# Lectures and notes: Mirzakhani’s volume recursion and approach for the
Witten-Kontsevich theorem on moduli tautological intersection numbers

Scott A. Wolpert111Partially supported by National Science Foundation grant
DMS - 1005852.

Figure 1: The boundary pants configurations for the length identity.

The following materials were presented in a short course at the 2011 Park City
Mathematics Institute, Graduate Summer School on Moduli Spaces of Riemann
Surfaces. Brad Safnuk assisted in the preparation for and running of the
course. It is my pleasure to thank Brad for his assistance.

##  1 Introduction.

The papers.

Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered
Riemann surfaces. Invent. Math., 167(1):179-222, 2007.

Weil-Petersson volumes and intersection theory on the moduli space of curves.
J. Amer. Math. Soc., 20(1):1-23, 2007.

The goals of the papers.

1\. Derive an explicit recursion for WP moduli space volume polynomials.

2\. Apply symplectic reduction to show that the polynomial coefficients are
intersection numbers.

3\. Show that the volume recursion satisfies the Virasoro relations - Witten’s
conjecture.

The trio of Maryam Mirzakhani papers [Mir07a, Mir07b, Mir08] are distinguished
for involving a large number of highly developed considerations. The first
work requires a detailed description of Teichmüller space, the action of the
mapping class group, formulas for Weil-Petersson (WP) symplectic geometry,
classification of simple geodesic arcs on a pair of pants, the length infinite
sum identity and exact calculations of integrals. The second work involves a
description of the moduli space tautological classes $\kappa_{1}$ and $\psi$,
as characteristic classes for $S^{1}$ principal bundles in hyperbolic
geometry, the moment map and exact symplectic reduction, as well as
combinatorial calculations. The third work uses the $PL$ structure of
Thurston’s space of measured geodesic laminations $\mathcal{MGL}$, the train-
track symplectic form and Masur’s result that the mapping class group acts
ergodically on $\mathcal{MGL}$. A fine feature of the works is that they
showcase important aspects of the geometry, topology and deformation theory of
Riemann/hyperbolic surfaces. Mirzakhani’s recursion for volume is applied in
all three works and in a current preprint. A discussion placing Mirzakhani’s
work into the context of approaches to Witten-Kontsevich theory is given in
the first section of the Mulase-Safnuk paper [MS08]. A brief exposition of
Kontsevich’s original solution [Kon92] of Witten’s conjecture, including the
basic geometry of tautological classes on moduli space and the ribbon graph
expansion of matrix integrals is given in the Bourbaki Seminar of Looijenga
[Loo93]. A brief exposition of Mirzakhani’s volume recursion, solution of
Witten-Kontsevich and applications of WP volume limits are given in [Do11]. An
overall exposition of Mirzakhani’s prime simple geodesic theorem [Mir08] is
given in [Wlp10, Chaps. 9, 10]. Mirzakhani’s work is just one part of a
subject with high activity and many active researchers; Google Scholar shows
200 citations to the three Mirzakhani works, and much more generally over 1100
citations to Witten’s original papers [Wit91, Wit92] on two-dimensional
gravity and gauge theories.

First paper brief. A symplectic fibered product decomposition for covers of
the moduli space of bordered Riemann/hyperbolic surfaces is combined with the
$d\ell\wedge d\tau$ formula for the symplectic form and a universal identity
for sums of geodesic-lengths to derive an explicit recursion for computing
volume. The volume of the moduli space of genus $g$, $n$ boundaries surfaces
is shown to be a polynomial with positive coefficients in surface boundary
lengths of total degree $6g-6+2n$.

Second paper brief. Twist Hamiltonian flows are combined with the boundary
length moment map to apply symplectic reduction for the family of moduli
spaces of bordered hyperbolic surfaces. The consequence is that the
coefficients of the volume polynomials are moduli space characteristic
numbers. A geometric construction shows that the characteristic numbers are
tautological intersection numbers. The volume recursion is shown to satisfy
Virasoro algebra constraints.

The following theorems are the immediate take away results of the papers.

######  Theorem.

The WP volume polynomials. The volume polynomials are determined recursively
from the volume polynomials of smaller total degree, [Mir07a, Formula (5.1) &
Theorem 8.1]. The volume $V_{g,n}(L_{1},\dots,L_{n})$ of the moduli space of
genus $g$, $n$ boundaries, hyperbolic surfaces with boundary lengths
$L=(L_{1},\dots,L_{n})$ is a polynomial

$V_{g,n}(L)\,=\,\sum_{\stackrel{{\scriptstyle\alpha}}{{|\alpha|\leq
3g-3+n}}}C_{\alpha}\,L^{2\alpha},$

for multi index $\alpha=(\alpha_{1},\dots,\alpha_{n})$ and where
$C_{\alpha}>0$ lies in $\pi^{6g-6+2n-2|\alpha|}\mathbb{Q}$, [Mir07a, Theorems
1.1 & 6.1]. The coefficients are intersection numbers given as

$C_{\alpha}=\frac{2^{\delta_{1g}\delta_{1n}}}{2^{|\alpha|}\alpha!(3g-3+n-|\alpha|)!}\,\int_{\overline{\mathcal{M}}_{g,n}}\psi_{1}^{\alpha_{1}}\cdots\psi_{n}^{\alpha_{n}}\omega^{3g-3+n-|\alpha|},$

where $\psi_{j}$ is the Chern class for the cotangent line along the $j^{th}$
puncture, $\omega$ is the symplectic form,
$\alpha!=\prod_{j=1}^{n}\alpha_{j}!$, and $\delta_{**}$ is the Kronecker
indicator delta, [Mir07b, Theorem 4.4].

######  Theorem.

[Mir07a, Theorems 6.3 & 6.4]. Recursive relations for the volume polynomial
leading coefficients. For a multi index $\alpha$, define

$(\alpha_{1},\dots,\alpha_{n})_{g}=C_{\alpha}\times
2^{-\delta_{1g}\delta_{1n}}\times\prod_{i=1}^{n}\alpha_{i}!\times
2^{|\alpha|},$

then for $n>0$ and $\sum_{i}\alpha_{i}=3g-3+n$,

$\mbox{the dilaton
equation}\qquad\quad(1,\alpha_{1},\dots,\alpha_{n})_{g}\,=\,(2g-2+n)(\alpha_{1},\dots,\alpha_{n})_{g}$

and for $n>0$ and $\sum_{i}\alpha_{i}=3g-2+n$,

$\mbox{the string
equation}\qquad(0,\alpha_{1},\dots,\alpha_{n})_{g}\,=\,\sum_{\alpha_{i}\neq
0}(\alpha_{1},\dots,\alpha_{i}-1,\dots,\alpha_{n})_{g}.$

For the intersection number generating function

$\mathbf{F}(\lambda,t_{0},t_{1},\dots)\,=\,\sum_{g=0}^{\infty}\lambda^{2g-2}\sum_{\\{d_{j}\\}}\,\langle\prod_{j=1}^{\infty}\tau_{d_{j}}\rangle_{g}\,\prod_{r\geq
0}t_{r}^{n_{r}}/n_{r}!\,,$

with $n_{r}=\\#\\{j\mid d_{j}=r\\}$, and

$\langle\tau_{d_{1}}\cdots\tau_{d_{n}}\rangle_{g}\,=\,\int_{\overline{\mathcal{M}}_{g,n}}\prod_{j=1}^{n}\psi_{j}^{d_{j}},$

for $\sum_{j=1}^{n}d_{j}=3g-3+n$ and the product $\langle\tau_{*}\rangle$
otherwise zero, then the exponential $e^{\mathbf{F}}$ satisfies Virasoro
algebra constraints, [Mir07b, Theorem 6.1].

##  2 The organizational outline and reading guide.

The following outline combines [Mir07a, Mir07b] with the exposition of [Wlp10,
Chapter 9]. The lectures are presented in the next section.

  * •

Teichmüller spaces, moduli spaces, mapping class groups and the symplectic
geometry.

    * –

The Teichmüller space $\mathcal{T}_{g,n}$ and moduli space
$\mathcal{M}_{g,n}$; the Teichmüller space
$\mathcal{T}_{g}(L_{1},\dots,L_{n})$ and moduli space
$\mathcal{M}_{g}(L_{1},\dots,L_{n})$ of prescribed length geodesic bordered
hyperbolic surfaces; the augmented Teichmüller space and Deligne-Mumford type
compactifications; Dehn twists and the mapping class group
$\operatorname{MCG}$ action.

    * –

The WP symplectic geometry, [Wlp10].

      * *

The symplectic form $\omega=2\omega_{\tiny{\mbox{WP\ K\"{a}hler}}}$ and
normalizations.

      * *

Geodesic-length functions $\ell_{\alpha}$, Fenchel-Nielsen infinitesimal twist
deformations $t_{\alpha}$ and the duality formula $\omega(\
,t_{\alpha})=d\ell_{\alpha}$.

      * *

Fenchel-Nielsen (FN) twist-length coordinates $(\tau_{j},\ell_{j})$ for
Teichmüller space and the formula $\omega=\sum_{j}d\ell_{j}\wedge d\tau_{j}$.

    * –

The intermediate moduli space $\mathcal{M}^{\gamma}_{g,n}$ of pairs
$(R,\gamma)$ \- a surface and a weighted multicurve
$\gamma=\sum_{j}c_{j}\gamma_{j}$.

      * *

The covering tower

$\mathcal{T}_{g,n}\longrightarrow\mathcal{M}^{\gamma}_{g,n}\longrightarrow\mathcal{M}_{g,n}.$

      * *

The stabilizer subgroup $\operatorname{Stab}(\gamma)\subset\operatorname{MCG}$
for a weighted multicurve. The $\operatorname{MCG}$ deck cosets for the
covering tower.

      * *

Symplectic structures for $\mathcal{M}^{\gamma}_{g,n}$ and
$\mathcal{M}_{g}(L)$.

    * –

The $\mathcal{T}_{g,n}$ and $\mathcal{M}^{\gamma}_{g,n}$ level sets of the
total length $\ell=\sum_{j}c_{j}\ell_{\gamma_{j}}$.

######  Lemma.

[Mir07a, Lemma 7.2]. Preparation for volume recursion and symplectic
reduction. A finite cover of $\mathcal{M}^{\gamma}_{g,n}$ is a fibered product
of symplectic planes and lower dimensional moduli spaces.

  * •

The McShane-Mirzakhani length identity.

    * –

The set $\mathcal{B}$ of homotopy classes rel boundary of simple arcs with
endpoints on the boundary. Classification of geodesic arcs normal to a
boundary: simple geodesics normal to a boundary at each endpoint
$\Longleftrightarrow$ disjoint pairs of boundary intervals
$\Longleftrightarrow$ wire frames for pants.

    * –

Birman-Series: simple geodesics have measure zero.

    * –

The rational exponential function $H$; the hyperbolic trigonometric functions
$\mathcal{D}$ and $\mathcal{R}$; relations.

    * –

The length identity.

######  Theorem.

[Mir07a, Theorem 1.3 & 4.2] and [TWZ06, Thrm. 1.8]. For a hyperbolic surface
$R$ with boundaries $\beta_{j}$ with lengths $L_{j}$,

$L_{1}\,=\,\sum_{\alpha_{1},\alpha_{2}}\,\mathcal{D}(L_{1},\ell_{\alpha_{1}}(R),\ell_{\alpha_{2}}(R))\,+\,\sum_{j=2}^{n}\sum_{\alpha}\,\mathcal{R}(L_{1},L_{j},\ell_{\alpha}(R)),$

where the first sum is over all unordered pairs of simple closed geodesics
with $\beta_{1},\alpha_{1},\alpha_{2}$ bounding an embedded pair of pants, and
the double sum is over simple closed geodesics with
$\beta_{1},\beta_{j},\alpha$ bounding an embedded pair of pants.

    * –

[Mir07a, Section 8] \- Recognizing and understanding the identity as a smooth
analog to a $\operatorname{MCG}$ fundamental domain. Reducing to the action of
smaller $\operatorname{MCG}$ groups.

  * •

The Mirzakhani volume recursion.

    * –

A covolume formula - writing a moduli integral as a length level set integral
- the role of the intermediate moduli space $\mathcal{M}_{g,n}^{\gamma}$ \-
[Mir07a, Theorem 7.1] and [Wlp10, Theorem 9.5].

    * –

The application for the McShane-Mirzakhani identity.

      * *

The integrals $\mathcal{A}_{g,n}^{connected}$,
$\mathcal{A}_{g,n}^{disconnected}$ and $\mathcal{B}_{g,n}$ of lower-
dimensional moduli volumes.

      * *

The corresponding connected and disconnected boundary pants configurations.

      * *

Combining the length identity and covolume formula for the main result; see
[Wlp10, pg. 91 bottom, pg. 92].

######  Theorem.

[Mir07a, Section 5 and Theorem 8.1]. For $(g,n)\neq(1,1),(0,3),$ the volume
$V_{g,n}(L)$ satisfies

$\frac{\partial\ }{\partial
L_{1}}L_{1}V_{g,n}(L)\,=\,\mathcal{A}_{g,n}^{connected}(L)\,+\mathcal{A}_{g,n}^{disconnected}(L)\,+\mathcal{B}_{g,n}(L).$

      * *

The integrals $\mathcal{A}^{*}_{*}(L)$ and $\mathcal{B}_{*}(L)$ are
polynomials in boundary lengths with coefficients sums of special values of
Riemann zeta; the coefficients are positive rational multiples of powers of
$\pi$.

  * •

Symplectic reduction and the Duistermaat-Heckman theorem.

    * –

The WP kappa equation $\omega=2\pi^{2}\kappa_{1}$ on Deligne-Mumford [Wlp90].

    * –

The geometry and topology of the Teichmüller space
$\widehat{\mathcal{T}}_{g}(L)=\widehat{\mathcal{T}}_{g}(L_{1},\dots,L_{n})$ of
hyperbolic surfaces with geodesic boundaries with points.

    * –

A symplectic structure for $\widehat{\mathcal{T}}_{g}(L)$ by summing on almost
tight pants. The boundary length moment map
$R\in\widehat{\mathcal{T}}_{g}(L)\mapsto L^{2}/2\in\mathbb{R}_{+}^{n}$.
Twisting boundary points as Hamiltonian flows.

    * –

Symplectic reduction, [FO06]. $\mathcal{T}_{g}(L)$ as the reduced space
$\widehat{\mathcal{T}}_{g}(L)/(S^{1})^{n}$. $\widehat{\mathcal{T}}_{g}(L)$ as
a principal $(S^{1})^{n}$ bundle over $\mathcal{T}_{g,n}(L)$ and the small $L$
equivalence $\mathcal{T}_{g}(L)\approx\mathcal{T}_{g,n}(0)$,

$\textstyle{(S^{1})^{n}\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{\mathcal{T}}_{g}(L)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{T}_{g}(L)\approx\mathcal{T}_{g,n}(0)\,.}$

    * –

Points on circles, cotangent $\mathbb{C}$-lines at punctures, circle bundles
and homotopic structure groups. The tautological cotangent line class $\psi$.

    * –

$\operatorname{MCG}$ equivariant maps and quotients. Deligne-Mumford type
compactifications and finite covers. The elliptic stack $2$.

    * –

The Duistermaat-Heckman normal form, [Mir07b, Theorem 3.2], [MS08, Section
2.5] and [Wlp10, pg. 95],

$2\omega_{\overline{\widehat{\mathcal{M}}}_{g}(L)/(S^{1})^{n}}\,\equiv\,2\omega_{\overline{\mathcal{M}}_{g,n}(0)}\
+\ \sum_{j}\frac{L_{j}^{2}}{2}\psi_{j}\ .$

    * –

The consequence of symplectic reduction, [Mir07b, Theorem 4.4].

######  Theorem.

The volume polynomial $V_{g,n}(L)$ coefficients are
$\overline{\mathcal{M}}_{g,n}$ intersection numbers given as

$C_{\alpha}=\frac{2^{\delta_{1g}\delta_{1n}}}{2^{|\alpha|}\alpha!(3g-3+n-|\alpha|)!}\,\int_{\overline{\mathcal{M}}_{g,n}}\psi_{1}^{\alpha_{1}}\cdots\psi_{n}^{\alpha_{n}}\omega^{3g-3+n-|\alpha|},$

where $\psi_{j}$ is the Chern class for the cotangent line along the $j^{th}$
puncture, $\omega$ is the symplectic form and $\delta_{**}$ is the Kronecker
indicator.

  * •

The pattern of intersection numbers.

    * –

The general intersection number symbol

$\langle\kappa_{1}^{d_{0}}\tau_{d_{1}}\cdots\tau_{d_{n}}\rangle_{g}\,=\,\int_{\overline{\mathcal{M}}_{g,n}}\prod_{j=1}^{n}\psi_{j}^{d_{j}}\kappa_{1}^{d_{0}}$

and volume polynomial expansions.

    * –

Examples: $V_{1,1}(L)=\frac{\pi^{2}}{6}\,+\,\frac{L^{2}}{24}$,
$V_{0,4}(L_{1},L_{2},L_{3},L_{4})=(4\pi^{2}+L_{1}^{2}+L_{2}^{2}+L_{3}^{2}+L_{4}^{2})/2$
and
$V_{2,1}(L)=\frac{1}{2211840}(L^{2}+4\pi^{2})(L^{2}+12\pi^{2})(5L^{4}+384\pi^{2}L^{2}+6960\pi^{4})$.

    * –

The partition function $\mathbf{F}=\sum_{g}\langle
e^{\sum_{j}t_{j}\tau_{j}}\rangle_{g}$ and Virasoro constraint differential
operators $\mathbf{L}_{k}$ [Mir07b]. The partition function
$\mathbf{G}=\sum_{g}\langle e^{s\kappa_{1}+\sum_{j}t_{j}\tau_{j}}\rangle_{g}$
and Virasoro constraint differential operators $\mathbf{V}_{k}$ [MS08, Theorem
1.1].

    * –

The Do remove a boundary relation

$\frac{\partial V_{g,n+1}}{\partial L_{n+1}}(L,2\pi i)\,=\,2\pi
i(2g-2+n)V_{g,n}(L),\qquad\cite[cite]{[\@@bibref{}{Dothe,DoNo}{}{}]}.$

    * –

The Manin-Zograf volumes generating function [MZ00]. The punctures asymptotic
- for positive constants $c,\,a_{g}$, genus fixed and large $n$, then

$V_{g,n}\,=\,c^{n}n!\,n^{(5g-7)/2}(a_{g}\,+\,O(1/n)),\qquad\cite[cite]{[\@@bibref{}{MZ}{}{}]}.$

    * –

The Schumacher-Trapani genera asymptotic - for $n$ fixed, there are positive
constants, then

$c_{1}^{\,g}(2g)!\,<\,V_{g,n}\,<\,c_{2}^{\,g}(2g)!,\qquad\cite[cite]{[\@@bibref{}{Gru,
SchTr}{}{}]}.$

    * –

The Zograf conjecture

$V_{g,n}\,=\,(4\pi^{2})^{2g+n-3}(2g+n-3)!\frac{1}{\sqrt{g\pi}}\big{(}1\,+\,\frac{c_{n}}{g}\,+\,O(1/g^{2})\big{)},$

for fixed $n$ and $g$ tending to infinity, [Zog08]. Expected values of
geometric invariants [Mir10].

##  3 The lectures.

The following is an exposition of Mirzakhani’s proof of the Witten-Kontsevich
theorem, including the immediate background material on Teichmüller theory,
moduli space theory and on symplectic reduction. The lectures are presented as
from a graduate text - the exposition follows the development of concepts, and
does not consider the historical development of the material. The goals are
general treatment of the material and overall understanding for the reader. In
places, the approaches of several authors are combined for a simpler treatment
of the material. Only immediate references to the literature are included. The
reader should consult the literature for the historical development, for
complete references and for consequences of the material.

  1. 1.

The background and overview.

  2. 2.

The McShane-Mirzakhani identity.

  3. 3.

The covolume formula and recursion.

  4. 4.

Symplectic reduction, principal $S^{1}$ bundles and the normal form.

  5. 5.

The pattern of intersection numbers and Witten-Kontsevich.

Lecture 1: The background and overview.

General background.

By Uniformization, for a surface of negative Euler characteristic, a conformal
structure is equivalent to a complete hyperbolic structure. We consider
Riemann surfaces $R$ of finite topological type with hyperbolic metrics,
possibly with punctures and geodesic boundaries, if boundaries are non empty.
Fix a topological reference surface $F$, and consider a marking, an
orientation preserving homeomorphism $f:F\rightarrow R$ up to homotopy. If
boundaries are non empty, homotopy is rel boundary setwise. Write
$\mathcal{T}$ for the Teichmüller space of $R$ \- the space of equivalence
classes of pairs $\\{(f,R)\\}$, where pairs are equivalent if there is a
homotopy mapping triangle with a conformal map (a hyperbolic isometry) between
Riemann surfaces.

Figure 2: A genus $3$ surface with $2$ geodesic boundaries.

We consider the following Teichmüller spaces $\mathcal{T}$: $\mathcal{T}_{g}$
\- for compact genus $g$ surfaces; $\mathcal{T}_{g}(L)$ \- for genus $g$
surfaces with labeled geodesic boundaries of prescribed lengths
$L=(L_{1},\dots,L_{n})$; $\mathcal{T}_{g,n}$ \- for genus $g$ surfaces with
$n$ labeled punctures. In the case of $\mathcal{T}_{g}(L)$, homotopies of
surfaces are rel boundaries setwise. $\mathcal{T}_{g}$ and $\mathcal{T}_{g,n}$
are complex manifolds, while $\mathcal{T}_{g}(L)$ is a real analytic manifold.

A non trivial, non puncture peripheral, free homotopy class $\alpha$ on $F$
has a unique geodesic representative for $f(\alpha)$ on the surface $R$ \- the
geodesic length $\ell_{\alpha}(R)$ provides a natural function on Teichmüller
space. Collections of geodesic-length functions provide local coordinates and
global immersions to Euclidean space for $\mathcal{T}$. The differential of
geodesic-length for a simple curve is nowhere vanishing. At each point of
$\mathcal{T}$, the differentials of geodesic-lengths of simple curves are
dense in the cotangent bundle.

A surface can be cut open on a simple closed geodesic - the boundaries are
isometric circles. Since a neighborhood of a simple geodesic has an $S^{1}$
symmetry, the boundaries can be reassembled with a relative rotation to form a
new hyperbolic structure. The deformation is the Fenchel-Nielsen (FN) twist.

Figure 3: A positive Fenchel-Nielsen twist deformation.

The infinitesimal deformation for unit speed hyperbolic displacement of
initial adjacent points, is the Fenchel-Nielsen infinitesimal twist vector
field $t_{\alpha}$ on $\mathcal{T}$. (A positive twist corresponds to
displacing to the right when crossing the geodesic.)

Geodesic boundaries of hyperbolic surfaces of common length can be assembled
to form new surfaces. Given boundary reference points, the relative rotation
is measured in terms of arc length. A hyperbolic pair of pants is a genus zero
surface with three geodesic boundaries. For pants, boundary reference points
are provided by considering the unique orthogonal connecting geodesics between
boundaries. At a gluing, the common boundary length $\ell$ and relative
rotation, the FN twist parameter $\tau$, are unrestricted parameters ($\tau$
is defined by continuation from an initial configuration). The length $\ell$
varies in $\mathbb{R}_{>0}$ and twist $\tau$ varies in $\mathbb{R}$. Each
finite topological type hyperbolic surface can be assembled from pairs of
pants.

######  Theorem 1.

Fenchel-Nielsen coordinates. Fixing the topological type of a pants
decomposition and an initial configuration, the FN parameters
$\prod_{j=1}^{3g-3+n}(\ell_{j},\tau_{j})$ define a real analytic equivalence
of $\mathcal{T}$ to $\prod_{j=1}^{3g-3+n}\mathbb{R}_{>0}\times\mathbb{R}$.

The Weil-Petersson (WP) metric is Kähler. The symplectic geometry begins with
the symplectic form $\omega=2\omega_{\tiny{\mbox{WP\ K\"{a}hler}}}$ and the
basic twist-length duality

$\omega(\ ,t_{\alpha})=d\ell_{\alpha}.$

It follows from the Lie derivative equation $L_{X}\omega(\ ,\ )=d\omega(X,\ ,\
)+d(\omega(X,\ ))$ that the form $\omega$ is invariant under all twist flows.
It follows that geodesic-length functions are Hamiltonian potentials for $FN$
infinitesimal twists. Symmetry reasoning shows that $\ell$ and $\tau$ provide
action-angle coordinates for $\omega$.

######  Theorem 2.

(W), [Wlp10]. The $d\ell\wedge d\tau$ formula. The WP symplectic form is

$\omega\,=\,\sum_{j=1}^{3g-3+n}\,d\ell_{j}\wedge d\tau_{j}.$

Frontier spaces are adjoined to $\mathcal{T}$ corresponding to allowing
$\ell_{j}=0$ with the FN angle $\theta_{j}=2\pi\tau_{j}/\ell_{j}$ then
undefined (in polar coordinates, angle is undefined at the origin). The
vanishing length describes pairs of pants with corresponding boundaries
represented by punctures - the equation $\ell_{\alpha}=0$ describes hyperbolic
structures with $\alpha$ represented by pairs of punctures. For a subset of
indices $J\subset\\{1,\dots,3g-3+n\\}$, the $J$-null stratum is
$\mathcal{S}(J)\,=\,\\{R\mbox{ degenerate}\mid\ell_{j}(R)=0\mbox{ iff }j\in
J\\}$. Each null stratum is a product of lower dimensional Teichmüller spaces.
A basis of neighborhoods in $\mathcal{T}\cup\mathcal{S}(J)$ is defined in
terms of the parameters $(\ell_{k},\theta_{k},\ell_{j})_{k\notin J,\,j\in J}$.

The augmented Teichmüller space is the stratified space

$\overline{\mathcal{T}}=\mathcal{T}\cup_{\tiny{pants\ decompositions\
}\mathcal{P}}\cup_{J\subset\mathcal{P}}\,\mathcal{S}(J).$

The space $\overline{\mathcal{T}}$ is also described as the Chabauty topology
closure of the discrete faithful type-preserving representations of
$\pi_{1}(F)$ into $PSL(2;\mathbb{R})$, modulo $PSL(2;\mathbb{R})$ conjugation.
The augmentation construction is valid for
$\mathcal{T}_{g},\mathcal{T}_{g}(L)$ and $\mathcal{T}_{g,n}$.
$\overline{\mathcal{T}}$ is a Baily-Borel type partial compactification.
$\overline{\mathcal{T}}$ is never locally compact. The $d\ell\wedge d\tau$
formula provides for the extension of the symplectic structure to the
augmented Teichmüller space $\overline{\mathcal{T}}$. Each strata is
symplectic.

The mapping class group ($\operatorname{MCG}$) $Homeo^{+}(F)/Homeo_{0}(F)$
acts on markings by precomposition and thus acts on $\mathcal{T}$. For
$Homeo^{+}(F)$ we consider type-preserving (boundary point, boundary curve),
boundary label preserving, orientation preserving homeomorphisms.
$Homeo_{0}(F)$ is the normal subgroup of elements homotopic to the identity
rel boundary setwise. A Dehn twist is a homeomorphism that is the identity on
the complement of a tubular neighborhood of a simple closed curve, non trivial
in homotopy, and rotates one boundary of the tubular neighborhood relative to
the other. Dehn twist classes generate $\operatorname{MCG}$.
$\operatorname{MCG}$ acts properly discontinuously on $\mathcal{T}$ and by
biholomorphisms for $\mathcal{T}_{g}$ and $\mathcal{T}_{g,n}$. Except for a
finite number of topological types the action is effective. Finite
$\operatorname{MCG}$ subgroups act with fixed points. $\operatorname{MCG}$
acts on the stratified space $\overline{\mathcal{T}}$. Bers observed that
there are constants $b_{g,n}$, depending on topological type, such that a
genus $g$, $n$ punctured hyperbolic surface has a pants decomposition with
seam lengths at most $b_{g,n}$. It follows that the domain $\\{\ell_{j}\leq
b_{g,n},\,0<\tau_{j}\leq\ell_{j}\\}$ in FN coordinates is a rough fundamental
set - each $\operatorname{MCG}$ orbit intersects the domain a bounded positive
number of times. $\overline{\mathcal{T}}/\operatorname{MCG}$ is a compact real
analytic orbifold; $\overline{\mathcal{T}}_{g}/\operatorname{MCG}$ and
$\overline{\mathcal{T}}_{g,n}/\operatorname{MCG}$ are topologically the
Deligne-Mumford stable curve compactifications of $\mathcal{M}_{g}$ and
$\mathcal{M}_{g,n}$. The Bers fundamental set observation combines with the
$d\ell\wedge d\tau$ formula to provide that the WP volume of $\mathcal{M}$ is
finite.

The Bers fiber space $\mathcal{C}$ (specifically $\mathcal{C}_{g}$ and
$\mathcal{C}_{g,n}$) is the complex disc holomorphic bundle over $\mathcal{T}$
with fiber over $\\{(f,R)\\}$ the universal cover $\widetilde{R}$. A point on
a fiber can be considered as a puncture and determines a curve from basepoint
for the fundamental group - so $\mathcal{C}_{g}\approx\mathcal{T}_{g,1}$ and
$\mathcal{C}_{g,n}\approx\mathcal{T}_{g,n+1}$. An extension
$\operatorname{MCG}_{\mathcal{C}}$ of $\operatorname{MCG}(F)$ by the
fundamental group $\pi_{1}(F)$ acts properly discontinuously and
holomorphically on $\mathcal{C}$. The group $\operatorname{MCG}_{\mathcal{C}}$
is isomorphic to $\operatorname{MCG}_{g,n+1}$. For the epimorphism from
$\operatorname{MCG}_{\mathcal{C}}$ to $\operatorname{MCG}(F)$, the first group
acts equivariantly on the fibration of $\mathcal{C}$ over $\mathcal{T}$. The
resulting map
$\pi:\mathcal{C}/\operatorname{MCG}\rightarrow\mathcal{T}/\operatorname{MCG}$
describes an orbifold bundle, the universal curve, with orbifold fibers - the
fibers are Riemann surfaces modulo their full automorphism group. Manifold
finite local covers and the quotient can be described by starting with a
surface with locally maximal symmetries and introducing a local trivialization
of the bundle by canonical (extremal, harmonic) maps of surfaces.

The augmentation construction applies to the Bers fiber space to give
$\overline{\mathcal{C}}$. $\operatorname{MCG}_{\mathcal{C}}$ acts on the
stratified space $\overline{\mathcal{C}}$.

Figure 4: A puncture section of the universal curve
$\overline{\mathcal{C}}/\operatorname{MCG}_{\mathcal{C}}$ over
$\overline{\mathcal{T}}/\operatorname{MCG}(F)$.

The augmentation quotient $\overline{\mathcal{C}}/\operatorname{MCG}$ is an
orbifold and almost an orbifold bundle over $\overline{\mathcal{M}}$ \- at a
node (a pair of punctures) of a Riemann surface, the fiber becomes vertical -
the local model of the fibration is the germ at the origin of the projection
$\\{(z,w)\\}\rightarrow\\{t=zw\\}$, the family of complex hyperbolas. The
turning of the fibers of the almost orbifold bundle
$\overline{\mathcal{C}}/\operatorname{MCG}\rightarrow\overline{\mathcal{M}}$
is measured by line bundles on $\overline{\mathcal{M}}$. The family of tangent
$\mathbb{C}$-lines to the fibers $(Ker\,d\pi)$ is the tangent bundle along a
smooth Riemann surface and the relative dualizing sheaf along a noded Riemann
surface. The hyperbolic metrics of the individual fibers provide a line bundle
metric for $(Ker\,d\pi)$ on $\overline{\mathcal{C}}$, that although not smooth
is sufficiently regular for calculation of the Chern form $\mathbf{c}_{1}$.
The kappa forms/cohomology classes
$\kappa_{k}=\int_{\pi^{-1}(\\{R\\})}\mathbf{c}_{1}^{k+1}$ given by integration
over fibers are basic to moduli geometry. The geometry and algebra of the
kappa classes is studied in the Carel Faber lectures. Explicit calculation of
the Chern form and integration provides the following.

Figure 5: Tangents along a fiber of the universal curve
$\overline{\mathcal{C}}/\operatorname{MCG}_{\mathcal{C}}$.

######  Theorem 3.

(W), [Wlp90]. For the hyperbolic metrics on fibers,
$2\pi^{2}\kappa_{1}=\omega$ pointwise on $\mathcal{M}$ and in cohomology on
$\overline{\mathcal{M}}$.

A conformal structure has a unique extension to fill in a puncture. A labeled
puncture defines a section $s$ of
$\overline{\mathcal{C}}_{g,n}/\operatorname{MCG}\rightarrow\overline{\mathcal{M}}_{g,n}$.
A section satisfies $\pi\circ s=id$, differentiating gives $d\pi\circ
ds=d\,id$. At a node, $d\pi$ vanishes in the node opening direction - for
$\pi(z,w)=t$ then $d\pi=wdz\,+\,zdw$ vanishes at the origin. Sections of
$\overline{\mathcal{C}}/\operatorname{MCG}$ over $\overline{\mathcal{M}}$ are
consequently disjoint from nodes. Along a puncture section
$s:\overline{\mathcal{M}}\rightarrow\overline{\mathcal{C}}/\operatorname{MCG}$,
we consider the family of tangent lines $(Ker\,d\pi)|_{s}$ or the dual family
$(Ker\,d\pi)^{*}|_{s}$. In the Carel Faber lectures, the Chern class is
denoted as $K$. -The pullback to $\overline{\mathcal{M}}$ by a puncture
section $s$ of the dual family $(Ker\,d\pi)^{*}$ is the moduli geometry
canonical psi class $\psi$ .-

In these lectures, to emphasize concepts and the underlying geometry, we will
at times informally interchange a line bundle and its Chern class, informally
refer to the moduli space as a manifold, the universal curve as a fiber
bundle, and at times refer to the open moduli space when actually the
augmentation quotient is required. Our goal is to discuss the central matters.
In spite of the informal approach, an experienced reader will find that the
treatment is complete.

Basic references for the above material are [Bus92] and [Wlp10].

Volume results overview.

Mirzakhani shows that the WP volume
$V_{g,n}(L)\,=\,V(\overline{\mathcal{M}}_{g}(L))$ is a polynomial in $L$, with
coefficients given by the intersection numbers of powers of $\kappa_{1}$ and
powers of $\psi$. She further shows that her recursion for determining the
volume polynomials satisfies the defining relations for the Witten-Kontsevich
conjecture. The following theorems are the immediate results of the two
papers.

######  Theorem 4.

The WP volume polynomials. The volume polynomials are determined recursively
from the volume polynomials of smaller total degree, [Mir07a, Formula (5.1) &
Theorem 8.1]. The volume $V_{g,n}(L_{1},\dots,L_{n})$ of the moduli space of
genus $g$, $n$ boundaries, hyperbolic surfaces with boundary lengths
$L=(L_{1},\dots,L_{n})$ is a polynomial

$V_{g,n}(L)\,=\,\sum_{\stackrel{{\scriptstyle\alpha}}{{|\alpha|\leq
3g-3+n}}}C_{\alpha}\,L^{2\alpha},$

for multi index $\alpha=(\alpha_{1},\dots,\alpha_{n})$ and where
$C_{\alpha}>0$ lies in $\pi^{6g-6+2n-2|\alpha|}\mathbb{Q}$, [Mir07a, Theorems
1.1 & 6.1]. The coefficients are intersection numbers given as

$C_{\alpha}=\frac{2^{\delta_{1g}\delta_{1n}}}{2^{|\alpha|}\alpha!(3g-3+n-|\alpha|)!}\,\int_{\overline{\mathcal{M}}_{g,n}}\psi_{1}^{\alpha_{1}}\cdots\psi_{n}^{\alpha_{n}}\omega^{3g-3+n-|\alpha|},$

where $\psi_{j}$ is the Chern class for the cotangent line along the $j^{th}$
puncture, $\omega$ is the symplectic form,
$\alpha!=\prod_{j=1}^{n}\alpha_{j}!$, and $\delta_{**}$ is the Kronecker
indicator delta, [Mir07b, Theorem 4.4].

######  Theorem 5.

[Mir07a, Theorems 6.3 & 6.4]. Recursive relations for the volume polynomial
leading coefficients. For a multi index $\alpha$, define

$(\alpha_{1},\dots,\alpha_{n})_{g}=C_{\alpha}\times
2^{-\delta_{1g}\delta_{1n}}\times\prod_{i=1}^{n}\alpha_{i}!\times
2^{|\alpha|},$

then for $n>0$ and $\sum_{i}\alpha_{i}=3g-3+n$,

$\mbox{the dilaton
equation}\qquad\quad(1,\alpha_{1},\dots,\alpha_{n})_{g}\,=\,(2g-2+n)(\alpha_{1},\dots,\alpha_{n})_{g}$

and for $n>0$ and $\sum_{i}\alpha_{i}=3g-2+n$,

$\mbox{the string
equation}\qquad(0,\alpha_{1},\dots,\alpha_{n})_{g}\,=\,\sum_{\alpha_{i}\neq
0}(\alpha_{1},\dots,\alpha_{i}-1,\dots,\alpha_{n})_{g}.$

For the intersection number generating function

$\mathbf{F}(\lambda,t_{0},t_{1},\dots)\,=\,\sum_{g=0}^{\infty}\lambda^{2g-2}\sum_{\\{d_{j}\\}}\,\langle\prod_{j=1}^{\infty}\tau_{d_{j}}\rangle_{g}\,\prod_{r\geq
0}t_{r}^{n_{r}}/n_{r}!\,,$

with $n_{r}=\\#\\{j\mid d_{j}=r\\}$, and

$\langle\tau_{d_{1}}\cdots\tau_{d_{n}}\rangle_{g}\,=\,\int_{\overline{\mathcal{M}}_{g,n}}\prod_{j=1}^{n}\psi_{j}^{d_{j}},$

for $\sum_{j=1}^{n}d_{j}=3g-3+n$ and the product $\langle\tau_{*}\rangle$
otherwise zero, then the exponential $e^{\mathbf{F}}$ satisfies Virasoro
algebra constraints, [Mir07b, Theorem 6.1].

A fine structure for volumes is suggested by the Zograf conjecture

$V_{g,n}\,=\,(4\pi^{2})^{2g+n-3}(2g+n-3)!\frac{1}{\sqrt{g\pi}}\big{(}1\,+\,\frac{c_{n}}{g}\,+\,O(1/g^{2})\big{)},$

for fixed $n$ and $g$ tending to infinity.

As an application of the method for recursion of volumes and intersection
numbers, Do derives a remove a boundary relation for the volume polynomials

$\frac{\partial V_{g,n+1}}{\partial L_{n+1}}(L,2\pi i)\,=\,2\pi
i(2g-2+n)V_{g,n}(L),\qquad\cite[cite]{[\@@bibref{}{Dothe}{}{}]}.$

The relation gives the compact case volume $V_{g}$.

Statement of the volume recursion [Mir07a, Sec. 5]. The WP volume
$V_{g}(L_{1},\dots,L_{n})$ of the moduli space
$\mathcal{T}_{g}(L_{1},\dots,L_{n})/\operatorname{MCG}$ is a symmetric
function of boundary lengths as follows.

  * •

For $L_{1},L_{2},L_{3}\geq 0$, formally set

$V_{0,3}(L_{1},L_{2},L_{3})=1$

and

$V_{1,1}(L_{1})=\frac{\pi^{2}}{12}+\frac{L_{1}^{2}}{48}.$

  * •

For $L=(L_{1},\dots,L_{n})$, let $\widehat{L}=(L_{2},\dots,L_{n})$ and for
$(g,n)\neq(1,1)$ or $(0,3)$, the volume satisfies

$\frac{\partial\ }{\partial
L_{1}}L_{1}V_{g}(L)=\mathcal{A}_{g}^{con}(L)+\mathcal{A}_{g}^{dcon}(L)+\mathcal{B}_{g}(L)$

where

$\mathcal{A}_{g}^{*}(L)=\frac{1}{2}\int_{0}^{\infty}\int_{0}^{\infty}\widehat{\mathcal{A}}_{g}^{*}(x,y,L)\,xy\,dxdy$

and

$\mathcal{B}_{g}(L)=\int_{0}^{\infty}\widehat{\mathcal{B}}_{g}(x,L)\,x\,dx.$

The quantities
$\widehat{\mathcal{A}}_{g}^{con},\,\widehat{\mathcal{A}}_{g}^{dcon}$ are
defined in terms of the function

$H(x,y)=\frac{1}{1+e^{\frac{x+y}{2}}}+\frac{1}{1+e^{\frac{x-y}{2}}}$

and moduli volumes for subsurfaces

$\widehat{\mathcal{A}}_{g}^{con}(x,y,L)=H(x+y,L_{1})V_{g-1}(x,y,\widehat{L})$

and surface decomposition sum

$\quad\quad\quad\widehat{\mathcal{A}}_{g}^{dcon}(x,y,L)=\sum\limits_{\stackrel{{\scriptstyle
g_{1}+g_{2}=g}}{{I_{1}\amalg
I_{2}=\\{2,\dots,n\\}}}}H(x+y,L_{1})V_{g_{1}}(x,L_{I_{1}})V_{g_{2}}(y,L_{I_{2}}),$

where in the second sum only decompositions for pairs of hyperbolic structures
are considered and the unordered sets $I_{1},I_{2}$ provide a partition. The
third quantity $\widehat{\mathcal{B}}_{g}$ is defined by the sum

$\quad\quad\quad\quad\frac{1}{2}\sum_{j=1}^{n}\big{(}H(x,L_{1}+L_{j})+H(x,L_{1}-L_{j})\big{)}\,V_{g}(x,L_{2},\dots,\widehat{L_{j}},\dots,L_{n}),$

where $L_{j}$ is omitted from the argument list of $V_{g}$.

The basic point: the volume $V_{g}(L_{1},\dots,L_{n})$ is an appropriate
integral of volumes for surfaces formed with one fewer pairs of pants.

Lecture 2: The McShane-Mirzakhani identity.

In 1991, Greg McShane discovered a universal identity for a sum of lengths of
simple geodesics for a once punctured torus, [McS98]. A generalization of the
identity serves as the analog of a partition of unity for the action of the
mapping class group. The identity enables reduction of the action to the
actions of smaller mapping class groups. Consideration of the identity begins
with a surface with geodesic boundaries and a study of arcs from the boundary
to itself.

Introduce $\mathcal{B}$, the set of non trivial free homotopy classes of
simple curves from the boundary to the boundary, homotopy rel the boundary. We
illustrate the approach by considering simple curves with endpoints on a
common boundary $\beta$; the analysis is similar for simple curves connecting
distinct boundaries. Each homotopy class contains a unique shortest geodesic,
orthogonal to $\beta$ at end points - refer to these geodesics as ortho
boundary geodesics. If the surface is doubled across its boundary, then the
ortho boundary geodesics double to simple closed geodesics.

The set $\mathcal{B}$ is in bijection to the set of topological pants embedded
in the surface with $\beta$ as one boundary - refer to these pants as
$\beta$-cuff pants. First note that the endpoints of an ortho boundary
geodesic $\gamma,[\gamma]\in\mathcal{B}$ are distinct. A small
neighborhood/thickening of $\gamma\cup\beta$ is the corresponding topological
pair of pants. Geometrically, the curve $\gamma$ separates $\beta$ into proper
sub arcs; the union of each sub arc with $\gamma$ is a simple curve, that
defines a free homotopy class containing a unique geodesic. The corresponding
geometric pair of pants $\mathcal{P}$ has boundaries $\beta$ and the two
determined geodesics. We will see below that a geometric pair of pants
contains a unique ortho boundary geodesic. The unions of ortho boundary
geodesics and $\beta$ are the spines, the wire frames, for the embedded
geometric $\beta$-cuff pants.

We now describe how the behavior of geodesics emanating orthogonally from
$\beta$ defines a Cantor subset of $\beta$. The Cantor set will have measure
zero and the length identity is simply the sum of lengths of the complementary
intervals. The following description follows the analysis by Tan-Wong-Zhang,
[TWZ06].

Consider the maximal continuations of geodesics emanating orthogonally from
$\beta$ \- refer to these geodesics as ortho emanating geodesics. In addition
to the ortho boundary geodesics, there are three types of ortho emanating
geodesics: non simple, simple infinite length and simple crossing the boundary
obliquely at a second endpoint. We will see that the types are detected by
considering initial segments in a pair of pants.

Consider the geometric pants $\mathcal{P}$, obtained from an ortho boundary
geodesic $\gamma$ (see Figure 6). The boundaries are $\beta$ and the two
defined geodesics $\alpha$ and $\lambda$ (in the special case $(g,n)=(1,1)$
then $\alpha=\lambda$). A spiral is an infinite simple geodesic ray that
accumulates to a simple closed geodesic. Two ortho emanating geodesics are
spirals with accumulation set $\alpha$ and two are spirals with accumulation
set $\lambda$. The two spirals accumulating to a boundary wind in opposite
directions around the boundary. The $\mathcal{P}$ main gaps are the two
disjoint subarcs of $\beta$ that each contain in their interior an endpoint of
$\gamma$ and have spiral initial points as endpoints. The main gaps will be
the components of the Cantor set complement corresponding to the pants
$\mathcal{P}$. From the geometry of pants, geodesics ortho emanating from the
main gaps are either the spiral endpoints, $\gamma$, non simple with self
intersection in $\mathcal{P}$ or simple crossing $\beta$ obliquely at a second
endpoint. The complement in $\beta$ of the main gaps are a pair of open
intervals. From the geometry of pants, for a given open interval all ortho
emanating geodesics exit the pants by crossing one of the boundaries $\alpha$
or $\lambda$. For a given open interval, the initial segments in $\mathcal{P}$
are simple and these geodesics are classified by their subsequent behavior
elsewhere on the surface, by their behavior on some other pair of pants. The
ortho boundary geodesic connecting $\beta$ to $\alpha$ is contained in one of
the open intervals, and the ortho boundary geodesic from $\beta$ to $\lambda$
is contained in the other. A pair of pants has an equatorial reflection,
stabilizing each boundary. The equatorial reflection acts naturally on the
decomposition of $\beta$, interchanging or stabilizing elements.

Figure 6: A pair of pants with equators, main gaps, two spirals and an
orthoboundary geodesic $\gamma$.

In the above, associated to a main gap are the ortho emanating geodesics that
self intersect in the pants, and the simple geodesics that obliquely cross
$\beta$ a second time - refer to the second type geodesics as boundary
oblique. The next observation is that the associations can be reversed, the
associations define bijections between main gaps and geodesics with particular
behaviors on the surface.

A boundary oblique geodesic and $\beta$ form a crooked wire frame that
determines a pair of pants, similar to an ortho boundary geodesic determining
a pair of pants. Boundary oblique geodesics come in continuous families with
each family limiting to an ortho boundary geodesic. A family and its limit
determine the same pair of pants. The initial points (the $\beta$ orthogonal
points) of family elements lie in a common main gap interval - this
observation reverses the association of segments of main gaps to boundary
oblique geodesics.

Next we describe reversing the association of segments of main gaps to non
simple ortho emanating geodesics. The first self intersection of such a
geodesic is contained in a unique embedded pair of pants. To see this,
consider the lasso subarc beginning at $\beta$ and ending where the geodesic
passes through its first self intersection point a second time. The boundary
of a small neighborhood/thickening of the lasso is the union of a simple
closed curve and an element of $\mathcal{B}$. A geometric argument shows that
the self intersection point is contained in the pants determined by the
element of $\mathcal{B}$ and the lasso initial point lies in the main gap for
the pants. This observation reverses the association of segments of main gaps
to non simple ortho emanating geodesics.

Figure 7: A small neighborhood of a lasso.

We recall that non simple with interior intersection is an open condition on
the space of geodesics and an open condition on the space of ortho emanating
geodesics. By considering the double of the surface, simple with all boundary
intersections orthogonal is a closed condition on the space of geodesics. The
set simple with orthogonal single boundary intersection is a Cantor set. The
classification of ortho emanating geodesics is complete.

######  Theorem 6.

(Tan-Wong-Zhang [TWZ06], and Mirzakhani [Mir07a], all following McShane
[McS98].) There is a Cantor set partition of boundary points by the behavior
of ortho emanating geodesics:

$\beta\,=\,\\{\mbox{simple with orthogonal single boundary intersection}\\}\\\
\,\cup\,\\{\mbox{ortho boundary geodesics}\\}\,\cup\,\\{\mbox{simple boundary
oblique geodesics}\\}\,\cup\,\\{\mbox{non simple}\\}.$

An $\epsilon$-neighborhood of the simple complete geodesics, orthogonal to the
boundary at intersections, is a countable union of thin corridors. In the
universal cover the corridors are described by reduced bi infinite words in
the fundamental group. By analyzing the number and width of corridors, Birman-
Series show that the set is very thin.

######  Theorem 7.

(Birman-Series, [BS85].) Simple geodesics have measure zero. The set
$\mathcal{S}$ of simple complete geodesics, orthogonal to the boundary at
intersections, has Hausdorff dimension $1$. The intersection of $\mathcal{S}$
and the boundary has Hausdorff dimension and measure $0$.

The basic summand for the length identity is a rational exponential function.
Define the function $H$ on $\mathbb{R}^{2}$ by

$H(x,y)=\frac{1}{1+e^{\frac{x+y}{2}}}+\frac{1}{1+e^{\frac{x-y}{2}}}$ (1)

and the corresponding functions $\mathcal{D},\mathcal{R}$ on $\mathbb{R}^{3}$
by

$\begin{split}\mathcal{D}(x,y,z)\,&=\,2\log\Bigg{(}\frac{e^{\frac{x}{2}}\,+\,e^{\frac{y+z}{2}}}{e^{\frac{-x}{2}}\,+\,e^{\frac{y+z}{2}}}\Bigg{)}\quad\mbox{and}\\\
\mathcal{R}(x,y,z)\,&=\,x\,-\,\log\Bigg{(}\frac{\cosh\frac{y}{2}\,+\cosh\frac{x+z}{2}}{\cosh\frac{y}{2}\,+\cosh\frac{x-z}{2}}\Bigg{)}.\end{split}$
(2)

The functions $\mathcal{D}$ and $\mathcal{R}$ are related to $H$ as follows,

$\begin{split}\frac{\partial\ }{\partial
x}\mathcal{D}(x,y,z)&=H(y+z,x),\quad\mathcal{D}(0,0,0)=0\quad\mbox{and}\\\
2\,\frac{\partial\ }{\partial
x}\mathcal{R}(x,y,z)&=H(z,x+y)+H(z,x-y),\quad\mathcal{R}(0,0,0)=0.\end{split}$
(3)

######  Theorem 8.

[Mir07a, Theorem 1.3 & 4.2] and [TWZ06, Thrm. 1.8]. The Mirzakhani-McShane
identity. For a hyperbolic surface $R$ with boundaries $\beta_{j}$ with
lengths $L_{j}$,

$L_{1}\,=\,\sum_{\alpha_{1},\alpha_{2}}\,\mathcal{D}(L_{1},\ell_{\alpha_{1}}(R),\ell_{\alpha_{2}}(R))\,+\,\sum_{j=2}^{n}\sum_{\alpha}\,\mathcal{R}(L_{1},L_{j},\ell_{\alpha}(R)),$

where the first sum is over all unordered pairs of simple closed geodesics
with $\beta_{1},\alpha_{1},\alpha_{2}$ bounding an embedded pair of pants, and
the double sum is over simple closed geodesics with
$\beta_{1},\beta_{j},\alpha$ bounding an embedded pair of pants.

###### Proof.

By the above Theorems, $\ell_{\beta}$ equals the sum over embedded
$\beta$-cuff pants of main gap lengths and the counterpart lengths for double
boundary cuff pants. To find the main gap lengths, begin with a formula for
the lengths of the complementary intervals. In a pair of pants, the ortho
boundary geodesic $\delta_{\alpha}$ from $\beta$ to $\alpha$ bisects a
complementary interval (see Figure 6). Let $\beta_{\alpha}$ be one of the
resulting half intervals. The segment $\beta_{\alpha}$ has the geodesic
$\delta_{\beta}$ emanating at one end and a spiral $\sigma$ to $\alpha$
emanating at the other end. In the universal cover, consider contiguous lifts
$\widetilde{\delta_{\alpha}}$, $\widetilde{\beta_{\alpha}}$ and
$\widetilde{\sigma}$. The three lifts and a half infinite ray lift
$\widetilde{\alpha}$ of $\alpha$, combine to form a quadrilateral
$\widetilde{\alpha},\widetilde{\delta_{\alpha}},\widetilde{\beta_{\alpha}},\widetilde{\sigma}$
with angles $\pi/2,\pi/2,\pi/2$ and $0$ between $\widetilde{\sigma}$ and
$\widetilde{\alpha}$. By hyperbolic trigonometry of quadrilaterals [Bus92], it
follows that

$\tanh\ell_{\beta_{\alpha}}\,=\,\operatorname{sech}\delta_{\alpha}\,=\,\frac{\sinh(\ell_{\beta}/2)\,\sinh(\ell_{\alpha}/2)}{\cosh(\ell_{\lambda}/2)\,+\,\cosh(\ell_{\beta}/2)\,\cosh(\ell_{\alpha}/2)}\,.$

The complementary interval length is
$2\ell_{\beta_{\alpha}}=\ell_{\beta}-\mathcal{R}(\ell_{\beta},\ell_{\lambda},\ell_{\alpha})$.
The formula for main gap lengths now follows from the general relation
$\mathcal{R}(x,y,z)\,+\,\mathcal{R}(x,z,y)=x\,+\,\mathcal{D}(x,y,z)$. For
double boundary cuff pants, the main gap lengths are added to the
complementary interval length. The result is
$\mathcal{R}(\ell_{\beta},\ell_{\alpha},\ell_{\lambda})$. ∎

Lecture 3: The covolume formula and recursion.

The main step is application of the length identity to reduce the action of
the mapping class group to an action of smaller mapping class groups, and
consequently express the volume as an integral over a length level set. The
result is an integral of products of lower dimensional volume functions - the
recursion.

The approach is illustrated by computing the genus one, one boundary, volume.
The length identity is

$L\,=\,\sum_{\alpha\
\tiny{simple}}\mathcal{D}(L,\ell_{\alpha},\ell_{\alpha}).$

Introduce $\operatorname{Stab}(\alpha)\subset\operatorname{MCG}$, the
stabilizer for $\operatorname{MCG}$ acting on free homotopy classes. A torus
is an elliptic curve with universal cover $\mathbb{C}$ with involution
$z\rightarrow-z$ stabilizing the deck transformation lattice. The involution
acts on tori and tori with one puncture or boundary. The involution reverses
orientation for the free homotopy class of each simple closed geodesic and the
stabilizer $\operatorname{Stab}(\alpha)$ is the semi direct product of the
Dehn twists by the involution $\mathbb{Z}/2\mathbb{Z}$ subgroup. The
involution acts trivially on Teichmüller space. (The torus is one of the
exceptional cases where the $\operatorname{MCG}$ action on $\mathcal{T}$ is
not effective. We will also discuss the torus case below, where a multiplicity
is involved.) A Dehn twist acts on the Teichmüller space in FN coordinates by
$(\ell,\tau)\rightarrow(\ell,\tau+\ell)$. The sector $\\{0\leq\tau<\ell\\}$ is
a fundamental domain for the $\operatorname{Stab}(\alpha)$ action. A mapping
class $h\in\operatorname{MCG}$ acts on a geodesic-length function by
$\ell_{\alpha}\circ h^{-1}=\ell_{h(\alpha)}$.

Write the length identity as

$L\,=\,\sum_{\alpha}\,\mathcal{D}(L,\ell_{\alpha},\ell_{\alpha})\,=\,\sum_{h\in\operatorname{MCG}/\operatorname{Stab}(\alpha)}\mathcal{D}(L,\ell_{h(\alpha)},\ell_{h(\alpha)}),$

use the $\operatorname{MCG}$ action on geodesic-length functions, to find

$LV(L)\,=\,\int_{\mathcal{T}(L)/\operatorname{MCG}}\sum_{\operatorname{MCG}/\operatorname{Stab}(\alpha)}\mathcal{D}(L,\ell_{\alpha}\circ
h^{-1},\ell_{\alpha}\circ h^{-1})\,\omega,$

change variables on $\mathcal{T}$ by $p=h(q)$ to find

$\sum_{h\in\operatorname{MCG}/\operatorname{Stab}(\alpha)}\int_{h(\mathcal{T}(L)/\operatorname{MCG})}\mathcal{D}(L,\ell_{\alpha},\ell_{\alpha})\,d\tau
d\ell\,=\,\int_{\mathcal{T}(L)/\operatorname{Stab}(\alpha)}\mathcal{D}(L,\ell_{\alpha},\ell_{\alpha})\,d\tau
d\ell,$

and use the $\operatorname{Stab}(\alpha)$ fundamental domain, to obtain the
integral

$\int_{0}^{\infty}\int_{0}^{\ell}\,\mathcal{D}(L,\ell,\ell)\,d\tau d\ell.$

The integral in $\tau$ gives a factor of $\ell$.

The derivatives $\partial\mathcal{D}(x,y,z)/\partial x$ and
$\partial\mathcal{R}(x,y,z)/\partial x$ are simpler than the original
functions $\mathcal{D}$ and $\mathcal{R}$ \- apply this observation and
differentiate in $L$ to obtain a formula for the derivative of $LV(L)$,

$\frac{\partial}{\partial
L}LV(L)\,=\,\int_{0}^{\infty}\frac{1}{1+e^{\ell+\frac{L}{2}}}\,+\,\frac{1}{1+e^{\ell-\frac{L}{2}}}\,\ell
d\ell\,=\,\frac{\pi^{2}}{6}\,+\,\frac{L^{2}}{8}.$

The formula $V(L)=\frac{\pi^{2}}{6}+\frac{L^{2}}{24}$ results.

We prepare for the general case. In algebraic geometry intersection theory,
the elliptic involution gives rise to multiplying elliptic intersection counts
by a factor of $1/2$. The factor corresponds to the generic fiber of the
universal elliptic curve being the quotient of the elliptic curve by its
involution. Along this line, the general volume recursion is simplified if
$V_{1,1}(L)$ is formally defined to be $1/2$ of the given value $V(L)$. In
mapping class group theory, the elliptic involution appears as the half Dehn
twist for simple closed curves bounding a torus. In particular, consider the
fundamental group $\pi_{1}(R)$ of a surface, with the standard presentation
$a_{1}b_{1}a_{1}^{-1}b_{1}^{-1}\cdots
a_{g}b_{g}a_{g}^{-1}b_{g}^{-1}c_{1}\cdots c_{n}=1$, with $c_{j}$ a loop about
the $j^{th}$ boundary. The half Dehn twist about the curve
$a_{1}b_{1}a_{1}^{-1}b_{1}^{-1}$ is the automorphism of $\pi_{1}(R)$ given by:
$a_{1}\rightarrow b_{1}a_{1}^{-1}b_{1}^{-1},b_{1}\rightarrow
b_{1}^{-1};a_{j}\rightarrow a_{1}^{-1}a_{j}a_{1},b_{j}\rightarrow
a_{1}^{-1}b_{j}a_{1},\,\mbox{for }j=2\dots g,\mbox{ and }c_{j}\rightarrow
a_{1}^{-1}c_{j}a_{1},\,\mbox{for }j=1\dots n$. The square of a half Dehn twist
is a Dehn twist and a half Dehn twist acts on the associated FN parameters by
$(\ell,\tau)\rightarrow(\ell,\tau+\ell/2)$.

We now set up for the covolume formula. Let $R$ be a hyperbolic surface with
geodesic boundaries $\beta_{1},\,\dots,\beta_{n}$. Consider a weighted
multicurve

$\gamma\,=\,\sum_{j=1}^{m}a_{j}\gamma_{j},$

where $a_{j}$ are real weights and $\gamma_{j}$ are distinct, disjoint, simple
closed geodesics. Define
$\operatorname{Stab}(\gamma)\subset\operatorname{MCG}$ to be the mapping
classes stabilizing the collection of unlabeled, weighted geodesics - elements
of $\operatorname{Stab}(\gamma)$ may permute components of the multicurve with
equal weights. Write $\operatorname{Stab}(\gamma_{j})$ for the stabilizer of
an individual geodesic and $\operatorname{Stab}_{0}(\gamma_{j})$ for the
subgroup of elements preserving orientation.

Write $R(\gamma)$ for the surface cut open along the $\gamma$ \- each
$\gamma_{j}$ gives rise to two new boundaries - $R(\gamma)$ may be
disconnected. Write
$\mathcal{T}(R(\gamma);\mathbf{x}),\,\mathbf{x}=(x_{1},\dots,x_{m})$ for the
(product) Teichmüller space of the cut open surface with the pair of
boundaries for $\gamma_{j}$ having length $x_{j}$. Denote by
$\operatorname{MCG}(R(\gamma))$ the product of mapping class groups of the
components of $R(\gamma)$ and by
$\mathcal{T}(R(\gamma);\mathbf{x})/\operatorname{MCG}(R(\gamma))$ the
corresponding product of moduli spaces. For the product of symplectic forms on
$\mathcal{T}(R(\gamma);\mathbf{x})$ corresponding to the components of
$R(\gamma)$, the volume $V(R(\gamma);\mathbf{x})$ is the product of volumes of
the component moduli spaces, where again the pair of boundaries for
$\gamma_{j}$ have common length $x_{j}$. Considerations also involve the
finite symmetry group

$\operatorname{Sym}(\gamma)\,=\,\operatorname{Stab}(\gamma)/\cap_{j}\operatorname{Stab}_{0}(\gamma_{j})$

of mapping classes that possibly permute and reverse orientation of the
$\gamma$ elements.

Summing the translations of a function over a group gives a group action
invariant function. Begin with a function $f$, suitably small at infinity, and
introduce the $\operatorname{MCG}$ sum

$f_{\gamma}(R)=\sum_{\operatorname{MCG}/\operatorname{Stab}(\gamma)}f\big{(}\sum_{j=1}^{m}a_{j}\ell_{h(\gamma_{j})}(R)\big{)}.$
(4)

The next theorem expresses the moduli space integral

$\int_{\mathcal{M}(R)}\,f_{\gamma}\,dV$

as a weighted integral of lower dimensional moduli space volumes.

######  Theorem 9.

[Mir07a, Thrm. 7.1] The covolume formula. For a weighted
$\gamma=\sum_{j=1}^{m}a_{j}\gamma_{j}$ and the $\operatorname{MCG}$ sum of a
function $f$, small at infinity, then

$\int_{\mathcal{T}(R)/\operatorname{MCG}}f_{\gamma}\,dV=(|\mbox{Sym}(\gamma)|)^{-1}\int_{\mathbb{R}_{>0}^{m}}f(|\mathbf{x}|)V(R(\gamma);\mathbf{x})\,\mathbf{x}\cdot
d\mathbf{x}$

where $|\mathbf{x}|=\sum_{j}a_{j}x_{j}$ and $\mathbf{x}\cdot
d\mathbf{x}=x_{1}\cdots x_{m}dx_{1}\cdots dx_{m}$.

###### Proof.

Corresponding to the components $R^{\prime}$ of the cut open surface
$R(\gamma)$, consider the short exact sequence for mapping class groups,

$1\longrightarrow\prod_{j}\mbox{Dehn}(\gamma_{j})\longrightarrow\bigcap_{j}\operatorname{Stab}_{0}(\gamma_{j})\longrightarrow\prod_{R(\gamma)\
components}\operatorname{MCG}(R^{\prime})\longrightarrow 1,$

and the associated fibration of Teichmüller spaces from Fenchel-Nielsen
coordinates,

$\textstyle{\prod_{R(\gamma)\ components}\mathcal{T}(R^{\prime})\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{T}(R)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod_{\gamma_{j}}\mathbb{R}_{>0}\times\mathbb{R}\,.}$

(The short exact sequence places half Dehn twists in the mapping class groups
of the tori with single boundaries.) The $d\ell\wedge d\tau$ formula provides
that the fibration is a fibration of symplectic manifolds.

To establish the formula, first write for coset sums

$\sum\limits_{\operatorname{MCG}/\cap_{j}\operatorname{Stab}_{0}(\gamma_{j})}f\,=\,\sum\limits_{\operatorname{MCG}/\operatorname{Stab}(\gamma)}\
\sum\limits_{\operatorname{Stab}(\gamma)/\cap_{j}\operatorname{Stab}_{0}(\gamma_{j})}f\,=\,|\operatorname{Sym}(\gamma)|\,f_{\gamma},$

using that $f_{\gamma}$ is $\operatorname{Sym}(\gamma)$ invariant for the
second equality. Substitute the resulting formula for $f_{\gamma}$ into the
integral, and unfold the sum (express the
$\operatorname{MCG}/\cap_{j}\operatorname{Stab}_{0}(\gamma_{j})$ translation
sum as a sum of translates of a $\operatorname{MCG}$ fundamental domain) to
obtain the equality

$\int_{\mathcal{T}(R)/\operatorname{MCG}}f_{\gamma}\,dV=(|\operatorname{Sym}(\gamma)|)^{-1}\int_{\mathcal{T}(R)/\cap_{j}\operatorname{Stab}_{0}(\gamma_{j})}f\,dV.$

Substitute the fibration

$\textstyle{\prod_{R(\gamma)\
components}\mathcal{T}(R^{\prime})/\operatorname{MCG}(R^{\prime})\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{T}(R)/\bigcap_{j}\operatorname{Stab}_{0}(\gamma_{j})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod_{\gamma_{j}}(\mathbb{R}_{>0}\times\mathbb{R})/\mbox{Dehn}_{*}(\gamma_{j})\,,}$

where $\mbox{Dehn}_{*}(\gamma_{j})$ is generated by a half twist if the curve
bounds a torus with a single boundary and otherwise is generated by a simple
twist. Substitute the factorization of the volume element

$dV=\prod_{R(\gamma)\
components}dV(R^{\prime})\times\prod_{\gamma_{j}}d\ell_{j}\wedge d\tau_{j}.$

The function $f$ depends only on the values $\mathbf{x}$. For the values
$\mathbf{x}$ fixed, perform the
$\prod\mathcal{T}(R^{\prime})/\operatorname{MCG}(R^{\prime})$ integration to
obtain the product volume $V(R(\gamma);\mathbf{x})$. Finally
$\mbox{Dehn}_{*}(\gamma_{j})$ acts only on the variable $\tau_{j}$ with
fundamental domain $0<\tau_{j}<\ell_{j}/2$ if $\gamma_{j}$ bounds a torus with
a single boundary or otherwise with fundamental domain $0<\tau_{j}<\ell_{j}$.
For a torus with a single boundary, the action is accounted for by using the
volume value that is $1/2$ the original $V(L)$. The right hand side of the
formula is now established. ∎

We are now ready to apply the covolume formula to the length identity. The
application follows the genus one example. Again - it is essential to use the
$\operatorname{MCG}$ geodesic-length function action $\ell_{\alpha}\circ
h^{-1}=\ell_{h(\alpha)}$, to consider a sum over topological configurations as
a sum of $\operatorname{MCG}$ translates of a function. Consider the action on
configurations. The mapping class group naturally acts on $\mathcal{B}$, the
set of non trivial free homotopy classes of simple curves with endpoints on
the boundary, rel the boundary. Recall the correspondences: elements of
$\mathcal{B}$ $\Longleftrightarrow$ wire frames $\Longleftrightarrow$ boundary
pants $\mathcal{P}$. Viewing Figure 1, the $\operatorname{MCG}$ orbits on
$\mathcal{B}$ are of three types, describing location of the boundary pants
$\mathcal{P}$.

  * •

Orbits for simple curves from $\beta_{1}$ to $\beta_{1}$.

    * –

A single orbit for $R-\mathcal{P}$ connected, with
$\operatorname{Sym}(\beta_{1},\alpha_{1},\alpha_{2})=\mathbb{Z}/2\mathbb{Z}$.

    * –

A collection of orbits for $R-\mathcal{P}$ disconnected. The orbits are
classified by the joint partitions of genus $g=g_{1}+g_{2}$ and of labeled
boundaries $\\{\beta_{2},\dots,\beta_{n}\\}$, with each resulting component
with negative Euler characteristic. In general
$\operatorname{Sym}(\beta_{1},\alpha_{1},\alpha_{2})=1$, except in the special
case of one boundary and $g_{1}=g_{2}$.

  * •

A collection of orbits, one for each choice of a second boundary. In
particular, an orbit for simple curves from $\beta_{1}$ to $\beta_{j},\,j\neq
1$. A resulting surface $R-\mathcal{P}$ is connected, with
$\operatorname{Sym}(\beta_{1},\beta_{j},\alpha)=1$.

Consider Theorem 8, and integrate each side of the identity

$L_{1}\,=\,\sum_{\alpha_{1},\alpha_{2}}\,\mathcal{D}(L_{1},\ell_{\alpha_{1}}(R),\ell_{\alpha_{2}}(R))\,+\,\sum_{j=2}^{n}\sum_{\alpha}\,\mathcal{R}(L_{1},L_{j},\ell_{\alpha}(R)),$

over the moduli space of $R$ relative to the volume $dV$. Form the $L_{1}$
partial derivative of each side to simplify the quantities $\mathcal{D}$ and
$\mathcal{R}$. Apply formulas (3) for the right hand side. Express the right
hand side as individual sums for given orbit types. Apply Theorem 9 for each
orbit type to find integrals in terms of lower dimensional moduli volumes as
follows.

  * •

For the sum over simple curves from $\beta_{1}$ to $\beta_{1}$ with
$R-\mathcal{P}$ connected, the summand function is $\mathcal{D}$ with

$\frac{\partial\mathcal{D}}{\partial
L_{1}}\,=\,H(\ell_{\alpha_{1}}+\ell_{\alpha_{2}},L_{1}),$

a function of length of a multicurve, and the resulting integral is

$\int_{0}^{\infty}\int_{0}^{\infty}H(x+y,L_{1})V_{g-1}(x,y,\widehat{L})\,xy\,dxdy,$

for $\widehat{L}=(L_{2},\dots,L_{n})$.

  * •

For the sum over simple curves from $\beta_{1}$ to $\beta_{1}$ with
$R-\mathcal{P}$ disconnected, the summand function is $\mathcal{D}$ with

$\frac{\partial\mathcal{D}}{\partial
L_{1}}\,=\,H(\ell_{\alpha_{1}}+\ell_{\alpha_{2}},L_{1}),$

a function of length of a multicurve, and the resulting integral is

$\int_{0}^{\infty}\int_{0}^{\infty}\sum_{\stackrel{{\scriptstyle
g_{1}+g_{2}=g}}{{I_{1}\amalg
I_{2}=\\{2,\dots,n\\}}}}H(x+y,L_{1})V_{g_{1}}(x,L_{I_{1}})V_{g_{2}}(y,L_{I_{2}})\,xy\,dxdy.$

  * •

For the sum over simple curves from $\beta_{1}$ to $\beta_{j},\,j\neq 1$, the
summand function is $\mathcal{R}$ with

$\frac{\partial\mathcal{R}}{\partial
L_{1}}\,=\,\frac{1}{2}H(\ell_{\alpha},L_{1}+L_{j})\,+\,\frac{1}{2}H(\ell_{\alpha},L_{1}-L_{j}),$

a sum of functions of weighted length of a multicurve, and the resulting
integral is

$\int_{0}^{\infty}\frac{1}{2}\sum_{j=2}^{n}\big{(}H(x,L_{1}+L_{j})+H(x,L_{1}-L_{j})\big{)}V_{g}(x,L_{2},\dots,\widehat{L_{j}},\dots,L_{n})\,x\,dx,$

where $L_{j}$ is omitted from the argument list of $V_{g}$.

Compare to the end of Lecture 1 \- the volume recursion is established. The
volume function $V_{g,n}(L)$ is recursively determined.

What type of function is $V_{g,n}(L)$? The recursion involves two elementary
integrals, see formula (1) above for the definition of $H$,

$\int_{0}^{\infty}x^{2j+1}H(x,t)\,dx\qquad\mbox{and}\qquad\int_{0}^{\infty}\int_{0}^{\infty}x^{2j+1}y^{2k+1}H(x+y,t)\,dxdy.$

By direct calculation, see [Mir07a, formula (6.2) and Lemma 6.2], each
integral is a polynomial in $t^{2}$ with each coefficient a product of
factorials and the Riemann zeta function at a non negative even integer, each
coefficient is a positive rational multiple of an appropriate power of $\pi$.
The first polynomial has degree $j+1$ in $t^{2}$, while the second has degree
$i+j+2$. The first part of Theorem 4 is now established. The second part is
the subject of the next lecture.

Lecture 4: Symplectic reduction, principal $S^{1}$ bundles and the normal
form.

The goal of the lecture is to establish the following theorem. The formula
combines with Theorem 3, the WP kappa equation $\omega=2\pi^{2}\kappa_{1}$, to
provide that the coefficients of the volume polynomial $V_{g}(L)$ are
tautological intersection numbers. The result completes the proof of Theorem
4.

######  Theorem 10.

For
$d=\frac{1}{2}\dim_{\mathbb{R}}\mathcal{T}_{g}(L)=\dim_{\mathbb{C}}\mathcal{T}_{g,n}$,
then

$V_{g}(L)\ =\
\frac{1}{d!}\int_{\mathcal{T}_{g}(L)/\operatorname{MCG}}\omega_{\mathcal{T}_{g}(L)}^{d}\
=\
\frac{1}{d!}\int_{\mathcal{T}_{g,n}/\operatorname{MCG}}\big{(}\omega\,+\,\sum_{j=1}^{n}\frac{L_{j}^{2}}{2}\psi_{j}\big{)}^{d}.$

The proof is essentially by establishing a cohomology equivalence between
symplectic spaces - combining symplectic reduction, the Duistermaat-Heckman
theorem and explicit geometry to obtain the formula.

The considerations of the lecture are presented for the appropriate
Teichmüller spaces $\mathcal{T}$ and the open moduli spaces
$\mathcal{T}/\operatorname{MCG}$. The constructions are compatible with the
augmentation construction. The results are valid for the appropriate
Teichmüller spaces and compactified moduli spaces. The compactified moduli
spaces are orbifolds. The following results are for cohomology statements over
$\mathbb{Q}$; cohomology arguments over $\mathbb{Q}$ for manifolds are in
general also valid for orbifolds. Alternatively, the orbifold matter can be
bypassed by applying the general result that compactified moduli spaces have
manifold finite covers, [BP00, Loo94].

The Teichmüller spaces. We consider the trio.

  * •

$\mathcal{T}_{g}(L)$ \- the space of marked genus $g$ hyperbolic surfaces,
with geodesic boundaries $\beta_{1},\dots,\beta_{n}$ of prescribed lengths
$L_{1},\dots,,L_{n}$. A hyperbolic surface can be doubled across its geodesic
boundary to obtain a compact hyperbolic surface of higher genus. Accordingly,
$\mathcal{T}_{g}(L)$ can be considered as a locus in $\mathcal{T}_{2g+n-1}$.
The symplectic form $\omega$ of the image Teichmüller space restricts to the
locus and defines a symplectic form on the locus. A pants decomposition for a
surface with boundary, can be doubled to give a pants decomposition for a
doubled surface. The doubled pants decomposition is characterized by
containing the geodesics $\beta_{1},\dots,\beta_{n}$ and being symmetric.
Fenchel-Nielsen coordinates and the $d\ell\wedge d\tau$ formula are applied
for doubled decompositions to obtain a description of the locus
$\mathcal{T}_{g}(L)\subset\mathcal{T}_{2g+n-1}$, and to define a symplectic
from $\omega_{\mathcal{T}_{g}(L)}$. The symplectic form is given as
$\sum_{j}d\ell_{j}\wedge d\tau_{j}$ (without boundary parameters) for any
pants decomposition of a surface with boundaries. $\operatorname{MCG}$
invariance is immediate.

  * •

$\widehat{\mathcal{T}}_{g,n}$ \- the space of marked genus $g$ hyperbolic
surfaces, with pointed geodesic boundaries $\beta_{1},\dots,\beta_{n}$ \-
boundary lengths are allowed to vary and a variable point is given on each
boundary. The $\mathbb{R}$ dimension of $\widehat{\mathcal{T}}_{g,n}$ is $2n$
greater than the $\mathbb{R}$ dimension of $\mathcal{T}_{g}(L)$. Symplectic
reduction requires a symplectic form on $\widehat{\mathcal{T}}_{g,n}$, that is
equivalent to $\omega_{\mathcal{T}_{g}(L)}$ on $L$ level sets and is invariant
under rotating points on boundaries. A form is given by describing
$\widehat{\mathcal{T}}_{g,n}$ as a higher dimensional Teichmüller space.

Figure 8: A genus $2$ surface with three pointed boundaries.

To this purpose, introduce almost tight pants - pairs of pants with two
labeled boundaries being punctures (length zero) and a third boundary of
prescribed length. An almost tight pants will be glued to each surface
boundary $\beta_{j}$. The pants equatorial reflection defines symmetric points
on the pants boundary; the puncture labeling uniquely determines an equatorial
boundary point. A standard model for a surface with pointed geodesic
boundaries is given by gluing on almost tight pants - at each boundary glue on
an almost tight pants with matching boundary length and the equatorial point
aligned with the point on the boundary. The construction does not involve
choices, so is natural with respect to marking homeomorphisms and the
$\operatorname{MCG}$ action.

Figure 9: Aligning boundary and equatorial points to glue on almost tight
pants.

For a surface $R$ with labeled, pointed boundaries, write $\widehat{R}$ for
the standard model surface with glued on almost tight pants. The punctures of
$\widehat{R}$ are labeled in pairs. By the general hyperbolic collar result,
small length geodesics are necessarily disjoint [Bus92, Wlp10]. For the
lengths $L_{1},\dots,L_{n}$ suitably small, the labeled geodesics
$\beta_{1},\dots,\beta_{n}$ are uniquely determined on the surface
$\widehat{R}$ by having small length and bounding labeled punctures. The
pointed boundary, marked surface $R$ is equivalent to the marked surface
$\widehat{R}$ modulo Dehn twists about the $\beta_{j}$ (Dehn twists, since the
boundary points are given on a circle). In particular, for $c$ suitably small,
the equivalence is between the open subset $\\{L<c\\}$ of
$\widehat{\mathcal{T}}_{g,n}$ and the open subset $\\{L<c\\}$ in
$\mathcal{T}_{g,2n}/\prod_{j}\mbox{Dehn}(\beta_{j})$.

Definition and equivalence of $S^{1}$ principal bundles are next.
Considerations begin with the short exact sequence from Theorem 9,

$1\longrightarrow\mbox{Dehn}(\beta)=\prod_{j}\mbox{Dehn}(\beta_{j})\longrightarrow\mbox{Stab}(\beta)=\bigcap_{j}\operatorname{Stab}(\beta_{j})\longrightarrow\operatorname{MCG}(R)\longrightarrow
1,$

(now $\operatorname{Stab}(\beta_{j})=\operatorname{Stab}_{0}(\beta_{j})$,
since an orientation preserving pants homeomorphism preserves boundary
orientation). The geodesics $\beta_{1},\dots,\beta_{n}$ define subsets of the
Riemann surface bundles (the universal curves) over
$\widehat{\mathcal{T}}_{g,n}$ and over
$\mathcal{T}_{g,2n}/\mbox{Dehn}(\beta)$. The subsets define oriented circle
bundles, provided automorphisms of the Riemann surfaces act at most as
rotations on the individual geodesics. The small lengths $L_{1},\dots,L_{n}$
and labeled boundaries provide the condition. The geodesics define circle
bundles over Teichmüller bases. We see below that rotation along geodesics
defines an $S^{1}$ principal structure. Next, from the above short exact
sequence and the definition of marking - the equivalence between geodesics
$\beta\subset R$ and $\beta\subset\widehat{R}$, and the projections of circle
bundles to bases, commute with the actions of
$\operatorname{MCG}(R)\approx\operatorname{Stab}(\beta)/\mbox{Dehn}(\beta)$.
The geodesics define equivalent orbifold $S^{1}$ principal bundles over the
quotients
$\\{L<c\\}\subset\widehat{\mathcal{T}}_{g,n}/\operatorname{MCG}(R)\times(S^{1})^{n}$
(the $\operatorname{MCG}(R)$ and $(S^{1})^{n}$ actions on
$\widehat{\mathcal{T}}_{g,n}$ commute) and
$\\{L<c\\}\subset\mathcal{T}_{g,2n}/\operatorname{Stab}(\beta)$.

The symplectic form of $\mathcal{T}_{g,2n}$ defines a symplectic form
$\omega_{\widehat{\mathcal{T}}_{g,n}}$ on the open subset $\\{L<c\\}$.
Fenchel-Nielsen coordinates and the $d\ell\wedge d\tau$ formula are applied.
The form $\omega_{\widehat{\mathcal{T}}_{g,n}}$ is given as
$\sum_{k}d\ell_{k}\wedge d\tau_{k}$ for any pants decomposition of
$\widehat{R}$ containing the multicurve $\beta$. Importantly, the form
$\omega_{\widehat{\mathcal{T}}_{g,n}}$ is given for surfaces $R$ with pointed
boundaries $\beta_{j}$, by an extended interpretation of the $d\ell\wedge
d\tau$ formula, with a term for each boundary, now with the interpretation
that $\tau(\beta_{j})$ parameterizes the location of the specified point. See
Figure 3, the parameter $\tau(\beta_{j})$ increasing corresponds to the point
moving on the boundary with the surface interior on the right.
$\operatorname{MCG}(R)$ invariance of the symplectic form is immediate.
Restriction of the form to $L$ level sets and invariance under rotating
boundary points are discussed below.

  * •

$\mathcal{T}_{g,n}$ \- the space of marked genus $g$ hyperbolic surfaces with
$n$ punctures. $\mathcal{T}_{g,n}$ has the $\operatorname{MCG}$ invariant
symplectic form $\omega$. $\mathcal{T}_{g,n}$ coincides with the Teichmüller
space $\mathcal{T}_{g}(0)$, where surface boundary lengths are zero.

We will relate the three symplectic manifolds.

Symplectic reduction for $\widehat{\mathcal{T}}_{g,n}$. We consider the
Hamiltonian geometry of FN twists, geodesic-lengths and especially the moment
map

$\widehat{\mathcal{T}}_{g,n}\ \stackrel{{\scriptstyle\mu}}{{\longrightarrow}}\
\widehat{L}=(L^{2}_{1}/2,\dots,L^{2}_{n}/2)\in\mathbb{R}_{\geq 0}^{n}.$

Write $t_{j}$ for the unit speed infinitesimal rotation of the point on the
boundary $\beta_{j}$; $t_{j}$ is a vector field on
$\widehat{\mathcal{T}}_{g,n}$. In terms of the standard model surfaces,
$t_{j}$ is the FN infinitesimal twist vector field for $\beta_{j}$, and
$t_{j}$ is the infinitesimal rotation of the $j^{th}$ almost tight pants. By
twist-length duality, we have $\omega_{\widehat{\mathcal{T}}_{g,n}}(-t_{j},\
)=dL_{j}$ and the scaled $-L_{j}t_{j}$ is unit infinitesimal rotation (unit
time flow is a full rotation). The function $\frac{1}{2}L_{j}^{2}$ is the
corresponding Hamiltonian potential, since
$\omega_{\widehat{\mathcal{T}}_{g,n}}(-L_{j}t_{j},\ )=d(\frac{1}{2}L_{j}^{2})$
(the momentum $\frac{1}{2}L^{2}$ determines the twist sign/orientation). The
vector fields $-L_{j}t_{j}$ are the infinitesimal generators for the
$(S^{1})^{n}$ action on $\widehat{\mathcal{T}}_{g,n}$, given by rotating the
boundary points. The symplectic form $\omega_{\widehat{\mathcal{T}}_{g,n}}$ is
twist invariant and we are ready for symplectic reduction, ready to consider
the quotient $\widehat{\mathcal{T}}_{g,n}/(S^{1})^{n}$.

Figure 10: Positive rotations for an $(S^{1})^{3}$ action.

A level set of the moment map
$\mu:\widehat{\mathcal{T}}_{g,n}\longrightarrow\mathbb{R}^{n}$ is a locus of
prescribed $\beta$ length hyperbolic surfaces. The group $(S^{1})^{n}$ acts on
level sets by rotating almost tight pants. The quotient of a level set by the
group is naturally $\mathcal{T}_{g}(L)$ \- the level set prescribes the
boundary lengths and the group action removes the location information for the
points.

######  Proposition 11.

Symplectic reduction. For
$\widehat{\mathcal{T}}_{g,n}/(S^{1})^{n}\approx\mathcal{T}_{g}(L)$, then

$\omega_{\widehat{\mathcal{T}}_{g,n}}\big{|}_{\mu^{-1}(\widehat{L})}/(S^{1})^{n}\
\approx\ \omega_{\mathcal{T}_{g}(L)}.$

###### Proof.

The form $\omega_{\widehat{\mathcal{T}}_{g,n}}$ is given by the $d\ell\wedge
d\tau$ formula for any pants decomposition of a standard model surface
containing the multicurve $\beta$. The differentials $dL_{j}$ vanish on $\mu$
level sets and the formula reduces to the sum for a pants decomposition of a
surface with boundary, a sum without boundary parameters - the
$\omega_{\mathcal{T}_{g}(L)}$ formula. ∎

$S^{1}$ principal bundles. We review basics about characteristic classes.

######  Definition 12.

Let $\pi:P\longrightarrow M$ be a smooth circle bundle over a smooth compact
manifold $M$. The bundle is $S^{1}$ principal provided,

  1. 1.

$S^{1}$ acts freely on $P$,

  2. 2.

$\pi(p_{1})=\pi(p_{2})$ if and only if there exists $s\in S^{1}$, such that
$p_{1}\cdot s=p_{2}$.

A connection for an $S^{1}$ principal bundle is a smooth distribution
$\mathcal{H}\subset\mathbf{T}P$ of tangent subspaces such that,

  1. 1.

$\mathbf{T}_{p}P=\mathcal{H}_{p}\oplus\ker\pi_{*}\big{|}_{p}$, for each $p\in
P$,

  2. 2.

$s^{*}\mathcal{H}_{p}=\mathcal{H}_{p\cdot s}$.

A connection is uniquely given as $\mathcal{H}=\ker A$, for a $1$-form $A$ on
$P$, provided $A$ is $S^{1}$ invariant and $A(\dot{s})=1$. An $S^{1}$
invariant inner product $\langle\ ,\ \rangle$ provides an example of an
invariant $1$-form by $A(v)=\langle
v,\dot{s}\rangle/\langle\dot{s},\dot{s}\rangle$. The curvature $2$-form on $P$
for a connection is $\Phi(v,w)=dA(\operatorname{hor}v,\operatorname{hor}w)$,
for $\operatorname{hor}$ the horizontal projection of $\mathbf{T}P$ to
$\mathcal{H}$.

######  Theorem 13.

[MS74]. There exists a unique closed $2$-form $\Omega$ on $M$, such that
$\Phi=\pi^{*}\Omega$. The cohomology class of $\Omega$ is independent of the
choice of $S^{1}$ principal connection for $P$ and the first Chern class is
$c_{1}(P)\,=\,[\Omega]\in H^{2}(M,\mathbb{Z})$.

As above, the variable point on the boundary $\beta_{j}$ of the surface $R$
defines an $S^{1}$ principal bundle $\widehat{\beta}_{j}$ over
$\mathcal{T}_{g}(L)$; $S^{1}$ acts by moving the point with the surface
interior on the left. A choice of connection for the bundle gives a first
Chern class $c_{1}(\widehat{\beta}_{j})$.

Applying the Duistermaat-Heckman theorem. We extend the definition of
$\widehat{\mathcal{T}}_{g,n}$ to include $L=0$; geodesic boundaries of $R$ can
be replaced with punctures. Hyperbolic structures converge for boundary
lengths tending to zero; in particular collar regions converge to cusp
regions. The extension of $\widehat{\mathcal{T}}_{g,n}$ is given by
parameterizing boundary points by points on a collar/cusp region boundary.

We recall basics about collars and cusps. For a geodesic $\alpha$ of length
$\ell_{\alpha}$, the standard collar in the upper half plane $\mathbb{H}$ is
$\mathcal{C}(\ell_{\alpha})=\\{d(z,i\mathbb{R}_{+})\leq w(\alpha)\\}$, for the
half width $w(\alpha)$ given by $\sinh w(\alpha)\sinh\ell_{\alpha}/2=1$. The
quotient cylinder $\\{d(z,i\mathbb{R}_{+})\leq w(\alpha)\\}/\langle z\mapsto
e^{\ell_{\alpha}}z\rangle$ embeds into $R$ to give a collar neighborhood of
the geodesic. For a cusp, the standard cusp in $\mathbb{H}$ is
$\mathcal{C}_{\infty}=\\{\Im z\geq 1/2\\}$. The quotient cylinder $\\{\Im
z\geq 1/2\\}/\langle z\mapsto z+1\rangle$ embeds into $R$ to give a cusp
region. The boundary of a collar, for $\ell_{\alpha}$ bounded, and boundary of
a cusp region have length approximately 2. Collars and cusp regions are
foliated by geodesics normal to the boundary. For geodesic-lengths tending to
zero, half collar neighborhoods Gromov-Hausdorff converge to a cusp region
(convergence is uniform on bounded distance neighborhoods of the boundary);
boundaries and geodesics normal to the boundary converge.

Figure 11: Projecting along geodesics to a collar and a cusp region boundary.

The geodesics normal to the boundary of a collar provide a projection from the
core geodesic to each collar boundary. The projection is used to note that
prescribing a point on a geodesic boundary of $R$ is equivalent to prescribing
a point on the boundary of the half collar neighborhood of the geodesic. Since
collars and their boundaries converge to a cusp region and its boundary, for
core geodesic length tending to zero, we have a description for the extension
of the definition of $\widehat{\mathcal{T}}_{g,n}$ to include surfaces with
collections of lengths $L_{j}$ zero.

The standard cusp region is uniformized by the variable $w=e^{2\pi iz}$. A
point on the cusp region boundary $\Im z=1/2$ corresponds to a point on
$|w|=e^{-\pi}$ and given the factor $e^{-\pi}$, a point on the circle
corresponds to a tangent vector at the origin. The variable $w$ is unique
modulo multiplication by a unimodular number; the identification of the circle
with tangent vectors at the origin is canonical. For an $S^{1}$ infinitesimal
generator $-L_{j}t_{j}$, displacement is to the left when crossing the
geodesic, (compare to the Figure 3 positive twist, right displacement) the
reference point moves with the surface interior on its left, the tangent
vector at the origin rotates clockwise, and a dual cotangent vector rotates
counter clockwise (the positive direction for a $\mathbb{C}$-line). Combining
equivalences, the $S^{1}$ principal bundle of a point on a cusp region
boundary is equivalent to a non zero vector in the cotangent line for the
puncture.

Proof of Theorem 10. The $\operatorname{MCG}(R)$ and $(S^{1})^{n}$ actions on
$\widehat{\mathcal{T}}_{g,n}$ commute; consider the quotient
$\widehat{\mathcal{T}}_{g,n}/\operatorname{MCG}(R)\times(S^{1})^{n}$. By the
Duistermaat-Heckman theorem, [CdS01, Chapter 30, Theorem 30.8], for small
values of $\widehat{L}$, including $0$, the reduced level sets
$\mu^{-1}(\widehat{L})/(S^{1})^{n}$ are mutually diffeomorphic. Furthermore by
Duistermaat-Heckman, the $L$ level set reduced symplectic form
$\omega_{\widehat{\mathcal{T}}_{g,n}}\big{|}_{\mu^{-1}(\widehat{L})}/(S^{1})^{n}$
is cohomologous to the sum of the $0$ level set reduced form
$\omega_{\widehat{\mathcal{T}}_{g,n}}\big{|}_{\mu^{-1}(0)}/(S^{1})^{n}$ and
the contributions $(L_{j}^{2}/2)\,c_{1}(\widehat{\beta}_{j})$, for
$c_{1}(\widehat{\beta}_{j})$ the first Chern class for the $S^{1}$ principal
bundle of a point varying on the $j^{th}$ cusp region boundary. Combining with
Proposition 11, gives the desired cohomology equivalence,

$\omega_{\mathcal{T}_{g}(L)}\ \equiv\ \omega_{\mathcal{T}_{g}(0)}\ +\
\sum_{j=1}^{n}\frac{L_{j}^{2}}{2}c_{1}(\widehat{\beta}_{j}).$

By the description of collars and cotangent lines at punctures, the circle
bundle $\widehat{\beta}_{j}$ is topologically equivalent to the psi line
bundle $\psi_{j}$ (see Lecture 1) with equality of first Chern classes. The
proof is finished.

Lecture 5: The pattern of intersection numbers and Witten-Kontsevich.

We begin with the discussion of Harris-Morrison [HM98, pgs. 71-75]. For a
finite sequence of non negative integers $\\{\alpha_{j}\\}$, define the top
$\psi$-intersection number by

$\langle\tau_{\alpha_{1}}\tau_{\alpha_{2}}\cdots\tau_{\alpha_{n}}\rangle_{g}\,=\,\int_{\overline{\mathcal{M}}_{g,n}}\psi_{1}^{\alpha_{1}}\psi_{2}^{\alpha_{2}}\cdots\psi_{n}^{\alpha_{n}}.$

For a non trivial pairing, the genus $g$, number of punctures $n$, and
exponents $\alpha_{j}$ are related by $3g-3+n=\sum_{j=1}^{n}\alpha_{j}$,
otherwise the pairing is defined as zero. More generally using exponents to
denote powers (repetitions) of the variables $\tau$, define

$\langle\tau_{0}^{d_{0}}\tau_{1}^{d_{1}}\cdots\tau_{m}^{d_{m}}\rangle_{g}\,=\,\int_{\overline{\mathcal{M}}_{g,d}}\prod_{j=0}^{m}\,\prod_{k=1}^{d_{j}}\,\psi_{(j,k)}^{\,j}.$

($\tau_{j}^{d_{j}}$ denotes that for $d_{j}$ punctures, the associated $\psi$
is raised to the $j^{th}$ power; the subscripts $(j,k)$ are distinct puncture
labels.) For the second pairing, the formal count of punctures is
$d=\sum_{j=0}^{m}d_{j}$, and the formal degree of the product is the count of
$\psi$ factors $\sum_{j=0}^{m}jd_{j}$. For a non trivial pairing, the genus,
number of punctures and degree are related by $3g-3+d=\sum_{j=0}^{m}jd_{j}$,
otherwise the pairing is zero. The psi classes are known to be positive -
integrals of products over subvarieties are positive; the non trivial pairings
are positive, matching Mirzakhani’s positivity of volume polynomial
coefficients, see Theorem 4.

Witten considered a partition function (probability of states), for two-
dimensional gravity. For an infinite vector
$\mathbf{t}=(t_{0},t_{1},\dots,t_{n},\dots)$, and $\gamma$ the formal sum
$\gamma=\sum_{j=0}^{\infty}t_{j}\tau_{j}$, Witten introduced a genus $g$
generating function for $\tau$ products,

$F_{g}(\mathbf{t})=\sum_{n=0}^{\infty}\frac{\langle\gamma^{n}\rangle_{g}}{n!},$

in which the numerator is defined by monomial expansion, resulting in the
formal power series

$F_{g}(\mathbf{t})=\sum_{\\{d_{j}\\}}\
\langle\prod_{j=0}^{\infty}\tau_{j}^{d_{j}}\rangle_{g}\,\prod_{j=0}^{\infty}\frac{t_{j}^{d_{j}}}{d_{j}!},$

where the sum is over all sequences of non negative integers $\\{d_{j}\\}$
with only finitely many non zero terms. –By Theorem 4, the intersection
numbers
$\langle\tau_{0}^{d_{0}}\tau_{1}^{d_{1}}\cdots\tau_{m}^{d_{m}}\rangle_{g}$ are
the coefficients of the leading terms of the volume polynomials $V_{g,n}(L)$.–

The quantum gravity partition function is

$\mathbf{F}(\lambda,\mathbf{t})\,=\,\sum_{g=0}^{\infty}\lambda^{2g-2}F_{g}(\mathbf{t}).$

Based on a realization of the function in terms of matrix integrals, Witten
conjectured that the partition function should satisfy two forms of the
Korteweg-deVries (KdV) equations. Kontsevich gave a proof of the conjecture
using a cell decomposition of the moduli spaces $\mathcal{M}_{g,n}$, [Kon92].
Cells are enumerated by ribbon graphs/fat graphs. Kontsevich encoded the
intersection numbers in an enumeration of trivalent ribbon graphs. He then
used Feynman diagram techniques and a matrix Airy integral to establish
Witten’s conjectures.

Two basic relations for the intersection numbers are: for $n>0$ and
$\sum_{i}\alpha_{i}=3g-2+n>0$, the

$\mbox{string
equation}\qquad\langle\tau_{0}\tau_{\alpha_{1}}\cdots\tau_{\alpha_{n}}\rangle_{g}\,=\,\sum_{\alpha_{i}\neq
0}\langle\tau_{\alpha_{1}}\cdots\tau_{\alpha_{i}-1}\cdots\tau_{\alpha_{n}}\rangle_{g},$

and for $n\geq 0$ and $\sum_{i}\alpha_{i}=3g-3+n\geq 0$, the

$\mbox{dilaton
equation}\qquad\quad\langle\tau_{1}\tau_{\alpha_{1}}\cdots\tau_{\alpha_{n}}\rangle_{g}\,=\,(2g-2+n)\langle\tau_{\alpha_{1}}\cdots\tau_{\alpha_{n}}\rangle_{g}.$

The first equation is for adding a new puncture without an associated factor
of $\psi$ in the product, while the second equation is for adding a new
puncture with a single associated factor of $\psi$.

Similar to setting $V_{0,3}(L)=1$, the intersection symbol for the thrice
punctured sphere is normalized to $\langle\tau_{0}^{3}\rangle_{0}=1$. The
general genus $0$ formula is

$\langle\tau_{\alpha_{1}}\cdots\tau_{\alpha_{n}}\rangle_{0}\,=\,\bigg{(}\frac{n-3}{\alpha_{1}!\cdots\alpha_{n}!}\bigg{)},$

with the right hand side a multinomial coefficient for $n-3$. The genus $0$
string equation is simply Pascal’s multinomial neighbor relation.

For genus $1$, Theorem 10 and the WP kappa equation, Theorem 3, give
$V(L)=\int_{\mathcal{M}_{1,1}}2\pi^{2}\kappa_{1}+\frac{L^{2}}{2}\psi$. The
formula combines with the Lecture 3 calculation
$V_{1,1}(L)=\frac{\pi^{2}}{12}+\frac{L^{2}}{48}$ (now including the elliptic
involution $\frac{1}{2}$ factor) to provide the evaluations,

$\frac{1}{2}\int_{\mathcal{M}_{1,1}}\kappa_{1}\,=\,\langle\tau_{1}\rangle_{1}\,=\,\frac{1}{24}.$

General genus $1$ evaluations are found from the single evaluation by applying
the string and dilaton equations. A consequence of the Witten conjecture is
that all $\langle\tau\rangle$ intersections can be calculated from the initial
values $\langle\tau_{0}^{3}\rangle_{0}=1$ and
$\langle\tau_{1}\rangle_{1}=\frac{1}{24}$, using the Virasoro equations
$L_{n}(e^{\mathbf{F}})=0$ for the partition function described below.

In the volume recursion, leading coefficients are obtained from leading
coefficients - the recursion specializes to leading coefficients, see [Mir07b,
Lemma 5.3]. We now sketch the application of the specialized recursion to
relations for the partition function and a solution of Witten’s conjecture.

Relations come from the Virasoro Lie algebra. The Witt subalgebra is generated
by the differential operators $\mathcal{L}_{n}=-z^{n+1}\partial/\partial
z,\,n\geq-1$, with commutators
$[\mathcal{L}_{n},\mathcal{L}_{m}]=(n-m)\mathcal{L}_{n+m}$. The string and
dilaton equations can be written as linear homogeneous differential equations
for the exponential $e^{\mathbf{F}}$ of the partition function. The
differential operator for the string equation is

$L_{-1}\,=\,-\frac{\partial}{\partial
t_{0}}\,+\,\frac{\lambda^{-2}}{2}\,t_{0}^{2}\,+\,\sum_{j=0}^{\infty}t_{j+1}\frac{\partial}{\partial
t_{j}},$

and the differential operator for the dilaton equation is

$L_{0}\,=\,-\frac{3}{2}\frac{\partial\quad}{\partial
t_{1}}\,+\,\sum_{j=0}^{\infty}\frac{2j+1}{2}t_{j}\frac{\partial}{\partial
t_{j}}\,+\,\frac{1}{16}.$

With simple conditions, there is a unique way to extend operator definitions
to obtain a representation of the $\\{\mathcal{L}_{n}\\}$ subalgebra. The
general operator is

$L_{n}\,=\,-\frac{(2n+3)!!}{2^{n+1}}\frac{\partial\quad}{\partial
t_{n+1}}\,+\,\sum_{j=0}^{\infty}\frac{(2j+2n+1)!!}{(2j-1)!!\,2^{n+1}}\,t_{j}\,\frac{\partial\quad}{\partial
t_{j+n}}\\\
+\,\frac{\lambda^{2}}{2}\,\sum_{j=0}^{n-1}\frac{(2j+1)!!(2n-2j-1)!!}{2^{n+1}}\,\frac{\partial^{2}\qquad\quad}{\partial
t_{j}\partial t_{n-j-1}},$

with commutator $[L_{n},L_{m}]\,=\,(n-m)L_{n+m}$.

######  Theorem 14.

[Mir07b, Theorem 6.1]. The Witten-Kontsevich conjecture: Virasoro constraints.
For $n\geq-1$, then $L_{n}(e^{\mathbf{F}})\,=\,0.$

###### Proof.

For an exponents multi index $\mathbf{k}=(k_{1},\dots,k_{n})$, the volume
recursion formula becomes the coefficient relation

$(2k_{1}+1)V_{g,n}(L)[\mathbf{k}]\,=\,\mathcal{A}_{g,n}^{con}(L)[\mathbf{k}]\,+\,\mathcal{A}_{g,n}^{dcon}(L)[\mathbf{k}]\,+\,\mathcal{B}_{g,n}(L)[\mathbf{k}].$

The leading coefficient relation takes the following explicit form (following
the labeling of boundaries, the punctures are labeled $1,\dots,n$)

$(2k_{1}+1)!!\langle\tau_{k_{1}}\cdots\tau_{k_{n}}\rangle\\\
=\frac{1}{2}\,\sum_{i+j=k_{1}-2}(2i+1)!!(2j+1)!!\sum_{I\subset\\{2,\dots,n\\}}\langle\tau_{i}\tau_{\mathbf{k}_{I}}\rangle\,\langle\tau_{j}\tau_{\mathbf{k}_{I^{c}}}\rangle\\\
+\,\frac{1}{2}\sum_{i+j=k_{1}-2}(2i+1)!!(2j+1)!!\langle\tau_{i}\tau_{j}\tau_{k_{2}}\cdots\tau_{k_{n}}\rangle\\\
+\,\sum_{j=2}^{n}\frac{(2k_{1}+2k_{j}-1)!!}{(2k_{j}-1)!!}\langle\tau_{k_{2}}\cdots\tau_{k_{1}+k_{j}-1}\cdots\tau_{k_{n}}\rangle.$

Rearranging the explicit relation provides that
$L_{k_{1}-1}(e^{\mathbf{F}})=0$. ∎

Mulase and Safnuk consider a generating function for the intersections of
combinations of the $\kappa_{1}$ and $\psi$ classes [MS08]

$\mathbf{G}(s,t_{0},t_{1},\dots)\,=\,\sum_{g}\,\langle e^{s\kappa_{1}+\sum
t_{j}\tau_{j}}\rangle_{g}\,=\,\sum_{g}\sum_{m,\\{d_{j}\\}}\langle\kappa_{1}^{m}\tau_{0}^{d_{0}}\tau_{1}^{d_{1}}\cdots\rangle_{g}\frac{s^{m}}{m!}\prod_{j=0}^{\infty}\frac{t_{j}^{d_{j}}}{d_{j}!},$

where again products, other than $3g-3+n$-products, are defined as zero.
Mulase and Safnuk use the volume recursion and rearrangement of terms to prove
the following.

######  Theorem 15.

[MS08, Thrm. 1.1] Virasoro constraints. For each $k\geq-1$, define

$\mathcal{V}_{k}=-\frac{1}{2}\sum_{i=0}^{\infty}(2(i+k)+3)!!\frac{(-2s)^{i}}{(2i+1)!}\frac{\partial\qquad}{\partial
t_{i+k+1}}+\frac{1}{2}\sum_{j=0}^{\infty}\frac{(2(j+k)+1)!!}{(2j-1)!!}t_{j}\frac{\partial\
}{\partial t_{j+k}}\\\ +\frac{1}{4}\sum_{\stackrel{{\scriptstyle
d_{1}+d_{2}=k-1}}{{d_{1},d_{2}\geq
0}}}(2d_{1}+1)!!(2d_{2}+1)!!\frac{\partial^{2}\quad}{\partial
t_{d_{1}}\partial
t_{d_{2}}}+\frac{\delta_{k,-1}t_{0}^{2}}{4}+\frac{\delta_{k,0}}{48},$

for the double factorial and Kronecker delta function $\delta_{*,*}$. Then

  * •

the operators $\mathcal{V}_{k}$ satisfy the Virasoro commutator relations
$[\mathcal{V}_{n},\mathcal{V}_{m}]=(n-m)\mathcal{V}_{n+m}$;

  * •

the generating function $\mathbf{G}$ satisfies
$\mathcal{V}_{k}(e^{\mathbf{G}})=0$ for $k\geq-1$.

The initial conditions and second system of equations uniquely determine the
generating function.

In a direct display that the intersection numbers for $\kappa_{1}$ and $\psi$
classes are equivalent intersection numbers for $\psi$ classes, Mulase and
Safnuk show that

$\mathbf{G}(s,t_{0},t_{1},t_{2},t_{3}\dots)\,=\,\mathbf{F}(t_{0},t_{1},t_{2}+\gamma_{2},t_{3}+\gamma_{3},\dots),$

where $\gamma_{j}=-(-s)^{j-1}/(2j+1)j!$ [MS08, Thrm. 1.2]. An explicit proof
of the relation also comes from a formula of Faber, expressing kappa classes
in terms of psi classes on moduli spaces for a greater number of punctures.

In his thesis [Do08], Norman Do presents a

$\mbox{generalized string equation}\qquad V_{g,n+1}(L,2\pi
i)\,=\,\sum_{k=1}^{n}\int L_{k}V_{g,n}(L)\,dL_{k},$

and

$\mbox{generalized dilaton equation}\qquad\frac{\partial V_{g,n+1}}{\partial
L_{n+1}}(L,2\pi i)\,=\,2\pi i(2g-2+n)V_{g,n}(L),$

where on the left hand side, the value $2\pi i$ is substituted for the
$(n+1)^{st}$ boundary length and $L=(L_{1},\dots,L_{n})$. By Theorem 4, the
second equation, for appropriate non negative multi indices
$\alpha=(\alpha_{1},\alpha_{2},\dots,\alpha_{n})$ and integers $m$, is
equivalent to the relations

$\int_{\overline{\mathcal{M}}_{g,n+1}}\psi_{1}^{\alpha_{1}}\psi_{2}^{\alpha_{2}}\cdots\psi_{n}^{\alpha_{n}}\psi_{n+1}(\kappa_{1}-\psi_{n+1})^{m}=(2g-2+n)\int_{\overline{\mathcal{M}}_{g,n}}\psi_{1}^{\alpha_{1}}\psi_{2}^{\alpha_{2}}\cdots\psi_{n}^{\alpha_{n}}\kappa_{1}^{m}.$

A proof of the generalized equations is based on the pullback relations for
psi and kappa classes, and general considerations for images of classes. In
particular for
$\pi:\overline{\mathcal{M}}_{g,n+1}\longrightarrow\overline{\mathcal{M}}_{g,n}$,
the morphism of forgetting the last puncture, then the classes
$\widetilde{\kappa}_{m},\ \widetilde{\psi}_{k}$ on
$\overline{\mathcal{M}}_{g,n+1}$ and $\kappa_{m},\ \psi_{k}$ on
$\overline{\mathcal{M}}_{g,n}$, satisfy
$\widetilde{\kappa}_{m}=\pi^{*}\kappa_{m}+\psi_{n+1}^{m}$ and
$\widetilde{\psi}_{k}=\pi^{*}\psi_{k},+D_{k}$ (for $D_{k}$ the divisor of the
$k^{th}$ puncture on the universal curve over $\overline{\mathcal{M}}_{g,n}$),
[HM98].

A second proof of the equations is based on exact formulas for the operations
in the volume recursion. For the generalized dilaton equation, consider the
following four operators acting on the ring
$\mathbb{C}[x^{2},y^{2},L_{1}^{2},\dots,L_{n}^{2},\dots]$,

$2\frac{\partial\ }{\partial
L_{1}}L_{1}[\cdot],\qquad\frac{\partial\quad}{\partial
L_{n+1}}[\cdot]\bigg{|}_{L_{n+1}=2\pi
i},\qquad\int_{0}^{\infty}\int_{0}^{\infty}xyH(x+y,L_{1})[\cdot]dxdy,\\\
\mbox{and}\quad\int_{0}^{\infty}x\big{(}H(x,L_{1}+L_{k})+H(x,L_{1}-L_{k})\big{)}[\cdot]dx.\qquad$

Formulas for the operators are developed. For the proof, the generalized
dilaton equation is written using the second operator, the volume recursion is
applied for the left hand side, operator formulas are applied, and terms are
gathered to give the right hand side. The generalized dilaton equation gives
WP volumes for the compact case, including the following examples

$\displaystyle V_{2,0}=\frac{43\pi^{6}}{2160},$ $\displaystyle V_{3,0}$
$\displaystyle=\frac{176557\pi^{12}}{1209600},$ $\displaystyle
V_{4,0}=\frac{1959225867017\pi^{18}}{493807104000}\qquad\mbox{and}\ $
$\displaystyle V_{5,0}$
$\displaystyle=\frac{84374265930915479\pi^{24}}{355541114880000}.$

We close the lecture by noting that there is extensive research on Witten’s
conjecture for the moduli space. The Kontsevich and Mirzakhani approaches are
analytic in nature. Okounkov and Pandharipande [OP09] transformed the question
to counting Hurwitz numbers, topological types of branched covers of the
sphere, and used a combinatorial approach to count factorizations of
permutations into transpositions. Their combinatorial approach gives
Kontsevich’s formula. There is an even greater body of research on
intersection numbers and relations on the moduli space [HM98].

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