Categorical enumerative invariants of the ground field

Categorical enumerative invariants (CEI) were introduced by Costello [5] and recently made explicit in [2] and [3]. These are invariants associated with a pair $(A,s)$ where

• •

  $A$ is a cyclic $A_{\infty}$ algebra of dimension $d$ over a field $\mathbb{K}$ (of characteristic zero) that is proper, smooth and satisfies the Hodge-to-de-Rham degeneration.

• •

  $s$ is a choice of splitting of the non-commutative Hodge filtration of $A$.

Explicitly, CEI take the following form

$$\langle\alpha_{1},\cdots,\alpha_{n}\rangle_{g,n}^{A,s}\in\mathbb{K},\;\;\;\;\forall(g,n),\;\;2g-2+n>0$$

with $\alpha_{1},\ldots,\alpha_{n}\in HH_{*-d}(A)[u]$ in the $d$-shifted Hochschild homology of $A$ adjoining a formal variable $u$ of homological degree $-2$. It was proved in [3] that CEI is only possibly non-zero if

(1)

$$\sum_{j=1}^{n}(-|\alpha_{j}|)=2(g-1)(3-d)+2n$$

In the cohomological degree convention, this is precisely the dimension axiom in Gromov-Witten theory. Indeed, the original motivation to define and study such invariants came from mirror symmetry following Kontsevich’s proposal [14]. It is expected that these invariants, when applied to Fukaya categories, should recover Gromov-Witten invariants at all genera.

At first glance, comparing these categorical invariants with Gromov-Witten invariants may look rather difficult since CEI are defined purely algebraically while the GW invariants involves moduli spaces of stable maps. The purpose of this paper is to demonstrate otherwise. We begin with the case when the ground field $\mathbb{K}=\mathbb{Q}$ and the algebra $A$ is also $\mathbb{Q}$. In this case, the dimension $d$ is zero, and the algebra $A$ has a unique homogeneous splitting of the Hodge filtration which we still denote by $s$. Furthermore, its Hochschild homology group is $HH_{*}(\mathbb{Q})=\mathbb{Q}$ and so the insertions lie in the space $HH_{*}(\mathbb{Q})[u]=\mathbb{Q}[u]$.

In order to prove the above theorem, it is necessary to have a better understanding of Sen-Zwiebach’s string vertices [18], since they play a central role in the construction of CEI.

We need to introduce some notations. For a topological space $X$, denote by $C_{*}(X)$ its normalized singular chain complex with coefficients in $\mathbb{Q}$. For each pair $(g,n)\in\mathbb{N}\times\mathbb{N}$ such that $2g-2+n>0$, denote by $M_{g,n}^{\sf fr}$ the moduli space of tuples $(\Sigma,p_{1},\ldots,p_{n},\varphi_{1},\ldots,\varphi_{n})$ where $\Sigma$ is a Riemann surface of genus $g$; $p_{1},\ldots,p_{n}$ are $n$ marked points in $\Sigma$; and for each $1\leq i\leq n$, $\varphi_{i}:D^{2}\rightarrow U(p_{i})\subset\Sigma$ is a holomorphic chart such that it extends to an open neighborhood of the unit disk in $\mathbb{C}$. We also require that the pair-wise intersections of the closures of the framed disks be empty, i.e. $\overline{U(p_{i})}\cap\overline{U(p_{j})}=\emptyset,\;\;\forall 1\leq i<j\leq n$. As in Segal [17], sewing along the coordinate charts defines two types of composition maps:

	$\displaystyle M_{g,n}^{\sf fr}\times M_{g^{\prime},n^{\prime}}^{\sf fr}\rightarrow M_{g+g^{\prime},n+n^{\prime}-2}^{\sf fr},\;\;\;1\leq i\leq n,1\leq j\leq n^{\prime}$
	$\displaystyle M_{g,n+2}^{\sf fr}\rightarrow M^{\sf fr}_{g+1,n},\;\;\;1\leq i<j\leq n+2$

We shall denote both of them by $c_{ij}$, as no confusion can arise in this way. The collection of topological spaces $\{M_{g,n}^{\sf fr}\}_{(g,n)}$ together with the two sewing operations form a modular operad defined by Getzler-Kapranov [10] in the category of topological spaces.

Applying the normalized singular chain functor $C_{*}$ yields a differential graded modular operad $\{C_{*}(M_{g,n}^{\sf fr})\}_{(g,n)}$. For the first type composition, note that the functor $C_{*}$ is lax monoidal, i.e. for two topological spaces $X$ and $Y$, we have a naturally defined map $C_{*}(X)\otimes C_{*}(Y)\rightarrow C_{*}(X\times Y)$, the Alexander-Whitney map. This enables us to define a composition map still denoted by

$$c_{ij}:C_{*}(M_{g,n}^{\sf fr})\otimes C_{*}(M_{g^{\prime},n^{\prime}}^{\sf fr})\stackrel{{\scriptstyle{\sf AW}}}{{\longrightarrow}}C_{*}(M_{g,n}^{\sf fr}\times M_{g^{\prime},n^{\prime}}^{\sf fr})\rightarrow C_{*}(M_{g+g^{\prime},n+n^{\prime}-2}^{\sf fr}).$$

Denote this differential graded modular operad by $\mathbb{M}^{\sf fr}$, with $\mathbb{M}^{\sf fr}(g,n):=C_{*}(M_{g,n}^{\sf fr})$.

The modular operad $\mathbb{M}^{\sf fr}$ is in fact an $S^{1}$-framed modular operad with the circle actions given by rotations of the local coordinate charts. Explicitly, the circle actions on $\mathbb{M}^{\sf fr}$ are given by degree one operators

$$B_{i}:\mathbb{M}^{\sf fr}(g,n)\rightarrow\mathbb{M}^{\sf fr}(g,n),\;\;1\leq i\leq n.$$

We refer to Section 2 for more details on the definition of $S^{1}$-framed modular operad and examples. Denote the homotopy quotient complex of the $(S^{1})^{n}$-action by

$$\mathbb{M}^{\sf fr}_{S^{1}}(g,n):=\big{(}\mathbb{M}^{\sf fr}(g,n)[u_{1}^{-1},\ldots,u_{n}^{-1}],\partial+\sum_{i=1}^{n}u_{i}\cdot B_{i}\big{)}$$

with $u_{1},\ldots,u_{n}$ circle parameters of homological degree $-2$. Observe that the symmetric group $S_{n}$ still acts on $\mathbb{M}^{\sf fr}_{S^{1}}(g,n)$. Denote its further quotient space by $\mathbb{M}^{\sf fr}_{S^{1}}(g,n)_{S_{n}}$.

With these preparations, we are ready to introduce one of the main constructions in defining CEI: Sen-Zwiebach’s differential graded Lie algebra (DGLA) [2, Section 3]. As a graded vector space, this DGLA is given by

$$\mathfrak{g}:=\big{(}\bigoplus_{(g,n)}\mathbb{M}^{\sf fr}_{S^{1}}(g,n)_{S_{n}}[1]\big{)}[[\hbar,\lambda]]$$

where $\hbar$, $\lambda$ are two formal variables both of homological degree $-2$. The differential of $\mathfrak{g}$ is of the form $\partial+\sum_{i=1}^{n}u_{i}\cdot B_{i}+\hbar\Delta$ with the operator $\Delta$ defined by all possible ways of “twisted sewing”:

$$\Delta(\alpha\cdot u_{1}^{-k_{1}}\cdots u_{n}^{-k_{n}}):=\sum_{1\leq i<j\leq n}\delta_{k_{i}=0}\cdot\delta_{k_{j}=0}\cdot c_{ij}(B_{i}\alpha)$$

The Lie bracket $\{-,-\}$ of $\mathfrak{g}$ is defined in a similar way:

$$\{\alpha\cdot u_{1}^{-k_{1}}\cdots u_{n}^{-k_{n}},\beta\cdot v_{1}^{-l_{1}}\cdots v_{n^{\prime}}^{-l_{n^{\prime}}}\}:=\sum_{1\leq i\leq n,1\leq j\leq n^{\prime}}\delta_{k_{i}=0}\cdot\delta_{l_{j}=0}\cdot c_{ij}(B_{i}\alpha,\beta)$$

Observe that the circle operator $B_{i}$ has homological degree one. This explains the shift by $[1]$ in the definition of $\mathfrak{g}$: to make the Lie bracket of homological degree zero. We refer to [2, Section 3] for more details of the construction of $\mathfrak{g}$.

By definition, the string vertex is a Maurer-Cartan element of the form

$$\mathcal{V}:=\sum_{(g,n)}\mathcal{V}_{g,n}\hbar^{g}\lambda^{2g-2+n}$$

in the DGLA $\mathfrak{g}$ satisfying an initial condition of $\mathcal{V}_{0,3}$. The Maurer-Cartan equation satisfied by $\mathcal{V}$ is equivalent to the following system of equations usually known as the quantum master equation:

(2)

$$(\partial+\sum_{i=1}^{n}u_{i}\cdot B_{i})\mathcal{V}_{g,n}+\Delta\mathcal{V}_{g-1,n+2}+\frac{1}{2}\sum_{g^{\prime}+g^{\prime\prime}=g,n^{\prime}+n^{\prime\prime}=n+2}\{\mathcal{V}_{g^{\prime},n^{\prime}},\mathcal{V}_{g^{\prime\prime},n^{\prime\prime}}\}=0$$

Costello proves that the above equations together with the initial condition of $\mathcal{V}_{0,3}$ characterize $\mathcal{V}$ up to homotopy ( [5, Theorem 1]).

In the homological convention, a Maurer-Cartan element has degree $-1$, which implies that $\mathcal{V}_{g,n}$ is of degree $6g-6+2n$ inside the equivariant chain complex $\mathbb{M}^{\sf fr}(g,n)[u_{1}^{-1},\ldots,u_{n}^{-1}]$. Thus explicitly, the string vertex at position $(g,n)$ is of the form

$$\mathcal{V}_{g,n}=\sum_{(k_{1},\ldots,k_{n})}\mathcal{V}_{g,n}^{k_{1},\ldots,k_{n}}u_{1}^{-k_{1}}\cdots u_{n}^{-k_{n}},$$

with the coefficient chains

$$\mathcal{V}_{g,n}^{k_{1},\ldots,k_{n}}\in C_{6g-6+2n-2k_{1}-\cdots-2k_{n}}(M_{g,n}^{\sf fr}).$$

By definition, it is also symmetric under permutations of the indices. Then, after unwinding the definition of CEI in [2, 3], in the case of Theorem A we have

$$\langle u^{k_{1}},\ldots,u^{k_{n}}\rangle^{\mathbb{Q},s}_{g,n}=\begin{cases}0,&\;\;\;k_{1}+\cdots+k_{n}\neq 3g-3+n,\\ n!\cdot|\mathcal{V}_{g,n}^{k_{1},\ldots,k_{n}}|,&\;\;\;k_{1}+\cdots+k_{n}=3g-3+n.\end{cases}$$

Here in the second case, note that $\mathcal{V}_{g,n}^{k_{1},\ldots,k_{n}}$ is a zero chain, hence it is a linear combination of points. The notation $|\cdot|$ stands for the sum of its coefficients.

To prove Theorem A, it remains to relate these numbers with integrals of $\psi$-classes on the Deligne-Mumford compactification $\overline{M}_{g,n}$. It is natural to do this on the level of operads.

In Section 2 we define the Feynman compactification of the $S^{1}$-framed modular operad $\mathbb{M}^{\sf fr}$. Denote the resulting operad by $F\mathbb{M}^{\sf fr}$. Intuitively speaking, this is an ordinary modular operad such that the circle actions on the composition maps of $\mathbb{M}^{\sf fr}$ have been universally trivialized. More precisely, an element of $F\mathbb{M}^{\sf fr}(g,n)$ is given by a stable graph $G\in\Gamma((g,n))$ together with decorations:

• •

  At a vertex $v$, the decoration is by an element in the homotopy quotient $\mathbb{M}_{S^{1}}^{\sf fr}(g(v),{\sf Leg}(v)):=\mathbb{M}^{\sf fr}(g(v),{\sf Leg}(v))_{(S^{1})^{|{\sf Leg}(v)|}}$.

• •

  At an edge $e$, it is decorated by a homological degree $2$ element $D_{e}$.

We think of the edge decoration $D_{e}$ as giving a universal trivialization of the circle action on the composition map, by setting its boundary $\partial D_{e}$ to be exactly the twisted sewing operation along the edge $e$, see Equation (6) for a precise definition.

To this point, we say a word about the proof of Theorem B. The idea is to exhibit a sequence of quasi-isomorphisms:

$$\begin{CD}F{\mathbb{M}}^{\sf fr}(g,n)@>{i^{\sharp}}>{}>{\sf Tot}\big{(}F{\mathbb{M}}^{\sf fr}(g,n)_{\bullet}\big{)}@>{\mathbb{I}}>{}>{\sf Tot}\big{(}F\widehat{\mathbb{M}}^{\sf fr}(g,n)_{\bullet}\big{)}@>{p}>{}>C_{*}(\overline{M}_{g,n})\end{CD}$$

Here $F{\mathbb{M}}^{\sf fr}(g,n)_{\bullet}$ is a simplicial resolution of $F{\mathbb{M}}^{\sf fr}(g,n)$ which we refer to as Mondello’s resolution as it’s first constructed by Mondello [15]. The hatted version $\widehat{\mathbb{M}}^{\sf fr}(g,n)$ is given by the normalized singular chain complex of the space $\widehat{M}_{g,n}^{\sf fr}$ constructed in [13]. Basically, this is the framed version of the oriented real blowup of $\overline{M}_{g,n}$ along its boundary divisors, see Section 2 for more details. The first map $i^{\sharp}$ is constructed explicitly in Section 3, see Equation (13). Our main construction is the second map $\mathbb{I}$ defined in Section 4. The natural inclusion map $M_{g,n}^{\sf fr}\hookrightarrow\widehat{M}_{g,n}^{\sf fr}$, being a homotopy equivalence, does not respect the operad structures in the strict sense. The map $\mathbb{I}$ is precisely to interpolate between the two operadic compositions using higher coherent homotopies. The last map $p$ is a canonical projection map, see Paragraph 4. We prove in Section 4 that the composition $p\mathbb{I}i^{\sharp}$ induces an isomorphism in homology that also respect the operadic composition on both sides.

Using Theorem B, we can finally unlock the mystery of string vertices and the quantum master equation (2). In Section 5, we obtain a simple formula expressing the orbifold fundamental class of the symmetric Deligne-Mumford space $\overline{M}_{g,n}/S_{n}$ in terms “disk bundles” over string vertices $\{\mathcal{V}_{g,n}\}_{(g,n)}$:

(3)

$$~{}[\overline{M}_{g,n}/S_{n}]=\sum_{G\in\Gamma((g,n))}\frac{1}{|{\sf Aut}(G)|}\prod_{e\in E_{G}}D_{e}\otimes\prod_{v\in V_{G}}\mathcal{V}^{\operatorname{Sym}}_{g(v),|{\sf Leg}(v)|}.$$

where the summation is over isomorphism classes of stable graphs of type $(g,n)$, $D_{e}$ is a degree $2$ element associated to an edge of $G$, and the superscript $\operatorname{Sym}$ denotes the symmetrization map. Note that the closedness of the right hand side follows from the quantum master equation (2). Note that the above equation makes sense under the isomorphism in Theorem B: the fundamental cycle $[\overline{M}_{g,n}/S_{n}]$ lies in the $S_{n}$-invariants of $H_{*}(\overline{M}_{g,n},\mathbb{Q})$, while the expression on the right hand side of Equation (3) lies inside the $S_{n}$-invariants of $H_{*}(F\mathbb{M}^{\sf fr}(g,n))$.

Using the above identity of the fundamental classes, Theorem $A$ follows immediately after unwinding the definitions of CEI.

Geometrically, Equation (3) may be viewed as a decomposition of the fundamental class of the symmetric Deligne-Mumford space $\overline{M}_{g,n}/S_{n}$ according to its boundary strata labeled by stable graphs. String vertices are simply a coherent choice of the complement of a tubular neighborhood of the boundary divisor $\partial\overline{M}_{g,n}/S_{n}$ in the ambient space $\overline{M}_{g,n}/S_{n}$ for all $(g,n)$ such that $2g-2+n>0$. The coherence equation is precisely the quantum master equation in the Sen-Zwiebach DGLA. Once such a coherent choice is made, Equation (3) simply corresponds to the decomposition of the fundamental class $[\overline{M}_{g,n}/S_{n}]$ in terms disk bundles over products of string vertices.

We illustrate this point of view in the following figures. The left figure corresponds to the cases when $(g,n)=(0,4)$ or $(1,1)$. For example, the symmetric quotient $\overline{M}_{0,4}/S_{4}$ is topologically a sphere. In it there is a point (denoted by $\infty$) representing a nodal sphere with $4$ marked points. Thus the fundamental class is decomposed into a (closed) disk bundle $D_{\infty}$ over the point $\infty$ and $\mathcal{V}_{0,4}$ which is the closure of $\overline{M}_{0,4}/S_{4}-D_{\infty}$. The same decomposition holds in the case of $\overline{M}_{1,1}$ with the point $\infty$ given by the once punctured nodal elliptic curve. The figure on the right illustrates the situation when two boundary divisors cross, producing a codimension $2$ strata in the moduli space. For example, the symmetric quotient $\overline{M}_{0,5}/S_{5}$ is decomposed into $\mathcal{V}_{0,5}$, disk bundles and double disk bundles.

This geometric point of view was already implicit in the original work of Sen-Zwiebach [18] and more recently the work of Costello-Zwiebach [6]. From this perspective, it is appropriate to think of the CEI at $(g,n)$ as a way of decomposing an integral over $\overline{M}_{g,n}$ into integrals over poly-disk bundles on each degenerate strata whose combinatorial types are given by stable graphs in $\Gamma((g,n))$.

In Section 2 we define Feynman compactifications of $S^{1}$-framed modular operads. In Section 3 we introduce a simplicial resolution of the Feynman compactification $F\mathbb{M}^{\sf fr}$, which is a minor modification of the one constructed by Mondello [15]. In Section 4, we prove Theorem B. Section 5 is devoted to prove the main identity (3). In Section 6 we obtain Theorem A, as well as clarify some technicalities in generalizing it to Fukaya categories with semi-simple cohomology.

Throughout the paper, we work with chain complexes and homological degree convention. Unless otherwise stated, we take the ground field to be $\mathbb{Q}$, the field of rationals. For a topological space $X$, the notation $C_{*}(X)$ stands for its normalized singular chain complex, with coefficients in $\mathbb{Q}$.

When working with graphs, we follow notations in [10, Section2]. By a labeled graph, we shall mean a graph $G$ that is endowed with a genus labeling $g:V_{G}\rightarrow\mathbb{N}$ on its set of vertices $V_{G}$. The genus of a labeled graph is defined by

$$\sum_{v\in V_{G}}g(v)+\dim H^{1}(G).$$

If $G$ is a labeled graph, denote by

• •

  $V_{G}$: the set of vertices of $G$.

• •

  $E_{G}$: the set of edges of $G$.

• •

  ${\mathsf{Leg}}(v)$: the set of half-edges at a vertex $v\in V_{G}$.

• •

  ${\mathsf{Leg}}(G)$: the set of all half-edges of $G$.

• •

  ${\sf Leaf}(G)$: the set of leaves of $G$.

Denote by $\Gamma(g,n)$ the set of isomorphism classes of labeled graphs of genus $g$ and with $n$ leaves. For a vertex $v\in V_{G}$ of a graph, its valency is denoted by $n(v)$. A graph is called stable if $2g(v)-2+n(v)>0$ holds at every vertex $v\in V_{G}$. The set of isomorphism classes of stable graphs is denoted by $\Gamma((g,n))$. If $h:G\rightarrow G^{\prime}$ is a morphism of graphs, denote by $E_{G\rightarrow G^{\prime}}\subset E_{G}$ the set of edges that are contracted by $h$, and denote by $E_{G\rightarrow G^{\prime}}^{c}=E_{G}\backslash E_{G\rightarrow G^{\prime}}$ the complement.

The author is grateful for Lino Amorim, Andrei Caldararu, and Kevin Costello for discussions around the topic.

Recall from [10] the notion of a stable $\mathbb{S}$-module given by a collection $\{P(g,n)|n,g\geq 0,2g+n-2<0\}$ of chain complexes with an action of the symmetric group $S_{n}$ on $P(g,n)$. For a finite set $I$ of cardinality $n$, we set

$$P(g,I):=\big{(}\bigoplus_{f:\{1,\ldots,n\}\rightarrow I}P(g,n)\big{)}_{S_{n}},$$

with the direct sum taken over all bijections from $\{1,\ldots,n\}$ to $I$. For each stable graph $G$, denote by

$$P(G):=\bigotimes_{v\in V_{G}}P(g(v),{\mathsf{Leg}}(v)).$$

A modular operad structure on the stable $\mathbb{S}$-module $P$ is given by an extension of $P$ to a functor on the category of stable graphs, that is, for each $h:G\rightarrow G^{\prime}$ we have a morphism $P(h):P(G)\rightarrow P(G^{\prime})$ such that $P(h_{1}h_{0})=P(h_{1})P(h_{0})$ whenever $h_{1}$ and $h_{0}$ are two composable morphisms of stable graphs.

We shall work with a $S^{1}$-framed version of modular operads. First, we introduce some terminologies. Let $(C,\partial)$ be a chain complex. By an $S^{1}$-action on $C$, we mean a homological degree one operator $B:C_{\bullet}\rightarrow C_{\bullet+1}$ such that $B^{2}=0$ and $\partial B+B\partial=0$. Let $n\geq 1$, by an $(S^{1})^{n}$-action, we mean $n$ mutually commuting circle actions $B_{j}\;(j=1,\ldots,n)$, i.e. $B_{i}B_{j}+B_{j}B_{i}=0$ for all $1\leq i,j\leq n$. Its homotopy quotient is defined as the chain complex

$$C_{(S^{1})^{n}}:=\Big{(}C[u_{1}^{-1},\ldots,u_{n}^{-1}],\partial+\sum_{j=1}^{n}B_{j}u_{j}\Big{)}$$

with the $u_{j}$’s of homological degree $-2$. The $\mathbb{Q}[u_{1},\ldots,u_{n}]$-module structure is given by the quotient of $C[u_{1}^{\pm},\ldots,u_{n}^{\pm}]$ by the subspace $\sum_{j}u_{j}\cdot C[u_{1},\ldots,u_{n}]$. Observe that the inclusion map

$$C\hookrightarrow C[u_{1}^{-1},\ldots,u_{n}^{-1}]=C_{(S^{1})^{n}}$$

is a map of chain complexes, which we refer to as the canonical quotient map.

With this preparation, an $S^{1}$-framed modular operad is given by a modular operad $P$ together with the following additional data:

• •

  There is an action of $S_{n}\rtimes(S^{1})^{n}$ on each $P(g,n)$, extending the $S_{n}$-action.

• •

  The morphism $P(h):P(G)\rightarrow P(G^{\prime})$ associated with a morphism $h:G\rightarrow G^{\prime}$ with $G,G^{\prime}\in\Gamma(g,n)$ is $S_{n}\rtimes(S^{1})^{|{\mathsf{Leg}}(G^{\prime})|}$-equivariant. Here we implicitly identified ${\mathsf{Leg}}(G^{\prime})$ as a subset of ${\mathsf{Leg}}(G)$ using the morphism $h$ (see [10, Section 2.13]).

• •

  The morphism $P(h)$ is invariant under simultaneous rotation at the two ends of an edge. More precisley, we require that

  $$P(h)(B_{e,+}\alpha)=P(h)(B_{e,-}\alpha).$$

  Here $\alpha\in P(G)$ is of the form $\otimes_{v\in V_{G}}\alpha_{v}$ and $B_{e,+}$ and $B_{e,-}$ are the two circle actions at the two half edges of a contracted edge $e\in E_{G\rightarrow G^{\prime}}$.

For each $(g,n)$, let $\widehat{M}_{g,n}$ be the oriented real blowup along the boundary divisors in the Deligne-Mumford space $\overline{M}_{g,n}$. An element in $\widehat{M}_{g,n}$ is given by a stable curve $(\Sigma,p_{1},\ldots,p_{n})\in\overline{M}_{g,n}$ together with a decoration at each nodal point by a unit tangent vector :

$$v_{\alpha}\in T_{x_{\alpha}}\Sigma^{n}\otimes T_{y_{\alpha}}\Sigma^{n},$$

where $\Sigma^{n}\rightarrow\Sigma$ is the normalization of $\Sigma$ and $\{x_{\alpha},y_{\alpha}\}$ is the preimage of a nodal point labeled by $\alpha$. We also consider a framed version of it denoted by $\widehat{M}_{g,n}^{\sf fr}$. An element of this moduli space $\widehat{M}_{g,n}^{\sf fr}$ consists of a decorated stable Riemann surface $(\Sigma,p_{1},\ldots,p_{n},\prod_{\alpha}v_{\alpha})\in\widehat{M}_{g,n}$, together with a framing around each marked point. That is, for each $1\leq i\leq n$, a local coordinate system $\varphi_{i}:\mathbb{D}^{2}\rightarrow U(p_{i})\subset\Sigma$ such that the biholomorphic maps $\varphi$’s extend to an open neighborhood of $\mathbb{D}^{2}$. We also require that the pair-wise intersection of the closures of the framed disks be empty, i.e. $\overline{U(p_{i})}\cap\overline{U(p_{j})}=\emptyset$. The circle $S^{1}$ acts on a local coordinate system by rotation: $\varphi\mapsto\varphi\circ e^{i\theta}$. This action is clearly a free action. It is also well-known that the set of local coordinate systems around any point $p_{i}$ is homotopy equivalent to $S^{1}$. Thus, the forgetful map

$$\widehat{M}_{g,n}^{\sf fr}\rightarrow\widehat{M}_{g,n}$$

is a homotopy $(S^{1})^{n}$-bundle.

We define a $S^{1}$-equivariant modular operad $\widehat{\mathbb{M}}^{\sf fr}$ by setting

$$\widehat{\mathbb{M}}^{\sf fr}(g,n):=C_{*}(\widehat{M}_{g,n}^{\sf fr}).$$

Let $h:G\rightarrow G^{\prime}$ be a morphism of stable graphs, the composition morphism is defined as follows. The map on the underlying stable curve is the product of inclusions

$$\prod_{v\in V_{G}}\overline{M}_{g(v),{\mathsf{Leg}}(v)}=\prod_{v^{\prime}\in V_{G^{\prime}}}\prod_{v\in h^{-1}(v^{\prime})}\overline{M}_{g(v),{\mathsf{Leg}}(v)}\rightarrow\prod_{v^{\prime}\in V_{G^{\prime}}}\overline{M}_{g(v^{\prime}),{\mathsf{Leg}}(v^{\prime})}$$

We keep the same framings at marked points, and the same decorations of tangent vectors at nodes corresponding to the edges of $E^{c}_{G\rightarrow G^{\prime}}$. For the decoration at a node corresponding to a contracted edge $e_{\alpha}\in E_{G\rightarrow G^{\prime}}$, we decorate by the unit tangent direction of the vector

$$v_{\alpha}:=d\varphi_{x_{\alpha}}(\frac{d}{dz})\otimes d\varphi_{y_{\alpha}}(\frac{d}{dw})\in T_{x_{\alpha}}\Sigma^{n}\otimes T_{y_{\alpha}}\Sigma^{n},$$

where $\varphi_{x_{\alpha}}:\mathbb{D}_{z}\rightarrow U(x_{\alpha})$ and $\varphi_{y_{\alpha}}:\mathbb{D}_{w}\rightarrow U(y_{\alpha})$ are the two framings at the two legs of the edge $e_{\alpha}$. This defines a map

$$\xi_{G\rightarrow G^{\prime}}:\prod_{v\in V_{G}}\widehat{M}_{g(v),{\mathsf{Leg}}(v)}^{\sf fr}\rightarrow\prod_{v^{\prime}\in V_{G^{\prime}}}\widehat{M}_{g(v^{\prime}),{\mathsf{Leg}}(v^{\prime})}^{\sf fr}$$

which, by taking normalized singular chain functor, induces the desired map $\widehat{\mathbb{M}}^{\sf fr}(h)$.

We define another example of $S^{1}$-equivariant modular operad $\mathbb{M}^{\sf fr}$ by setting

$$\mathbb{M}^{\sf fr}(g,n):=C_{*}(M_{g,n}^{\sf fr})$$

given by the moduli space of smooth curves of type $(g,n)$ together with a framing at each marked point. The composition morphism

$$\mathbb{M}^{\sf fr}(h):\mathbb{M}^{\sf fr}(G)\rightarrow\mathbb{M}^{\sf fr}(G^{\prime})$$

associated to a contraction map $h:G\rightarrow G^{\prime}$ is defined as follows. Again we keep the same framing at the leaves of $G$ and $G^{\prime}$. For a contracted edge $e\in E_{G\rightarrow G^{\prime}}$, we sew the Riemann surfaces at the two ends of $e$ using the two framings $\varphi_{x_{\alpha}}:\mathbb{D}_{z}\rightarrow U(x_{\alpha})$ and $\varphi_{y_{\alpha}}:\mathbb{D}_{w}\rightarrow U(y_{\alpha})$. More precisely, we cut out $U(x_{\alpha})$ and $U(y_{\alpha})$ from the two surfaces, then identify a neighborhood of the boundary circle by the equation $zw=1$. This defines a map

$$\eta_{G\rightarrow G^{\prime}}:\prod_{v\in V_{G}}M_{g(v),{\mathsf{Leg}}(v)}^{\sf fr}\rightarrow\prod_{v\in V_{G^{\prime}}}M_{g(v^{\prime}),{\mathsf{Leg}}(v^{\prime})}^{\sf fr}$$

which induces the desired map $\mathbb{M}^{\sf fr}(h)$.

It is a general fact of the oriented real blowup construction that the natural inclusion map $M_{g,n}^{\sf fr}\hookrightarrow\widehat{M}_{g,n}^{\sf fr}$ is a homotopy equivalence, which induces a homotopy equivalence of chain complexes for each $(g,n)$:

$$I_{g,n}:\mathbb{M}^{\sf fr}(g,n)\hookrightarrow\widehat{\mathbb{M}}^{\sf fr}(g,n).$$

However, observe that the operad structure of $\widehat{\mathbb{M}}^{\sf fr}$ does not restrict to that of $\mathbb{M}^{\sf fr}$ since when composing, the former is by forming a nodal curve while the latter is by sewing using framings.

Let $P$ be an $S^{1}$-equivariant modular operad. We may form the homotopy quotients: for each pair $(g,n)$ we set

$$P(g,n)_{(S^{1})^{n}}=\Big{(}P(g,n)[u_{1}^{-1},\ldots,u_{n}^{-1}],\partial+\sum_{j=1}^{n}B_{j}u_{j}\Big{)}$$

This collection of stable $\mathbb{S}$-module does not form a modular operad. This is evident in the previous two examples: for instance in the case of $\mathbb{M}^{\sf fr}$, the sewing operation of two Riemann surfaces is ambiguously defined if we first quotient out the circle actions on the framings.

Nevertheless, there does exist a degree one composition by allowing a full $S^{1}$-twist of the composition map of $P$. There are two types of compositions depending on whether we compose along a loop edge or a non-loop edge. We begin with the loop edge case as illustrated in Figure $1$.

Indeed, we write down $\rho_{ij}:P(g,n)_{(S^{1})^{n}}\rightarrow P(g+1,n-2)_{(S^{1})^{n-2}}$ by setting

(4)

$$~{}\rho_{ij}(x\cdot u_{1}^{-k_{1}}\cdots u_{n}^{-k_{n}}):=\begin{cases}0&k_{i}\neq 0,\mbox{ or }\;k_{j}\neq 0\\ P(c_{ij})(B_{i}x)u_{1}^{-k_{1}}\cdots u_{n}^{-k_{n}}&k_{i}=k_{j}=0\end{cases}$$

The case of a non-loop edge contraction is similar, see Figure $2$. Explicitly we define a map of chain complexes $\rho_{ij}:P(g,n)_{(S^{1})^{n}}\otimes P(h,m)_{(S^{1})^{m}}\rightarrow P(g+h,n+m-2)_{(S^{1})^{n+m-2}}$ for each $1\leq i\leq n,1\leq j\leq m$ by

(5)

$\displaystyle~{}\begin{split}&\rho_{ij}(x\cdot u_{1}^{-k_{1}}\cdots u_{n}^{-k_{n}},y\cdot v_{1}^{-l_{1}}\cdots v_{m}^{-l_{m}})\\ =&\begin{cases}0&k_{i}\neq 0,\mbox{ or }l_{j}\neq 0\\ P(c_{ij})(B_{i}x\otimes y)u_{1}^{-k_{1}}\cdots u_{n}^{-k_{n}}\cdot v_{1}^{-l_{1}}\cdots v_{m}^{-l_{m}}&k_{i}=l_{j}=0\end{cases}\end{split}$

where $h$ is the composition map of $c_{ij}$ is the graph map that contracts the edge connecting $i$ and $j$.

Generalizing the above, for each stable graph $G$, we put

$$P_{S^{1}}(G):=\bigotimes_{v\in V_{G}}P(g(v),{\sf Leg}(v))_{(S^{1})^{|{\sf Leg}(v)|}}$$

to be the homotopy quotient version of $P(G)$, i.e. on each vertex of $G$ we put the homotopy quotient by the circle action. Furthermore, associated to an edge $e\in G$ we also have the corresponding twisted composition map defined using $\rho_{ij}$’s constructed above:

$$\rho_{e}:P_{S^{1}}(G)\rightarrow P_{S^{1}}(G/e)$$

Note that $\rho_{e}$ is of homological degree one.

We now proceed to define the Feynman compactification $FP$ of an $S^{1}$-framed modular operad $P$. The underlying stable $\mathbb{S}$-module of $FP$ is given by

$$FP(g,n):=\bigoplus_{G\in\Gamma(g,n)}\big{(}D(G)\otimes P_{S^{1}}(G)\big{)}_{{\sf Aut}(G)},$$

where $D(G)$ is the ”cocycle” (see [10, Section 4]) defined by

$$D(G):=\bigotimes_{e\in E_{G}}\mathbb{Q}\cdot D_{e}.$$

The notation $\mathbb{Q}\cdot D_{e}$ stands for the one-dimensional vector space generated by $D_{e}$ at homological degree $2$. Hence $D(G)$ is also one-dimensional, at homological degree $2|E_{G}|$, generated by the tensor product

$$D_{G}:=\bigotimes_{e\in E_{G}}D_{e}.$$

The differential on $FP(g,n)$ is the given by $\partial+\delta$ with $\partial$ the differential of $P_{S^{1}}(G)$, while the extra map $\delta$ is given by

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$\displaystyle~{}\delta(D_{G}\otimes\alpha)=\sum_{e\in E_{G}}D_{G/e}\otimes\rho_{e}(\alpha),\;\;\alpha\in P_{S^{1}}(G).$

Observe that since $\rho_{e}$ has degree one while $D_{e}$ each has degree two, the degree of $\delta$ is $-1$ as required. The idea behind this definition of $\delta$ is that each $D_{e}$ should be thought of as a universal trivialization of the twisted sewing operation along $e$.

Associated with a morphism $h:G\rightarrow G^{\prime}$ of stable graphs, we shall define a composition map $FP(h):FP(G)\rightarrow FP(G^{\prime})$ so that the stable $\mathbb{S}$-module $\{FP(g,n)\}_{(g,n)}$ forms a modular operad. By the structure of modular operads [10, Section 3], it suffices to construct $FP(c_{ij})$ for an edge contraction map $c_{ij}:G\rightarrow G/e$ as in Figure $1$ and Figure $2$. In the case of Figure $1$, by definition of $FP$, we have

	$\displaystyle FP(G)$	$\displaystyle=\bigoplus_{H\in\Gamma_{g,n+2}}\big{(}D(H)\otimes P_{S^{1}}(H)\big{)}_{{\sf Aut}(H)}$
	$\displaystyle FP(G/e)$	$\displaystyle=\bigoplus_{K\in\Gamma_{g+1,n}}\big{(}D(K)\otimes P_{S^{1}}(K)\big{)}_{{\sf Aut}(K)}$

For a stable graph $H\in\Gamma_{g,n+2}$, we denote by $H_{ij}\in\Gamma_{g+1,n}$ the stable graph obtained from $H$ by forming a loop $e$ using the two legs labeled by $i$ and $j$. Observe that $D(H_{ij}/e)\cong D(H)$ since both graph have isomorphic set of edges. Then we set the composition map as

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$\displaystyle~{}\begin{split}&FP(c_{ij})\big{(}D_{H}\otimes xu_{1}^{-k_{1}}\cdots u_{n+2}^{-k_{n+2}}\big{)}\\ =&\begin{cases}D_{H_{ij}}\otimes(x\cdot u_{1}^{-k_{1}}\cdots u_{n+2}^{-k_{n+2}})\cdot(u_{i}+u_{j})&k_{i}\neq 0,\mbox{ or }\;k_{j}\neq 0\\ D_{H_{ij}/e}\otimes P(c_{ij})(x)\cdot u_{1}^{-k_{1}}\cdots u_{n+2}^{-k_{n+2}}&k_{i}=k_{j}=0\end{cases}\end{split}$

{Lemma}

The composition map $FP(c_{ij})$ is a chain map.