A The operadic framework

2 Reminder on cyclic and modular operads

Operads were defined by Boardman–Vogt and May [Reference Boardman and VogtBV68, Reference MayMay72, Reference Boardman and VogtBV73] to describe algebraic structures featuring operations with multiple inputs and one output. For an operad $\mathcal {O}$ with values in a (higher) symmetric monoidal category $\mathcal {S}$ and $n\ge 0$ , we denote by $\mathcal {O}(n)\in \mathcal {S}$ the object of n-ary operations, i.e., operations with n inputs and one output. Through the permutation of the n inputs, the permutation group on n letters acts on $\mathcal {O}(n)$ . There is also an operadic composition $\circ _i : \mathcal {O}(n) \otimes \mathcal {O}(m) \longrightarrow \mathcal {O}(n+m-1)$ inserting an operation of arity m into the i-th slot of an n-ary operation, where $1\le i\le n$ , and an operadic unit in $\mathcal {O}(1)$ acting neutrally with respect to operadic composition. We will not discuss the compatibility of all this data and instead refer e.g., to [Reference FresseFre17, Chapters 1–3].

An operad that will be of particular relevance in this article is the operad $\mathsf {f}E_2$ of framed little disks whose space $\mathsf {f}E_2(n)$ of arity n operations is given by the space of embeddings of n two-dimensional disks into a disk that are composed of translations, rescalings, and rotations. This operad can be defined for disks of any dimension and is an example of an operad that largely motivated the invention of operads in [Reference Boardman and VogtBV68, Reference MayMay72, Reference Boardman and VogtBV73]. Each space $\mathsf {f}E_2(n)$ is aspherical; more precisely, it is the classifying space of the framed braid group on n strands. This implies that we may see $\mathsf {f}E_2$ as a groupoid-valued operad. This statement and an explicit model is discussed in [Reference WahlWah01, Reference Salvatore and WahlSW03].

The $\mathsf {f}E_2$ -operad is an example of a cyclic operad in the sense of Getzler and Kapranov [Reference Getzler, Kapranov, Bott and YauGK95]. This means that it comes with additional structure that allows us to permute the inputs with the output. Seen differently, this means that the distinction between inputs and the output is erased in a consistent way. The cyclic operad $\mathsf {f}E_2$ -operad is equivalent to the operad of oriented genus zero surfaces.

When considering surfaces of all genus, we see that another operadic structure is present, namely, a self-composition of operations that glues two boundary components of a connected surface together. This gives us on the surface operad the structure of a modular operad in the sense of [Reference Getzler and KapranovGK98]. For the surface operad, we will content ourselves throughout with its groupoid-version because we will restrict later to bicategorical algebras. The surface operad $\mathsf {Surf}$ has as its groupoid $\mathsf {Surf}(n)$ of arity n operations the groupoid of surfaces (in this article, these are always compact, oriented, smooth) that are connected and have $n+1$ parametrized boundary components, with the morphisms being isotopy classes of diffeomorphisms preserving the orientation and the boundary parametrizations. Phrased differently, morphisms are mapping classes between surfaces, see [Reference Farb and MargalitFM12] for an introduction to mapping class groups. The gluing of surfaces gives us the operadic composition.

We will now turn to a precise description of cyclic and modular operads, in the context that is needed for this article. To this end, we will use the description from [Reference CostelloCos04] based on categories of graphs. A graph is a set H of half edges, a set V of vertices, a source map $s : H \longrightarrow V$ sending a half edge to the vertex that it is attached to, and a $\mathbb {Z}_2$ -action on H sending a half edge to the half edge that it is glued to. One calls the $\mathbb {Z}_2$ -orbits edges. An orbit with two elements is called internal edge; an orbit with one element is called (external) leg. One defines now the category $\mathsf {Graphs}$ : Objects are finite disjoint unions of corollas, i.e., graphs with one vertex and a finite, possibly empty set of legs glued to it. The morphisms $T\longrightarrow T'$ are equivalence classes of finite graphs (graphs that have finitely many vertices and half edges) with an identification of T with the graph obtained by cutting open all internal edges of and an identification between $T'$ and the graphs constructed by contracting all internal edges of . Here, an identification between graphs is a bijection between the sets of half edges and vertices commuting with the source maps and the $\mathbb {Z}_2$ -actions. Two such triples and are considered equivalent if there is an identification $T\cong T'$ compatible with $\alpha _1,\alpha _2,\alpha _1',\alpha _2'$ . We refer to [Reference CostelloCos04] for the definition of the composition (roughly, it is given by the insertion of graphs into each other). Disjoint union gives us a symmetric monoidal structure.

Inside $\mathsf {Graphs}$ , there is the symmetric monoidal subcategory $\mathsf {Forests}$ with the same objects, but with only those morphisms coming from graphs that are forests, i.e., graphs all of whose connected components are contractible. Now a cyclic operad with values in a symmetric monoidal bicategory $\mathcal {S}$ is defined as a symmetric monoidal functor $\mathcal {O}:\mathsf {Forests} \longrightarrow \mathcal {S}$ while a modular operad with values in $\mathcal {S}$ is defined as a symmetric monoidal functor $\mathcal {O}:\mathsf {Graphs}\longrightarrow \mathcal {S}$ . Here, we understand “symmetric monoidal functor between symmetric monoidal bicategories” ( $\mathsf {Graphs}$ is a 1-category, but can be seen as bicategory) always in the weak sense, i.e., up to coherent homotopy, see [Reference Schommer-PriesSP09, Section 2] for the appropriate framework. The details of bicategorical operads up to coherent homotopy are developed in [Reference Müller and WoikeMW23a, Section 2.1].

The connection to the more traditional description is as follows: Let $\mathcal {O}$ be an $\mathcal {S}$ -valued cyclic or modular operad and T a corolla with $n\ge 0$ legs. Then, $\mathcal {O}(T)\in \mathcal {S}$ is the object of operations of arity $n-1$ (which means $n-1$ inputs and one output; we speak in this case of total arity n). The permutation action on the n legs of T induces a homotopy coherent action of the permutation group on n letters on $\mathcal {O}(T)$ . Since modular operads, as opposed to cyclic ones, are defined on $\mathsf {Graphs}$ , we can evaluate on non-contractible morphisms; this gives us exactly the self-composition of operations.

Note that neither for cyclic nor modular operads, the distinction between inputs and outputs is made; in order to make such a choice, one would have to introduce for each corolla a preferred leg called root. This gives us the category $\mathsf {RForests}$ of rooted forests. A non-cyclic operad is a symmetric monoidal functor $\mathsf {RForests} \longrightarrow \mathcal {S}$ .

With the definition of cyclic and modular operads given so far, operadic identities (operations of total arity two behaving neutrally with respect to composition) are not yet included, but they can be defined directly [Reference Müller and WoikeMW23a, Definition 2.3], and we assume throughout the article that operads (non-cyclic, cyclic or modular) come with operadic identities.

3 Flat vector bundles over moduli spaces: a description using modular operads

For any cyclic or modular operad, we will define in this section the notion of a flat vector bundle over the “moduli space of operations of the operad”. Basically, this amounts to setting up a categorical or rather operadic version of the Riemann–Hilbert correspondence to bridge between the description of spaces of conformal blocks in [Reference Müller and WoikeMW23a, Reference Müller and WoikeMW24b, Reference Brochier and WoikeBW22], the description of correlators in [Reference Fuchs and SchweigertFS17] and the microcosm principle [Reference Baez and DolanBD95].

To this end, recall e.g., from [Reference Mac Lane and MoerdijkMLM92, Section I.5] that for a functor $F:\mathcal {C}\longrightarrow \mathsf {Cat}$ the Grothendieck construction is the category $\int F$ of pairs $(c,x)$ with $c\in \mathcal {C}$ and $x \in F(c)$ . A morphism $(c,x)\longrightarrow (c',x')$ is a pair $(f,\alpha )$ of a morphism $f:c \longrightarrow c'$ and a morphism $\alpha : F(f)x \longrightarrow x'$ . We will also apply the Grothendieck construction to homotopy coherent functors $F:\mathcal {C}\longrightarrow \mathsf {Cat}$ (here $\mathcal {C}$ is still a 1-category, but seen as a bicategory; $\mathsf {Cat}$ is seen as a bicategory). Note that, in that case, their coherence data will enter into the composition of $\int F$ .

This definition is made analogously for cyclic and ordinary operads instead of modular ones by replacing $\mathsf {Graphs}$ with $\mathsf {Forests}$ and $\mathsf {RForests}$ , respectively.

Definition 3.2 (The trivial operad over a modular operad)

For a $\mathsf {Cat}$ -valued modular operad $\mathcal {O}$ (or a cyclic or ordinary operad) and any symmetric monoidal category $\mathcal {K}$ , we define the trivial $\mathcal {O}$ -operad with values in the symmetric monoidal category $\mathcal {K}$ as the symmetric monoidal functor

$$ \begin{align*} \star : \int\mathcal{O}\longrightarrow \mathcal{K} \end{align*} $$

sending a pair $(T,o)$ of a corolla T and an operation $o\in \mathcal {O}(T)$ to the monoidal unit $I\in \mathcal {K}$ and any morphism to the identity of I.

Formally, $\star $ depends on $\mathcal {O}$ and $\mathcal {K}$ , but we suppress this in the notation because it will always be clear from the context.

In the sequel, we denote by $\mathsf {vect}$ the symmetric monoidal category of finite-dimensional vector spaces over our fixed ground field k.

4 Self-dual objects

In this section, we will use self-dual objects inside modular (and cyclic) algebras to construct flat vector bundles over modular (and cyclic) operads. This definition will need, for any symmetric monoidal bicategory $\mathcal {S}$ , the concept of an $\mathcal {S}$ -valued algebra over a cyclic or modular operad $\mathcal {O}$ with values in the category $\mathsf {Cat}$ of categories. The detailed definition is given in [Reference Müller and WoikeMW23a, Section 2.2-2.4]. We recall here the main points: For a non-cyclic operad $\mathcal {O}$ , an algebra structure on some object is a map from $\mathcal {O}$ to the endomorphism operad on that object. For cyclic or modular operads, the endomorphism operad needs to be made cyclic or modular, respectively, by means of a non-degenerate symmetric pairing on the underlying object. For an object $\mathcal {A} \in \mathcal {S}$ in a symmetric monoidal bicategory $\mathcal {S}$ , a non-degenerate symmetric pairing is a map $\kappa : \mathcal {A}\boxtimes \mathcal {A}\longrightarrow I$ (here $\boxtimes $ is the monoidal product of $\mathcal {S}$ , and I is the monoidal unit) that exhibits $\mathcal {A}$ as its own dual in the homotopy category of $\mathcal {S}$ (this is the non-degeneracy) and is equipped with the structure of a homotopy fixed point with respect to the $\mathbb {Z}_2$ -action on $\kappa $ via the symmetric braiding (this is the symmetry). Thanks to non-degeneracy, there exists an essentially unique map $\Delta : I \longrightarrow \mathcal {A}\boxtimes \mathcal {A}$ , the coevaluation map, that together with $\kappa $ satisfies the zigzag identities up to isomorphism. This allows us to define a symmetric monoidal functor $ \mathsf {End}_\kappa ^{\mathcal {A}}: \mathsf {Graphs} \longrightarrow \mathsf {Cat}$ via $\mathsf {End}_\kappa ^{\mathcal {A}}(T):= \mathcal {S}( \mathcal {A}^{\boxtimes \mathsf {Legs}(T)} , I)$ for any corolla T, the endomorphism operad of $(\mathcal {A},\kappa )$ , a $\mathsf {Cat}$ -valued modular operad that can be seen as cyclic operad if needed. Here, $\mathcal {A}^{\boxtimes \mathsf {Legs}(T)}$ is the unordered monoidal product of $\mathcal {A}$ with index set $\mathsf {Legs}(T)$ . The definition of $\mathsf {End}_\kappa ^{\mathcal {A}}$ on morphisms uses the coevaluation. We may now define the structure of a cyclic or modular $\mathcal {O}$ -algebra on $\mathcal {A}\in \mathcal {S}$ as the choice of a non-degenerate symmetric pairing $\kappa $ on $\mathcal {A}$ and a map $\mathcal {O}\longrightarrow \mathsf {End}_\kappa ^{\mathcal {A}}$ of $\mathsf {Cat}$ -valued cyclic or modular operads. We note that bicategorical cyclic or modular algebras over a cyclic or modular operad form a 2-groupoid [Reference Müller and WoikeMW23a, Proposition 2.18].

Our main example for the symmetric monoidal bicategory $\mathcal {S}$ is the symmetric monoidal bicategory $\mathsf {Lex}^{\mathsf {f}}$ of finite linear categories (or just finite categories) over our algebraically closed field k that we fix throughout:

• • The objects are finite k-linear categories [Reference Etingof and OstrikEO04, Reference Etingof, Gelaki, Nikshych and OstrikEGNO15], i.e., linear abelian categories with finite-dimensional morphism spaces, finitely many isomorphism classes of simple objects, enough projective objects and finite length for every object.

• • The 1-morphisms are left exact functors, i.e., functors preserving finite limits.

• • The 2-morphisms are natural linear transformations.

The monoidal product of $\mathsf {Lex}^{\mathsf {f}}$ is the Deligne product with monoidal unit $\mathsf {vect}$ , the category of finite-dimensional vector spaces over our fixed algebraically closed field k.

If $\mathcal {O}$ is a $\mathsf {Cat}$ -valued modular operad, a $\mathsf {Lex}^{\mathsf {f}}$ -valued modular $\mathcal {O}$ -algebra has an underlying finite category $\mathcal {A} \in \mathsf {Lex}^{\mathsf {f}}$ with a non-degenerate symmetric pairing $\kappa : \mathcal {A}\boxtimes \mathcal {A} \longrightarrow \mathsf {vect}$ . Since $\kappa $ is left exact, it induces an equivalence $D:\mathcal {A} \longrightarrow \mathcal {A}^{\mathrm {op}}$ determined by $\kappa (X,Y)\cong \mathcal {A}(DX,Y)$ . The equivalence D, also referred to as duality functor, comes with an isomorphism $D^2 \cong \mathrm {id}_{\mathcal {A}}$ thanks to the symmetry of $\kappa $ . The coevaluation $\Delta : \mathsf {vect} \longrightarrow \mathcal {A}\boxtimes \mathcal {A}$ can be identified with an object in $\mathcal {A}\boxtimes \mathcal {A}$ , namely, the coend $\Delta = \int ^{X \in \mathcal {A}} DX \boxtimes X$ . For an introduction to coends in finite categories, we refer to [Reference Fuchs, Schaumann and SchweigertFSS20]. For each operation $o\in \mathcal {O}(T)$ for a corolla T, the map of modular operads from $\mathcal {O}$ to the endomorphism operad of $(\mathcal {A},\kappa )$ gives us a left exact functor $\mathcal {A}_o :\mathcal {A}^{\boxtimes \mathsf {Legs}(T)}\longrightarrow \mathsf {vect}$ . We will not spell out here the compatibility with gluing and instead refer to [Reference Müller and WoikeMW23a, Section 2].

We will now prepare the definition of a self-dual object. First, recall that, for a symmetric monoidal bicategory $\mathcal {S}$ and $\mathcal {A}\in \mathcal {S}$ , a (generalized) object in $\mathcal {A}$ is a 1-morphism $X:I\longrightarrow \mathcal {A}$ . A generalized object in a finite category $\mathcal {A}\in \mathsf {Lex}^{\mathsf {f}}$ is a left exact functor $X:\mathsf {vect} \longrightarrow \mathcal {A}$ . By additivity, this functor is determined by its value $X(k)$ on the ground field k. Therefore, it amounts to an object in $\mathcal {A}$ in the usual sense.

Definition 4.5 (Self-dual object)

Let $\mathcal {A}$ be an object in $\mathcal {S}$ with a non-degenerate symmetric pairing by $\kappa : \mathcal {A}\boxtimes \mathcal {A}\longrightarrow I$ and coevaluation $\Delta : I\longrightarrow \mathcal {A}\boxtimes \mathcal {A}$ . A self-dual object in $\mathcal {A}$ is an object $X:I\longrightarrow \mathcal {A}$ together with a symmetric 2-pairing $\beta : X\boxtimes X \longrightarrow \Delta $ that is non-degenerate in the sense that there exists a map $\delta : \mathrm {id}_I \longrightarrow \kappa \circ (X\boxtimes X)$ , called symmetric 2-copairing, such that

$$ \begin{align*} X \xrightarrow{\ \delta \boxtimes X \ } \kappa (X\boxtimes X) \boxtimes X \xrightarrow{\ X \boxtimes \beta \ } (\kappa \boxtimes \mathrm{id}) (X \boxtimes \Delta) \cong X \end{align*} $$

and

$$ \begin{align*} X \xrightarrow{\ X \boxtimes \delta \ } X \boxtimes \kappa (X \boxtimes X) \xrightarrow{\ \beta \boxtimes X \ } (\mathrm{id}\boxtimes \kappa) (\Delta \boxtimes X) \cong X \end{align*} $$

are the identity of X.

The main result of this section is that self-dual objects produce flat vector bundles.

Proposition 4.7 Let $\mathcal {O}$ be a category-valued modular operad and $\mathcal {A}$ a modular $\mathcal {O}$ -algebra in a symmetric monoidal bicategory $\mathcal {S}$ . Moreover, let $X:I\longrightarrow \mathcal {A}$ be a self-dual object and $\beta : X \boxtimes X \longrightarrow \Delta $ its symmetric 2-pairing, where $\Delta $ is the coevaluation object for $\mathcal {A}$ . Then,

(4.1) $$ \begin{align} \mathbb{V}_X^{\mathcal{A}} : \int \mathcal{O} \longrightarrow \mathcal{S}(I,I) \ , \quad (T,o) \longmapsto \mathcal{A}_o\circ X^{\boxtimes \mathsf{Legs}(T)} \quad \text{for a corolla}\ T \end{align} $$

is a symmetric monoidal functor, i.e., an $\mathcal {O}$ -operad with values in $\mathcal {S}(I,I)$ .

If $\mathcal {S}=\mathsf {Lex}^{\mathsf {f}}$ , then $\mathcal {S}(I,I)=\mathsf {vect}$ , so the construction produces flat vector bundles over $\mathcal {O}$ in that case. The functor $\mathbb {V}_X^{\mathcal {A}}$ depends on $\beta $ , but this is suppressed in the notation because we see the pairing as part of the self-dual object X.

The proof of Proposition 4.7 will make use of the calculus construction from [Reference Müller and WoikeMW23a, Section 6] that we briefly recall now; for details, we refer to the original article: For the non-degenerate symmetric pairing $\kappa : \mathcal {A} \boxtimes \mathcal {A}\longrightarrow I$ of $\mathcal {A}$ , there is a category $\mathcal {E}_\kappa ^{\mathcal {A}}$ of pairs of $T=\sqcup _{j\in J} T^{(j)} \in \mathsf {Graphs}$ (J is a finite set, and the $T^{(j)}$ are corollas) and families $(Z_j)_{j\in J}$ of 1-morphisms $Z_j : I \longrightarrow \mathcal {A}^{\boxtimes \mathsf {Legs} \left ( T^{(j)}\right )}$ . A morphism

is a morphism in $\mathsf {Graphs}$ and for any $j\in J$ a 2-morphism between 1-morphisms $I \longrightarrow \boxtimes _{\ell \in L_j} \mathcal {A}^{\boxtimes \mathsf {Legs} \left (S^{(\ell )}\right )}$ , where $L_j$ is the preimage of j under the map $L \longrightarrow J$ induced by , and is defined via

The category $\mathcal {E}_\kappa ^{\mathcal {A}}$ comes with a projection $\mathcal {E}_\kappa ^{\mathcal {A}}\longrightarrow \mathsf {Graphs}$ . The pullback $\left (\int \mathcal {O}\right )\times _{\mathsf {Graphs}} \mathcal {E}_\kappa ^{\mathcal {A}}$ (it is also a homotopy pullback because $\int \mathcal {O}\longrightarrow \mathsf {Graphs}$ is a Grothendieck fibration) is symmetric monoidal and comes with a symmetric monoidal functor

(4.2) $$ \begin{align} \mathsf{Calc} _{\mathcal{A}}: \left(\int \mathcal{O}\right)\times_{\mathsf{Graphs}} \mathcal{E}_\kappa^{\mathcal{A}} \longrightarrow \mathcal{S}(I,I)\ , \end{align} $$

the so-called calculus functor sending

$$ \begin{align*}(T,O,Z)=\left( \sqcup_{j\in J} T^{(j)}, \left(o_j \in \mathcal{O}(T^{(j)})\right)_{j\in J} , (Z_j)_{j\in J} \right)\in \left(\int \mathcal{O}\right)\times_{\mathsf{Graphs}} \mathcal{E}_\kappa^{\mathcal{A}} \end{align*} $$

to

$$ \begin{align*} \mathsf{Calc} _{\mathcal{A}}(T,O,Z)= \bigotimes_{j\in J} \mathcal{A}_{o_j} \circ Z_j . \end{align*} $$

Proof of Proposition 4.7

First let us define a symmetric monoidal functor $K: \int \mathcal {O} \longrightarrow \mathcal {E}_\kappa ^{\mathcal {A}}$ . It sends a corolla T and $o\in \mathcal {O}(T)$ to $(T, X^{\boxtimes \mathsf {Legs}(T)})$ . Let now

be a morphism in $\int \mathcal {O}$ (we can without loss of generality assume that T is a corolla). We define to be the morphism in $\mathsf {Graphs}$ together with a 2-morphism (to be specified) running from $ \boxtimes _{\ell \in L} X^{\boxtimes \mathsf {Legs} \left (S^{(\ell )}\right )}$ to the same object but with each copy of $X\boxtimes X$ belonging to a pair of edges collapsed by being replaced with $\Delta $ . Therefore, we can use $\beta : X \boxtimes X \longrightarrow \Delta $ to define K on morphisms. It is a straightforward verification that K is in fact a symmetric monoidal functor.

Now by the universal property of the (homotopy) pullback $\left (\int \mathcal {O}\right )\times _{\mathsf {Graphs}} \mathcal {E}_\kappa ^{\mathcal {A}}$ the functors $K: \int \mathcal {O}\longrightarrow \mathcal {E}_\kappa ^{\mathcal {A}}$ and $\mathrm {id}:\int \mathcal {O}\longrightarrow \int \mathcal {O}$ induce a functor $\int \mathcal {O}\longrightarrow \left (\int \mathcal {O}\right )\times _{\mathsf {Graphs}} \mathcal {E}_\kappa ^{\mathcal {A}}$ that is also symmetric monoidal. We restrict the calculus functor (4.2) along this functor and find the desired symmetric monoidal functor (4.1).

We finish this section by characterizing self-dual objects in a finite category explicitly.

Lemma 4.11 Let $\mathcal {A} \in \mathsf {Lex}^{\mathsf {f}}$ be a finite category with non-degenerate symmetric pairing $\kappa $ , moreover $D:\mathcal {A}\longrightarrow \mathcal {A}^{\mathrm {op}}$ the equivalence defined via $\kappa (X,Y)\cong \mathcal {A}(DX,Y)$ for all $X,Y \in \mathcal {A}$ which comes with a natural isomorphism $\omega : \mathrm {id}_{\mathcal {A}}\longrightarrow D^2$ induced by the symmetry isomorphisms of $\kappa $ . Then, the structure of a self-dual object on $X\in \mathcal {A}$ is an isomorphism $\psi : X \longrightarrow DX$ identifying X with its dual $DX$ such that

$$ \begin{align*} X \xrightarrow{\ \omega_X \ } D^2 X \xrightarrow{\ D\psi \ } DX \end{align*} $$

coincides with $\psi $ .

5 The cyclic and modular microcosm principle

The microcosm principle developed by Baez–Dolan [Reference Baez and DolanBD98, Section 4.3] says, roughly, that in order to be able to define an algebra A over an operad $\mathcal {O}$ inside a category $\mathcal {A}$ , the category $\mathcal {A}$ needs to be itself an algebra over $\mathcal {O}$ . In this situation, $\mathcal {A}$ is called the macrocosm in which the microcosm A lives. For instance, in the case $\mathcal {O}=\mathsf {As}$ , this tells us that in order to define what an associative algebra A in a category $\mathcal {A}$ should be, we need to equip $\mathcal {A}$ with a monoidal structure. Then, we can define what it means for $A \in \mathcal {A}$ to be an associative algebra relative to the monoidal structure of $\mathcal {A}$ .

The following precise definition of the microcosm principle builds on the original definition of Baez–Dolan, but includes cyclic and modular operads into the picture using the framework developed for the latter in [Reference CostelloCos04, Reference Müller and WoikeMW23a] and, of course, self-dual objects and their flat vector bundles. Recall that in order to define categorical algebras over cyclic and modular operads, we needed a non-degenerate symmetric pairing to make the endomorphism operad cyclic. In other words, the macrocosm needs a non-degenerate symmetric pairing. It will not come as surprise that also on the microcosmic level we will need a non-degenerate symmetric pairing relative to the macrocosmic pairing.

Definition 5.1 (Modular $\mathcal {O}$ -algebra with coefficients — Modular Microcosm Principle)

Let $\mathcal {O}$ be a $\mathsf {Cat}$ -valued modular operad and $\mathcal {A}$ an $\mathcal {S}$ -valued modular $\mathcal {O}$ -algebra. A modular $\mathcal {O}$ -algebra (with coefficients) in $\mathcal {A}$ (or relative to $\mathcal {A}$ ) is

• • a self-dual object $X:I\longrightarrow \mathcal {A}$ in $\mathcal {A}$ (Definition 4.5),

• • together with a monoidal transformation,

  

satisfying the unitality condition that the component

$$ \begin{align*} \xi_{(T_1,1_{\mathcal{O}})} : \star \longrightarrow \mathbb{V}_X^{\mathcal{A}} (1_{\mathcal{O}}) = \mathcal{A}_{1_{\mathcal{O}}} \circ (X\boxtimes X) \cong \kappa \circ (X\boxtimes X) \end{align*} $$

selects the 2-copairing of the self-dual object; here $1_{\mathcal {O}}$ is the operadic identity, and the isomorphism $\mathcal {A}_{1_{\mathcal {O}}}\cong \kappa $ is part of the data of $\mathcal {A}$ [Reference Müller and WoikeMW23a, Definition 2.13].

Remark 5.4 For any $o\in \mathcal {O}(T)$ , any morphism in $\mathsf {Graphs}$ induces a morphism in $\int \mathcal {O}$ that we just denote by again. For an $\mathcal {O}$ -algebra $\xi :\star \longrightarrow \mathbb {V}_X^{\mathcal {A}}$ with coefficients in $\mathcal {A}$ , the naturality of $\xi $ implies that the diagram

on $\mathcal {S}(I,I)$ commutes. This implies that $\xi $ is already determined by those components $\xi _{(T,o)}$ on operations $o\in \mathcal {O}(T)$ that generate the operations of $\mathcal {O}$ under operadic composition. In fact, we can further reduce this to the case where T is not an arbitrary object in $\mathsf {Graphs}$ , but a corolla. This is because of the monoidality of $\xi $ .

6 Microcosmic modular extension

In the subsequent sections, our goal will be to apply the modular microcosm principle in relevant cases. This means that, for a given modular operad $\mathcal {O}$ of interest and a modular algebra $\mathcal {A}$ over it, we would like to classify all modular $\mathcal {O}$ -algebras in $\mathcal {A}$ . This is a rather non-trivial task for which we need to establish an important technical result in this section.

First, recall that for a $\mathsf {Cat}$ -valued cyclic operad $\mathcal {O}:\mathsf {Forests} \longrightarrow \mathsf {Cat}$ , one can define following [Reference CostelloCos04] the modular envelope of $\mathcal {O}$ , a modular operad $\mathsf {U}_{\!\!\tiny \int } \mathcal {O}:\mathsf {Graphs} \longrightarrow \mathsf {Cat}$ sending $T\in \mathsf {Graphs}$ to the Grothendieck construction

$$ \begin{align*} \mathsf{U}_{\!\!\tiny\int} \mathcal{O}(T):= \int \left( \mathsf{Forests} / T \longrightarrow \mathsf{Forests} \xrightarrow{\ \mathcal{O} \ } \mathsf{Cat} \right), \end{align*} $$

where $\mathsf {Forests} / T$ is the slice category of the inclusion $\mathsf {Forests} \longrightarrow \mathsf {Graphs}$ over T, and $\mathsf {Forests} / T \longrightarrow \mathsf {Forests}$ is the forgetful functor. Intuitively, $\mathsf {U}_{\!\!\tiny \int } \mathcal {O}$ is the modular operad obtained by completing the cyclic operad $\mathcal {O}$ , in a homotopically correct way, to a modular operad. The version of the modular envelope used here, while still largely following [Reference CostelloCos04], is slightly adapted to $\mathsf {Cat}$ -valued operads and their bicategorical algebras (see [Reference Müller and WoikeMW23a, Section 7.1]. In particular, the construction is made such that a cyclic $\mathcal {O}$ -algebra gives rise to a modular $\mathsf {U}_{\!\!\tiny \int } \mathcal {O}$ -algebra $\widehat {\mathcal {A}}$ that is called the modular extension of $\mathcal {A}$ . On an operation , it is given by image of $\mathcal {A}_o \in \prod _{\ell \in L} \mathcal {S}\left (\mathcal {A}^{ \boxtimes \mathsf {Legs} \left ( \widetilde T^{(\ell )} \right ) },I\right )$ under the functor

obtained by the evaluation of the endomorphism operad for $\mathcal {A}$ on .

The following is the main technical result of this article:

Theorem 6.1 (Microcosmic version of modular extension)

For a $\mathsf {Cat}$ -valued cyclic operad $\mathcal {O}$ and a cyclic $\mathcal {O}$ -algebra $\mathcal {A}$ in a symmetric monoidal bicategory $\mathcal {S}$ , there is a canonical pair of inverse equivalences

between cyclic $\mathcal {O}$ -algebras in $\mathcal {A}$ and modular -algebras in the modular extension $\widehat {\mathcal {A}}$ .

Here, $B:\mathsf {Cat} \longrightarrow \mathsf {sSet}$ is the nerve, $|-|:\mathsf {sSet} \longrightarrow \mathsf {Top}$ the geometric realization and the fundamental groupoid.

Remark 6.2 The modular extension $\widehat {\mathcal {A}}$ of $\mathcal {A}$ is a modular $\mathsf {U}_{\!\!\tiny \int } \mathcal {O}$ -algebra sending all morphisms in the categories of operations to isomorphisms (see [Reference Müller and WoikeMW23a, Remark 7.2]. Therefore, it can be seen as a modular algebra over (the localization of $\mathsf {U}_{\!\!\tiny \int } \mathcal {O}$ at all morphisms in the categories of operations). This however does not influence the notion of a modular algebra with coefficients in $\widehat {\mathcal {A}}$ . In other words, , so that we can rephrase Theorem 6.1 equivalently as an equivalence

(6.1)

Proof of Theorem 6.1

By Remark 6.2 we can equivalently prove the equivalence (6.1).

We start by explaining, why there is a restriction functor $\mathsf {ModAlg}( \mathsf {U}_{\!\!\tiny \int } \mathcal {O}; \widehat {\mathcal {A}}) \longrightarrow \mathsf {CycAlg}(\mathcal {O};\mathcal {A})$ : First, we observe that being a self-dual object $X: I \longrightarrow \mathcal {A}$ in $\mathcal {A}$ is the same as being a self-dual object $X:I\longrightarrow \widehat {\mathcal {A}}$ in $\widehat {\mathcal {A}}$ because $\mathcal {A}$ and its modular extension $\widehat {\mathcal {A}}$ have the same underlying object in $\mathcal {S}$ and the same non-degenerate symmetric pairing. Moreover, there is a symmetric monoidal functor $J: \int \mathcal {O} \longrightarrow \int \mathsf {U}_{\!\!\tiny \int } \mathcal {O}$ sending $(T,o)$ with $o\in \mathcal {O}(T)$ to $(T,\mathrm {id}_T,o) \in \int \mathsf {U}_{\!\!\tiny \int } \mathcal {O}$ . Clearly, for the flat vector bundles from Proposition 4.7

(6.2) $$ \begin{align} \mathbb{V}_X^{\widehat{\mathcal{A}}} \circ J=\mathbb{V}_X^{\mathcal{A}} . \end{align} $$

The restriction functor

(6.3) $$ \begin{align} J^*: \mathsf{ModAlg}\left( \mathsf{U}_{\!\!\tiny\int} \mathcal{O}; \widehat{\mathcal{A}}\right) \longrightarrow \mathsf{CycAlg}(\mathcal{O} ;\mathcal{A}) \end{align} $$

takes an object $\xi \in \mathsf {ModAlg}( \mathsf {U}_{\!\!\tiny \int } \mathcal {O}; \widehat {\mathcal {A}})$ , i.e., a monoidal transformation $\xi : \star \longrightarrow \mathbb {V}_X^{\widehat {\mathcal {A}}}$ for some self-dual object X in $\widehat {\mathcal {A}}$ , and restricts along J. Since $\star \circ J=\star $ and (6.2), this produces a monoidal transformation $\star \longrightarrow \mathbb {V}_X^{\mathcal {A}}$ . After the straightforward verification that this restriction preserves the unitality condition from Definition 5.1, we obtain an object in $\mathsf {CycAlg}(\mathcal {O};\mathcal {A})$ . This establishes the restriction functor (6.3).

Now, suppose that we are given a cyclic algebra $\xi : \star \longrightarrow \mathbb {V}_X^{\mathcal {A}}$ with coefficients in $\mathcal {A}$ . An operation in $\mathsf {U}_{\!\!\tiny \int } \mathcal {O}(T')$ , where $T'$ is without loss of generality a corolla, is a morphism for some $T\in \mathsf {Graphs}$ and an operation $o\in \mathcal {O}(T)$ . We decompose T into corollas, $T = \sqcup _{\ell \in L} T^{(\ell )}$ , so that o corresponds to a family $(o_\ell )_{\ell \in L} \in \prod _{\ell \in L} \mathcal {O}\left (T^{(\ell )}\right ) \simeq \mathcal {O}(T)$ of operations. The graph provides a morphism in $\int \mathsf {U}_{\!\!\tiny \int } \mathcal {O}$ , again denoted by by slight abuse of notation. We now define $\widehat \xi : \star \longrightarrow \mathbb {V}_X^{\widehat {\mathcal {A}}}$ as the transformation whose component at is given by the commutativity of the diagram

(6.4)

This makes $\widehat \xi $ natural, and we extend it monoidally. Moreover, $\widehat \xi $ clearly extends $\xi $ , i.e., $J^* \widehat \xi = \xi $ . In fact, any extension of $\xi $ to a monoidal natural transformation $\widehat \xi : \star \longrightarrow \mathbb {V}_X^{\widehat {\mathcal {A}}}$ must make the triangle (6.4) commute as follows from Remark 5.4; in other words, $\widehat \xi $ is the unique extension. If we now define an extension functor $E: \mathsf {CycAlg}(\mathcal {O};\mathcal {A}) \longrightarrow \mathsf {ModAlg}( \mathsf {U}_{\!\!\tiny \int } \mathcal {O}; \widehat {\mathcal {A}}) $ by $E \xi := \widehat \xi $ (the definition clearly extends to morphisms), then $J^* \circ E \cong \mathrm {id}_{ \mathsf {CycAlg}(\mathcal {O};\mathcal {A}) }$ because E provides, as just discussed, an extension, and $E \circ J^* \cong \mathrm {id}_{ \mathsf {ModAlg}( \mathsf {U}_{\!\!\tiny \int } \mathcal {O}; \widehat {\mathcal {A}}) }$ because this extension is unique.

7 Cyclic associative algebras

In this section, we apply the cyclic microcosm principle to the associative operad, the cyclic $\mathsf {Set}$ -valued operad $\mathsf {As}$ sending a corolla T to the set of cyclic orders on $\mathsf {Legs}(T)$ . If T has $n+1$ legs, $\mathsf {As}(T)$ can be non-canonically identified with the permutation group on n letters. Of course, we can see $\mathsf {As}$ as $\mathsf {Cat}$ -valued cyclic operad. The reader will not see any surprises in this section because cyclic associative algebras will have to amount to symmetric Frobenius algebras, with the right definition of symmetric Frobenius algebra that we will give below in Definition 7.2. In that regard, this section is rather a sanity check for the cyclic microcosm principle set up above.

By [Reference Müller and WoikeMW23a, Theorem 4.12] cyclic $\mathsf {As}$ -algebras in $\mathsf {Lex}^{\mathsf {f}}$ are pivotal Grothendieck–Verdier categories in $\mathsf {Lex}^{\mathsf {f}}$ , as defined by Boyarchenko–Drinfeld in [Reference Boyarchenko and DrinfeldBD13], using the notion of $\star $ -autonomous categories [Reference BarrBar79]. A Grothendieck–Verdier category in $\mathsf {Lex}^{\mathsf {f}}$ (note that our conventions are dual to the ones in [Reference Boyarchenko and DrinfeldBD13]) is a monoidal category $\mathcal {A}$ in $\mathsf {Lex}^{\mathsf {f}}$ with monoidal product $\otimes :\mathcal {A}\boxtimes \mathcal {A}\longrightarrow \mathcal {A}$ (this is exactly a non-cyclic associative algebra structure) together with an object $K\in \mathcal {A}$ , the dualizing object, such that the functors $\mathcal {A}(K,X\otimes -)$ become representable via $\mathcal {A}(K,X\otimes -)\cong \mathcal {A}(DX,-)$ with $D:\mathcal {A}\longrightarrow \mathcal {A}^{\mathrm {op}}$ being an equivalence. The functor D is called Grothendieck–Verdier duality; it sends the monoidal unit $I\in \mathcal {A}$ to K. Moreover, there are canonical isomorphisms $D^2I\cong I$ and $D^2 K\cong K$ . A pivotal structure on a Grothendieck–Verdier category is a monoidal isomorphism $\omega :\mathrm {id}_{\mathcal {A}}\cong D^2$ whose component at K is the canonical isomorphism $D^2 K\cong K$ . Under the equivalence from pivotal Grothendieck–Verdier categories in $\mathsf {Lex}^{\mathsf {f}}$ with cyclic $\mathsf {As}$ -algebras in $\mathsf {Lex}^{\mathsf {f}}$ , a pivotal Grothendieck–Verdier category $\mathcal {A}$ needs to be equipped with a non-degenerate pairing. This pairing is $\kappa (-,-)=\mathcal {A}(D-,-)$ with its symmetry isomorphism coming from the pivotal structure. A pivotal structure can be equivalently described through isomorphisms $\psi _{X,Y}:\mathcal {A}(K,X \otimes Y)\cong \mathcal {A}(K,Y\otimes X)$ squaring to the identity and subject to a cocycle condition [Reference Boyarchenko and DrinfeldBD13, Section 6].

We should point out that the Grothendieck–Verdier duality can be, but need not be a rigid duality. In the case of a rigid duality, D sends $X \in \mathcal {A}$ to an object $X^\vee $ for which we can find a morphism $d:X^\vee \otimes X \longrightarrow I$ as well as a morphism $b: I\longrightarrow X\otimes X^\vee $ such that the zigzag identities are satisfied. If the Grothendieck–Verdier duality of a pivotal Grothendieck–Verdier category in $\mathsf {Lex}^{\mathsf {f}}$ is rigid and if its unit is simple, it is a pivotal finite tensor category in the sense of [Reference Etingof and OstrikEO04, Reference Etingof, Gelaki, Nikshych and OstrikEGNO15]. Note that in principle, one can define two different versions of a rigid duality, a left, and a right one. For pivotal finite tensor categories, these canonically agree, so we refrain from making this distinction. It should be noted that, even in Grothendieck–Verdier categories, generalizations of the evaluation and coevaluation exist, but these satisfy substantially weaker properties (see [Reference Cockett and SeelyCS97, Theorem 4.5] and [Reference Fuchs, Schaumann, Schweigert and WoodFSSW24, Definition 3.33]).

By [Reference Boyarchenko and DrinfeldBD13, Section 3.1] a Grothendieck–Verdier category $\mathcal {A}$ has a second monoidal product defined via

(7.1) $$ \begin{align} X \odot Y := D^{-1}(DY \otimes DX) \end{align} $$

for all $X,Y \in \mathcal {A}$ . This is not necessarily a monoidal product in $\mathsf {Lex}^{\mathsf {f}}$ , but it is right exact. If D is rigid, then $\odot \cong \otimes $ .

With this second monoidal product, we can characterize self-dual objects in a cyclic associative algebra in $\mathsf {Lex}^{\mathsf {f}}$ . We find a structure that is dual to the notion of a Grothendieck–Verdier pairing in [Reference Fuchs, Schaumann, Schweigert and WoodFSSW24, Definition 4.7].

Lemma 7.1 (Characterization of self-dual objects with coefficients in a cyclic associative algebra / pivotal Grothendieck–Verdier category)

Let $\mathcal {A}$ be a pivotal Grothendieck–Verdier category in $\mathsf {Lex}^{\mathsf {f}}$ . The structure of a self-dual object on some $X\in \mathcal {A}$ amounts exactly to morphisms $\beta : X \odot X \longrightarrow I$ and $\delta : K \longrightarrow X \otimes X$ such that $\delta $ is a fixed point of the $\mathbb {Z}_2$ -action on $\mathcal {A}(K,X\otimes X)$ via the pivotal structure and such that

$$ \begin{align*} \mathcal{A}(K,X \otimes X)\otimes \mathcal{A}(X\odot X,I)\cong \mathcal{A}(DX,X)\otimes \mathcal{A}(X,DX) \xrightarrow{\ \circ_X \ } \mathcal{A}(DX,DX) \end{align*} $$

sends $\delta \otimes \beta $ to $\mathrm {id}_{DX}$ while

$$ \begin{align*} \mathcal{A}(X\odot X,I) \otimes \mathcal{A}(K,X \otimes X)\cong \mathcal{A}(X,DX) \otimes \mathcal{A}(DX,X) \xrightarrow{\ \circ_{DX} \ } \mathcal{A}(X,X) \end{align*} $$

sends $\beta \otimes \delta $ to $\mathrm {id}_X$ .

Justified by this result, we will refer to a map $\beta : X \odot X \longrightarrow I$ as in Lemma 7.1 as a non-degenerate symmetric pairing on X; we call $\delta : K \longrightarrow X \otimes X$ the associated non-degenerate symmetric copairing.

Proof We have $\kappa (X,X)\cong \mathcal {A}(K,X \otimes X)$ , and the symmetry isomorphism of $\kappa $ amounts to the $\mathbb {Z}_2$ -action on $\mathcal {A}(K,X \otimes X)$ by the pivotal structure of D. Similarly,

$$ \begin{align*} (\mathcal{A}\boxtimes \mathcal{A})(X \boxtimes X,\Delta)&=\mathcal{A}(X,\Delta')\otimes\mathcal{A}(X,\Delta") \qquad \text{with Sweedler notation } \Delta = \Delta' \boxtimes\Delta"\\& \cong \kappa(DX,\Delta')\otimes\kappa(DX,\Delta")\\ &\cong \kappa(DX,DX)\qquad [{MW23a},\ \mathrm{Remark}\ 2.25]\\ &\cong \mathcal{A}(D^2X,DX)\\ &\cong \mathcal{A}(K,DX \otimes DX)\\ & \cong \mathcal{A}(D(DX\otimes DX),I) \\ &\cong \mathcal{A}(X\odot X,I) . \end{align*} $$

From this, we can deduce that the description given in Definition 4.5 is equivalent the one given in the Lemma.

With Lemma 7.1, we are in the position to characterize cyclic associative algebras relative to cyclic associative algebras in $\mathsf {Lex}^{\mathsf {f}}$ , i.e., cyclic associative algebras in pivotal Grothendieck–Verdier categories. Needless to say, we expect some sort of Frobenius algebras, but we should highlight that the fact that these are the type of algebras that the cyclic microcosm principle produces is a statement that requires a proof. Viewed differently, it is of course also a test that the cyclic microcosm principle in this article is set up the way it should be.

Symmetric Frobenius algebras can be defined in pivotal rigid monoidal categories [Reference Fuchs and StignerFS08]; the fact that they can be defined through the microcosm principle in $\star $ -autonomous categories is mentioned in [nLa17, Section 2] (see, however, the warning in Remark 7.4 below). The notion also appears in [Reference EggerEgg10, Definition 2.3.2] and [Reference Fuchs, Schaumann, Schweigert and WoodFSSW24, Section 4.1]. For us, the following notion will turn out to be the “correct” one (where “correct” has the precise meaning that Proposition 7.3, that we will state afterwards, holds).

B Applications in quantum topology and conformal field theory

8 Classification of open correlators

We now turn to the applications in open conformal field theory. We denote by $\mathsf {O}$ the modular open surface operad, a groupoid-valued operad whose groupoid of operations of arity n (to be thought of as n inputs and one output) has as objects connected compact oriented surfaces with at least one boundary component and $n+1$ parametrized intervals embedded in its boundary. Morphisms of $\mathsf {O}(n)$ are mapping classes of diffeomorphisms between these surfaces preserving the orientation and the boundary parametrization. References for a more detailed definition of $\mathsf {O}$ include [Reference CostelloCos07b, Reference GiansiracusaGia11]; here we follow the definitions in [Reference Müller, Schweigert, Woike and YangMSWY23, Section 3], where the open-closed case is considered, and also [Reference Müller and WoikeMW24a].

This is in line with the notion of consistent systems of correlators in [Reference Fuchs and SchweigertFS17, Reference Fuchs, Schweigert, Jurčo, Sämann, Schreiber and WolfFS19] (that are also defined via monoidal natural transformations – so the connection is not surprising), but here we consider the open version. Let us unpack the data: Denote by $\mathcal {A}$ the underlying category for the open modular functor $\mathfrak {B}$ (one might call this the category of boundary labels). For a corolla T, and $X_i \in \mathcal {A}$ for $1\le i\le n:=|\mathsf {Legs}(T)|$ , the vector space is the space of conformal blocks for the surface and boundary labels $X_1,\dots ,X_n$ . A consistent system of open correlators $\xi \in \mathsf {ModAlg}(\mathsf {O};\mathfrak {B})$ has an underlying object $F\in \mathcal {A}$ and vectors in the spaces of conformal blocks that are invariant under the mapping class group actions and compatible with sewing along boundary intervals.

Corollary 8.2 For any cyclic associative algebra $\mathcal {A}$ in $\mathcal {S}$ , there is an equivalence

where $\mathcal {A}_{\text { !}}$ denotes the modular extension of the cyclic $\mathsf {As}$ -algebra $\mathcal {A}$ .

Proof Every open modular functor $\mathfrak {B}$ is equivalent to the modular extension of its restriction to disks with marked intervals, which is a cyclic $\mathsf {As}$ -algebra $\mathcal {A}$ , i.e., a pivotal Grothendieck–Verdier category (see [Reference Müller and WoikeMW24a, Theorem 2.2]. This implies that the consistent systems of open correlators for $\mathfrak {B}$ are described by the category $\mathsf {ModAlg}\left ( \mathsf {O}; \mathcal {A}_{\text { !}}\right )$ . Now the statement follows from

$$ \begin{align*} \mathsf{ModAlg}\left( \mathsf{O}; \mathcal{A}_{\text{ !}}\right) \stackrel{\text{Corollary {8.2}}}{\simeq}\mathsf{CycAlg}\left( \mathsf{As}; \mathcal{A}\right) \stackrel{\text{Proposition~{7.3}}}{\simeq} \mathsf{symmetric\ Frobenius\ algebras\ in} \ \mathcal{A} . \end{align*} $$

9 A calculation recipe for open correlators

In this section, we demonstrate how the open correlators from Theorem 8.3 can be calculated explicitly. Suppose that $\mathfrak {B}$ is an open modular functor and $\mathcal {A}$ its underlying pivotal Grothendieck–Verdier category, i.e., the restriction to disks $\mathbb {D}^2_n$ with intervals $n\ge 1$ in its boundary. By [Reference Müller and WoikeMW24a, Theorem 2.2], $\mathfrak {B}$ is, up to a canonical equivalence, the modular extension of $\mathcal {A}$ , i.e., $\mathfrak {B}\simeq \mathcal {A}_{\text { !}}$ as open modular functors ( $\simeq $ is an equivalence of modular algebras over the open surface operad). For this reason, we know [Reference Müller and WoikeMW23a, Example 7.3]

(9.1) $$ \begin{align} \mathfrak{B}( \mathbb{D}^2_n; X_1,\dots,X_n ) \cong \mathcal{A}(K,X_1\otimes \dots \otimes X_1) \end{align} $$

for $X_1,\dots ,X_n \in \mathcal {A}$ . Let us now describe in more detail the consistent system of open correlators for a symmetric Frobenius algebra F in $\mathcal {A}$ :

• • The multiplication, the unit and the pairing: For $n=3$ and since F is self-dual

  $$ \begin{align*} \mathfrak{B}( \mathbb{D}^2_3; F , F, F ) \cong \mathcal{A}(K, F \otimes F \otimes F) \cong \kappa(F,F\otimes F) \stackrel{{\scriptsize 7.2}}{\cong} \mathcal{A}(F \odot F, F) . \end{align*} $$

  The consistent system of open correlator $\xi ^F$ associated to a symmetric Frobenius algebra gives us a vector $\xi _{\mathbb {D}^2_3}^F \in \mathfrak {B}( \mathbb {D}^2_3; F , F, F )$ , and by definition this is the multiplication $\mu : F \odot F \longrightarrow F$ . Pictorially, this means that the disk with three intervals in its boundary, each labeled with F, is “decorated” with the multiplication operation:

  (9.2)

  

  The boundary intervals are drawn in blue. The decoration is the red graph whose legs are colored with F; the multiplication operation should be imagined as sitting at the trivalent vertex. Note, however, that the legs are not incoming or outgoing. It is the advantage of working with cyclic operads and their algebras that we do not (and should not) make such distinctions. For $n=1$ , $\mathfrak {B}( \mathbb {D}^2_1; F )\cong \mathcal {A}(K,F)$ , and the distinguished vector is the unit $K\longrightarrow F$ . For $n=2$ , $\mathfrak {B}( \mathbb {D}^2_2; F ,F )\cong \mathcal {A}(K,F\otimes F)$ , and the distinguished vector is the copairing $\delta : K \longrightarrow F \otimes F$ by the unitality requirement in Definition 5.1. Since this copairing is part of a self-duality, it is by construction non-degenerate. This is simply because the non-degeneracy of the pairing on the microcosmic level needs to match the non-degeneracy of the macrocosmic one. In [Reference Fjelstad, Fuchs, Runkel and SchweigertFFRS08, Section 4.5], the non-degeneracy is an extra assumption called non-degeneracy of the disk two-point function.

  Thanks to the self-duality of F, we can see $D\eta $ as a morphism $F\longrightarrow I$ , the counit of F. Its evaluation on the unit $K\longrightarrow F$ is the distinguished vector in the space of conformal blocks $\mathfrak {B}(\mathbb {D}^2_0)\cong \mathcal {A}(K,I)$ for the disk without marked intervals.

• • A graphical presentation via Poincaré duality: We can draw (9.2) as a triangle, with the embedded intervals as segments on the edges. The decoration with the multiplication then describes a partition of the triangle that is Poincaré dual to the original one:

  (9.3)

  

  In other words, we recover the labeling principle underlying the constructions in [Reference Fuchs, Runkel and SchweigertFRS02, Reference Fuchs, Runkel and SchweigertFRS04a, Reference Fuchs, Runkel and SchweigertFRS04b, Reference Fuchs, Runkel and SchweigertFRS05, Reference Fjelstad, Fuchs, Runkel and SchweigertFFRS06a].

• • Disks with more boundary intervals: For $n\ge 4$ , the vector $\xi _{\mathbb {D}^2_n}^F$ is obtained by gluing disks with three intervals on their boundary together; pictorially:

  (9.4)

  

  The gluing is along the gray dashed lines. Under the isomorphism

  $$ \begin{align*} \mathcal{A}(K,F ^{\otimes n})\cong \mathcal{A}(DF,F^{\otimes (n-1)})\cong \mathcal{A}((DF)^{\odot (n-1)},F)\cong \mathcal{A}(F^{\odot (n-1)},F), \end{align*} $$
  the vector $\xi _{\mathbb {D}^2_5}^F$ is the arity $n-1$ multiplication $F^{\odot (n-1)}\longrightarrow F$ that multiplies elements together from left to right. If we think of this surface as obtained by gluing three triangles (9.3) together, the decoration is again obtained by passing to the Poincaré dual.

• • More complicated surfaces: Beyond disks with marked intervals, $\xi $ is completely determined by the fact that $\mathcal {A}_{\text { !}}$ is a modular algebra. The detailed treatment is in [Reference Müller and WoikeMW23a, Example 7.3] (see also [Reference Müller and WoikeMW24a, Equation (4.1)]; we discuss here an example: Suppose that we glue in (9.4) the two rightmost boundary intervals together, thereby creating an annulus with three intervals embedded in its outer boundary circle:

  (9.5)

  

The fact that $\mathcal {A}_{\text { !}}$ is a modular algebra, in particular, the first square in [Reference Müller and WoikeMW23a, Definition 2.13], tells us how to obtain from this triangulation of : We need to fill the coevaluation object $\Delta =\int ^{X \in \mathcal {A}} DX \boxtimes X \in \mathcal {A}\boxtimes \mathcal {A}$ into the arguments of the functor $\mathcal {A}_{\text { !}}(\mathbb {D}_5^2; -)$ in which the gluing occurs. This gives us

With (9.1), we arrive at

Since the consistent system of open correlators $\xi ^F$ solves the sewing constraints for gluing along intervals, is the image of $\xi _{\mathbb {D}_5^2}^F$ under

where the third map is induced by the map

(9.6) $$ \begin{align} DF \otimes F \longrightarrow \mathbb{F}\end{align} $$

obtained by applying $\otimes $ to the structure map $DF \boxtimes F \longrightarrow \int ^{X \in \mathcal {A}} DX\boxtimes X$ of the coend. Note that by definition the map $F \boxtimes F \longrightarrow \int ^{X \in \mathcal {A}}DX\boxtimes X$ obtained from the self-duality $DF\cong F$ and structure map of the coend is exactly the symmetric 2-pairing (this is how we defined self-duality in the first place, see Definition 4.5). We can symbolically represent this vector through the red graph in (9.5). The two arrows landing on the dashed cut symbolize how the two copies of F are mapped to the coend through the structure map and the self-duality.

Example 9.1 As we have seen above, the open correlators can be calculated directly from the operations of the symmetric Frobenius algebra $F\in \mathcal {A}$ and the map (9.6) “sorting $F \otimes F$ into the coend”. In Hopf-algebraic examples, the latter part can be made more explicit: Suppose that H is a pivotal finite-dimensional Hopf algebra, and $\mathcal {A}$ the category of finite-dimensional H-modules. Then, $\mathbb {F}=\int ^{X \in \mathcal {A}} X^\vee \otimes X$ (because $\otimes $ is exact), and we can make the following further statements that follow from [Reference Kerler and LyubashenkoKL01, Theorem 7.4.13]: The object $\mathbb {F}$ is given by the coadjoint representation $H^*_{\text {coadj}}$ , which is the dual of the adjoint H-action on itself given by $h.x := h'x S(h")$ , where we have used the Sweedler notation for the coproduct of H. For a symmetric Frobenius algebra F in H-modules, the map $F\otimes F \longrightarrow H_{\text {coadj}}^*$ from (9.6) is given by

$$ \begin{align*} F \otimes F \ni x \otimes y \longmapsto \left( H\ni h \longmapsto \psi(x)(h.y) \right), \end{align*} $$

with the self-duality $\psi :F \longrightarrow F^*$ .

10 Classification of ansular correlators

By [Reference Müller and WoikeMW23a, Theorem 5.13], $\mathsf {Lex}$ -valued cyclic algebras over the $\mathsf {f}E_2$ -operad are equivalent to ribbon Grothendieck–Verdier categories as defined in [Reference Boyarchenko and DrinfeldBD13], i.e., braided Grothendieck–Verdier categories $\mathcal {A}$ equipped with a natural automorphism $\theta _X:X\longrightarrow X$ called balancing such that $\theta _I=\mathrm {id}_I$ , $\theta _{X \otimes Y}=c_{Y,X}c_{X,Y}(\theta _X \otimes \theta _Y)$ for $X,Y \in \mathcal {A}$ , with $c_{X,Y}:X\otimes Y \xrightarrow {\ \cong \ } Y \otimes X$ being the braiding, such that additionally the compatibility $D\theta _X=\theta _{DX}$ with the Grothendieck–Verdier duality holds. As explained in [Reference Boyarchenko and DrinfeldBD13], any ribbon Grothendieck–Verdier category has an underlying pivotal Grothendieck–Verdier category, i.e., it is an enhancement of the structure discussed in the previous two sections. If the Grothendieck–Verdier duality is rigid and the unit simple, we obtain the notion of a finite ribbon category [Reference Etingof, Gelaki, Nikshych and OstrikEGNO15].

By replacing in the definition of the surface operad all surfaces with three-dimensional handlebodies (as always compact oriented; instead of boundary components, we consider disks embedded in the boundary surface of the handlebody), one obtains the modular operad $\mathsf {Hbdy}$ of handlebodies (see [Reference GiansiracusaGia11, Section 4.3] and [Reference Müller and WoikeMW23a, Section 7.2] for more details. A ( $\mathsf {Lex}^{\mathsf {f}}$ -valued) $\mathsf {Hbdy}$ -algebra is called ansular functor, and by the main result of [Reference Müller and WoikeMW24b] building on [Reference GiansiracusaGia11, Reference Müller and WoikeMW23a, Reference Müller and WoikeMW23c] the restriction to genus zero produces an equivalence to cyclic framed $E_2$ -algebras:

Corollary 10.5 For any cyclic framed $E_2$ -algebra $\mathcal {A}$ in $\mathcal {S}$ , there is an equivalence

(10.1)

If $\mathcal {A}$ is a finite ribbon category, consistent systems of ansular correlators in genus zero agree exactly with the consistent systems of genus zero correlators that are being classified by symmetric commutative Frobenius algebras in [Reference Fuchs and SchweigertFS17, Proposition 4.7]. Therefore, Theorem 10.6 immediately implies.

Let us briefly highlight why this is actually a non-obvious statement: For a handlebody $H_g$ of genus g, [Reference Fuchs and SchweigertFS17, Proposition 4.7] easily gives us the invariance under Dehn twists. But unlike for mapping class group of surfaces, handlebody groups are not generated by Dehn twists. Instead, the Dehn twists generate a normal subgroup $\mathsf {Tw}(H_g)\subset \mathsf {Map}(H_g)$ , the twist group, that fits into a short exact sequence

(10.2) $$ \begin{align} 1 \longrightarrow \mathsf{Tw}(H_g)\longrightarrow\mathsf{Map}(H_g)\longrightarrow\mathsf{Out}(F_g)\longrightarrow 1, \end{align} $$

where $\mathsf {Out}(F_g)$ is the group of outer automorphisms of the free group $F_g$ on g generators (see [Reference LuftLuf78]). In other words, $\mathsf {Map}(H_g)$ is much larger than $\mathsf {Tw}(H_g)$ .

In a finite ribbon category $\mathcal {A}$ , the monoidal unit I is obviously a symmetric commutative Frobenius algebra. Theorem 10.6 then implies.

11 A multiplicative invariant for modular algebras

The corolla $T_1$ with two legs is sent by any $\mathsf {Grpd}$ -valued modular operad $\mathcal {O}:\mathsf {Graphs} \longrightarrow \mathsf {Cat}$ to the groupoid $\mathcal {O}(T_1)$ of unary operations. The line with two vertices on it gives us a morphism $T_1 \sqcup T_1 \longrightarrow T_1$ endowing $\mathcal {O}(T_1)$ with the structure of a $\mathsf {Grpd}$ -valued algebra (that can be seen as a topological algebra if needed), the algebra $\mathfrak {A}_{\mathcal {O}}$ of unary operations [Reference Müller and WoikeMW24b, Definition 3.5]. Its unit is the operadic identity. The $\mathbb {Z}_2$ -action coming from the cyclic symmetry endows $\mathfrak {A}_{\mathcal {O}}$ with an anti involution of algebras. We denote the multiplication in $\mathfrak {A}_{\mathcal {O}}$ by $\star $ . It makes $\mathfrak {A}_{\mathcal {O}}$ into a monoidal category.

Theorem 11.1 Let $\mathcal {B}$ be an $\mathcal {S}$ -valued modular $\mathcal {O}$ -algebra. Then, for any self-dual object F in $\mathcal {B}$ ,

$$ \begin{align*} R: \mathfrak{A}_{\mathcal{O}} \longrightarrow \mathcal{S}(I,I) \ , \quad o \longmapsto R_o := \mathcal{B}_o (F,F) \end{align*} $$

is a lax symmetric monoidal functor. This means in particular:

1. (i) We have products

   $$ \begin{align*} \star : R_o \otimes R_{o'} \longrightarrow R_{o\star o'} \end{align*} $$
   graded over unary operations, associative and unital relative to the associators and unitors of $\mathfrak {A}_{\mathcal {O}}$ and $\mathcal {S}(I,I)$ , and natural in $o, o' \in \mathfrak {A}_{\mathcal {O}}$ .

2. (ii) Every $R_o$ is an $R_1$ -module, where $1 \in \mathfrak {A}_{\mathcal {O}}$ is the unit, and the $\operatorname {Aut}(o)$ -action on $R_o$ is through $R_1$ -module maps.

3. (iii) The operations $R_o \otimes R_{o'} \longrightarrow R_{o\star o'}$ are $\mathsf {Map}(o) \times \mathsf {Map}(o')$ -equivariant.

Proof The functor $J: \mathfrak {A}_o \longrightarrow \int \mathcal {O}$ sending $o \in \mathfrak {A}_o$ to $(T_1 , o )$ is lax symmetric monoidal because the gluing morphism $g:T_1 \sqcup T_1 \longrightarrow T_1$ gives us the structure maps

$$ \begin{align*} J(o) \sqcup J(o') =(T_1,o)\sqcup (T_1,o') \xrightarrow{\ ( g,\mathrm{id} ) \ } (T_1,o\star o')=J(o\star o') . \end{align*} $$

The functor R is just the composition

(11.1) $$ \begin{align} \mathfrak{A}_{\mathcal{O}} \xrightarrow{\ J \ } \int \mathcal{O} \xrightarrow{\ \mathbb{V}_F^{\mathcal{B}} \ }\mathcal{S}(I,I) \end{align} $$

of the lax symmetric monoidal functor J with the symmetric monoidal functor $\mathbb {V}_F^{\mathcal {B}}$ from Proposition 4.7. The statements (i)–(iii) just partially spell out what the lax symmetric monoidality of R entails.

Corollary 11.2 Let in Theorem 11.1 additionally a modular $\mathcal {O}$ -algebra structure $\xi : \star \longrightarrow \mathbb {V}_F^{\mathcal {B}}$ on F with coefficients in $\mathcal {B}$ be given. Then, for $o,o' \in \mathfrak {A}_{\mathcal {O}}$ ,

(11.2) $$ \begin{align} \xi_o \star \xi_{o'} = \xi_{o\star o'} . \end{align} $$

The following gives us a far-reaching generalization of the multiplicative structures built in [Reference JuhászJuh18, Section 5.2] on the state spaces of a three-dimensional topological field theory:

Corollary 11.3 Let $\mathcal {B}$ be a $\mathsf {Lex}^{\mathsf {f}}$ -valued ansular functor such that the ribbon Grothendieck–Verdier category obtained by evaluation on the circle comes with a self-dual unit. Then, there are natural unital and associative product maps

$$ \begin{align*} \star : \mathcal{B}(H) \otimes \mathcal{B}(H') \longrightarrow \mathcal{B}(H \# H') \end{align*} $$

for handlebodies without embedded disks, where $\#$ is the connected sum.

The spaces of conformal blocks appearing here are for handlebodies without embedded disks in their boundary, but in order to have the operation $\star $ , one still needs to fix disks at which the connected sum is taken. The equivariance of the $\star $ -operation in the sense of Theorem 11.1 (iii) is for the handlebodies groups of the handlebodies with these disks.

For the direct sum $\bigoplus _{g\ge 0} \widehat {\mathcal {A}}(H_g)$ , this implies the following consequence.

Proposition 11.6 Let $\mathcal {A}$ be a finite ribbon category. Then, the products

(11.3) $$ \begin{align} \star : \widehat{\mathcal{A}}(H) \otimes \widehat{\mathcal{A}}(H') \longrightarrow \widehat{\mathcal{A}}(H\# H') \end{align} $$

are injective.

Without the hypotheses that $\mathcal {A}$ is finite ribbon, this is generally wrong: In the Example 11.5, the products (11.3) are given by the multiplication operation $F \otimes F \longrightarrow F$ of the commutative Frobenius algebra F. Clearly, this map is not injective if $\mathrm {dim}\, F \ge 2$ .

Proof of Proposition 11.6

We use [Reference Müller and WoikeMW23a, Theorem 7.8] to see that, after choosing a cut system for H with genus g and $H'$ with genus $g'$ , the map $\star $ is given by the map

Here, $\oint $ is the left exact coend [Reference LyubashenkoLyu96] (see [Reference Müller and WoikeMW23a, Theorem 6.4] for the connection to the gluing of modular algebras. From this, we deduce that the multiplication map $\star $

(11.4) $$ \begin{align} \mathcal{A}(I,\mathbb{F}^{\otimes g}) \otimes \mathcal{A}(I,\mathbb{F}^{\otimes g'})\longrightarrow \mathcal{A}(I,\mathbb{F}^{\otimes (g+g')}) \end{align} $$

is just the map induced by $\otimes $ . Since I is simple, this map is injective by [Reference Müller and WoikeMW23b, Lemma 4.6].

Note that the multiplicative structure can, after choosing a cut system, be written down directly via (11.4). With this purely algebraic description, the independence of the cut system and the equivariance with respect to the mapping class group actions that is a part of Corollary 11.3 seem hard to see. The question whether there are products on spaces of conformal blocks with respect to the connected sum – a structure that is not part of the axioms of ansular or modular functors – was posed to us by Simon Lentner and Christoph Schweigert. The results of this section, in particular Corollary 11.3, provide a purely topological construction of these operations.

Proposition 11.7 For a finite ribbon category $\mathcal {A}$ , the distinguished $\mathsf {Map}(H)$ -invariant vectors $\xi _H \in \widehat {\mathcal {A}}(H)$ are non-zero for all handlebodies and satisfy

(11.5) $$ \begin{align} \xi_H \star \xi_{H'} = \xi_{H \# H'}. \end{align} $$

One might first think that $\mathcal {A}=0$ has to be excluded, but the trivial linear monoidal category is not a finite ribbon category because $\mathsf {End}_{\mathcal {A}}(I)\neq k$ in that case.

Lemma 11.8 Let $\mathcal {A}$ be a unimodular finite ribbon category. If for any handlebody H with genus g

$$ \begin{align*} \mathrm{dim}\, \widehat{\mathcal{A}}(H)< 2^g \end{align*} $$

for the space of conformal blocks, then $\mathcal {A}\simeq \mathsf {vect}$ .

Clearly, if $\mathcal {A}=\mathsf {vect}$ or $H=\mathbb {B}^3$ , the representation would be k with trivial action and hence irreducible.

Example 11.10 (Symmetric categories)

Let $\mathcal {A}$ be a finite ribbon category. By Proposition 11.6 we have an injective map

$$ \begin{align*} \star : \widehat{\mathcal{A}}(H_1) ^{\otimes g} \longrightarrow \widehat{\mathcal{A}}(H_g) . \end{align*} $$

Let us think of the handlebody $H_g$ as embedded in $\mathbb {R}^3$ . The map $\star $ becomes $\mathbb {Z}_g$ -equivariant if we let the generator $1 \in \mathbb {Z}_g$ act on $\widehat {\mathcal {A}}(H_1) ^{\otimes g}$ by $v_1 \otimes v_2 \otimes \dots \otimes v_g \longmapsto v_g \otimes v_1 \otimes \dots \otimes v_{g-1}$ (cyclic permutation), and act on $\widehat {\mathcal {A}}(H_g)$ by a rotation of $H_g$ by $2\pi /g$ as indicated in the following picture (see also [Reference Farb and MargalitFM12, Section 13.2.1]):

Here, the numbering of the copies of $H_1$ and the sense of the rotation have to be chosen accordingly of course. If $\mathcal {A}$ is not $\mathsf {vect}$ , then the $\mathbb {Z}_g$ -action on $\widehat {\mathcal {A}}(H_1) ^{\otimes g}$ is faithful thanks to Lemma 11.8. By injectivity of $\star ,$ the same is true for the rotation action on $ \widehat {\mathcal {A}}(H_g)$ .

Suppose now that our finite ribbon category $\mathcal {A}$ is symmetric, and we choose the identity as ribbon structure. By what we have just seen the handlebody group action on $\widehat {\mathcal {A}}(H_g)$ cannot be trivial unless $\mathcal {A}=\mathsf {vect}$ . Clearly, all elements in the twist group (see (10.2)) act trivially, but the action of $\mathsf {Out}(F_g)$ can still be non-trivial! If the action on $\widehat {\mathcal {A}}(H_g)$ extended to the mapping class group of $\partial H_g$ , then this could only be a trivial action because $\mathsf {Map}(\partial H_g)$ is generated by Dehn twists, and all of these act trivially [Reference Müller and WoikeMW23b, Theorem 4.8] – a contradiction. We conclude that a finite symmetric tensor category that is not $\mathsf {vect}$ does not extend to a modular functor. This might be a little bit surprising because a first naïve intuition might tell us that we can just build a modular functor with trivial mapping class group actions. Note that by Example 11.5, there are actually symmetric ribbon Grothendieck–Verdier categories that extend to a modular functor, but rigid ones with simple unit do not, unless they are $\mathsf {vect}$ !

12 A moduli space for open conformal field theories

In this last section, we explain how the modular microcosm principle can be used to construct and study moduli spaces of conformal field theories. This will be worked out here for open conformal field theories; extensions to other situations will be considered elsewhere. An idea developed most notably in [Reference Fuchs, Runkel and SchweigertFRS02, Reference Fuchs, Runkel and SchweigertFRS04a, Reference Fuchs, Runkel and SchweigertFRS04b, Reference Fuchs, Runkel and SchweigertFRS05, Reference Fjelstad, Fuchs, Runkel and SchweigertFFRS06a] is that conformal field theories are described as a pair of a category encoding the monodromy data (this can be for example a tensor category coming from a vertex operator algebra) and a consistent system of correlators (that takes the form of some sort of Frobenius algebra). These pairs can be organized in a moduli space as follows (we consider the open case): By sending an open modular functor $\mathfrak {B}$ to the modular algebras with coefficients in $\mathfrak {B}$ , we obtain a functor $\mathsf {ModAlg}(\mathsf {O};-):\mathsf {ModAlg}(\mathsf {O}) \longrightarrow \mathsf {Cat}$ . This allows us to define a moduli space of open conformal field theories (in $\mathsf {Lex}^{\mathsf {f}}$ ) by taking the nerve and realization of the Grothendieck construction of this functor

$$ \begin{align*} \mathsf{OpenCFT} := \left|B\int \left( \mathsf{ModAlg}(\mathsf{O};-):\mathsf{ModAlg}(\mathsf{O}) \longrightarrow \mathsf{Cat} \right)\right| . \end{align*} $$

By its definition and Theorem 8.3, $\mathsf {OpenCFT}$ is a space whose points are pairs $C=(\mathcal {A},F)$ of the monodromy data of the open conformal field theory (which is a pivotal Grothendieck–Verdier category $\mathcal {A}$ ) and a consistent system of open correlators (which is a symmetric Frobenius algebra F in $\mathcal {A}$ ). We denote by $\mathsf {OpenCFT}|_{\mathcal {A}} \subset \mathsf {OpenCFT}$ the subspace of $\mathsf {OpenCFT}$ consisting of those open conformal field theories whose monodromy data is (equivalent to) $\mathcal {A}$ .

Theorem 12.1 Let $C=(\mathcal {A},F)$ be an open conformal field theory given by its monodromy data, namely, a pivotal Grothendieck–Verdier category $\mathcal {A}$ in $\mathsf {Lex}^{\mathsf {f}}$ and a symmetric Frobenius algebra F in $\mathcal {A}$ . Then, there is the following seven-term exact sequence

in which the last two non-trivial terms are pointed sets.

We will prove this result by realizing that this seven-term exact sequence is the long exact sequence of homotopy groups for the following fiber sequence.

Proposition 12.2 For any pivotal Grothendieck–Verdier category $\mathcal {A}$ in $\mathsf {Lex}^{\mathsf {f}}$ , there is a homotopy fiber sequence

$$ \begin{align*} |B\mathsf{SymFrobAlg}(\mathcal{A})|\longrightarrow \mathsf{OpenCFT}|_{\mathcal{A}} \longrightarrow |B \mathsf{AUT}(\mathcal{A})| \end{align*} $$

Proof Thanks to [Reference ThomasonTho79, Theorem 1.2], there is a homotopy equivalence

$$ \begin{align*} \mathsf{OpenCFT} \simeq \underset{\mathfrak{B} \in \mathsf{ModAlg}(\mathsf{O}) }{\operatorname{hocolim}}\, |B\mathsf{ModAlg}(\mathsf{O};\mathfrak{B})| . \end{align*} $$

From Theorem 8.3, we can now conclude that $\mathsf {OpenCFT}|_{\mathcal {A}}$ is homotopy equivalent to the homotopy quotient $|B\mathsf {SymFrobAlg}(\mathcal {A})| // \mathsf {AUT}(\mathcal {A})$ of the the space $|B\mathsf {SymFrobAlg}(\mathcal {A})|$ by the 2-group $\mathsf {AUT}(\mathcal {A})$ . This gives us the desired homotopy fiber sequence.