted to each \(\mathcal{E}_{A}\).

For \([z,w] \in \mathcal{E}_{A}\),

\[ [z;\tau_{1} w_{1}, \ldots, \tau_{n} w_{n}] = [\tau_{1}^{-1}z_{1} , \ldots, \tau_{n}^{-1}z_{n}; \tau_{1} w_{1}, \ldots, \tau_{n}w_{n}], \]

\[ \text{where } \tau_{i} = \begin{cases} \tau, \quad & \text{if } i \in A, \\ 1, \quad & \text{if } i \not\in A. \end{cases} \]

Symplectic Cut

\(S^{1}\)-action on \(M\) has proper moment map \(\Phi[z,w] = \tfrac{1}{2}\|w\|^{2}\).

Extend it to \(M \times \mathbb{C}\) via

\[ e^{i\theta} \cdot (m,\xi) = ( e^{i\theta} \cdot m, e^{i\theta} \xi), \] with moment map

\[ \rho_{\text{cut}} : M \times \mathbb{C}\rightarrow\mathbb{R}; \quad (m,\xi) \mapsto \Phi(m) + \tfrac{1}{2}|\xi|^{2}. \]

The symplectic cut is the quotient

\[ M_{\epsilon-\text{cut}} := \rho_{\text{cut}}^{-1}(\epsilon)/S^{1} \cong \{m \in M : \Phi(m) < \epsilon\} \sqcup (\Phi^{-1}(\epsilon)/S^{1}). \]

Compactification of \(M\)

\(S^{1}\) acts on \(M\), depending combinatorially on \(A \subseteq \{1,\ldots, n\}\). Recall: \[ S^{1}_{A} = (\tau_{1}, \ldots, \tau_{n}), \text{ with } \tau_{i} = \tau\text{ if } i \in A; \tau_{i} = 1 \text{ otherwise}. \]

Action comes from inclusion \(S^{1}_{A} \hookrightarrow T^{n} \rightarrow T^{d}\), with moment map

\[ [z,w] \mapsto \Big\langle \phi_{\mathbb{R}}[z,w], \sum_{i \in A} u_{i} \Big\rangle. \]

Cutting introduces new half-spaces

\[ \Delta_{A}^{(\epsilon)} := \Delta_{A} \cap \bigg\{ y \in \Delta_{A} : \Big\langle y,\, \sum_{i \in A} u_{i} \Big\rangle + \epsilon\geq 0 \bigg\}. \]

Outlook From Here

  • Hypertoric manifolds with non-compact cores;
  • Applying localisation formulae;
  • Geometric quantisation and lattice point enumeration?

References

  • Delzant, T. – Hamiltoniens périodiques et images convexes de l’application moment, Bull. Soc. Math. France, 116, 1988.

  • Bielawski, R. and Dancer, A. – The geometry and topology of toric hyperkähler manifolds, Comm. Anal. Geom., 8, 2000.

  • Proudfoot, N. – Hyperkahler analogues of Kahler quotients, Ph.D. Thesis, Harvard University, 2004.

  • Lerman, E. – Symplectic Cuts, Math. Res. Lett., 2, 1995.