--- date: 2021-10-27 19:35 modification date: Sunday 3rd April 2022 22:40:39 title: "2021-07-02" aliases: [2021-07-02] --- Last modified date: <%+ tp.file.last_modified_date() %> --- - Tags: - #projects/research/topics-with-phil - Refs: - #todo/add-references - Links: - #todo/create-links --- # 2021-07-02 ## Notes - $\bar\mgn$ denotes the DM compactification. - Goal: look at moduli spaces of hyperbolic surfaces and equip it with a symplectic structure using the Weil-Petersson form. - Interesting because polfinalynomials involving volumes compute intersection numbers on $\mgn$. - Curves with marked points = punctured Riemann surfaces = punctured surfaces with complete constant curvature Riemannian metrics. Almost all admit a hyperbolic metric. - $\mgn(\vector L)$: moduli of hyperbolic surfaces with $n$ geodesic boundary components with lengths prescribed by $L_i$, up to boundary-label-preserving isometries. - Construct as quotient of $\tgn$ to get an orbifold structure, use Fenchel-Nielsen coordinates to descend a Weil-Petersson symplectic form $\omega$, making $\mgn(\vector L)$ symplectic. :::{.definition title="Weil-Petersson volume"} \[ V_{g, n}(\mathbf{L})=\int_{\mathcal{M}_{g, n}(\mathbf{L})} \frac{\omega^{3 g-3+n}}{(3 g-3+n) !} .\] ::: :::{.theorem title="Mirzakhani"} Letting $\psi_i \in H^2(\bar\mgn; \QQ)$ be $\psi\dash$classes and $\kappa_1$ be the first Mumford-Morita-Miller class. \[ V_{g, n}(\vector L) = \sum_{m + \sum a_i = 3g -3 + n} { \qty{2\pi^2}^m \prod L_i^{2a_i} \over 2^{\sum a_i } m! \prod a_i! } \int_{\bar\mgn} \kappa_1^m \prod \psi_i^{a_i} .\] ::: - Gauss-Bonnet: for $K$ the Gaussian curvature, \[ \int_\Sigma K \dV = 2\pi \chi(\Sigma) .\] - For a metric of constant curvature $K=-1$ to exist, need $\chi(\Sigma) < 0$, so $2-2g-n<0$ where $(g, n)$ are the genus and number of punctures. - Hyperbolic surfaces: $\HH^2 / G$ for $G\leq \PSL_2(\RR)$. - Equivalence of categories between smooth algebraic curves and compact Riemann surfaces. - Uniformization: Every Riemannian metric on $\Sigma$ is conformally equivalent to $g$, a complete constant curvature metric. $g$ is unique if $\chi(\Sigma)<0$ and the curvature $K=-1$. So Riemann surfaces biject with hyperbolic surfaces. - So every conformal class of metrics contains a hyperbolic metric when $\chi(\Sigma)<0$. - If $\Sigma$ is complete, the punctures become hyperbolic cusps. - $\mgn(\vector 0)$ defined as hyperbolic surfaces with $n$ labeled cusps, homeomorphic as an orbifold to $\mgn$. - Interpret length zero geodesic boundaries as cusps. - $\tgn(\vector L)$: moduli of marked hyperbolic surfaces $(S, f)$ where the marking is a diffeomorphism $f: \Sigma_{g, n}\to S$, up to isometries over $\Sigma$ that commute up to isotopy. - Example: hyperbolic Dehn twists cut along simple closed geodesics, twist, and glue back. Yields a loop in $\mgn(\vector L)$, since one full twist is a self-isometry, but lifts to a nontrivial path in $\tgn(\vector L)$ - Building blocks of surfaces with $\chi(\Sigma) < 0$: pairs of pants, i.e. $S^2$ with 3 punctures. - Pants decomposition: disjoint SCCs $\gamma$ where $\Sigma\sm \gamma = \disjoint P_i$ with $P_i$ a pair of pants. - Number of pants: exactly $-\chi(\Sigma) = 2g-2+n$ pants and $3g-3+n$ SCCs. - Every SCC is homotopic to a unique simple closed geodesic. - There is a fibering: \begin{tikzcd} {\MCG_{g, n} \da \Diff^+(\Sgn) / \Diff_0^+(\Sgn)} && {\tgn(\vector L)} \\ \\ && {\mgn(\vector L)} \arrow[from=1-1, to=1-3] \arrow[from=1-3, to=3-3] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMCwwLCJcXE1DR197Zywgbn0gXFxkYSBcXE1DR197Zywgbn0gXFxkYSBcXERpZmZeKyhcXFNnbikgLyBcXERpZmZfMF4rKFxcU2duKSJdLFsyLDAsIlxcdGduKFxcdmVjdG9yIEwpIl0sWzIsMiwiXFxtZ24oXFx2ZWN0b3IgTCkiXSxbMCwxXSxbMSwyXV0=) - The action is $[\phi] \actson (X, f) \da (X, f\circ \phi)$. This is properly discontinuous but not necessarily free, making $\dim \mgn(\vector L) = 6g-6+2n$. - There is a homeomorphism $\tgn(\vector L) \to \RR_{\geq 0}^{m}\cross \RR^m$ for $m \da 3g-3+n$ where a surface is sent to its Fenchel-Nielsen coordinates (length and twist parameters). - Yields a symplectic form $\omega \da \sum_{k=1}^{\ell} d\ell_k \wedge d\tau_k$ with is remarkably $\MCG_{g, n}\dash$invariant. - By uniformization, as $g, n$ vary, the moduli spaces $\mgn$ are all diffeomorphic *but* not necessarily symplectomorphic. - There is a finite cover $\tilde\mgn(\vector L)\to \mgn(\vector L)$ by a smooth manifold. - Compactify by introducing *stable* curves everywhere: nodes on curves should correspond to degenerating the hyperbolic length of a SCC to zero. - Stable hyperbolic surface $\approx$ a collection of hyperbolic surfaces whose cusps have been formally paired up - $\bar\mgn(\vector 0)$ admits a complex structure, but $\bar\mgn(\vector L)$ generally does not, although it admits a real-analytic structure. - $[\omega] = 2\pi^2 \kappa_1$ in $H^2(\bar\mgn; \RR)$. - $\mgn(\vector L)$ is homeomorphic as an orbifold to moduli of ribbon graphs - Raise $\omega$ to enough powers to get \[ \frac{\omega^{3 g-3+n}}{(3 g-3+n) !}=d \ell_{1} \wedge d \tau_{1} \wedge d \ell_{2} \wedge d \tau_{2} \wedge \cdots \wedge d \ell_{3 g-3+n} \wedge d \tau_{3 g-3+n} .\] where $\tgn$ has infinite volume with respect to this form, but $\mgn$ has finite volume. - Note $\mcm_{1, 1}(0)$ is exceptional since a generic point is an orbifold point with orbifold group $\ZZ/2$, since every hyperbolic torus with a cusp admits an elliptic involution. :::{.example title="Volumes"} Some volumes: \[ V_{g, n}(\vector L) \da \int_{\mgn(\vector L)} {\omega^m \over m!} && m \da 3g-3+n .\] - $V_{0, 1}(x) = 0$ - $V_{0, 2}(x, y) = 0$ - $V_{0, 3}(x,y,z) = 1$ - $V_{0, 4}(0,0,0,0) = 2\pi^2$ - $V_{0, 4}(x,y,z,w) = {1\over 2}\qty{x^2 + y^2 + z^2 + w^2 + 4\pi^2 }$ - $V_{0, n}(\vector 0)$ admits a recursive formula in $n$. - $V_{1, 1}(0) = \pi^2/12$ - $V_{1, 1}(x) = {1\over 48}\qty{x^2 + 4\pi^2}$. - $V_{1, 2}(0, 0) = \pi^4/4$ - $V_{1, 2}(x, y) = {1\over 192} x^4 + {1\over 96} x^2 y^2 + {1\over 192} y^4 + {\pi^2 \over 12}x^2 + {\pi^2 \over 12} y^2 + {\pi^4 \over 4}$. - $V_{g, n}(\vector 0) = q (2\pi^2)^{3g-3+n}$ - $V_{g, n}(\vector L) \in \RR\adjoin{L_i^2}$. A striking observation: $V(g, 1)(2\pi i) = 0$ for all $g$. ::: :::{.proposition title="Computing $\mcm_{1, 1}(0)$ "} ![](Archive/0100_Reading_Phil_Summer_2021/figures/2021-07-02_20-04-28.png) [2021-07-02_20-04-28](Archive/0100_Reading_Phil_Summer_2021/figures/2021-07-02_20-04-28.png) ::: :::{.theorem title="Mirzakhani recursion"} ![](Archive/0100_Reading_Phil_Summer_2021/figures/2021-07-02_20-13-10.png) ![](Archive/0100_Reading_Phil_Summer_2021/figures/2021-07-02_20-19-58.png) ::: :::{.remark} \envlist - Important definition: this $H$ function \[ H(x, y)=\frac{1}{1+\exp \frac{x+y}{2}}+\frac{1}{1+\exp \frac{x-y}{2}} .\] - Stable curve: at worst nodal singularities and finite automorphism group. :::