# Deligne-Mumford space

in progress!

Here we describe the domain moduli spaces needed for the construction of the composition operations $\mu ^{d}$ described in Polyfold Constructions for Fukaya Categories#Composition Operations; we refer to these as Deligne-Mumford spaces. In some constructions of the Fukaya category, e.g. in Seidel's book, the relevant Deligne-Mumford spaces are moduli spaces of trees of disks with boundary marked points (with one point distinguished). We tailor these spaces to our needs:

• In order to ultimately produce a version of the Fukaya category whose morphism chain complexes are finitely-generated, we extend this moduli spaces by labeling certain nodes in our nodal disks by numbers in $[0,1]$. This anticipates our construction of the spaces of maps, where, after a boundary node forms with the property that the two relevant Lagrangians are equal, the node is allowed to expand into a Morse trajectory on the Lagrangian.
• Moreover, we allow there to be unordered interior marked points in addition to the ordered boundary marked points.

These interior marked points are necessary for the stabilization map, described below, which serves as a bridge between the spaces of disk trees and the Deligne-Mumford spaces. We largely follow Chapters 5-8 of [LW [[1]]], with two notable departures:

• In the Deligne-Mumford spaces constructed in [LW], interior marked points are not allowed to collide, which is geometrically reasonable because Li-Wehrheim work with aspherical symplectic manifolds. Therefore Li-Wehrheim's Deligne-Mumford spaces are noncompact and are manifolds with corners. In this project we are dropping the "aspherical" hypothesis, so we must allow the interior marked points in our Deligne-Mumford spaces to collide -- yielding compact orbifolds.
• Li-Wehrheim work with a single Lagrangian, hence all interior edges are labeled by a number in $[0,1]$. We are constructing the entire Fukaya category, so we need to keep track of which interior edges correspond to boundary nodes and which correspond to Floer breaking.

## Some notation and examples

For $k\geq 1$, $\ell \geq 0$ with $k+2\ell \geq 2$ and for a decomposition $\{0,1,\ldots ,k\}=A_{1}\sqcup \cdots \sqcup A_{m}$, we will denote by $DM(k,\ell ;(A_{j})_{{1\leq j\leq m}})$ the Deligne-Mumford space of nodal disks with $k$ boundary marked points and $\ell$ unordered interior marked points and with numbers in $[0,1]$ labeling certain nodes. (This anticipates the space of nodal disk maps where, for $i\in A_{j},i'\in A_{{j'}}$, $L_{i}$ and $L_{{i'}}$ are the same Lagrangian if and only if $j=j'$.) It will also be useful to consider the analogous spaces $DM(k,\ell ;(A_{j}))^{{{\text{ord}}}}$ where the interior marked points are ordered. As we will see, $DM(k,\ell ;(A_{j}))^{{{\text{ord}}}}$ is a manifold with boundary and corners, while $DM(k,\ell ;(A_{j}))$ is an orbifold; moreover, we have surjections

$DM(k,\ell ;(A_{j}))^{{{\text{ord}}}}\quad {\stackrel {\simeq }{\longrightarrow }}\quad DM(k,\ell ;(A_{j})),$

which are defined by forgetting the ordering of the interior marked points (there is an $S_{\ell }$-action on the domain, and this map takes orbits).

Before precisely constructing and analyzing these spaces, let's sneak a peek at some examples.

Here is $DM(3,0;(\{0,2,3\},\{1\}))$.

TO DO

Here is $DM(4,0;(\{0,1,2,3,4\}))$, with the associahedron $DM(4,0;(\{0\},\{1\},\{2\},\{3\},\{4\}))$ appearing as the smaller pentagon. Note that the hollow boundary marked point is the "distinguished" one.

need to edit to insert incoming/outgoing Morse trajectories

And here are $DM(0,2;(\{0\},\{1\},\{2\}))^{{{\text{ord}}}}\to DM(0,2;(\{0\},\{1\},\{2\}))$ (lower-left) and $DM(1,2;(\{0\},\{1\},\{2\}))^{{{\text{ord}}}}\to DM(1,2;(\{0\},\{1\},\{2\}))$. $DM(0,2;(\{0\},\{1\},\{2\}))$ and $DM(1,2;(\{0\},\{1\},\{2\}))$ have order-2 orbifold points, drawn in red and corresponding to configurations where the two interior marked points have bubbled off onto a sphere. The red bits in $DM(0,2;(\{0\},\{1\},\{2\}))^{{{\text{ord}}}},DM(1,2;(\{0\},\{1\},\{2\}))^{{{\text{ord}}}}$ are there only to show the points with nontrivial isotropy with respect to the ${\mathbf {Z}}/2$-action which interchanges the labels of the interior marked points.

## Definitions

Our Deligne-Mumford spaces model trees of disks with boundary and interior marked points, where in addition there may be trees of spheres with marked points attached to interior points of the disks. We must therefore work with ordered resp. unordered trees, to accommodate the trees of disks resp. spheres.

An ordered tree $T$ is a tree satisfying these conditions:

• there is a designated root vertex ${\text{rt}}(T)\in T$, which has valence $|{\text{rt}}(T)|=1$; we orient $T$ toward the root.
• $T$ is ordered in the sense that for every $v\in T$, the incoming edge set $E^{{{\text{in}}}}(v)$ is equipped with an order.
• The vertex set $V$ is partitioned $V=V^{m}\sqcup V^{c}$ into main vertices and critical vertices, such that the root is a critical vertex.

An unordered tree is defined similarly, but without the order on incoming edges at every vertex; a labeled unordered tree is an unordered tree with the additional datum of a labeling of the non-root critical vertices. For instance, here is an example of an ordered tree $T$, with order corresponding to left-to-right order on the page:

INSERT EXAMPLE HERE

We now define, for $m\geq 2$, the space $DM(m)^{{{\text{sph}}}}$ resp. $DM(m)^{{{\text{sph,ord}}}}$ of nodal trees of spheres with $m$ incoming unordered resp. ordered marked points. (These spaces can be given the structure of compact manifolds resp. orbifolds without boundary -- for instance, $DM(2)^{{{\text{sph,ord}}}}={\text{pt}}$, $DM(3)^{{{\text{sph,ord}}}}\cong {\mathbb {CP}}^{1}$, and $DM(4)^{{{\text{sph,ord}}}}$ is diffeomorphic to the blowup of ${\mathbb {CP}}^{1}\times {\mathbb {CP}}^{1}$ at 3 points on the diagonal.)

$DM(m)^{{{\text{sph}}}}:=\{(T,\underline O)\;|\;{\text{(1),(2),(3) satisfied}}\}/_{\sim },$

where (1), (2), (3) are as follows:

1. $T$ is an unordered tree, with critical vertices consisting of $m$ leaves and the root.
2. For every main vertex $v\in V^{m}$ and each edge $e\in E(v)$, $O_{{v,e}}$ is a marked point in $S^{2}$ satisfying the requirement that for $v$ fixed, the points $(O_{{v,e}})_{{e\in E(v)}}$ are distinct. We denote this (unordered) set of marked points by $\underline o_{v}$.
3. The tree $(T,\underline O)$ satisfies the stability condition, i.e. for every $v\in V^{m}$, we have $\#E(v)\geq 3$.

The equivalence relation $\sim$ is defined by declaring that $(T,\underline O)$, $(T',\underline O')$ are equivalent if there exists an isomorphism of labeled trees $\zeta :T\to T'$ and a collection $\underline \psi =(\psi _{v})_{{v\in V^{m}}}$ of holomorphism automorphisms of the sphere such that $\psi _{v}(\underline O_{v})=\underline O_{{\zeta (v)}}'$ for every $v\in V^{m}$.

The spaces $DM(m)^{{{\text{sph,ord}}}}$ are defined similarly, but where the underlying combinatorial object is an unordered labeled tree.

Next, we define, for $k\geq 0,m\geq 1,k+2m\geq 2$ and a decomposition $\{1,\ldots ,k\}=A_{1}\sqcup \cdots \sqcup A_{n}$, the space $DM(k,m;(A_{j}))$ of nodal trees of disks and spheres with $k$ ordered incoming boundary marked points and $m$ unordered interior marked points. Before we define this space, we note that if $T$ is an ordered tree with $V^{c}$ consisting of $k$ leaves and the root, we can associate to every edge $e$ in $T$ a pair $P(e)\subset \{1,\ldots ,k\}$ called the node labels at $e$:

1. Choose an embedding of $T$ into a disk $D$ in a way that respects the orders on incoming edges and which sends the critical vertices to $\partial D$.
2. This embedding divides $D^{2}$ into $k+1$ regions, where the 0-th region borders the root and the first leaf, for $1\leq i\leq k-1$ the $i$-th region borders the $i$-th and $(i+1)$-th leaf, and the $k$-th region borders the $k$-th leaf and the root. Label these regions accordingly by $\{0,1,\ldots ,k\}$.
3. Associate to every edge $e$ in $T$ the labels of the two regions which it borders.

With all this setup in hand, we can finally define $DM(k,m;(A_{j}))$:

$DM(k,m;(A_{j})):=\{(T,\underline \ell ,\underline x,\underline O,\underline \beta ,\underline z)\;|\;{\text{(1)--(6) satisfied}}\}/_{\sim },$

where (1)--(6) are defined as follows:

1. $T$ is an ordered tree with critical vertices consisting of $k$ leaves and the root.
2. $\underline \ell =(\ell _{e})_{{e\in E^{{{\text{Morse}}}}}}$ is a tuple of edge lengths with $\ell _{e}\in [0,1]$, where the Morse edges $E^{{{\text{Morse}}}}$ are those satisfying $P(e)\subset A_{j}$ for some $j$. Moreover, for a Morse edge $e\in E^{{{\text{Morse}}}}$ incident to a critical vertex, we require $\ell _{e}=1$.
3. For every main vertex $v\in V^{m}$ and each edge $e\in E(v)$, $x_{{v,e}}$ is a marked point in $\partial D$ such that, for any $v\in V^{m}$, the sequence $(x_{{v,e^{0}(v)}},\ldots ,x_{{v,e^{{|v|-1}}(v)}})$ is in (strict) counterclockwise order. Similarly, $\underline O$ is a collection of unordered interior marked points -- i.e. for $v\in V^{m}$, $\underline O_{v}$ is an unordered subset of $D^{0}$.
4. $\underline \beta =(\beta _{i})_{{1\leq i\leq t}},\beta _{i}\in DM(p_{i})^{{{\text{sph,ord}}}}$ is a collection of sphere trees such that the equation $m'-t+\sum _{{1\leq i\leq t}}p_{i}=m$ holds.
5. $\underline z=(z_{1},\ldots ,z_{t}),z_{i}\in \underline O$ is a distinguished tuple of interior marked points, which corresponds to the places where the sphere trees are attached to the disk tree.
6. The disk tree satisfies the stability condition, i.e. for every $v\in V^{m}$ with $\underline O_{v}=\emptyset$, we have $\#E(v)\geq 3$.

The definition of the equivalence relation is defined similarly to the one in $DM(m)^{{{\text{sph}}}}$. $DM(k,m;(A_{j}))^{{{\text{ord}}}}$ is defined similarly.

## Gluing and the definition of the topology

We will define the topology on $DM(k,m;(A_{j}))$ (and its variants) like so:

• Fix a representative ${\hat \mu }$ of an element $\sigma \in DM(k,m;(A_{j}))$.
• Associate to each interior nodes a gluing parameter in $D_{\epsilon }$; associate to each Morse-type boundary node a gluing parameter in $(-\epsilon ,\epsilon )$; and associate to each Floer-type boundary node a gluing parameter in $[0,\epsilon )$. Now define a subset $U_{\epsilon }(\sigma ;{\hat \mu })\subset DM(k,m;(A_{j}))$ by gluing using the parameters, varying the marked points by less than $\epsilon$ in $\ell ^{\infty }$-norm (for the boundary marked points) resp. Hausdorff distance (for the interior marked points), and varying the lengths $\ell _{e}$ by less than $\epsilon$.
• We define the collection of all such sets $U_{\epsilon }(\sigma ;{\hat \mu })$ to be a basis for the topology.

To make the second bullet precise, we will need to define a gluing map

$[\#]:U_{\epsilon }^{{{\text{Floer}}}}(\underline 0)\times U_{\epsilon }^{{{\text{Morse}}}}(\underline 0)\times U_{\epsilon }^{{{\text{int}}}}(\underline 0)\times \prod _{{1\leq i\leq t}}U_{\epsilon }^{{{\hat {\beta _{i}}}}}(\underline 0)\times U_{\epsilon }({\hat \mu })\to DM(k,m;(A_{j})),$

${\bigl (}\underline r,(\underline s_{i}),({\hat T},\underline \ell ,\underline x,\underline O,\underline \beta ,\underline z){\bigr )}\mapsto [\#_{{\underline r}}({\hat T}),\#_{{\underline r}}(\underline \ell ),\#_{{\underline r}}(\underline x),\#_{{\underline r}}(\underline O),\#_{{\underline s}}(\underline \beta ),\#_{{\underline r}}(\underline z)].$

The image of this map is defined to be $U_{\epsilon }(\sigma ;{\hat \mu })$.

Before we can construct $[\#]$, we need to make some choices:

• Fix a gluing profile, i.e. a continuous decreasing function $R:(0,1]\to [0,\infty )$ with $\lim _{{r\to 0}}R(r)=\infty$, $R(1)=0$. (For instance, we could use the exponential gluing profile $R(r):=\exp(1/r)-e$.)
• For each main vertex ${\hat v}\in {\hat V}^{m}$ and edge ${\hat e}\in {\hat E}({\hat v})$, we must choose a family of strip coordinates $h_{{{\hat e}}}^{\pm }$ near ${\hat x}_{{{\hat v},{\hat e}}}$ (this notion is defined below).
• For each main vertex ${\hat v}$ and point in $\underline O_{v}$, we must choose a family of cylinder coordinates near that point. (We do not make precise this notion, but its definition should be evident from the definition of strip coordinates.) We must also choose cylinder coordinates within the sphere trees which are attached to the disk tree.

Here we define the notion of (positive resp. negative) strip coordinates.

Given $x\in \partial D$, we call biholomorphisms
$h^{+}:{\mathbb {R}}^{+}\times [0,\pi ]\to N(x)\setminus \{x\},$
$h^{-}:{\mathbb {R}}^{-}\times [0,\pi ]\to N(x)\setminus \{x\}$
into a closed neighborhood $N(x)\subset D$ of $x$ positive resp. negative strip coordinates near $x$ if it is of the form $h^{\pm }=f\circ p^{\pm }$, where
• $p^{\pm }:{\mathbb {R}}^{\pm }\times [0,\pi ]\to \{0<|z|\leq 1,\;\Im z\geq 0\}$ are defined by
$p^{+}(z):=-\exp(-z),\quad p^{-}(z):=\exp(z);$
• $f$ is a Moebius transformation which maps the extended upper half plane $\{z\;|\;\Im z\geq 0\}\cup \{\infty \}$ to the $D$ and sends $0\mapsto x$.

(Note that there is only a 2-dimensional space of choices of strip coordinates, which are in bijection with the complex automorphisms of the extended upper half plane fixing $0$.)

Now that we've defined strip coordinates, we say that for ${\hat x}\in \partial D$ and $x$ varying in $U_{\epsilon }({\hat x})$, a family of positive resp. negative strip coordinates near ${\hat x}$ is a collection of functions
$h^{+}(x,\cdot ):{\mathbb {R}}^{+}\times [0,\pi ]\to D,$
$h^{-}(x,\cdot ):{\mathbb {R}}^{-}\times [0,\pi ]$
if it is of the form $h^{\pm }(x,\cdot )=f_{x}\circ p^{\pm }$, where $f_{x}$ is a smooth family of Moebius transformations sending $\{z\;|\;\Im z\geq 0\}\cup \{\infty \}$ to the disk, such that $f_{x}(0)=x$ and $f_{x}(\infty )$ is independent of $x$.

With these preparatory choices made, we can define the sets $U_{\epsilon }^{{{\text{Floer}}}}(\underline 0),U_{\epsilon }^{{{\text{Morse}}}}(\underline 0),U_{\epsilon }^{{\beta _{i}}}(\underline 0),U_{\epsilon }({\hat \mu })$ appearing in the definition of $[\#]$.

• $U_{\epsilon }^{{{\text{Floer}}}}(\underline 0)$ is a product of intervals $[0,\epsilon )$, one for every Floer-type nodal edge in ${\hat T}$ ("Floer-type" meaning an edge whose adjacent regions are labeled by a pair not contained in a single $A_{j}$, "nodal" meaning that the edge connects two main vertices).
• $U_{\epsilon }^{{{\text{Morse}}}}(\underline 0)$ is a product of intervals $(-\epsilon ,\epsilon )$, one for every Morse-type nodal edge in ${\hat T}$ ("Morse-type meaning an edge whose adjacent regions are labeled by a pair contained in a single $A_{j}$, "nodal" meaning ${\hat \ell }_{{{\hat e}}}=0$).

$U_{\epsilon }^{{{\text{int}}}}(\underline 0)$ is a product of disks $D_{\epsilon }(0)$, one for every point in ${\hat {\underline O}}$ to which a sphere tree is attached.

• $U_{\epsilon }^{{{\hat \beta }_{i}}}(\underline 0)$ is a product of disks $D_{\epsilon }(0)$, one for every interior edge in the unordered tree underlying the sphere tree ${\hat \beta }_{i}$.
• $U_{\epsilon }({\hat \mu })$ is the $\epsilon$-neighborhood of ${\hat \mu }$, i.e. the set of $({\hat T},\underline \ell ,\underline x,\underline O,\underline \beta ,\underline z)$ satisfying these conditions:
• For every Morse-type edge ${\hat e}$, $\ell _{{{\hat e}}}$ lies in $({\hat \ell }_{{{\hat e}}}-\epsilon ,{\hat \ell }_{{{\hat e}}}+\epsilon )$. Moreover, for every Morse-type nodal edge resp. edge adjacent to a critical vertex, we fix $\ell _{{{\hat e}}}=0$ resp. $\ell _{{{\hat e}}}=1$.
• For every main vertex ${\hat v}\in {\hat V}^{m}$, the marked points $(\underline x_{{{\hat v}}},\underline O_{{{\hat v}}})$ must be within $\epsilon$ of $(\underline {{\hat x}}_{{{\hat v}}},\underline {{\hat O}}_{{{\hat v}}})$ (under $\ell ^{\infty }$-norm for the boundary marked points and Hausdorff distance for the interior marked points).
• Similar restrictions apply to $\underline \beta$.
• $\underline z$ must be within $\epsilon$ of ${\hat {\underline z}}$ in Hausdorff distance.

We can now construct the gluing map $[\#]$. We will not include all the details.

• The glued tree $\#_{{\underline r}}({\hat T})$ is defined by collapsing the gluing edges ${\hat E}_{{\underline r}}^{g}:=\{{\hat e}\in {\hat E}^{{{\text{nd}}}}\;|\;r_{{{\hat e}}}>0\}$, where the nodal edges ${\hat E}^{{{\text{nd}}}}$ are the Floer-type nodal edges and Morse-type nodal edges (see above). For any vertex $v\in \#_{{\underline r}}({\hat V})$, we denote by ${\hat V}_{v}^{g}\subset {\hat V}$ the vertices in ${\hat T}$ which are identified to form $v$. These vertices form a subtree of ${\hat T}$; denote the edges in this subtree by ${\hat E}_{{\underline r}}^{g}$.
• To define the objects $\#_{{\underline r}}(\underline \ell ),\#_{{\underline r}}(\underline x),\#_{{\underline r}}(\underline O),\#_{{\underline s}}(\underline \beta ),\#_{{\underline r}}(\underline z)$, we will "glue" certain disks and spheres together, using the choices of strip coordinates and cylinder coordinates made above. The crucial constructions will be, for each main vertex $v\in \#_{{\underline r}}({\hat V})$, a glued disk $\#_{{\underline r}}(\underline D)_{v}$ formed by gluing together the disks corresponding to the vertices in ${\hat E}_{v}^{g}$. We now make a few definitions towards the construction of $\#_{{\underline r}}(\underline D)_{v}$:
• For ${\hat e}\in {\hat E}_{v}^{g}$, define the gluing length $R_{{{\hat e}}}:=R(r_{{{\hat e}}})$.
• Denote by $L_{{R_{{{\hat e}}}}}$ the left translation of $s$ by $R_{{{\hat e}}}$,
$L_{{R_{{{\hat e}}}}}:[0,R_{{{\hat e}}}]\times [0,\pi ]\to [-R_{{{\hat e}}},0]\times [0,\pi ].$
• We define our glued surface $\#_{{\underline r}}(\underline D)_{v}$ by using strip coordinates to identify neighborhoods of each pair of gluing edges in ${\hat E}_{v}^{g}$ with $[0,\infty )\times [0,\pi ]$ (in the component closer to the root) resp. $(-\infty ,0]\times [0,\pi ]$ (in the component further from the root), and then replacing these two ends by the quotient space
$([0,R_{{{\hat e}}}]\times [0,\pi ])\sqcup ([-R_{{{\hat e}}},0]\times [0,\pi ])/L_{{R_{{{\hat e}}}}}.$
The resulting Riemann surface $\#_{{\underline r}}(\underline D)_{v}$ is biholomorphic to a disk, and all the information (boundary marked points, interior marked points, etc.) persists in the glued disk.
• We perform a similar gluing construction at each of the interior nodes, using the families of cylinder coordinates we chose earlier.

## The atlas of Deligne-Mumford space

We define charts for $DM(k,m;(A_{j}))^{{{\text{ord}}}}$ like so:

• Fix an element $\sigma \in DM(k,m;(A_{j}))^{{{\text{ord}}}}$ and a representative ${\hat \mu }=({\hat T},{\hat {\underline \ell }},{\hat {\underline x}},{\hat {\underline o}})$ of $\sigma$.
• Choose a local slice near ${\hat \mu }$:
• for every disk component with $\geq 3$ boundary marked points, fix the first three boundary marked points;
• for every disk component with $\leq 2$ boundary marked points, fix the distinguished boundary marked points and the first interior marked point;
• for every sphere component, fix the first three marked points.
• The restriction to this local slice of the gluing map $[\#]$ defined above is our chart.

old stuff

For $d\geq 2$, the moduli space of domains

${\mathcal {M}}_{{d+1}}:={\frac {{\bigl \{}\Sigma _{{\underline {z}}}\,{\big |}\,\underline {z}=\{z_{0},\ldots ,z_{d}\}\in \partial D\;{\text{pairwise disjoint}}{\bigr \}}}{\Sigma _{{\underline {z}}}\sim \Sigma _{{\underline {z}'}}\;{\text{iff}}\;\exists \psi :\Sigma _{{\underline {z}}}\to \Sigma _{{\underline {z}'}},\;\psi ^{*}i=i}}$

can be compactified to form the Deligne-Mumford space $\overline {\mathcal {M}}_{{d+1}}$, whose boundary and corner strata can be represented by trees of polygonal domains $(\Sigma _{v})_{{v\in V}}$ with each edge $e=(v,w)$ represented by two punctures $z_{e}^{-}\in \Sigma _{v}$ and $z_{e}^{+}\in \Sigma _{w}$. The thin neighbourhoods of these punctures are biholomorphic to half-strips, and then a neighbourhood of a tree of polygonal domains is obtained by gluing the domains together at the pairs of strip-like ends represented by the edges.

The space of stable rooted metric ribbon trees, as discussed in [BV], is another topological representation of the (compactified) Deligne Mumford space $\overline {\mathcal {M}}_{{d+1}}$.