POINTED ADMISSIBLE G-COVERS AND G-EQUIVARIANT COHOMOLOGICAL FIELD THEORIES

TYLER J. JARVIS, RALPH KAUFMANN, AND TAKASHI KIMURA

Abstract. For any finite group G we define the moduli space of pointed admissible G-covers and the concept of a G-equivariant cohomological field theory (G-CohFT), which, when G is the trivial group, reduce to the moduli space of stable curves and a cohomological field theory (CohFT), respectively. We prove that taking the “quotient” by G reduces a G-CohFT to a CohFT. We also prove that a G-CohFT contains a G-Frobenius algebra, a G-equivariant generalization of a Frobenius algebra, and that the “quotient” by G agrees with the obvious Frobenius algebra structure on the space of G-invariants, after rescaling the metric. We then introduce the moduli space of G-stable maps into a smooth, projective variety X with G action. Gromov-Witten-like invariants of these spaces provide the primary source of examples of G-CohFTs. Finally, we explain how these constructions generalize (and unify) the Chen-Ruan orbifold Gromov-Witten invariants of [X/G] as well as the ring H^{•}(X, G) of Fantechi and Göttsche.

1. Introduction

The purpose of this paper is to introduce a generalization of Kontsevich and Manin’s notion of a cohomological field theory (or CohFT) [KoMa], in the presence of a finite group G, which we call a G-equivariant cohomological field theory (or G-CohFT). Examples of (usual) CohFTs include the Gromov-Witten invariants of a smooth, projective variety (c.f., [Ma]) and the r-spin CohFT [JKV, PoVa, Po2]. A G-CohFT provides a framework for studying the physical procedure of orbifolding [Kau, Mo], as well as a structure for understanding both Chen-Ruan orbifold Gromov-Witten invariants of global quotients by a finite group [CR1, CR2, AGV] and the non-commutative ring structure of Fantechi and Göttsche [FG]. We now describe in some detail the motivation for studying G-CohFTs.

The first motivation comes from topological field theory. Recall that a Frobenius algebra H is a finite-dimensional, commutative, associative, unital algebra with an invariant metric. It can be regarded as a two-dimensional topological field theory, in the sense of Atiyah-Segal, associated to a cobordism category of two (real) dimensional, compact, oriented surfaces with boundary. A CohFT is a generalization of the above, but where the role of the cobordism category is replaced by \{H_{r}(\overline{M}_{g,n} )\}

Date: May 15, 2003. 2000 Mathematics Subject Classification. Primary: 14N35, 53D45. Secondary: 14H10, 14L30,

14A20, 57M12, Research of the first author was partially supported by NSF grant DMS-0105788. Research of the second author was partially supported by NSF grant DMS-0070681. Research of the third author was partially supported by NSF grants DMS-9803427 and DMS-

9729992.

for all r, where \overline{M}_{g,n} is the moduli space of stable curves of genus g with n marked points. By specializing to r = 0, one finds that the state space of the theory H recovers the structure of a Frobenius algebra.

For every finite group G Turaev [Tu] introduced a G-equivariant topological field theory (which he called a homotopy field theory) whose state space H is a (nonprojective) G-Frobenius algebra associated to a cobordism category of principal G-bundles over two (real) dimensional, compact, oriented surfaces with boundary. A (non-projective) G-Frobenius algebra (borrowing terminology from [Kau]) is a finite-dimensional, G-graded G-module with a G-equivariant associative multiplication, metric, and unit, and whose multiplication is braided commutative, satisfying an additional genus-one compatibility condition (called the trace axiom). By braided commutative we mean that the multiplication commutes with the action of the generator of the braid group which acts on tensor products of G-graded G-modules. If G is the trivial group, then a G-Frobenius algebra is a Frobenius algebra. Furthermore, the space of G-invariants \overline{H} of a G-Frobenius algebra inherits the structure of a Frobenius algebra graded by \overline{G}, the set of conjugacy classes of G. Kaufmann [Kau] considered a generalization of the above construction which allowed for projective factors.

This procedure of restricting to the space of invariants can be interpreted as a kind of orbifolding procedure from physics [Kau, Mo] where the subspace of \overline{H} graded by 1 in \overline{G} is called the untwisted sector, and the subspaces graded by nontrivial elements in \overline{G} are called twisted sectors.

Question 1. Is there a generalization of a CohFT, called a G-CohFT, where \overline{M}_{g,n} is replaced by another moduli space \overline{M}_{g,n}^{G} , such that for all r, the collec-

tion \{H_{r}(\overline{M}_{g,n}^{G} )\} endows the state space H of the theory with an algebraic structure whose specialization to r = 0 induces the structure of a G-Frobenius algebra on H? A G-CohFT should also have the property that when G is the trivial group, a G-CohFT reduces to a CohFT. Furthermore, by performing the correct “quotient” by G, the space of G-invariants \overline{H} should inherit the structure of a CohFT graded by \overline{G}.

G-Frobenius algebra H From H_{0}(\overline{M}_{g,n}^{G} )

G-CohFT H
From H_{•}(\overline{M}_{g,n}^{G} )

\overline{G}-graded Frobenius algebra \overline{H} From H_{0}(\overline{M}_{g,n} )

\begin{aligned}/G \\ & \end{aligned}

\overline{G}-graded CohFT \overline{H}
From H_{•}(\overline{M}_{g,n} )

\begin{aligned}/G \\ & \end{aligned}

Figure 1. Schematic of Question (1) where each box contains an algebraic structure and the responsible homology group, the horizontal arrows are restrictions, and the vertical arrows are “quotients” by G. When G is the trivial group, the two rows coincide.

The second motivation for studying G-CohFTs comes from orbifold Gromov-Witten invariants and is about how to construct certain examples of G-CohFTs associated to a smooth, projective variety X with an action of a finite group G.

G-EQUIVARIANT COHOMOLOGICAL FIELD THEORIES 3

Consider the G-graded G-module H(X) := ⨁_{m∈G} H^{•}(X^{m} ), where X^{m} denotes the fixed-point set in X of m, and let \overline{H}(X) denote its space of G-invariants. Chen and Ruan [CR1, CR2] introduced the notion of Gromov-Witten invariants for orbifolds, which, when applied to the global quotient [X/G], has a state space isomorphic to \overline{H}(X). An algebro-geometric version of this theory was introduced by [AGV]. The key geometric object in these constructions was \overline{M}_{g,n} ([X/G]), the moduli space of orbifold stable maps into the quotient [X/G]. The state space \overline{H}(X) of this theory is graded by \overline{G}, and the Gromov-Witten invariants are expected to yield a CohFT associated to each [X/G]. An important special case arises by considering only those contributions from \overline{M}_{g,n} ([X/G], 0), the moduli of orbifold stable maps which have degree zero. This endows \overline{H}(X) with the structure of a Frobenius algebra graded by \overline{G}, called variously stringy orbifold cohomology, Chen-Ruan cohomology, or just orbifold cohomology of [X/G].

When G is a trivial group, \overline{M}_{g,n} ([X/G]) reduces to the usual moduli space \overline{M}_{g,n} (X) of stable maps into X, and the Gromov-Witten invariants of X make H^{•}(X) into a CohFT. Restricting to contributions from \overline{M}_{g,n} ([X/G], 0) alone, one obtains the usual cohomology ring of X, which is a Frobenius algebra. “Forgetting” the stable map yields a morphism \overline{M}_{g,n} (X) \overline{M}_{g,n} for all stable pairs (g, n), which is an isomorphism when X is a point.

Fantechi and Göttsche [FG] were able to obtain the structure of the Chen-Ruan orbifold cohomology on \overline{H}(X) by first introducing a certain ring structure with metric on H(X) and then taking G-invariants. In fact, their ring satisfies all of the axioms of a G-Frobenius algebra except, possibly, the trace axiom. However, their construction is not obviously part of a larger structure and does not explicitly involve the moduli space of orbifold stable maps.

Question 2. For any smooth, projective variety X with a G-action, does there exist a moduli space \overline{M}_{g,n}^{G} (X) of a G-equivariant version of stable maps such that “for-

getting” the map yields a morphism \overline{M}_{g,n}^{G} (X) \overline{M}_{g,n}^{G} for stable pairs (g, n)? This map should be an isomorphism when X is a point.

There should also exist G-equivariant Gromov-Witten invariants associated to \overline{M}_{g,n}^{G} (X) which yield a G-CohFT with state space H(X), generalizing the usual construction when G is the trivial group. Furthermore, by taking the appropriate “quotient” by G, one should recover the orbifold Gromov-Witten invariants of [X/G] as in [CR1, CR2, AGV] with associated state space \overline{H}(X).

Finally, by considering only those contributions from the moduli \overline{M}_{g,n}^{G} (X, 0) of stable maps of degree zero, one should be able to recover the G-Frobenius algebra structure in [FG] and prove that the trace axiom must hold.

This paper provides affirmative answers to both of these questions.

The first part of this paper is devoted to answering the first question. We introduce \overline{M}_{g,n}^{G} , the moduli space of n-pointed admissible G-covers of genus g.

Roughly speaking, it consists of a tuple (E ^{π} C; p̃_{1}, …, p̃_{n} ), where E and C are (at worst, nodal) curves, (C, p_{1}, …, p_{n} ) is a stable curve of genus g, where p̃_{i} are points in E and p_{i} := π(p̃_{i} ), and π maps nodes of E to nodes of C. Furthermore, we require that, away from π^{−1}(p_{i}) and nodes, E is a principal G-bundle; however, E is allowed to have ramification over the marked points and nodes. Our construction F-G G-Frobenius Algebra H(X)

From \overline{M}_{g,n}^{G} (X, 0)

G-Equivariant GW Theory H(X) From \overline{M}_{g,n}^{G} (X)

C-R Orbifold Cohomology \overline{H}(X) From \overline{M}_{g,n} ([X/G], 0)

\begin{aligned}/G \\ & \end{aligned}

C-R Orbifold GW Theory \overline{H}(X)
From \overline{M}_{g,n} ([X/G])

\begin{aligned}/G \\ & \end{aligned}

Figure 2. Schematic of Question (2) where each box contains an algebraic structure and the relevant moduli space, F-G denotes Fantechi-Göttsche, C-R denotes Chen-Ruan, the horizontal arrows denote restriction, and the vertical arrows denote taking “quotients” by G.

differs from the stack of admissible covers in [ACV], as we require the additional data of p̃_{i} in E over each marked point p_{i} in C.

By forgetting the data associated to the G-cover, one obtains a morphism st : \overline{M}_{g,n}^{G} \overline{M}_{g,n} , where \overline{M}_{g,n} is the moduli space of stable curves. We prove

that \overline{M}_{g,n}^{G} is a smooth, Deligne-Mumford stack, flat, proper, and quasi-finite (but

not representable) over \overline{M}_{g,n} . Furthermore, \overline{M}_{g,n}^{G} has an action of the symmetric group S_{n} by permuting the ordering of the marked points, and it has an action of G^{n} by translation of the marked points. In fact, \overline{M}_{g,n}^{G} admits an action of the braid group B_{n}, which factors through the S_{n} and G^{n} actions.

The collection \{\overline{M}_{g,n}^{G} \} possesses gluing morphisms, provided that the monodromies of the two marked points to be glued together are inverses of one another. These gluing morphisms are equivariant under the action of S_{n} and G^{n}. One may regard the collection \{\overline{M}_{g,n}^{G} \} as a G-equivariant colored modular operad, where the coloring is by elements of G. Furthermore, the morphism st respects the S_{n} actions and the gluing morphisms.

A G-CohFT is defined analogously to a CohFT, but where the role of \overline{M}_{g,n} is replaced by \overline{M}_{g,n}^{G} , and where G-equivariance is maintained throughout the construction. We prove that there is an external tensor product and a (usual) tensor product associated to equivariant CohFTs. We then define the correct notion of taking a “quotient” by G and prove that this procedure has the desired properties. The procedure of taking quotients involves an intermediate step on the stack \overline{M}_{g,n} (BG) of stable maps into the classifying stack BG (i.e., the stack of admissible covers without the additional points p̃_{i}). We show that in this intermediate step the stack \overline{M}_{g,n} (BG) can be replaced by the quotient [\overline{M}_{g,n}^{G} /G^{n} ], but that the resulting “quotient” CohFTs are isomorphic.

The last part of this paper will treat the second question. We introduce the moduli space of G-stable maps \overline{M}_{g,n}^{G} (X) and describe the G-equivariant Gromov-

Witten invariants. By restricting to contributions from \overline{M}_{g,n}^{G} (X, 0) alone, we prove that the state space H(X) inherits a G-Frobenius algebra structure which agrees with that from [FG], and in particular that the trace axiom holds for their ring. The proof consists of relating the virtual fundamental class to an analogous cohomology class in their construction.

The details of the construction of a G-CohFT for general equivariant Gromov-Witten invariants, properties of potential functions, and applications to higher spin curves will be explored elsewhere [IP].

Finally, we observe that orbifolding plays an important role in mirror symmetry and in certain Landau-Ginzburg theories (see, for example, [CoKat, Ma]). We expect our construction to play a similar role in the theory of CohFTs. There are also related notions of orbifolding which appear in the context of vertex algebras (see, for example, [Kir]). It would be enlightening to better understand their relationship.

The outline of this paper is as follows. In Section 2 we describe the moduli spaces \overline{M}_{g,n}^{G} , their associated forgetful and gluing morphisms, group actions, and automorphism groups. In Section 3 we briefly review important facts from the category of G-graded G-modules, including the braid group action and tensor products. In Section 4 we define G-CohFTs and their tensor products. We prove that a (nonprojective) G-CohFT always contains a G-Frobenius algebra. In Section 5 we define how to obtain a CohFT from a G-CohFT by taking the appropriate “quotient.” We prove that this is consistent with the obvious notion of taking a quotient for a G-Frobenius algebra, after rescaling the metric, and then work out the example of the orbifold cohomology of BG. In Section 6 we introduce the moduli space of G-stable maps and equivariant Gromov-Witten invariants, reproduce the ring of [FG] as a special case, and prove that the trace axiom is satisfied.

Remark 1.1. Unless otherwise specified, we assume that all cohomology rings are over the ground ring ℂ, although all constructions here are also valid over the rationals ℚ.

Also, unless otherwise specified, all groups which appear are finite and all group actions are right group actions.

Notation 1.2. The stack quotient of a variety X by G will be denoted [X/G] and the coarse moduli space of this quotient will be denoted X/G.

Acknowledgments. We would like to thank Heidi Jarvis for help with typesetting and proofing. We would also like to thank Jim Stasheff for his help with improving the exposition of the manuscript.

The first author would like to thank Boston University for its hospitality. The second author would like to thank the Max–Planck–Institut für Mathematik and the Institut des Hautes Études Scientifiques for their hospitality. And the third author would like to thank the Institute for Advanced Study, where much of this work was done, for their financial support under National Science Foundation agreement No. DMS-9729992. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

2. The moduli spaces

Let (C ^{ϖ} T, p_{1}, …, p_{n} ) be a stable curve over T of genus g, with marked points (sections) p_{1}, …, p_{n}. We want to study a variant of the space of admissible G-covers of C, as defined in [ACV, Def 4.3.1]. We recall the definition here:

Definition 2.1. A finite morphism π : E C to an n-pointed, genus-g, stable curve C ^{ϖ} T, p_{1}, …, p_{n} over T is an admissible G-cover if

(1) E/T is itself a nodal curve (not necessarily connected).

(2) Nodes of E map to nodes of C.

(3) There is a right action ρ_{E} of G on E preserving π, and such that

(4) the restriction of π to C_{gen} (the points of C which are neither marked points nor nodes) is a principal G-bundle.

(5) At points of E lying over nodes of C the structure of the maps E ^{π} C ^{ϖ} T is locally the same as (analytically isomorphic to) that of

Spec A[z, w]/(zw − t)  Spec A[x, y]/(xy − t^{r})  Spec A,

where we have t ∈ A, x = z^{r} and y = w^{r}, for some integer r > 0.

(6) At points of E lying over marked points of C the structure of the maps E ^{π} C ^{ϖ} T is locally the same as (analytically isomorphic to) that of

Spec A[z]  Spec A[x]  Spec A,

where x = z^{s} for some integer s > 0.

(7) The action of the stabilizer G_{q} ⊆ G at each node q of E is balanced; that is, the eigenvalues of the action on the tangent space at q are multiplicative inverses of each other.

Theorem 4.3.2 of [ACV] shows that the stack of admissible G-covers is isomorphic to the stack \overline{M}_{g,n} (BG) of balanced twisted stable maps into the classifying stack of G.

2.1. Definition, construction, and basic properties of \overline{M}_{g,n}^{G} .

Given an admissible G-cover (E ^{π} C, p_{1}, …, p_{n} ), let p̃_{i} ∈ π^{−1}(p_{i}) be a choice of a point in the fiber over p_{i} for all i = 1 …n.

Definition 2.2. Let \overline{M}_{g,n}^{G} denote the stack of admissible G-covers

(π : E C, p_{1}, …, p_{n} , p̃_{1}, …, p̃_{n} )

of n-pointed, genus-g, stable curves, together with a choice of n marked points p̃_{i} ∈ E such that π(p̃_{i} ) = p_{i} for all i = 1, …, n. We call such objects n-pointed admissible G-covers. A morphism of such objects is a G-equivariant fibered diagram; that is, a morphism of the underlying stable curves, together with a G-equivariant morphism of the induced admissible G-covers preserving the points p̃_{i}.

Because the curve C is oriented, a pointed admissible G-cover (E ^{π} C, p̃_{1}, …, p̃_{n} ) has a well-defined monodromy m_{i} at each marked point p̃_{i}; namely, E induces a principal G-bundle over C − {p_{1}, …, p_{n} }, and the orientation gives a small loop in C − {p_{1}, …, p_{n} } around each p_{i}, with a lift to a path in a small neighborhood of p̃_{i} in E − {p̃_{1}, …, p̃_{n} }. The lift is not uniquely determined, but the difference between the starting and ending sheets of the lifted path is given by a well-defined element m_{i} ∈ G.

Since the points p̃_{i} are determined up to a discrete choice by C, π, and the points p_{i}, the monodromy is invariant under deformation of the curve C, the cover E, and the points p_{i}. Also note that, while the action ρ_{E} acts on the points p̃_{i} by right multiplication, it acts on the holonomies by conjugation.

Let G_{A} denote the set G, considered as a right G-space under conjugation. Associated to any object (E ^{π} C, p̃_{1}, …, p̃_{n} ) there exists an element m = (m_{1}, …, m_{n} ) ∈ G_{A}^{n}; namely, m_{i} is the monodromy of E at the point p̃_{i}. Definition 2.3. Denote the canonical morphism we have just described by

(1)  e : \overline{M}_{g,n}^{G}  G_{A}^{n},

and let

(2)  \overline{M}_{g,n}^{G} (m) := e^{−1}(m)

denote the substack of objects in \overline{M}_{g,n}^{G} that map to m.

Since e is locally constant, we may write

\overline{M}_{g,n}^{G} = ∐_{m∈G_{A}^{n}} \overline{M}_{g,n}^{G} (m).

The stack \overline{M}_{g,n}^{G} and the substacks \overline{M}_{g,n}^{G} (m) can be explicitly constructed as follows.

Theorem 2.4. The stack \overline{M}_{g,n}^{G} and the substacks \overline{M}_{g,n}^{G} (m) are smooth Deligne-Mumford stacks, flat, proper, and quasi-finite over \overline{M}_{g,n} .

Proof. Let Adm_{g,n}^{G} be the stack of admissible G-covers of n-pointed, genus-g

curves, and let E ^{π} C ^{ϖ} Adm_{g,n}^{G} be the universal G-cover and stable curve, with universal sections p_{i} of ϖ. Let A_{i} be the image in C of the ith section p_{i}, and let

E_{i} := E ×_{C} A_{i}

be the restriction of E to A_{i}. Let

W := E_{1} ×_{Adm_{g,n}^{G}} E_{2} ×_{Adm_{g,n}^{G}} ⋯ ×_{Adm_{g,n}^{G}} E_{n}

be the fibered product of the E_{i}. It is straightforward to see that W is the stack of admissible G-covers, together with explicit choices of sections p̃_{i} ∈ E lying over the sections p_{i}; that is, W = \overline{M}_{g,n}^{G} .

Theorems 3.0.2 and 4.3.2 of [ACV] show that the space Adm_{g,n}^{G} is isomorphic to \overline{M}_{g,n} (BG), the stack of twisted stable maps to the classifying stack BG, and is a smooth DM stack, flat, proper, and quasi-finite over \overline{M}_{g,n} . Since these properties are preserved by the above-listed fibered products, the theorem follows for \overline{M}_{g,n}^{G} . The substacks \overline{M}_{g,n}^{G} (m) are finite disjoint unions of connected components

of \overline{M}_{g,n}^{G} , so the theorem also holds for them. □

2.2. Morphisms and group actions on \overline{M}_{g,n}^{G} .

There are several obvious morphisms on \overline{M}_{g,n}^{G} . First, there are the forgetful morphisms

\overline{M}_{g,n}^{G}  s̃t Adm_{g,n}^{G} ≅ \overline{M}_{g,n} (BG) ŝt \overline{M}_{g,n} ,

which were shown in Theorem 2.4 to be proper, flat, and quasi-finite. We denote the composition by

st := ŝt ∘ s̃t.

We also have the evaluation morphism (2):

e : \overline{M}_{g,n}^{G} G_{A}^{n}.

Recall (see [JK]) that while \overline{M}_{g,n} (BG) cannot be written as a disjoint union of substacks indexed by m ∈ G_{A}^{n}, it does have a decomposition indexed by conjugacy classes of G.

Definition 2.5. We denote the set of conjugacy classes of G by \overline{G} and the conjugacy class of m ∈ G by \overline{m}. Similarly, we denote by \overline{m} ∈ \overline{G}^{n} the n-tuple of conjugacy classes determined by m ∈ G^{n}.

As described in [JK], we have

\overline{M}_{g,n} (BG) = ∐_{\overline{m}∈\overline{G}^{n}} \overline{M}_{g,n} (BG; \overline{m}),

where some of the substacks may be empty.

Definition 2.6. We let \overline{M}_{g,n}^{G} (\overline{m}) denote the preimage s̃t^{−1}(\overline{M}_{g,n} (BG; \overline{m})), which is easily seen to be

\overline{M}_{g,n}^{G} (\overline{m}) = ∐_{m′∈\overline{m}} \overline{M}_{g,n}^{G} (m′).

The stack \overline{M}_{g,n}^{G} (\overline{m}) has a right G^{n} action

(3)  ρ(γ_{1}, …, γ_{n} ) : \overline{M}_{g,n}^{G} (m)  \overline{M}_{g,n}^{G} (γ_{1}^{−1}m_{1}γ_{1}, …, γ_{n}^{−1}m_{n} γ_{n} ),

which acts by right multiplication on the n marked points (p̃_{1}, …, p̃_{n} ) ) (p̃_{1} · γ_{1}, …, p̃_{n} · γ_{n} ). We sometimes write ρ_{i} for the action on the ith factor: ρ_{i}(γ) = ρ(1, …, γ, …1).

Of course, there is the natural symmetric group action on \overline{M}_{g,n}^{G} obtained by

relabeling points, but the stack \overline{M}_{g,n}^{G} also admits an action of the braid group B_{n}.

Definition 2.7. Let B_{1} be the trivial group. If n ≥ 2, let B_{n} be the group with generators {b_{1}, …, b_{n−1} } subject to the relations

(4)  b_{i}b_{i+1} b_{i} = b_{i+1} b_{i}b_{i+1}

for all i = 1, …, n − 1 and

(5)  b_{i}b_{j} = b_{j}b_{i}

if |i − j| > 1. B_{n} is called the braid group on n-letters.

For each generator b_{i} of the braid group B_{n}, there is an isomorphism (6) \overline{M}_{g,n}^{G} (m_{1}, …, m_{i}, m_{i+1} , …, m_{n} ) ^{b_{i}} \overline{M}_{g,n}^{G} (m_{1}, …, m_{i}m_{i+1} m_{i}^{−1}, m_{i}, …, m_{n} ).

These are given by b_{i} := ρ_{i}(m_{i}^{−1}) ∘ s_{i}, where s_{i} is the element (i, i + 1) in S_{n} which transposes i and i + 1, and ρ_{i} is the group action on \overline{M}_{g,n}^{G} obtained by right translation of the i-th marked point. It is straightforward to check that the induced isomorphisms satisfy the braid relations (4) and (5), thus they induce an action of B_{n} on \overline{M}_{g,n}^{G} .

Finally, there are the three fundamental morphisms: forgetting tails, gluing trees, and gluing loops.

Forgetting Tails: Let m be any n-tuple

m := (m_{1}, …, m_{n} ) ∈ G_{A}^{n},

and let 1 be the identity in G. Whenever the pair (g, n) is stable (i.e., 2g − 2+n > 0) there is a natural forgetting tails morphism τ̃ : \overline{M}_{g,n+1}^{G} (m, 1) \overline{M}_{g,n}^{G} (m) defined as follows.

First, simply forgetting the data associated to the (n+1)st marked point usually yields an object of \overline{M}_{g,n}^{G} (m), but if the resulting curve is unstable, then we need to contract the unstable component to a point p. In those cases it is true, but not immediately obvious, that we can produce a suitable G-cover E on the new curve, and where necessary, assign a point p̃ in E over p. We now describe how this works.

We have two cases: first, when the resulting unstable component D is a (genus-zero) −1-curve with one marked point p_{i}, and one node q; and second, when the unstable component D is a −2-curve with two nodes q and q′ and no marked points.

In either case, the unstable component D is a genus-zero curve with two special points (call them q and q′ for simplicity of notation). It is straightforward to see that for any q̃′ ∈ E over q′ with monodromy m, the connected component D̃ of E containing q̃′ is a finite cover of D with automorphism group Aut_{D} D̃ generated by m, which acts faithfully on all points but q and q′. In particular, it is fully ramified over q and q′, and unramified at all other points. Thus there is a canonical G-equivariant isomorphism φ : E|_{q} ^{∼} E|_{q′} . This shows in the first case, where p_{i} = q′, that there is a canonical choice of q̃ ∈ E|_{q} (namely, q̃ = φ^{−1}(p̃_{i} )), and thus a well-defined point of \overline{M}_{g,n}^{G} (m).

In the second case, the isomorphism φ allows the construction of a principal G-bundle on the curve with the unstable component D contracted. In this case, we need no point q̃—the data we already have will give a point of \overline{M}_{g,n}^{G} . Thus in every case the forgetting tails morphism exists.

Gluing Trees: Given any m ∈ G_{A}^{n_{1}} and m′ ∈ G_{A}^{n_{2}} , as well as an additional element μ ∈ G_{A}, let g := g_{1} + g_{2} and n := n_{1} + n_{2}. We have the gluing trees morphism:

(7)  ρ_{tree} : \overline{M}_{g_{1} ,n_{1} +1}^{G} (m, μ) × \overline{M}_{g_{2} ,n_{2} +1}^{G} (μ^{−1}, m′)  \overline{M}_{g,n}^{G} (m, m′)

given by attaching the universal G-covers E ^{π} C′ ^{ϖ} \overline{M}_{g_{1} ,n_{1} +1}^{G} (m, μ)

and E′ ^{π′} C ^{ϖ′} \overline{M}_{g_{2} ,n_{2} +1}^{G} (μ^{−1}, m′) along the sections p̃_{n_{1} +1} ∈ E and p̃′_{1} ∈ E′, and attaching the universal curves C and C′ along the sections p_{n_{1} +1} and p′_{1}. It is straightforward to see that, because the monodromies μ and μ^{−1} are inverses, the induced cover is indeed an admissible G-cover of the resulting stable curve, and thus gives an object in \overline{M}_{g,n}^{G} (m, m′).

More generally, let I = {i_{1}, …, i_{n_{1}} } and J = {j_{1}, …, j_{n_{2}} } be any disjoint subsets of {1, …, n} such that I ⊔ J = {1, …, n}. For any integers s, t with i ≤ s ≤ n_{1}, 1 ≤ t ≤ n_{2} there is a morphism

(8)  \overline{M}_{g_{1} ,n_{1} +1}^{G} (m_{i_{1}} , …, m_{i_{s−1}} , μ, m_{i_{s}} , …, m_{i_{n_{1}}} )

×\overline{M}_{g_{2} ,n_{2} +1}^{G} (m_{j_{1}} , …, m_{j_{t−1}} , μ^{−1}, m_{j_{t}} , …, m_{j_{n_{2}}} )
\overline{M}_{g,n}^{G} (m_{1}, …, m_{n} ).

Gluing loops: Given any m ∈ G_{A}^{n} and μ ∈ G_{A} we have the gluing loops morphism:

(9)  ρ_{loop} : \overline{M}_{g−1,n+2}^{G} (m, μ, μ^{−1})  \overline{M}_{g,n}^{G} (m),

defined in a manner similar to the gluing trees morphism; namely, one attaches the universal G-cover E to itself along the two sections p̃_{n+1} and p̃_{n+2} , and the universal curve C to itself along the sections p_{n+1} and p_{n+2} . As with gluing trees, the gluing loops morphism can be defined more generally for any two sections p̃_{i}, and p̃_{j}, provided they have inverse monodromies.

Remark 2.8. Even more generally, if two points do not have inverse monodromies, the braid group action may still allow one to glue them. For example, for any i_{1} < i_{2} with i_{1}, i_{2} ∈ {0, …, n+1} and σ = m_{i_{1} +1}^{−1} m_{i_{1} +2}^{−1} …m_{i_{2}}^{−1}μ^{−1}m_{i_{2}} …m_{i_{1} +1} , we have a morphism

\overline{M}_{g,n+2}^{G} (m_{1}, …, m_{i_{1}} , μ, m_{i_{1} +1} , …, m_{i_{2}} , σ, m_{i_{2} +1} , …, m_{n+2} )
b_{i_{2} +1} ∘b_{i_{1} +2} ∘···∘b_{i_{2}} \overline{M}_{g,n+2}^{G} (m_{1}, …, m_{i_{1}} , μ, μ^{−1}, m_{i_{1} +1} , …, m_{n} )
ρ_{loop} \overline{M}_{g+1,n}^{G} (m_{1}, …, m_{n} ).

Remark 2.9. Since the collection \{\overline{M}_{g,n}^{G} \} has gluing morphisms which are equi-

variant under the actions of G^{n} and S_{n}, one may regard \{\overline{M}_{g,n}^{G} \} as a G-equivariant colored modular operad where the set of colors is the G-set G_{A}. Since the action of the braid group B_{n} (see Equation (6)) on \overline{M}_{g,n}^{G} is constructed from the G^{n} and

S_{n} actions, one may also regard \{\overline{M}_{g,n}^{G} \} as a colored modular operad, but where the role of the permutation group is replaced by the braid group.

Remark 2.10. It is worth pointing out that the stack \overline{M}_{g,n+1}^{G} (m, 1) is not the

universal curve over \overline{M}_{g,n}^{G} (m), but rather the universal orbicurve.

2.3. Holonomy and other tools for studying G-covers.

Let G_{R} denote G considered as a right G-module. Note that the automorphism group Aut^{G}(G_{R}) of G_{R} is again G, acting by left multiplication. Given a pointed admissible cover (E ^{π} C, p̃_{1}, …, p̃_{n} ) and given any point p̃_{0} ∈ E_{gen} := π^{−1}(C_{gen} ) lying over p_{0} ∈ C_{gen} , we have an isomorphism of right G-modules νp̃_{0} : E|_{p_{0}} ^{∼} G_{R}, given by

νp̃_{0} (p̃_{0}γ) := γ.

Changing the base point p̃_{0} to p̃_{0}α changes the map νp̃_{0} by left multiplication by α^{−1}.

ν_{p̃_{0} α} = α^{−1}νp̃_{0} .

Definition 2.11. The choice of p̃_{0} ∈ E_{gen} gives a homomorphism from the fundamental group to G:

χp̃_{0} : π_{1}(C_{gen} , p_{0}) G,

which we call holonomy. One way to see this homomorphism explicitly is to pull E_{gen} back to the trivial admissible cover Ẽ of the universal cover U of C_{gen} . Automorphisms of U are precisely π_{1}(C_{gen} , p_{0}), and they induce automorphisms of Ẽ ≅ U × G_{R}, and therefore of G_{R}:

π_{1}(C_{gen} , p_{0}) ≅ Aut_{C_{gen}} U ^{χ} Aut^{G} G_{R} = G.

Conversely, given any homomorphism χ : π_{1}(C_{gen} , p_{0}) G, it is easy to see that we get a uniquely determined admissible G-cover of (C, p_{1}, …, p_{n} ) and a distinguished point p̃_{0,χ} over p_{0}. This G-cover is given by first taking the quotient of U × G_{R} by the action of π_{1}(C_{gen} , p_{0}) ^{id,χ} Aut_{C_{gen}} U × Aut^{G} G_{R} and then extending it to all of C. Such an extension is uniquely determined by the G-cover on C_{gen} . The point p̃_{0,χ} is the image of (p_{0}, 1) ∈ U × G_{R} under this quotient. We call this cover the admissible G-cover of C induced by χ and p_{0}, and we denote it E_{χ,p_{0}} , or E_{χ} if p_{0} is clear from context.

The following proposition is an immediate consequence of well-known corresponding results for principal G-bundles (see, for example, [Fu, Chapters 13–14]) and is straightforward to check.

Proposition 2.12. Given a homomorphism χ : π_{1}(C_{gen} , p_{0}) G, the induced E and p̃_{0,χ} have holonomy χp̃_{0} equal to χ, and conversely, given an E and p̃_{0} the bundle E_{χp̃_{0}} is canonically isomorphic to E, via an isomorphism identifying p̃_{0,χp̃_{0}} with the original p̃_{0}.

Thus the data of E, p̃_{0} is equivalent to a choice of homomorphism χ : π_{1}(C_{gen} , p_{0}) G. A different choice of point p̃_{0}, say p̃_{0}α, changes χ by conjugation χ_{p̃_{0} α} = α^{−1}χα.

Furthermore, given a path γ from p_{0} to p′_{0} in C_{gen} and the corresponding unique lift γ̃ of γ from p̃_{0} to p̃′_{0} ∈ E|_{p′_{0}} , the holonomy χp̃′_{0} is induced from χp̃_{0} by conjugation with γ.

[figure: π_{1}(C_{gen} , p_{0}); χp̃_{0}; G; π_{1}(C_{gen} , p_{0}); ad(γ); χp̃′_{0}]

And conversely, given any χ′ : π_{1}(C_{gen} , p_{0}) G determined from χ by conjugation by γ, the induced G-cover E_{χ′} is canonically isomorphic to E_{χ}, and the induced point p̃′_{0,χ′} is that obtained by parallel transporting p̃_{0} along γ (i.e., p̃′_{0,χ} is the endpoint of γ̃).

Definition 2.13. For any path d from p_{0} to p_{i} in C_{gen} (that is, a path in C such that the image of (0, 1) lies in C_{gen} and d(1) = p_{i} and d(0) = p_{0}), we have an induced element σ_{d} of π_{1}(C_{gen} , p_{0}) defined by following d from p_{0} to a little loop around p_{i}, tracing the loop out once counterclockwise, and then returning along d (or rather d^{−1}) to p_{0}.

Moreover, for any admissible G-cover with point p̃_{0} ∈ E|_{p_{0}} , the path d determines a point p̃(d) ∈ E|_{p_{i}} , which is the endpoint of the unique lift d̃ of d in E that begins at p̃_{0}.

Finally, given p̃_{0} and a path d from p_{0} to p_{i}, holonomy and the map νp̃_{0}

induce an isomorphism of right G-modules \overline{ν}_{d,p̃_{0}} : E|_{p_{i}} ^{∼} ⟨m_{i}⟩\G_{R}, where m_{i} := χp̃_{0} (σ_{d}), and ν̄_{d,p̃_{0}} maps the point p̃(d) to the coset ⟨m_{i}⟩.

Definition 2.14. In genus zero, a choice of paths d_{i} from the point p_{0} to the point p_{i} for each i ∈ {1, …, n} induces loops σ_{d_{i}} that generate π_{1}(C_{gen} , p_{0}). Not every choice of monodromy m ∈ G^{n} satisfies the same relations that the generators σ_{i} do, and thus not every choice of monodromy defines a holonomy χ, but for those m that do, there is a uniquely determined pointed admissible G-cover

ζ(d_{1}, …, d_{n} ; m) := (E_{χ} ℂℙ^{1}, p̃_{1}, …, p̃_{n} )

by defining the holonomy χ to be given by the monodromy

χ(σ_{d_{i}} ) = m_{i}, for i ∈ {1, …, n}

and letting the points p̃_{i} := p̃(d_{i}) be the points induced as in Definition 2.13. Since the loops σ_{d_{i}} generate the fundamental group of ℂℙ^{1} − {p_{1}, …, p_{n} }, this construction gives a well-defined pointed admissible G-cover.

It is clear from our discussion so far that every smooth, genus-zero, n-pointed, admissible G-cover (E ℂℙ^{1}, p̃_{1}, …, p̃_{n} ) that has all of its points p̃_{i} in the same connected component of E must be of the form ζ(d_{1}, …, d_{n} ; m) for some choice of p_{0}, paths (d_{1}, …, d_{n} ), and m ∈ G^{n}. Assume that the points p_{0}, …, p_{n} ∈ C are given. Of special interest is the case where the induced generators of the fundamental group have product equal to 1. We denote the subset of such n-tuples of paths by

(10)  P_{C} := {(d_{1}, …, d_{n} )|d_{i} a path from p_{0} to p_{i}, and ∏_{i=1}^{n} σ_{d_{i}} = 1},

and the corresponding pointed admissible G-covers of C by

(11)  ζ_{C} := {ζ(d; m)|d ∈ P_{C}, m ∈ G^{n} , ∏_{i=1}^{n} m_{i} = 1}.

Definition-Proposition 2.15. Given a choice of p_{0}, …, p_{n} ∈ C, with genus(C) = 0, there is a transitive action of the braid group on the set P_{C}, where

(12)  b_{i}d_{i} = d_{i+1}
(13)  b_{i}d_{i+1} = σ_{d_{i} +1} d_{i}
(14)  b_{i}d_{j} = d_{j} if j ≠ i, i + 1.

This action of B_{n} on the set P_{C} is compatible with the usual braid action on π_{1}(C_{gen} , p_{0}); that is, for each i we have σ_{d_{i}} ∈ π_{1}(C_{gen} , p_{0}), and

(15)  σ_{bd_{i}} = bσ_{d_{i}} .

Consequently, the braid action on P_{C} induces an action of the braid group on ζ_{C}, distinct from the braid action on all of \overline{M}_{g,n}^{G} that we defined earlier. To distinguish the two, we will denote this new action by β : B_{n} Aut(ζ_{C}).

Proof. The fact that the given equation defines an action and that the action is compatible with the usual action on the fundamental group is a straightforward calculation. That the action is transitive follows from the classical fact that the braid group generates all automorphisms of the fundamental group that preserve the property of the product of generators being trivial.

Since the product of generators and the product of monodromies are both trivial, the induced holonomy bχ : σ_{(bd_{i} )} ) m_{i} is still a well-defined homomorphism of groups. Thus for each admissible cover ζ(d; m) ∈ ζ_{C} and for each b ∈ B_{n} we can define

(16)  β(b)ζ(d; m) := ζ(bd; m).
□

2.4. Automorphisms, isomorphisms, and fibers.

Definition 2.16. Let Aut_{C}^{G} E denote the group of G-equivariant automorphisms of E over C. Any φ ∈ Aut_{C}^{G} E must induce a G-equivariant automorphism φ′ : G_{R} G_{R} of right G-modules by φ′ = νp̃_{0} ∘ φ ∘ ν−1 ; p̃_{0}. It is easy to see that if φ(p̃) = p̃_{0}g, then φ′ is simply left multiplication by g. This gives a homomorphism

Ψp̃_{0} : Aut_{C}^{G} E G.

Proposition 2.17. The homomorphism Ψp̃_{0} : Aut_{C}^{G} E G commutes with every element of im(χp̃_{0} ), and depends only upon the (path-)component of E_{gen} in which p̃_{0} lies. Moreover, if C is irreducible, then Ψp̃_{0} is an isomorphism to the centralizer of (i.e., the subgroup of G which commutes with every element of) the image of χp̃_{0} :

Ψp̃_{0} : Aut_{C}^{G} E ^{∼} C(im χp̃_{0} ).

Proof. It is straightforward to check that a change of base point from p̃_{0} to p̃′_{0} = p̃_{0}γ changes Ψp̃_{0} by conjugation.

Ψ_{p̃_{0} γ} = γ^{−1}Ψp̃_{0} γ.

On the other hand, given a path σ : [0, 1] E_{gen} from p̃_{0} to another point q̃_{0} we may parallel transport any point p̃_{0}γ of the fiber E|_{p_{0}} to the point q̃_{0} γ in the fiber E|_{q_{0}} , thus giving an isomorphism of right G-sets σ_{∗} : E|_{p_{0}} ^{∼} E|_{q_{0}} , and one can check that the induced homomorphisms Ψp̃_{0} and Ψq̃_{0} are the same:

Ψp̃_{0} = Ψq̃_{0} : Aut_{C}^{G} E G.

The first two claims of the proposition follow.

It is straightforward to check that if C_{gen} is path connected, then Ψp̃_{0} is injective, and surjectivity can be seen by uniformizing C_{gen} , pulling E back to a trivial bundle on the uniformizer, and checking that left multiplication by any element of G which commutes with holonomy descends to a G-equivariant automorphism of E over C. □

We now turn our attention to automorphisms of pointed admissible G-covers. For a pointed admissible G-cover (E ^{π} C, p̃_{1}, …, p̃_{n} ), we denote the group of Gequivariant automorphisms of E over C which fix the points p̃_{i} by Aut_{C}^{G}(E, p̃_{1}, …, p̃_{n} ).

Proposition 2.18. If C is an irreducible curve, and if m_{1}, …, m_{n} ∈ G_{A} are the monodromies of the admissible G-cover E at p̃_{1}, …, p̃_{n} , respectively, then for any elements γ_{1}, …, γ_{n} such that p̃_{0} ∈ E_{gen} lies in the same connected component of E_{gen} as p̃_{1}γ_{1}, …, p̃_{n} γ_{n}, the map Ψp̃_{0} induces an isomorphism

Ψp̃_{0} : Aut_{C}^{G}(E, p̃_{1}, …, p̃_{n} ) ^{∼} ⟨γ_{1}^{−1}m_{1}γ_{1}⟩ ∩ ⋯ ∩ ⟨γ_{n}^{−1}m_{n} γ_{n} ⟩ ∩ C(im(χp̃_{0} )),

where C(im(χp̃_{0} )) denotes the centralizer of the image of χp̃_{0} .

Proof. If p̃_{i} γ_{i} is in the same component of E_{gen} as p̃_{0}, then there is a path d in C_{gen} from p_{0} to p_{i} which lifts to a path d̃ from p̃_{0} to p̃_{i}, and we have an isomorphism \overline{ν}_{d,p̃_{0}} : E|_{p_{i}} ^{∼} ⟨γ_{i}^{−1}m_{i}γ_{i}⟩\G_{R} of right G-sets taking p̃_{i} γ_{i} to the coset ⟨γ_{i}^{−1}m_{i}γ_{i}⟩. An automorphism φ ∈ Aut_{C}^{G} E with Ψp̃_{0} (φ) = g takes the coset ⟨γ_{i}^{−1}m_{i}γ_{i}⟩ to itself if and only if g ∈ ⟨γ_{i}^{−1}m_{i}γ_{i}⟩. Thus φ fixes the points p̃_{i} γ_{i} and also p̃_{i} if and only if Ψp̃_{0} (φ) ∈ ⟨γ_{i}^{−1}m_{i}γ_{i}⟩ for every i. □

Of course, if p̃_{i} γ is in the same connected component of E_{gen} as p̃_{i} α, then, since Ψp̃_{0} (φ) commutes with holonomy, including γ^{−1}α, the condition Ψp̃_{0} (φ) ∈ ⟨γ^{−1}m_{i}γ⟩ is the same as the condition Ψp̃_{0} (φ) ∈ ⟨α^{′−1}m_{i}α⟩.

Proposition 2.19. For any smooth pointed curve (C, p_{1}, …, p_{n} ) having no nontrivial automorphisms, choose an admissible cover (E ^{π} C, p_{1}, …, p_{n} ) ∈ Adm_{g,n}^{G} . For any p̃_{0} ∈ E_{gen} and for any choice of paths d_{i} in C_{gen} from p_{0} = π(p̃_{0}) to p_{i}, let σ_{i} = σ_{d_{i}} be the corresponding element of π_{1}(C_{gen} , p_{0}). We can describe the fiber (s̃t)^{−1}([E ^{π} C, p_{1}, …, p_{n} ]) of the forgetful map s̃t : \overline{M}_{g,n}^{G} Adm_{g,n}^{G} as the quotient stack

(s̃t)^{−1}(E ^{π} C, p_{1}, …, p_{n} ) = [(∏_{i=1}^{n} ⟨χp̃_{0} (σ_{i})⟩\G_{R}) /C(χp̃_{0} )] = ∐_{Ip̃_{0}} BHp̃_{0} ,

where C(χp̃_{0} ) is the centralizer of the image of χp̃_{0} , acting diagonally on the product, the index set Ip̃_{0} is (∏ _{i=1}^{n} ⟨χp̃_{0} (σ_{i})⟩\G_{R}) / (C(χp̃_{0} )/Hp̃_{0} ), and the group Hp̃_{0} is the image under Ψp̃_{0} of the automorphism group of any pointing (p̃_{1}, …, p̃_{n} ) of E:

Hp̃_{0} = Ψp̃_{0} (Aut_{C}^{G}(E, p̃_{1}, …, p̃_{n} )) = C(χp̃_{0} ) ∩ ⟨χp̃_{0} (σ_{1})⟩ ∩ ⋯ ∩ ⟨χp̃_{0} (σ_{n} )⟩.

Proof. A choice of pointing p̃_{1}, …, p̃_{n} ∈ E is equivalent to a choice \overline{ν} p̃_{0} (p̃_{i} ) ∈ ⟨χp̃_{0} (σ_{i})⟩\G_{R} for each i ∈ {1, …, n}, and any isomorphism between two pointings (E, p̃_{1}, …, p̃_{n} ) and (E, p̃′_{1}, …, p̃′_{n} ) induces an automorphism of E. Conversely, the automorphisms of E act on the set of all pointings, thus Proposition 2.17 gives the first equality. For any pointing, the homomorphism Ψp̃_{0} takes the automorphism group Aut_{C}^{G}(E, p̃_{1}, …, p̃_{n} ) to Hp̃_{0} := C(χp̃_{0} ) ∩ ⟨χp̃_{0} (σ_{1})⟩ ∩ …, ∩⟨χp̃_{0} (σ_{n} )⟩ by Proposition 2.18. The second equality follows. □

Proposition 2.20. Let C = C^{1} ∪ C^{2} be the union of two irreducible curves joined at a single node q. Choose points p̃_{0}^{1}, p̃_{0}^{2} ∈ E_{gen} lying over C_{gen}^{1} and C_{gen}^{2} , respectively, and such that p̃_{0}^{1} and p̃_{0}^{2} lie in the same connected component of E. Also, choose a point q̃ ∈ E|_{q} of the fiber over q which lies in the same connected component of E as p̃_{0}^{1} and p̃_{0}^{2}. Let μ and μ^{−1} be the monodromy of E at q with respect to the orientations of C^{1} and C^{2} respectively. We have an injective homomorphism

Ψ := (Ψp̃_{0}^{1} , Ψp̃_{0}^{2} ) : Aut_{C}^{G}(E, p̃_{1}, …, p̃_{n} ) ) G × G,

which depends only on the connected component of E in which p̃_{0}^{1} and p̃_{0}^{2} lie, and

im Ψ = {(g_{1}, g_{2}) ∈ im Ψp̃_{0}^{1} × im Ψp̃_{0}^{2} |g_{1}g_{2}^{−1} ∈ ⟨μ_{1}⟩}.

Proof. The injectivity follows from arguments similar to the irreducible case. The condition on the elements (g_{1}, g_{2}) ∈ im Ψp̃_{0}^{1} × im Ψp̃_{0}^{2} comes from the fact that any automorphism of E must take both “sides” of the node q̃ to the same point: q̃g_{1} = q̃g_{2}, but q̃g_{i} is only determined up to a (left) coset of ⟨μ⟩. □

Let C be an irreducible curve with one node q obtained by attaching 2 points q_{+} and q_{−} of the normalized curve C^{ν}. An admissible G-cover E of C is obtained by attaching two points q̃_{+} ∈ E^{ν} |_{q_{+}} and q̃_{−} ∈ E^{ν} |_{q_{−}} of an admissible G-cover E^{ν} on C^{ν} which have monodromy μ and μ^{−1}, respectively, for some μ ∈ G. Let p̃_{0} ∈ E_{gen}^{ν} = E_{gen} be in the same connected component of E^{ν} as q̃_{+} is, and let γ ∈ G be chosen so that q̃_{−}γ^{−1} is in that same component of E^{ν}.

Proposition 2.21. Any automorphism φ ∈ Aut_{C}^{G}(E, p̃_{1}, …, p̃_{n} ) induces an automorphism N(φ) ∈ Aut_{C^{ν}}^{G} (E^{ν} , p̃_{1}, …, p̃_{n} ) by pullback to the normalization. For any p̃_{0} ∈ E_{gen} the homomorphism N is injective and is compatible with Ψp̃_{0} ; that is, the following diagram commutes:

[figure: Aut_{C}^{G}(E, p̃_{1}, …, p̃_{n} ); Ψp̃_{0}; G; Aut_{C^{ν}}^{G} (E^{ν} , p̃_{1}, …, p̃_{n} ); N; ∩; Ψp̃_{0}; G; ‖]

Moreover, we have

im(Ψp̃_{0} ∘ N) = {g ∈ Ψp̃_{0} (Aut_{C^{ν}}^{G} E^{ν} )|g ∈ C(γ)}.

Proof. Commutativity of the diagram is straightforward to check and injectivity of N follows from the fact that Ψp̃_{0} is injective. The fact that the image commutes with γ follows from an argument similar to that for holonomy in Proposition 2.17. □

2.5. Distinguished components of \overline{M}_{g,n}^{G} .

Several distinguished components of \overline{M}_{g,n}^{G} will be useful for our construction of G-CohFTs. We describe them and their basic properties in this subsection.

2.5.1. The substack ξ(m) of \overline{M}_{0,3}^{G} (m).

Definition 2.22. For any m := (m_{1}, m_{2}, m_{3}) ∈ G_{A}^{3}, if the product ∏ _{i=1}^{3} m_{i} is not 1, then we define the stack ξ(m) to be the empty stack. Otherwise, let C denote the sphere ℂℙ^{1} with special points p_{0} := 0 and p_{j} = exp(2πj\sqrt{−1}/3) for j ∈ \{1, 2, 3\}. For each j ∈ {1, 2, 3} let d_{i} be the path in C determined by following a straight line from p_{0} to p_{i}. These paths induce elements σ_{i} := σ_{d_{i}} ∈ π_{1}(ℂℙ^{1} − {p_{1}, p_{2}, p_{3}}, p_{0}) (see Definition 2.13), which generate the fundamental group and have trivial product: σ_{1}σ_{2}σ_{3} = 1. Thus the triple d = (d_{1}, d_{2}, d_{3}) is in P_{C}.

We define ξ(m) to be the connected component of \overline{M}_{0,3}^{G} (m) containing the geometric point ζ(d; m), as defined in Definition 2.14.

Remark 2.23. For any m ∈ G, it is clear that the component ξ(m, m^{−1}, 1) is the unique component of \overline{M}_{0,3}^{G} (m, m^{−1}, 1) such that all the points p̃_{i} lie in the same connected component of the admissible G-cover E.

Lemma 2.24. For any m ∈ G_{A}^{3} we have the following identities for the ξ(m). (1) ρ(γ, γ, γ)(ξ(m)) = ξ(γm_{1}γ^{−1}, γm_{2}γ^{−1}, γm_{3}γ^{−1}) for any γ ∈ G.

(2) ρ(m_{1}, 1, 1)(ξ(m)) = ρ(1, m_{2}, 1)(ξ(m)) = ρ(1, 1, m_{3})(ξ(m)) = ξ(m).
(3) For the generators b_{1}, b_{2} of the braid group B_{3}
b_{1}ξ(m) = ξ(m_{1}m_{2}m_{1}^{−1}, m_{1}, m_{3})
b_{2}ξ(m) = ξ(m_{1}, m_{2}m_{3}m_{2}^{−1}, m_{2}).

Thus for any element b ∈ B_{3}, we have

bξ(m) = ξ(bm),

where b acts on the triple m via the Hurwitz action (i.e., the obvious action where, for example, b_{1}(m_{1}, m_{2}, m_{3}) := (m_{1}m_{2}m_{1}^{−1}, m_{1}, m_{3})).

(4) Let s be an isomorphism induced from a cyclic permutation (also denoted

s) in S_{3}, then

sξ(m) = ξ(sm).

Proof. The first identity follows from the fact that the global right action translates all points in the admissible G-cover in ξ by γ. Under this action, the i-th monodromy m_{i} changes to γ^{−1}m_{i}γ for all i = 1, …, n.

The second statement follows from the fact that the action of ρ_{i} on the i-th point p̃_{i} is the same (via the map ν̄p̃_{0} ) as right multiplication acting on the right G-coset ⟨m_{i}⟩; that is, the action φ_{i}(m_{i}) is trivial.

The third statement follows from studying the results of sliding points p_{j} around p_{i}, which we now describe in the case of b_{2}. The case of b_{1} is essentially the same.

The transformation T : z ) 1/z takes p_{0} = 0 to , fixes p_{1}, and interchanges p_{2} and p_{3}. Let E′ = T_{∗}E := (T^{−1})^{∗}E, p̃′_{2} := T_{∗}(p̃_{3}), p̃′_{3} := T_{∗}(p̃_{2}), and p̃′_{1} := T_{∗}(p̃_{1}). The pointed cover (E′, p̃′_{1}, p̃′_{2}, p̃′_{3}) corresponds to the geometric point representing the image of ξ(m) under the action of the transposition s_{(2,3)} . Let γ be a straight path from p_{0} to that passes between p_{3} and p_{1}, e.g., the path γ(t) = −i/(1 − t). Note that via γ we have an isomorphism of (un-pointed) admissible G-covers E′ ≅ E_{χ,p_{0}} , where χ is the homomorphism π_{1}(C_{gen} , p_{0}) G, given by taking γT_{∗}σ_{i}γ^{−1} to m_{i} and with the induced p̃_{0,χ} being the “parallel transport” of T^{∗}(p̃_{0}) along γ. The loop γT_{∗}σ_{3}γ^{−1} (around T(p_{3}) = p_{2}) and the loop γT_{∗}σ_{1}γ^{−1} (around T(p_{1}) = p_{1}) are homotopic to the loops σ_{2} and σ_{1}, respectively. But the loop γT_{∗}σ_{2}γ^{−1} is homotopic to σ_{2}σ_{3}σ_{2}^{−1}. Thus the (un-pointed) G-cover E′ is isomorphic to E′′ := E_{χ,p_{0}} , where χ is the homomorphism taking σ_{1} to m_{1}, σ_{2} to m_{2}m_{3}m_{2}^{−1}, and σ_{3} to m_{2}, that is, to the G-cover E′′ associated to ξ(b_{2}m). And the points p̃′_{1} and p̃′_{3} are the same as those that are induced on ξ(b_{2}m). However, the point p̃′_{2} is not the same as the point p̃′′_{2} induced on ξ(b_{2}m); indeed, p̃′′_{2} is induced by parallel transport from p̃_{0} along the path d_{2}, whereas p̃′_{2} is induced by parallel transport from p̃_{0} along γT_{∗}d_{3}γ^{−1} = (σ_{3})^{−1}d_{2}. Thus they differ by the holonomy m_{2} of the loop σ_{3}; i.e., the claim of the third statement holds.

Finally, the last statement of the lemma follows from the fact that a rotation (multiplication by ± exp(2πi/3)) of ℂℙ^{1} will induce the permutation s on ξ. □

The construction of ξ(m) depends a priori on the choices of p_{i} and d_{i}, but we will see in Proposition 2.26 that it is independent of these choices.

Before we give that proposition, we need to understand better how the different braid actions interact. The fact that the braid action on paths agrees with the usual braid action on loops shows that for any b ∈ B_{3} we may write

bd = (ω_{1}d_{ψ(b(1))} , ω_{2}d_{ψ(b(2))} , ω_{3}d_{ψ(b(3))} )

for some choice of ω_{i} ∈ π_{1}(C_{gen} , p_{0}) (and ψ(b(i)) is the action on i of the permutation induced by the standard surjection B_{3} ^{ψ} S_{3}). Let bχ denote the induced holonomy σ_{bd_{i}} ) m_{i}, and let γ_{i} be the image of ω_{i} in G via bχ. It is clear that bχ is the same homomorphism as that induced by taking d_{i} ) bm_{i}, but the point p̃(bd_{i}) differs from that defined by p̃(d_{ψ(b(i))} by ρ(γ_{i}). That is, we have

β(b)ξ(m) = β(b)ζ(d; m)
= ζ(bd; m)
= ρ(γ)ζ(d; bm)
= ρ(γ)ξ(bm)

= ρ(γ)bξ(m). (17)

Lemma 2.25. The braid action β on ζ_{C} ⊂ \overline{M}_{0,3}^{G} factors through the standard symmetric group action on \overline{M}_{0,3}^{G} via the usual surjection ψ : B_{3} S_{3} to the symmetric group. That is, for any b ∈ B_{3}, d ∈ P_{C}, and m, such that ∏ m_{i} = 1, we have

β(b)ζ(d; m) = ψ(b)ζ(d; m).

Proof. By transitivity of the B_{n} action on P_{C}, for every ζ(d′; m) there exists a b′ ∈ B_{n} such that d′ = b′d, where d is the set of paths used to define ξ. So it suffices to check this only in the case of ξ(m); i.e., where the paths are the standard d. Checking the generators of B_{3} is now quite easy. For example, in the case of b = b_{1} the shift γ is simply (m_{1}, 1, 1) and so equation (17) and Lemma 2.24 item (3) gives

β(b_{1})ξ(m) = ρ(m_{1}, 1, 1)ξ(bm)
= ρ(m_{1}, 1, 1)bξ(m)
= s_{1,2} ξ,

as desired. □

Proposition 2.26. For any m ∈ G^{3} with ∏ _{i=1}^{3} m_{i} = 1, any choice of points p′_{0}, …, p′_{3} ∈ ℂℙ^{1}, and any choice of paths d′_{i} from p′_{0} to p′_{i} for each i ∈ {1, 2, 3} with trivial product (i.e., d′ = (d′_{1}, d′_{2}, d′_{3}) ∈ P_{C}), the geometric point of \overline{M}_{0,3}^{G} (m) defined by ζ(d′, m) lies in the component ξ(m).

Proof. Using the action of PGL(2, ℂ) we may assume that p′_{1} = p_{1}, p′_{2} = p_{2}, and p′_{3} = p_{3}.

Moreover, given any path δ from p_{0} to p′_{0}, we may replace the paths d′_{i} by d′_{i}δ. This gives an isomorphism between the n-pointed admissible G cover defined by the d′_{i} and that defined by the d′_{i}δ. Thus we may assume that p′_{0} = p_{0}.

Since both sets of paths d = (d_{1}, d_{2}, d_{3}) (from the definition of ξ(m)) and d′ = (d′_{1}, d′_{2}, d′_{3}) lie in the set P_{C}, and since the braid action on P_{C} is transitive (Definition-Proposition 2.15), there is an element b′ ∈ B_{3} such that

ζ(d′; m) = ζ(b′d; m) = β(b′)ξ(m).

Moreover, since the endpoint of each d′_{i} is p_{i}, we must have

b′ ∈ ker(ψ : B_{3} S_{3}),

that is, b′ lies in the pure braid group.

The proposition now follows from Lemma 2.25.

□

2.5.2. Distinguished components of \overline{M}_{0,4}^{G} .

Definition 2.27. Let m = (m_{1}, …, m_{4}) be chosen so that ∏ _{i=1}^{4} m_{i} = 1, and let

m_{+} := (m_{1}m_{2})^{−1}, m_{−} = m_{+}^{−1} .

We let ξ_{0,4} (m) denote the component of \overline{M}_{0,4}^{G} (m) which contains the image of ξ(m_{1}, m_{2}, m_{+}) × ξ(m_{−}, m_{3}, m_{4}) under the gluing map

ρ : \overline{M}_{0,3}^{G} (m_{1}, m_{2}, m_{+}) × \overline{M}_{0,3}^{G} (m_{−}, m_{3}, m_{4}) \overline{M}_{0,4}^{G} (m).

Definition 2.28. For any closed substack Q ⊆ \overline{M}_{g,n}^{G} , consider the homology class

[Q] in H_{•}(\overline{M}_{g,n}^{G} ). We define

[[Q]] := 0

when Q is empty, otherwise,

[[Q]] := \frac{1}{deg(st_{Q} )}[Q],

where deg(st_{Q} ) is the degree of the forgetful morphism restricted to Q:

st_{Q} : Q \overline{M}_{g,n} .

Lemma 2.29. Using the notation of Definition 2.27, let

m′_{+} := (m_{4}m_{1})^{−1}, m′_{−} := (m′_{+})^{−1}.

We further let ρ′ denote the gluing map composed with the cyclic permutation s = (4, 3, 2, 1) ∈ S_{4}, that is, ρ′ = s ∘ ρ_{tree} :

\overline{M}_{0,3}^{G} (m_{4}, m_{1}, m′_{+})×\overline{M}_{0,3}^{G} (m′_{−}, m_{2}, m_{3}) ^{ρ_{tree}} \overline{M}_{0,4}^{G} (m_{4}, m_{1}, m_{2}, m_{3}) ^{s} \overline{M}_{0,4}^{G} (m),

and we let ρ′′ denote the gluing map

ρ′′ : \overline{M}_{0,3}^{G} (m_{1}, m′_{+}, m_{4}) × \overline{M}_{0,3}^{G} (m_{2}, m′_{−}, m_{3}) \overline{M}_{0,4}^{G} (m).

(1) The component ξ_{0,4} (m) contains the image of ξ(m_{4}, m_{1}, m′_{+})×ξ(m′_{−}, m_{2}, m_{3}) under the map ρ′ and the image of ξ(m_{1}, m′_{+}, m_{4}) × ξ(m_{2}, m′_{−}, m_{3}) under

the map ρ′′.

(2) We have the following equalities in H_{2}(ξ_{0,4} (m)):

[[ρ(ξ(m_{1}, m_{2}, m_{+}) × ξ(m_{−}, m_{3}, m_{4})) ]] = [[ρ′(ξ(m_{4}, m_{1}, m′_{+}) × ξ(m′_{−}, m_{2}, m_{3})) ]],

and (3)

ρ_{∗}([[ξ(m_{1}, m_{2}, m_{+}) ]]⊗[[ξ(m_{−}, m_{3}, m_{4}) ]]) = ρ′_{∗}([[ξ(m_{4}, m_{1}, m′_{+}) ]]⊗[[ξ(m′_{−}, m_{2}, m_{3}) ]]).

Proof. For any choice m ∈ G_{A}^{4} with ∏ _{i=1}^{4} m_{i} = 1, a construction similar to that of ξ(m_{1}, m_{2}, m_{3}) on ℂℙ^{1} − \{p_{1}, p_{2}, p_{3}, p_{4}\}, with, say, p_{i} := (\sqrt{−1})^{i} and p_{0} := 0, and with straight-line paths d_{i} to each p_{i}, gives a pointed admissible G-cover of ℂℙ^{1}−{p_{1}, p_{2}, p_{3}, p_{4}} which has two obvious degenerations. The first degeneration is given by contracting the great circle defined by \{z = t(1+\sqrt{−1})|t ∈ ℝ∪ \}. This can easily be seen to be the image of ξ(m_{1}, m_{2}, m_{+}) × ξ(m_{−}, m_{3}, m_{4}) under the gluing map ρ_{tree} : \overline{M}_{0,3}^{G} (m_{1}, m_{2}, m_{+}) × \overline{M}_{0,3}^{G} (m_{−}, m_{3}, m_{4}) \overline{M}_{0,4}^{G} (m). Similarly, the second degeneration, given by contracting the great circle \{z = t(1 − \sqrt{−1})|t ∈ ℝ ∪ }, is the image of ξ(m_{1}, m′_{+}, m_{4}) × ξ(m_{2}, m′_{−}, m_{3}) under the gluing map \overline{M}_{0,3}^{G} (m_{1}, m′_{+}, m_{4}) × \overline{M}_{0,3}^{G} (m_{2}, m′_{−}, m_{3}) \overline{M}_{0,4}^{G} (m). The first claim follows from Lemma 2.24 item 4 and the fact that all these gluing morphisms are wellbehaved under cyclic permutations.

To see the second claim, consider the forgetful morphism

st : ξ_{0,4} (m) \overline{M}_{0,4} .

By pulling back the corresponding boundary divisors on \overline{M}_{0,4} , one obtains the equality

[ρ(ξ(m_{1}, m_{2}, m_{+})×ξ(m_{−}, m_{3}, m_{4}))]\frac{A}{B} = [ρ′(ξ(m_{4}, m_{1}, m′_{+})×ξ(m′_{−}, m_{2}, m_{3}))]\frac{A′}{B},

where A is the order of the automorphism group of ρ(ξ(m_{1}, m_{2}, m_{+})×ξ(m_{−}, m_{3}, m_{4})), A′ is the order of the automorphism group of ρ′(ξ(m_{4}, m_{1}, m′_{+})×ξ(m′_{−}, m_{2}, m_{3})), and B is the order of the automorphism group of a generic point in \overline{M}_{0,4}^{G} (m).

Finally, we observe that

ρ_{∗}([ξ(m_{1}, m_{2}, m_{+})]⊗[ξ(m_{−}, m_{3}, m_{4})]) = [ρ(ξ(m_{1}, m_{2}, m_{+})×ξ(m_{−}, m_{3}, m_{4}))]\frac{C}{D_{+}D_{−}} ,

where C is the order of the automorphism group of a generic point in ρ(ξ(m_{1}, m_{2}, m_{+})× ξ(m_{−}, m_{3}, m_{4})), D_{+} is the order of the automorphism group of ξ(m_{1}, m_{2}, m_{+}), and D_{−} is the order of the automorphism group of ξ(m_{−}, m_{3}, m_{4}). This equation, together with its counterpart from ρ_{∗}([ξ(m_{1}, m_{2}, m_{+})] ⊗ [ξ(m_{−}, m_{3}, m_{4})]) and the previously derived equation, yields the desired result. □

2.5.3. Distinguished components of \overline{M}_{1,1}^{G} .

Definition 2.30. Choose elements a, b, m_{1} ∈ G such that m_{1} = [a, b]. Let ρ_{b} be the composition of the morphisms

(18)  ξ(m_{1}, b, ab^{−1}a^{−1}) ^{ρ_{3} (a)} \overline{M}_{0,3}^{G} (m_{1}, b, b^{−1}) ^{ρ′_{b}}  \overline{M}_{1,1}^{G} (m_{1}),

where the first morphism is right action by a in the third factor, and the second morphism in the gluing morphism identifying the 2nd and 3rd marked points.

Similarly, let ρ_{a} be the composition of the morphisms

(19)  ξ(m_{1}, bab^{−1}, a^{−1}) ^{ρ_{2} (b)} \overline{M}_{0,3}^{G} (m_{1}, a, a^{−1}) ^{ρ′_{a}}  \overline{M}_{1,1}^{G} (m_{1}),

where the first morphism is right action by b in the second factor, and the second is again the gluing morphism identifying the 2nd and 3rd marked points.

We define ξ_{1,1} (m_{1}, a, b) to be the component of \overline{M}_{1,1}^{G} (m_{1}) containing the image of ρ_{b}.

Lemma 2.31. The images of ρ_{a} and ρ_{b} lie in the same connected component ξ_{1,1} (m_{1}, a, b) of \overline{M}_{1,1}^{G} (m_{1}). Moreover, the following equation holds in H_{2}(\overline{M}_{1,1}^{G} (m_{1})):

(20)  ρ′_{b∗}([[ρ_{3}(a)ξ(m_{1}, b, ab^{−1}a^{−1}) ]]) = ρ′_{a∗} ([[ρ_{2}(b)ξ(m_{1}, bab^{−1}, a^{−1}) ]]).

Proof. The images of ρ_{a} and ρ_{b} are degenerations of the same smooth admissible G-cover over a smooth torus. In particular, consider a smooth, one-pointed torus (T, p_{1}) with generators α, β, and γ of π_{1}(T, p_{0}) for some point p_{0}, with γ corresponding to the loop σ_{d} induced by a path d from p_{0} to p_{1} (as in Definition 2.13), and [α, β] = γ. The homomorphism χ : π_{1}(T, p_{0}) G that takes α, β , and γ to a, b, and m_{1}, respectively, defines an admissible G-cover E_{χ}, and a point p̃_{0,χ} . Parallel transport along d induces a point p̃_{1} with monodromy m_{1}, giving us a pointed admissible G-cover E_{χ}, p̃_{1}.

It is straightforward to see that the image of ρ_{a} corresponds to the α-cycle shrinking to become a node, while the image of ρ_{b} corresponds to the β-cycle shrinking to become a node. Thus both images lie in the same connected component ξ_{1,1} (m_{1}) of \overline{M}_{1,1}^{G} (m_{1}).

Equation (20) follows from the identity

ρ′_{b∗}([ρ_{3}(a)ξ(m_{1}, b, ab^{−1}a^{−1})])\frac{A}{B} = ρ′_{a∗} ([ρ_{2}(b)ξ(m_{1}, bab^{−1}, a^{−1})])\frac{A′}{B′},

where A is the order of the automorphism group of ρ_{3}(a)ξ(m_{1}, b, ab^{−1}a^{−1}), A′ is the order of the automorphism group of [ρ_{2}(b)ξ(m_{1}, bab^{−1}, a^{−1})], B is the order of the automorphism group of ρ′_{b}(ρ_{3}(a)ξ(m_{1}, b, ab^{−1}a^{−1})), and B′ is the order of the automorphism group of ρ′_{a} (ρ_{2}(b)ξ(m_{1}, bab^{−1}, a^{−1})).

However, B = B′, as their corresponding automorphism groups are both isomorphic to C(a, b) ⊆ G (see Proposition 2.21). □

3. The category of G-graded G-modules

In this section, we briefly review some well-known facts from the category of G-graded G-modules (see [Kas, BaKi]) which will be useful in the sequel.

3.1. G-graded G-modules and their G-coinvariants.

Definition 3.1. Let H := ⨁_{m∈G} H_{m} be a finite-dimensional G_{A}-graded vector

space which is endowed with the structure of a right G-module ρ(γ) : H ^{∼} H for all γ in G, with ρ(γ) taking H_{m} to H_{γ^{−1} mγ} for all m in G. (H, ρ) is said to be a G-graded G-module.

A G-invariant metric η on a G-graded G-module H is a symmetric, nondegenerate, bilinear form η on H which is G-invariant (under the diagonal G action) and which respects the grading, i.e., for all v_{m_{+}} in H_{m_{+}} and v_{m_{−}} in H_{m_{−}} we have η(v_{m_{+}} , v_{m_{−}} ) = 0 unless m_{+}m_{−} = 1.

G-graded G-modules form a category whose objects are G-graded G-modules and whose morphisms are homomorphisms of G-modules which respect the G-grading. Furthermore, the dual of a G-graded G-module inherits the structure of a G-graded G-module.

Example 3.2. Any finite-dimensional G-module V is a G-graded G-module where H_{1} := V and H_{m} := 0 for all m not equal to 1 in G.

Example 3.3. The simplest example of a nontrivial G-graded G-module is ℂ[G], the free vector space generated by G, with its natural G-grading, endowed with the G-action ρ(γ)m := γ^{−1}mγ for all γ, m in G.

Definition 3.4. Recall that \overline{G} is the set of conjugacy classes of G, the conjugacy class of m in G is denoted by \overline{m}, and the conjugacy class of m^{−1} is denoted by \overline{m}^{−1}.

A section s of the natural map G \overline{G} is said to be involutive if s(\overline{m}^{−1}) = s(\overline{m})^{−1} for all \overline{m}.

Definition 3.5. Let (H, ρ) be a G-graded G-module. Let π_{G} : H ) H be the averaging map

π_{G}(v) := \frac{1}{|G|} ∑_{γ∈G} ρ(γ)v

for all v in H. Let \overline{H} be the image of π_{G}. The vector space \overline{H} is called the space of G-coinvariants of H, and it inherits a grading by \overline{G}, denoted by

\overline{H} = ⨁_{\overline{γ} ∈\overline{G}} \overline{H}\overline{γ} .

If η is a metric on H, then let \overline{η} be the restriction of the metric \frac{1}{|G|} η to \overline{H}.

Remark 3.6. The reason for the factor of \frac{1}{|G|} in the definition of \overline{η} will become evident when we discuss the geometry of G-CohFTs.

Let us describe \overline{H} in terms of H.

Proposition 3.7. Let (H, ρ) be a G-graded G-module with a G-invariant metric η.

(1) Consider v\overline{m} in \overline{H}\overline{m}, where v\overline{m} = ∑ _{m′∈\overline{m}} v_{m′} . For all m′ in \overline{m}, v_{m′}

belongs to H_{m′}^{C(m′)} , the C(m′)-invariant subspace of H_{m′} . In particular,
for all v_{m} in H_{m},
π_{G}(v_{m} ) = π_{G}(π_{C(m)} (v_{m} )),
where π_{C(m)} : H_{m} H_{m}^{C(m)} is the averaging map

π_{C(m)} (v_{m} ) := \frac{1}{|C(m)|} ∑_{γ∈C(m)} ρ(γ)v_{m} .

(2) For all m in G, the map π_{m} : H_{m}^{C(m)} \overline{H}\overline{m}, defined as
π_{m} (v_{m} ) := π_{G}(v_{m} ),

is an isomorphism of vector spaces.

(3) For all m_{±} in G and v_{m_{±}} in H_{m_{±}}^{C(m_{±})} , where m_{+}m_{−} = 1, we have

η(π_{m_{+}} (v_{m_{+}} ), π_{m_{−}} (v_{m_{−}} )) = η(π_{G}(v_{m_{+}} ), π_{G}(v_{m_{−}} )) = \frac{|C(m_{+})|}{|G|}η(v_{m_{+}} , v_{m_{−}} ).

(4) If s is an involutive section of the natural map G \overline{G}, then

⨁_{\overline{m} ∈\overline{G}} H_{s(\overline{m} )}^{C(s(\overline{m} ))} \overline{H},

22 T. J. JARVIS, R. KAUFMANN, AND T. KIMURA

taking v_{s(\overline{m} )} ) π_{G}(v_{s(\overline{m} )}), is an isomorphism of vector spaces which is not an isometry.

(5) \overline{η} is nondegenerate, i.e., \overline{H} is a \overline{G}-graded vector space with metric \overline{η}. Proof. To prove part (1), consider w_{m} in H_{m}. We have

π_{G}(w_{m} ) = \frac{1}{|G|} ∑_{γ′∈G} ρ(γ′)w_{m}
= \frac{1}{|G|} ∑_{[γ]∈(C(m)\G)} ∑_{c∈C(m)} ρ(cγ)w_{m}
= \frac{|C(m)|}{|G|} ∑_{[γ]∈C(m)\G} ρ(γ)\frac{1}{|C(m)|} ∑_{c∈C(m)} ρ(c)w_{m}
= \frac{|C(m)|}{|G|} ∑_{[γ]∈C(m)\G} ρ(γ)π_{C(m)} (w_{m} ).

We conclude that

(21)  π_{G}(w_{m} ) = \frac{|C(m)|}{|G|} ∑_{[γ]∈C(m)\G} π_{C(γ^{−1} mγ)} (ρ(γ)w_{m} ),

which finishes the proof.

We prove part (2) by showing that the map f_{m} : \overline{H}\overline{m} H_{m}^{C(m)} , defined by

(22)  f_{m} ( ∑_{m′∈\overline{m}} v_{m′} ) := \frac{|G|}{|C(m)|}v_{m} ,

is the inverse of π_{m}. Notice that the right hand side is C(m)-invariant by part (1). Consider w_{m} in H_{m}^{C(m)} . We have

f_{m} (π_{G}(w_{m} )) = f_{m} (\frac{|C(m)|}{|G|} ∑_{[γ]∈C(m)\G} ρ(γ)w_{m} )

= \frac{|C(m)|}{|G|}f_{m} (w_{m} ) = \frac{|C(m)|}{|G|} \frac{|G|}{|C(m)|}w_{m} = w_{m} .

Therefore, π_{m} is an isomorphism.

To prove part (3), observe that

η(π_{G}(v_{m_{+}} ), π_{G}(v_{m_{−}} )) = \frac{1}{|G|^{2}} ∑_{γ_{±}∈G} η(ρ(γ_{+})v_{m_{+}} , ρ(γ_{−})v_{m_{−}} )

= \frac{1}{|G|^{2}} ∑_{γ_{±}∈G} η(ρ(γ_{−}^{−1} )ρ(γ_{+})v_{m_{+}} , v_{m_{−}} )

= \frac{1}{|G|^{2}} ∑_{γ_{±}∈G} η(ρ(γ_{+}γ_{−}^{−1} )v_{m_{+}} , v_{m_{−}} )

= \frac{1}{|G|^{2}} ∑_{γ∈G} ∑_{γ_{+}∈G} η(ρ(γ)v_{m_{+}} , v_{m_{−}} )

= \frac{1}{|G|} ∑_{γ∈G} η(ρ(γ)v_{m_{+}} , v_{m_{−}} )

= \frac{1}{|G|} ∑_{γ∈C(m_{+})} η(ρ(γ)v_{m_{+}} , v_{m_{−}} )

= \frac{|C(m_{+})|}{|G|}η(v_{m_{+}} , v_{m_{−}} ),

where we used the C(m_{+})-invariance of v_{m_{+}} in the last equality.

Part (4) follows immediately from (2) and (3). Involutivity of s is needed to insure that ⨁_{\overline{m} ∈\overline{G}} H_{s(\overline{m} )}^{C(s(\overline{m} ))} inherits a metric from H compatible with its grading.

To prove part (5), let m_{+}m_{−} = 1, so that η restricted to H_{m_{+}} ⊕ H_{m_{−}} is nondegenerate. H_{m_{±}} is a C(m_{±})-module, so one can write

(23)  H_{m_{±}} = H_{m_{±}}^{C(m_{±})} ⊕ H′_{m_{±}}

as C(m_{±})-modules, where H′_{m_{±}} is the direct sum of all nontrivial irreducible representations of C(m_{±}) appearing in H_{m_{±}} . Notice that since m_{+}m_{−} = 1, we have C(m_{+}) = C(m_{−}).

Since η is C(m_{+})-invariant, η restricted to H_{m_{+}} ⊕ H_{m_{−}} is the direct sum of η restricted to H_{m_{+}}^{C(m_{+})} ⊕ H_{m_{−}}^{C(m_{+})} and η restricted to H′_{m_{+}} ⊕ H′_{m_{−}} . Therefore, η restricted to H_{m_{+}}^{C(m_{+})} ⊕ H_{m_{−}}^{C(m_{−})} is nondegenerate.

Let v_{m_{+}} be in H_{m_{+}}^{C(m_{+})} . Suppose that η(π_{G}(v_{m_{+}} ), π_{G}(v_{m_{−}} )) = 0 for all v_{m_{−}} in H_{m_{−}}^{C(m_{−})} . By part (3), this is equivalent to the condition η(v_{m_{+}} , v_{m_{−}} ) = 0 for all v_{m_{−}} in H_{m_{−}}^{C(m_{−})} . However, η restricted to H_{m_{+}}^{C(m_{+})} ⊕ H_{m_{−}}^{C(m_{−})} is nondegenerate, therefore, v_{m_{+}} = 0. Thus η restricted to \overline{H} is also non-degenerate. □

3.2. Tensor products and the braid group.

As is usual in the representation theory of groups, there are two kinds of tensor products associated to G-graded G-modules,

Definition 3.8. Let H′ be a G′-graded G′-module and H′′ be a G′′-graded G′′- module. Their vector space tensor product H′ ⊗ H′′ is naturally a G′ × G′′-graded G′ × G′′-module called the external tensor product of H′ and H′′.

On the other hand, the category of G-graded G-modules has a natural tensor product which differs from the tensor product of their underlying vector spaces.

Definition 3.9. Let H′ := ⨁_{m∈G} H′_{m} and H′′ := ⨁_{m∈G} H′′_{m} be two G-graded G-modules. Let

H′ ⊙ H′′ := ⨁_{m∈G} H′_{m} ⊗ H′′_{m} ,

with the induced G-module structure, where G acts diagonally. We call H′ ⊙ H′′ the tensor product of H′ and H′′.

Remark 3.10. The G-graded G-module ℂ[G] has the important property that

H ⊙ ℂ[G] ≅ ℂ[G] ⊙ H ≅ H

for any G-graded G-module H.

Finally, we note that this category has a natural action of the braid group, which we now describe.

Definition 3.11. Let H be a G-graded G-module. Its n-fold tensor product H^{⊗n} inherits the structure of a right G^{n} -module. Furthermore, the permutation group S_{n} acts on H^{⊗n} by permuting its factors.

For all i = 1, …, n − 1, let b_{i} : H^{⊗n} H^{⊗n} be defined by

b_{i}(v_{m_{1}} ⊗⋯⊗v_{m_{i}} ⊗v_{m_{i+1}} ⋯⊗v_{m_{n}} ) := v_{m_{1}} ⊗⋯⊗(ρ(m_{i}^{−1})v_{m_{i+1}} )⊗v_{m_{i}} ⊗⋯⊗v_{m_{n}}

for all v_{m_{j}} in H_{m_{j}} , m_{j} in G, and j = 1, …, n − 1.

The following proposition is immediate.

Proposition 3.12. The elements {b_{1}, …, b_{n} } on H^{⊗n} give an action of B_{n} on H^{⊗n}.

4. G-equivariant cohomological field theories

In this section, we introduce the notion of a G-CohFT , defined in terms of \overline{M}_{g,n}^{G} , and prove some of its basic properties.

4.1. G-CohFTs and G-Frobenius algebras.

Definition 4.1. A tuple ((H, ρ), η, {Λ_{g,n} }, 1) is said to be a G-equivariant Cohomological Field Theory (G-CohFT) if the following axioms hold:

(1) (G-Graded G-module) (H, ρ) is a G-graded G-module. The subspace H_{1} is called the untwisted sector of the G-CohFT, and H_{m}, where m ≠ 1, is called a twisted sector of the G-CohFT.

(2) (G^{n} and S_{n} Invariance) For all m := (m_{1}, …, m_{n} ) in G^{n} and all stable pairs (g, n), if we denote H_{m} := ⨂_{i=1}^{n} H_{m_{i}} , then Λ_{g,n} is an element of ⨁_{m} H^{•}(\overline{M}_{g,n}^{G} (m)) ⊗ H_{m}^{∗} which is both S_{n} -invariant and G^{n} -invariant.

(3) (Identity) The element 1 in H_{1} is non-zero, and is called the flat identity or vacuum vector.

(a) (G-Invariance of the Identity) The vacuum vector 1 is G invariant, i.e., ρ(γ)1 = 1 for all γ in G.

(b) (Flat Identity) Under the forgetting tails morphism τ̃ : \overline{M}_{g,n+1}^{G} (m, 1)

\overline{M}_{g,n}^{G} (m), we have

Λ_{g,n+1} (v_{m_{1}} , …, v_{m_{n}} , 1) = τ̃^{∗}Λ_{g,n} (v_{m_{1}} , …, v_{m_{n}} )

for all m in G^{n}, and v_{m_{i}} in H_{m_{i}} for all i = 1, …, n.

(4) (Metric) η is a symmetric, nondegenerate, bilinear form on H such that

η(v_{m_{1}} , v_{m_{2}} ) := ∫_{[[ξ(m_{1} ,m_{2} ,1) ]]} Λ_{0,3} (v_{m_{1}} , v_{m_{2}} , 1).

It follows that η(v_{m_{1}} , v_{m_{2}} ) = 0 unless m_{1}m_{2} = 1. Recall that ξ is defined in Subsection 2.5.1 and the scaled class [[Q]] in Definition 2.28.

(5) (Factorization) Fix any m_{+} ∈ G and m_{−} := (m_{+})^{−1}. Let the set {e_{α} } be a basis for H_{m_{+}} , the set \{{ \atop e_{β}} \} be a basis for H_{m_{−}} , and η^{αβ} be the inverse of the metric

η : H_{m_{+}} ⊗ H_{m_{−}} ℂ

relative to these bases.

(a) For all stable pairs (g_{1}, n_{1} + 1) and (g_{2}, n_{2} + 1) let g = g_{1} + g_{2} and n = n_{1} + n_{2}. For all m in G^{n} and all (v_{m_{1}} …, v_{m_{n}} ) ∈ H_{m} we require

(ρ_{tree}^{∗} Λ_{g,n} )(v_{m_{1}} , …, v_{m_{n}} ) = ∑_{α,β} Λ_{g_{1} ,n_{1} +1} (v_{m_{i_{1}}} , …, v_{m_{in_{1}}} , e_{α} )η^{αβ} Λ_{g_{2} ,n_{2} +1} ({ \atop e_{β}} , v_{m_{j_{1}}} , …, v_{m_{jn_{2}}} )

for all partitions {i_{1}, …, i_{n_{1}} } ⊔ {j_{1}, …, j_{n_{2}} } of the set {1, …, n}.

(b) For all stable pairs (g−1, n+2), all m ∈ G^{n}, and all (v_{m_{1}} , …, v_{m_{n}} ) ∈ H_{m}, the classes Λ must satisfy

(ρ_{loop}^{∗} Λ_{g,n} )(v_{m_{1}} , …, v_{m_{n}} ) = ∑_{α,β} Λ_{g−1,n+2} (v_{m_{1}} , …, v_{m_{n}} , e_{α} , { \atop e_{β}} )η^{αβ} .

Remark 4.2. If G is the trivial group, then a G-CohFT coincides with a CohFT in the sense of Kontsevich-Manin [KoMa].

Example 4.3. The simplest example of a G-CohFT has as its state space H = ⨁_{m∈G} H_{m} := H^{•}(G) = H^{0}(G) ≅ ℂ[G] as G-graded G-modules, i.e., if {e_{m} }_{m∈G} denotes the obvious basis in H, then the G-action ρ(γ) : H_{m} ) H_{γ^{−1} mγ} is ρ(γ)(e_{m} ) := e_{γ^{−1} mγ} for all γ, m in G.

For all m = (m_{1}, …, m_{n} ) in G^{n}, let

Λ_{g,n} (e_{m_{1}} , …, e_{m_{n}} ) := e^{∗}1_{m},

where e : \overline{M}_{g,n}^{G} G^{n}, and 1_{m} in H^{0}(G^{n} ) denotes the fundamental class of the point m in G^{n}.

It follows that

η(e_{m_{1}} , e_{m_{2}} ) := ∫_{[[ξ(m_{1} ,m_{2} ,1) ]]} Λ_{0,3} (e_{m_{1}} , e_{m_{2}} , 1) = δ_{m_{1} ,m_{2}^{−1}} .

Definition 4.4. We will call the G-CohFT of the previous example the group algebra G-CohFT, and we will denote it simply by ℂ[G] whenever it is clear from context that we mean the group algebra G-CohFT and not just the algebra itself.

Remark 4.5. We will see in the next section that this G-CohFT induces the G-Frobenius algebra ℂ[G], and a standard argument (along the lines of [Tu]) shows that the two constructions are actually equivalent, thus we are justified in the terminology and notation of the previous definition.

4.2. Tensor products of equivariant CohFTs.

Given two equivariant CohFTs, one can construct a new one by taking their tensor product. As in the case of G-graded G-modules, there are two tensor products associated to G-CohFTs. The first, the external tensor product, associates to a G-CohFT and a G′-CohFT a G × G′-CohFT. The second is a tensor product in the category of G-CohFTs.

Proposition 4.6. For all m′ in G′^{n} and m′′ in G′′^{n}, let m′ × m′′ denote the element ((m′_{1}, m′′_{2}), …, (m′_{n} , m′′_{n})) in (G′ × G′′)^{n}. Consider the commuting diagram

[figure: \overline{M}_{g,n}^{G′×G′′} (m′ × m′′); Υ; \overline{M}_{g,n}^{G′} (m′) ×_{\overline{M}_{g,n}} \overline{M}_{g,n}^{G′′} (m′′); pr′′; \overline{M}_{g,n}^{G′′} (m′′)]

[figure: \overline{M}_{g,n}^{G′} (m′); pr′; st′; \overline{M}_{g,n}; st′′]

where st′ and st′′ forget the pointed admissible covers and \overline{M}_{g,n}^{G′} (m′)×_{\overline{M}_{g,n}} \overline{M}_{g,n}^{G′′} (m′′) is the fibered product with projections pr′ and pr′′. The map Υ takes an object (E C; p̃_{1}, …, p̃_{n} ) to ((E′ C; p̃′_{1}, …, p̃′_{n} ), (E′′ C; p̃′′_{1}, …, p̃′′_{n})), where E′ is the variety E/G′′ and p̃′_{i} is the marked point on E′ induced by p̃_{i}, E′′ is the variety E/G′ and p̃′′_{i} is the marked point on E′′ induced by p̃_{i}.

(1) Υ is an isomorphism, preserving the (G′ × G′′)^{n} and S_{n} actions.

(2) The morphism pr′ is G^{′n} -equivariant and pr′′ is G^{′′n} -equivariant.

(3) The morphisms pr′, pr′′, st′, st′′ are S_{n} -equivariant.

(4) The morphisms Υ, pr′, pr′′, st′, st′′ commute with the gluing morphisms.

Proof. Part (1) follows from the fact that both E′  C (resp. E′′  C) are

admissible G′-covers (resp. G′′-covers) with the proper monodromies. Conversely, given a pointed admissible G′-cover (E′ C; p̃′_{1}, …, p̃′_{n} ) and a pointed admissible G′′-cover (E′′ C; p̃′′_{1}, …, p̃′′_{n}), it is clear that (E′×_{C}E′′ C; (p̃′_{1}, p̃′′_{1}), …, (p̃′_{n} , p̃′′_{n})) is a pointed admissible G′ × G′′-cover, and thus Υ is an isomorphism. The equivariance under the actions of (G′ × G′′)^{n} and S_{n} is manifest.

Similarly, Parts (2) and (3) are manifest.

We now treat part (4) in the case of the loop for the morphism pr′. For all m′_{±} in G′ and m′′_{±} in G′′ such that m′_{+}m′_{−} = m′′_{+}m′′_{−} = 1, consider the diagram

(24)

\overline{M}_{g−1,n+2}^{G′×G′′} (m′ × m′′, (m′_{+}, m′′_{+}), (m′_{−}, m′′_{−})) \overline{ρ̃ ; }\overline{M}_{g,n}^{G′×G′′} (m′ × m′′)

[figure: \overline{M}_{g−1,n+2}^{G′} (m′, m′_{+}, m′_{−}); pr′_{cut}; ρ̃′; \overline{M}_{g,n}^{G′} (m′); pr′]

where ρ̃ and ρ̃′ are the gluing morphisms and pr′_{cut} and pr′ are the canonical projections. Part (4) states that this diagram commutes, which follows immediately from the definition of the morphisms involved. Similarly, the analogous diagrams for Υ, pr′′, st′, and st′′ also commute. The proof in the case of the tree is identical and will be omitted. □

Corollary 4.7. Let (H′, η′, {Λ′_{g,n} }, 1′) be a G′-CohFT and (H′′, η′′, {Λ′′_{g,n} }, 1′′) be a G′′-CohFT. If we define (25) Λ_{g,n} (v′_{m′_{1}} ⊗v′′_{m′′_{1}} , …, v′_{m′_{n}} ⊗v′′_{m′′_{n}} ) := Υ^{∗}((pr^{′∗}Λ′_{g,n} (v′_{m′_{1}} , …, v′_{m′_{n}} ))∪(pr^{′′∗}Λ′′_{g,n} (v′′_{m′′_{1}} , …, v′′_{m′′_{n}} )))

for all v′_{m′_{i}} in H′_{m′_{i}} and v′′_{m′′_{i}} in H′′_{m′′_{i}} , where the morphisms pr′ and pr′′ are defined as in Proposition 4.6, then (H′ ⊗H′′, η′ ⊗η′′, {Λ_{g,n} }, 1′ ⊗1′′) is a G′ ×G′′-CohFT.

Proof. Let G := G′ × G′′. Using the tensor product of a G′-graded G′-module and G′′-graded G′′-module, H′ ⊗ H′′ inherits the structure of a G-graded G-module. The G-invariance of 1′ ⊗ 1′′ follows.

The G^{n} - and S_{n} -invariance follow from Proposition 4.6(2) and (3), respectively.

The flatness of the identity follows immediately from the definition of Λ_{g,n} .

The metric axiom follows from observation that since \overline{M}_{0,3} is a point, the fibered product \overline{M}_{0,3}^{G′} (m′) ×_{\overline{M}_{0,3}} \overline{M}_{0,3}^{G′′}(m′′) is equal to \overline{M}_{0,3}^{G′} (m′) × \overline{M}_{0,3}^{G′′}(m′′).

We prove the factorization axiom in the case of the loop—the case of the tree is similar. Let us adopt the notation from Proposition 4.6 and define v′_{m′} × v′′_{m′′} to be (v′_{m′_{1}} ⊗ v′′_{m′′_{1}} , …, v′_{m′_{n}} ⊗ v′′_{m′′_{n}} ) for all v′_{m′} in H′_{m′} and v′′_{m′′} in H′′_{m′′}.

¿From the definition of Λ we have

ρ̃^{∗}Λ_{g,n} (v′_{m′} × v′′_{m′′}) = ((pr′ × pr′′) ∘ Δ̃ ∘ Υ ∘ ρ̃)^{∗}(Λ′_{g,n} (v′_{m′}) ⊗ Λ′′_{g,n} (v′′_{m′′})),

where Δ̃ is the diagonal morphism

Δ̃ : \overline{M}_{g,n}^{G′} (m′)×_{\overline{M}_{g,n}} \overline{M}_{g,n}^{G′′} (m′′) \overline{M}_{g,n}^{G′} (m′)×_{\overline{M}_{g,n}} \overline{M}_{g,n}^{G′′} (m′′)×\overline{M}_{g,n}^{G′} (m′)×_{\overline{M}_{g,n}} \overline{M}_{g,n}^{G′′} (m′′).

Let Δ denote the diagonal morphism

Δ : \overline{M}_{g,n}^{G} (m × m′) \overline{M}_{g,n}^{G} (m × m′) × \overline{M}_{g,n}^{G} (m × m′)

and Δ_{cut} denote the diagonal morphism associated to \overline{M}_{g−1,n+2}^{G} (m′×m′′, (m′_{+}, m′′_{+}), (m′_{−}, m′′_{−})) for any m′_{±} in G′ and m′′_{±} in G′′, such that m′_{+}m′_{−} = m′′_{+}m′′_{−} = 1. We have

(pr′ × pr′′) ∘ Δ̃ ∘ Υ ∘ ρ̃ = (pr′ × pr′′) ∘ (Υ × Υ) ∘ Δ̃ ∘ ρ̃
= (pr′ × pr′′) ∘ (Υ × Υ) ∘ (ρ̃× ρ̃) ∘ Δ_{cut}
= (ρ̃′ × ρ̃′′) ∘ (pr′ × pr′′) ∘ (Υ × Υ) ∘ Δ_{cut},

where the first equality follows from the identity (Υ × Υ) ∘ Δ = Δ̃ ∘ Υ, the second from the identity (ρ̃ × ρ̃) ∘ Δ_{cut} = Δ ∘ ρ̃, and the third from Proposition 4.6(4). Putting these together, we obtain

ρ̃^{∗}Λ_{g,n} (v′_{m′} × v′′_{m′′}) = ((pr′ ∘ Υ)^{∗}ρ^{′∗}Λ′_{g,n} (v′_{m′})) ∪ ((pr′′ ∘ Υ)^{∗}ρ^{′′∗}Λ′′_{g,n} (v′′_{m′′}))
= ((pr′ × pr′′) ∘ (Υ × Υ) ∘ Δ)^{∗}(ρ^{′∗}Λ′_{g,n} (v′_{m′})) ⊗ ρ^{′′∗}Λ′′_{g,n} (v′′_{m′′})))
= ((pr′ × pr′′) ∘ Δ̃ ∘ Υ)^{∗}(ρ^{′∗}Λ′_{g,n} (v′_{m′})) ⊗ ρ^{′′∗}Λ′′_{g,n} (v′′_{m′′})))
= Υ^{∗}((pr^{′∗}ρ^{′∗}(Λ′_{g,n} (v′_{m′}))) ∪ (pr^{′′∗}ρ^{′′∗}(Λ′′_{g,n} (v′′_{m′′}))))
= Υ^{∗}((pr^{′∗}Λ′_{g−1,n+2} (v′_{m′}, e′_{α[m′_{+}]} , e′_{α[m′_{−}]} )) ∪
(pr^{′′∗}Λ′′_{g−1,n+2} (v′′_{m′′}, e′′_{β[m′′_{+}]} , e′′_{β[m′′_{−}]} ))η^{′α[m′_{+}]α[m′_{−}]} η^{′′β[m′′_{+}]β[m′′_{−}]}

as desired, where {e′_{α[m′_{±}]} } is a basis for H′_{m′_{±}} and {e′′_{β[m′′_{±}]} } is a basis for H′′_{m′′_{±}} .

This completes the case of the loop. The case of the tree is identical and will be omitted. □

Definition 4.8. Let 𝔊′ = (H′, η′, {Λ′_{g,n} }, 1′) be a G′-CohFT and 𝔊′′ := (H′′, η′′, {Λ′′_{g,n} }, 1′′) be a G′′-CohFT. Their external tensor product 𝔊′ ⊗ 𝔊′′ is the G′ × G′′-CohFT (H′ ⊗ H′′, η′ ⊗ η′′, {Λ_{g,n} }, 1′ ⊗ 1′′), where Λ_{g,n} is defined by Equation (25).

The category of G-CohFTs also has a tensor product induced from the diagonal morphism on \overline{M}_{g,n}^{G} .

Definition 4.9. Let 𝔊′ = (H′, η′, {Λ′_{g,n} }, 1′) and 𝔊′′ = (H′′, η′′, {Λ′′_{g,n} }, 1′′) be G-CohFTs, then consider the tuple (H, η, {Λ_{g,n} }, 1) given by

(1) H = H′ ⊙ H′′ as G-graded G-modules,

(2) For all v′_{m_{1}} ⊗ v′′_{m_{1}} in H_{m_{1}} and v′_{m_{2}} ⊗ v′′_{m_{2}} in H_{m_{2}} ,

η(v′_{m_{1}} ⊗ v′′_{m_{1}} , v′_{m_{2}} ⊗ v′′_{m_{2}} ) := η′(v′_{m_{1}} , v′_{m_{2}} )η′′(v′′_{m_{1}} , v′′_{m_{2}} ),

(3) 1 := 1′ ⊗ 1′′, and

(4)

Λ_{g,n} (v′_{m_{1}} ⊗ v′′_{m_{1}} , …, v′_{m_{n}} ⊗ v′′_{m_{n}} ) := Λ′_{g,n} (v′_{m_{1}} , …, v′_{m_{n}} ) ∪ Λ′′_{g,n} (v′′_{m_{1}} , …, v′′_{m_{n}} ).

(H, η, {Λ_{g,n} }, 1) is said to be the tensor product of the G-CohFTs (H′, η′, {Λ′_{g,n} }, 1′) and (H′′, η′′, {Λ′′_{g,n} }, 1′′) and is denoted 𝔊′ ⊙ 𝔊′′.

Proposition 4.10. The tensor product of two G-CohFTs is a G-CohFT.

Proof. The proof follows, first, from the fact that the diagonal morphism

Δ : \overline{M}_{g,n}^{G} (m) ) \overline{M}_{g,n}^{G} (m) × \overline{M}_{g,n}^{G} (m)

induces a morphism

H_{•}(\overline{M}_{g,n}^{G} (m)) ) H_{•}(\overline{M}_{g,n}^{G} (m)) ⊗ H_{•}(\overline{M}_{g,n}^{G} (m)),

which respects the gluing, the S_{n} actions, and the G^{n} action, and second, from the fact that the cup product is induced via pullback of the diagonal morphism. The definitions of the flat identity and the metric are easily verified. □

Remark 4.11. Let H′ and H′′ be two G-graded G-modules. The G-module structure on H′ ⊙H′′ is induced from the G×G-module structure on the external tensor product H′⊗H′′ via the diagonal homomorphism G ^{⊂} G×G. An analogous phenomenon occurs in the category of G−CohFTs, where the role of the homomorphism G ^{⊂} G × G is replaced by a natural inclusion \overline{M}_{g,n}^{G} (m) ^{⊂} \overline{M}_{g,n}^{G×G} (m × m) for all stable pairs (g, n) and m in G^{n}. This inclusion respects the actions of G^{n} and S_{n} as well as the gluing morphisms. Consequently, the tensor product in the category of G-CohFTs “factors through” the external tensor product.

This natural inclusion is obtained as follows. The diagonal morphism Δ : \overline{M}_{g,n}^{G} (m) ^{⊂} \overline{M}_{g,n}^{G} (m)×

\overline{M}_{g,n}^{G} (m) can be written as the composition

\overline{M}_{g,n}^{G} (m) ^{⊂} Δ̂ \overline{M}_{g,n}^{G} (m) ×_{\overline{M}_{g,n}} \overline{M}_{g,n}^{G} (m) ^{⊂} ^{j} \overline{M}_{g,n}^{G} (m) × \overline{M}_{g,n}^{G} (m),

where Δ̂ is the diagonal morphism into the fibered product, and j is the obvious inclusion. However, \overline{M}_{g,n}^{G} (m) ×_{\overline{M}_{g,n}} \overline{M}_{g,n}^{G} (m) is isomorphic to \overline{M}_{g,n}^{G×G} (m × m)

via Υ. Observe that Δ̂ and ĵboth preserve the actions of S_{n} and G^{n} and the gluing operations.

The G-CohFT ℂ[G] is initial among all G-CohFTs, in the following sense.

Proposition 4.12. Let 𝔊 := (H, η, {Λ_{g,n} }, 1) be any G-CohFT. The tensor product of ℂ[G] with 𝔊 satisfies

ℂ[G] ⊙ 𝔊 ≅ 𝔊 ⊙ ℂ[G] ≅ 𝔊.

The proof is immediate from the definition of tensor product.

4.3. G-Frobenius algebras.

Recall that a Frobenius algebra is a special CohFT. This statement admits a generalization to G-CohFTs and G-Frobenius algebras, as we will see in Theorem 4.15.

Definition 4.13. Let us adopt the notation that v_{m} is a vector in H_{m} for any m ∈ G. A tuple ((H, ρ), ·, 1, η) is said to be a (non-projective) G-Frobenius algebra [Kau, Tu] provided that the following hold:

(1) (G-graded G-module) (H, ρ) is a G-graded G-module.

(2) (Self-invariance) For all γ in G, ρ(γ) : H_{γ} ) H_{γ} is the identity map.

(3) (Metric) η is a symmetric, nondegenerate, bilinear form on H such that η(v_{m_{1}} , v_{m_{2}} ) is nonzero only if m_{1}m_{2} = 1.

(4) (G-graded Multiplication) The binary product (v_{1}, v_{2}) ) v_{1} · v_{2}, called the multiplication on H, preserves the G-grading (i.e., the multiplication takes H_{m_{1}} ⊗ H_{m_{2}} to H_{m_{1} m_{2}} ) and is distributive over addition.

(5) (Associativity) The multiplication is associative; i.e.,

(v_{1} · v_{2}) · v_{3} = v_{1} · (v_{2} · v_{3})

for all v_{1}, v_{2}, and v_{3} in H.

(6) (Braided Commutativity) The multiplication is invariant with respect to the braiding:

v_{m_{1}} · v_{m_{2}} = (ρ(m_{1}^{−1})v_{m_{2}} ) · v_{m_{1}} ,

for all m_{i} ∈ G and all v_{m_{i}} ∈ H_{m_{i}} with i = 1, 2.

(7) (G-equivariance of the Multiplication)

(ρ(γ)v_{1}) · (ρ(γ)v_{2}) = ρ(γ)(v_{1} · v_{2})

for all γ in G, and all v_{1}, v_{2} ∈ H.

(8) (G-invariance of the Metric)

η(ρ(γ)v_{1}, ρ(γ)v_{2}) = η(v_{1}, v_{2})

for all γ in G, and all v_{1}, v_{2} ∈ H.

(9) (Invariance of the Metric)

η(v_{1} · v_{2}, v_{3}) = η(v_{1}, v_{2} · v_{3})

for all v_{1}, v_{2}, v_{3} ∈ H.

(10) (G-invariant Identity) The element 1 in H_{1} is the identity element under the multiplication, and which satisfies

ρ(γ)1 = 1

for all γ in G.

(11) (Trace Axiom) For all a, b in G and v in H_{[a,b]} , let L_{v} denote left multiplication by v:

Tr_{H_{a}} (L_{v}ρ(b^{−1})) = Tr_{H_{b}} (ρ(a)L_{v}).

Remark 4.14. When G is the trivial group, a G-Frobenius algebra is a Frobenius algebra, a unital, commutative, associative algebra with an invariant metric. Given a general G-Frobenius algebra H, there are two ways that one can construct a Frobenius algebra from it. The first Frobenius algebra is obtained by considering the subalgebra H_{1}. The second approach is to consider \overline{H}, the algebra of G-coinvariants of H, with its induced multiplication and identity. The metric on H induces a metric on \overline{H} which makes \overline{H} into a Frobenius algebra.

Theorem 4.15. Let ((H, ρ), η, {Λ_{g,n} }, 1) be a G-CohFT. Define a multiplication · on H as follows: For any m_{1}, m_{2} ∈ G, let m_{3} = (m_{1}m_{2})^{−1}. For all v_{m_{1}} in H_{m_{1}} and v_{m_{2}} in H_{m_{2}} , define

v_{m_{1}} · v_{m_{2}} := ∫_{[[ξ(m_{1} ,m_{2} ,m_{3} ) ]]} Λ_{0,3} (v_{m_{1}} , v_{m_{2}} , e_{α} )η^{αβ} f_{β} ,

where {e_{α} } is a basis for H_{m_{3}} , {f_{β} } is a basis for H_{m_{3}^{−1}} , and η^{αβ} is the inverse of the metric in those bases.

The tuple ((H, ρ), ·, 1, η) is a G-Frobenius algebra.

Proof. The G-module (H, ρ), the metric η, and the identity element 1 in the G-CohFT are the same for the G-Frobenius algebra.

The invariance of the metric follows from the fact that

sξ(m_{1}, m_{2}, m_{3}) = ξ(m_{2}, m_{3}, m_{1}),

where s is the isomorphism induced from the cyclic permutation in S_{3} (this is proved in Lemma 2.24).

Notice also that since ξ(m_{1}, m_{2}, m_{3}) is empty unless m_{1}m_{2}m_{3} = 1, the product is naturally graded.

The product is not commutative, in general, because ξ(m_{1}, m_{2}, m_{3}) ≠ ξ(m_{2}, m_{1}, m_{3}). However, it is braided commutative, because (26) ξ(m_{1}, m_{2}, m_{3}) = b_{1}^{−1}ξ(m_{1}m_{2}m_{1}^{−1}, m_{1}, m_{3}) = σρ_{1}(m_{1})ξ(m_{1}m_{2}m_{1}^{−1}, m_{1}, m_{3}),

with σ the transposition (1, 2) ∈ S_{3}, as shown in Lemma 2.24. The relation (26) on ξ implies the braided commutativity via the equation

v_{m_{1}} · v_{m_{2}} = ∫_{[[ξ(m_{1} ,m_{2} ,m_{3} ) ]]} Λ_{0,3} (v_{m_{1}} , v_{m_{2}} , e_{α} )η^{αβ} f_{β}
= ∫_{[[(σρ_{1} (m_{1} ))ξ(m_{1} m_{2} m_{1}^{−1} ,m_{1} ,m_{3} )  ]]} Λ_{0,3} (v_{m_{1}} , v_{m_{2}} , e_{α} )η^{αβ} f_{β}
= ∫_{[[ξ(m_{1} m_{2} m_{1}^{−1} ,m_{1} ,m_{3} ) ]]}(σρ_{1}(m_{1}))^{∗}(Λ_{0,3} (v_{m_{1}} , v_{m_{2}} , e_{α} ))η^{αβ} f_{β}
= ∫_{[[ξ(m_{1} m_{2} m_{1}^{−1} ,m_{1} ,m_{3} ) ]]} ρ_{1}(m_{1})^{∗}σ^{∗}(Λ_{0,3} (v_{m_{1}} , v_{m_{2}} , e_{α} ))η^{αβ} f_{β}
= ∫_{[[ξ(m_{1} m_{2} m_{1}^{−1} ,m_{1} ,m_{3} ) ]]} ρ_{1}(m_{1})^{∗}(Λ_{0,3} (v_{m_{2}} , v_{m_{1}} , e_{α} ))η^{αβ} f_{β}
= ∫_{[[ξ(m_{1} m_{2} m_{1}^{−1} ,m_{1} ,m_{3} ) ]]} Λ_{0,3} (ρ_{1}(m_{1}^{−1})v_{m_{2}} , v_{m_{1}} , e_{α} )η^{αβ} f_{β}
= (ρ(m_{1}^{−1})v_{m_{2}} ) · v_{m_{1}} .

Again, using the braided commutativity for the classes ξ and the invariance of 1, we can show that 1 is indeed a unit for the multiplication, since

v_{m_{1}} · 1 = ∫_{[[ξ(m_{1} ,1,m_{1}^{−1} ) ]]} Λ_{0,3} (v_{m_{1}} , 1, e_{α} )η^{αβ} f_{β}
= ∫_{[[ξ(m_{1} ,m_{1}^{−1} ,1) ]]} Λ_{0,3} (v_{m_{1}} , e_{α} , 1)η^{αβ} f_{β}
= η(v_{m_{1}} , 1)η^{αβ} f_{β} = v_{m_{1}} ,

where we introduced a basis (e_{α} ) of H_{m_{1}^{−1}} and a basis (f_{β} ) of H_{m_{1}} .

The property that 1 is a unit implies that the invariance of the metric follows from

(27)  η(v_{m_{1}} , v_{m_{2}} ) = η(v_{m_{1}} · v_{m_{2}} , 1).

Equation (27) in turn follows from

η(v_{m_{1}} · v_{m_{2}} , 1) = ∫_{[[ξ(m_{1} ,m_{2} ,m_{3} ) ]]} Λ_{0,3} (v_{m_{1}} , v_{m_{2}} , e_{α} )η^{αβ} ∫_{[[ξ(m_{3} ,1,1) ]]} Λ_{0,3} (f_{β} , 1, 1)
= ∫_{[[ξ(m_{1} ,m_{2} ,m_{3} ) ]]} Λ_{0,3} (v_{m_{1}} , v_{m_{2}} , e_{α} )η^{αβ} η(f_{β} , 1)
= ∫_{[[ξ(m_{1} ,m_{2} ,m_{3} ) ]]} Λ_{0,3} (v_{m_{1}} , v_{m_{2}} , 1)
= η(v_{m_{1}} , v_{m_{2}} ),

where we use the notation m_{3} := (m_{1}m_{2})^{−1}, and we let {e_{α} } be a basis of H_{m_{3}} and {f_{β} } be a basis of H_{m_{3}^{−1}} .

The ρ(γ)-invariance of H_{γ} follows from the second part of Lemma 2.24:

ρ(γ)v_{γ} = ρ(γ)v_{γ} · 1 = ∫_{[[ξ(γ,1,γ^{−1} ) ]]} Λ_{0,3} (ρ(γ)v_{γ} , 1, e_{α} )η^{αβ} f_{β}
= ∫_{[[ρ(γ,1,1)ξ(γ,1,γ^{−1} ) ]]} Λ_{0,3} (ρ(γ)v_{γ} , 1, e_{α} )η^{αβ} f_{β}
= ∫_{[[ξ(γ,1,γ^{−1} ) ]]} Λ_{0,3} (v_{γ} , 1, e_{α} )η^{αβ} f_{β}
= v_{γ} .

Again we use bases {e_{α} } of H_{γ^{−1}} and {f_{β} } of H_{γ}.

The self invariance, together with the invariance of the metric, imply the symmetry of the metric:

η(v_{m} , v_{m^{−1}} ) = η(v_{m} v_{m^{−1}} , 1) = η(ρ(m^{−1})(v_{m^{−1}} )v_{m} , 1) = η(v_{m^{−1}} , v_{m} ).

The G-invariance of the metric follows from the G^{n} -invariance of Λ and the ρ-invariance of the unit 1 via

η(ρ(γ)v_{m_{1}} , ρ(γ)v_{m_{2}} ) = ∫_{[[ξ(γ^{−1} m_{1} γ,γ^{−1} m_{2} γ,1)  ]]} Λ_{0,3} (ρ(γ)v_{m_{1}} , ρ(γ)v_{m_{2}} , 1)
= ∫_{[[ρ(γ,γ,γ)ξ(m_{1} ,m_{2} ,1)  ]]} Λ_{0,3} (ρ(γ)v_{m_{1}} , ρ(γ)v_{m_{2}} , ρ(γ)1)
= ∫_{[[ξ(m_{1} ,m_{2} ,1) ]]} Λ_{0,3} (v_{m_{1}} , v_{m_{2}} , 1)
= η(v_{m_{1}} , v_{m_{2}} ),

where we used the first property of ξ of Lemma 2.24.

The above in turn gives the G-equivariance of the multiplication

ρ(γ)v_{m_{1}} · ρ(γ)v_{m_{2}} = ∫_{[[ξ(γ^{−1} m_{1} γ,γ^{−1} m_{2} γ,(γ^{−1} m_{3} γ)^{−1} )  ]]} Λ_{0,3} (ρ(γ)v_{m_{1}} , ρ(γ)v_{m_{2}} , e_{α} )η^{αβ} f_{β}
= ∫_{[[ρ(γ,γ,γ)(ξ(m_{1} ,m_{2} ,m_{3} ))  ]]} Λ_{0,3} (ρ(γ)v_{m_{1}} , ρ(γ)v_{m_{2}} , ρ(γ)e′_{α} )η^{αβ} ρ(γ)f′_{β}
= ∫_{[[ξ(m_{1} ,m_{2} ,m_{3} ]]} Λ_{0,3} (v_{m_{1}} , v_{m_{2}} , e′_{α} )η^{′αβ} ρ(γ)f′_{β}
= ρ(γ)(v_{m_{1}} · v_{m_{2}} ),

where we used m_{3} := (m_{1}m_{2})^{−1}, a basis {e_{α} } of H_{γ^{−1} m_{3} γ} , {f_{β} } of H_{γ^{−1} m_{3}^{−1} γ} , and the transformed bases {e′_{α} := ρ(γ^{−1})e_{α} } of H_{m_{3}} and {f′_{β} := ρ(γ^{−1})f_{β} } of H_{m_{3}^{−1}} . Also, we used the notation η^{′αβ} for the inverse metric of η_{αβ} = η(e′_{α} , f′_{β} ), the G-invariance of the metric η′_{αβ} = η_{αβ} , and the first property of Lemma 2.24.

Associativity follows from Lemma 2.29 in the following way:

(v_{m_{1}} · v_{m_{2}} ) · v_{m_{3}} = ∫_{[[ξ(m_{1} ,m_{2} ,m_{+} ]]} Λ_{0,3} (v_{m_{1}} , v_{m_{2}} , e_{α} )η^{αβ} ∫_{[[ξ(m_{−},m_{3} ,m_{4} ) ]]} Λ_{0,3} (f_{β} , v_{m_{3}} , k_{γ} )η^{γδ} l_{δ}
= ∫_{[[ξ(m_{1} ,m_{2} ,m_{+})×ξ(m_{−},m_{3} ,m_{4} )  ]]} ρ^{∗}Λ_{0,4} (v_{m_{1}} , v_{m_{2}} , v_{m_{3}} , k_{γ} )η^{γδ} l_{δ}
= ∫_{[[ρ(ξ(m_{1} ,m_{2} ,m_{+})×ξ(m_{−},m_{3} ,m_{4} )  ]]} Λ_{0,4} (v_{m_{1}} , v_{m_{2}} , v_{m_{3}} , k_{γ} )η^{γδ} l_{δ}
= ∫_{[[ρ′(ξ(m_{4} ,m_{1} ,m′_{+})×ξ(m′_{−},m_{2} ,m_{3} ))  ]]} Λ_{0,4} (v_{m_{1}} , v_{m_{2}} , v_{m_{3}} , k_{γ} )η^{γδ} l_{γ}
= ∫_{[[ξ(m_{4} ,m_{1} ,m′_{+})×ξ(m′_{−},m_{2} ,m_{3} )  ]]} ρ^{′∗}Λ_{0,4} (v_{m_{1}} , v_{m_{2}} , v_{m_{3}} , k_{γ} )η^{γδ} l_{δ}
= ∫_{[[ξ(m_{4} ,m_{1} ,m′_{+}) ]]} Λ_{0,3} (k_{γ} , v_{m_{1}} , e_{α} )η^{αβ} ∫_{[[ξ(m′_{−},m_{2} ,m_{3} ) ]]} Λ_{0,3} (f_{β} , v_{m_{2}} , v_{m_{3}} )η^{γδ} l_{δ}
= ∫_{[[ξ(m_{2} ,m_{3} ,m′_{−}) ]]} Λ_{0,3} (v_{m_{2}} , v_{m_{3}} , f_{β} )η^{βα} ∫_{[[ξ(m_{1} ,m′_{+},m_{4} ]]} Λ_{0,3} (v_{m_{1}} , e_{α} , k_{γ} )η^{γδ} l_{δ}
= v_{m_{1}} · (v_{m_{2}} · v_{m_{3}} ),

where we used the S_{3}-invariance of Λ, the fourth property of 2.24, and the symmetry of the metric. Also, we introduced the notation m_{4} = (m_{1}m_{2}m_{3})^{−1} and used the notations of Lemma 2.29 for m_{±}, m′_{±}, ρ, ρ′ ; i.e., m_{+} := (m_{1}m_{2})^{−1}, m_{−} := m_{1}m_{2}, m′_{+} = m_{2}m_{3}, and m′_{−} := (m_{2}m_{3})^{−1}. Furthermore, {e_{α} } is a basis of H_{m_{+}} , {f_{β} } is a basis of H_{m_{−}} , {k_{γ} } is a basis of H_{m_{4}} , and {l_{δ}} is a basis of H_{m_{4}^{−1}} .

Lastly, the proof of the trace axiom follows using Lemma 2.31:

Tr_{H_{a}} (L_{v}ρ(b^{−1})) = η(η^{αβ} f_{β} , v_{aba^{−1} b^{−1}} · ρ(b^{−1})e_{α} )
= η(η^{αβ} f_{β} , ∫_{[[ξ(m_{1} ,bab^{−1} ,a^{−1} ) ]]} Λ_{0,3} (v_{m_{1}} , ρ(b^{−1})e_{α} , f_{γ} )η^{γδ} e_{δ})
= ∫_{[[ξ(a^{−1} ,a,1) ]]} Λ_{0,3} (f_{β} , e_{δ}, 1)η^{αβ} ∫_{[[ξ(m_{1} ,bab^{−1} ,a^{−1} ) ]]} Λ_{0,3} (v_{m_{1}} , ρ(b^{−1})e_{α} , f_{γ} )η^{γδ}
= ∫_{[[ξ(m_{1} ,bab^{−1} ,a^{−1} ) ]]} Λ_{0,3} (k_{λ}, ρ(b^{−1})e_{α} , f_{β} )η^{αβ} ∫_{[[ξ(1,m_{1} ,m_{1}^{−1} ) ]]} Λ_{0,3} (1, v_{m_{1}} , l_{μ} )η^{λμ}
= ∫_{[[ρ_{b∗} ξ(m_{1} ,bab^{−1} ,a^{−1} ) ]]} Λ_{1,1} (k_{λ}) ∫_{[[ξ(1,m_{1} ,m_{1}^{−1} ) ]]} Λ_{0,3} (1, v_{m_{1}} , l_{μ} )η^{λμ}
= ∫_{[[ρ_{a∗} ξ(m_{1} ,b,ab^{−1} a^{−1} ) ]]} Λ_{1,1} (k_{λ}) ∫_{[[ξ(1,m_{1} ,m_{1}^{−1} ) ]]} Λ_{0,3} (1, v_{m_{1}} , l_{μ} )η^{λμ}
= ∫_{[[ξ(m_{1} ,b,ab^{−1} a^{−1} ) ]]} Λ_{0,3} (k_{λ}, g_{γ} , ρ(a)^{−1}h_{δ})η^{γδ} ∫_{[[ξ(1,aba^{−1} b^{−1} ,m_{1}^{−1} ) ]]} Λ_{0,3} (1, v_{m_{1}} , l_{μ} )η^{λμ}
= ∫_{[[ξ(b^{−1} ,b,1) ]]} Λ_{0,3} (h_{τ}, g_{γ} , 1)η^{στ} ∫_{[[ξ(m_{1} ,b,ab^{−1} a^{−1} ) ]]} Λ_{0,3} (v_{m_{1}} , g_{σ} , ρ(a^{−1})h_{δ})η^{γδ}
= Tr_{H_{b}} (ρ(a)L_{v}),

where we used Lemma 2.29, as well as Lemma 2.31 with its notation for the maps ρ_{a} , ρ_{b} and m_{1} = [a, b], and introduced the bases {e_{α} } of H_{a}, {f_{β} } of H_{a^{−1}} , {g_{γ} } of H_{b}, {h_{δ}} of H_{b^{−1}} , {k_{λ}} of H_{m_{1}} , and {l_{μ} } of H_{m_{1}^{−1}} .

□

We can now justify naming the G-CohFT ℂ[G] of Example 4.3 the group ring G-CohFT.

Proposition 4.16. In the group ring G-CohFT, the metric η on H = ⨁_{g∈G} ℂ satisfies

η(e_{m_{1}} , e_{m_{2}} ) = δ_{m_{1} ,m_{2}^{−1}}

for all m_{1}, m_{2} in G.

The multiplication is given by

e_{m_{1}} · e_{m_{2}} = e_{m_{1} m_{2}}

for all m_{1}, m_{2} in G. The identity element is 1 := e_{1}. The resulting G-Frobenius algebra is isomorphic to the group ring ℂ[G].

Proof. The multiplication operation is

e_{m_{1}} · e_{m_{2}} = ∫_{[[ξ(m_{1} ,m_{2} ,(m_{1} m_{2} )^{−1} ) ]]} Λ_{0,3} (e_{m_{1}} , e_{m_{2}} , e_{(m_{1} m_{2} )^{−1}} )e_{m_{1} m_{2}} = e_{m_{1} m_{2}} .

The metric and identity element follow by a similar calculation. □

5. CohFTs and Quotients of G-CohFTs

In this subsection, we explain how to obtain a CohFT from a G-CohFT by taking the appropriate quotient with respect to G. Geometrically, going from a G-CohFT to a CohFT corresponds to going from \overline{M}_{g,n}^{G} to \overline{M}_{g,n} , where the Λ_{g,n} are allowed to only act upon elements of \overline{H}. We perform this procedure in two steps. The first step is to go from \overline{M}_{g,n}^{G} to \overline{M}_{g,n} (BG). The second step is to go from \overline{M}_{g,n} (BG) to \overline{M}_{g,n} .

5.1. From \overline{M}_{g,n}^{G} to \overline{M}_{g,n} (BG).

We begin with a useful lemma.

Lemma 5.1. For all \overline{m} in \overline{G}^{n}, the forgetful morphism s̃t\overline{m} : \overline{M}_{g,n}^{G} (\overline{m}) \overline{M}_{g,n} (BG; \overline{m})

induces a ring isomorphism s̃t∗ ; \overline{m} : H^{•}(\overline{M}_{g,n} (BG; \overline{m})) H^{•}(\overline{M}_{g,n}^{G} (\overline{m}))^{G^{n}} .

Proof. Let C be the constant sheaf of complex numbers on \overline{M}_{g,n}^{G} (\overline{m}), and let C′ be the constant sheaf on \overline{M}_{g,n} (BG; \overline{m}).

Since s̃t is finite, the Leray spectral sequence degenerates, giving

H^{p}(\overline{M}_{g,n}^{G} (\overline{m}), C) = H^{p}(\overline{M}_{g,n} (BG; \overline{m}), s̃t_{∗}(C)).

Since these sheaves are all sheaves of vector spaces over ℂ, they are all divisible, hence the coinvariant map π_{G^{n}} is well defined and preserves invariants; i.e., if i : (s̃t_{∗}C)^{G^{n}} s̃t_{∗}C is the natural inclusion, then π_{G^{n}} ∘ i = 1. Thus taking G^{n} -invariants is the same as applying the map π_{G^{n}} , and is exact. So a general homological argument gives that

(H^{p}(\overline{M}_{g,n} (BG; \overline{m}), s̃t_{∗}C))^{G^{n}} = H^{p}(\overline{M}_{g,n} (BG; \overline{m}), (st_{∗}C)^{G^{n}} ),

and we have

(H^{p}(\overline{M}_{g,n}^{G} (\overline{m}), C))^{G^{n}} = H^{p}(\overline{M}_{g,n} (BG; \overline{m}), (s̃t_{∗}C)^{G^{n}} ).

On the other hand, we have s̃t^{∗}(C′) = C, so by adjointness we have a map j : C′ s̃t_{∗}C. Composing with π_{G^{n}} , we get a map of sheaves π_{G^{n}} ∘ j : C′ (s̃t_{∗}C)^{G^{n}} . On each fiber this map is an isomorphism, since for a fixed admissible cover E C the fiber F := s̃t^{−1}([E C]) is a disjoint union of points with transitive G^{n} -action inducing the G^{n} -action on s̃t_{∗}C = ⨁_{f∈F} C′, and j is just given by q ) (q, q, ..., q). Some straightforward work shows that for any vector space V and any set F with transitive G^{n} -action, the vector space V × F has as its G^{n} -invariants exactly the image of the map j : V V × F, taking v to (v, v, ..., v). In particular, this holds for V = C′. Since the fiber F and the G^{n} - action on F are unchanged under small deformation, this shows that the morphism of sheaves π_{G^{n}} ∘ j induces an isomorphism on stalks, and thus is an isomorphism of sheaves. So we have

H^{p}(\overline{M}_{g,n} (BG; \overline{m}), C′) = H^{p}(\overline{M}_{g,n} (BG; \overline{m}), (s̃t_{∗}(C))^{G^{n}} )
= H^{p}(\overline{M}_{g,n} (BG; \overline{m}), s̃t_{∗}C)^{G^{n}} = H^{p}(\overline{M}_{g,n}^{G} , C)^{G^{n}}

as desired.

□

Proposition 5.2. Let (H, η, Λ_{g,n} , 1) be a G-CohFT. There exist uniquely-determined

classes Λ̂_{g,n} in ⨁_{\overline{m}∈\overline{G}^{n}} H^{•}(\overline{M}_{g,n} (BG; \overline{m})) ⊗ \overline{H}∗ ; \overline{m} such that

s̃t∗ ; \overline{m}Λ̂_{g,n} (v\overline{m}) = Λ_{g,n} (v\overline{m})

for all v\overline{m} in \overline{H}\overline{m}.

Proof. Consider v\overline{m} in \overline{H}\overline{m} for \overline{m} in \overline{G}^{n}. For all γ in G^{n} we have

ρ(γ)^{∗}(Λ_{g,n} (v\overline{m})) = Λ_{g,n} (ρ(γ^{−1})_{∗}v\overline{m})) = Λ_{g,n} (v\overline{m})),

where the first equality is by the (diagonal) G^{n} -invariance of Λ_{g,n} and the second is by the definition of \overline{H}. Therefore, Λ_{g,n} (v\overline{m}) belongs to H^{•}(\overline{M}_{g,n} (\overline{m}))^{G^{n}} , and we are done by the previous lemma. □

Fix an element m_{+} in G and let m_{−} := m_{+}^{−1}. To each such choice, we have the following associated commutative diagram which we will use extensively hereafter, and whose morphisms and other terms we explain below:

(28)

[figure: FG ; Γ̂; μ̃; \overline{M}G ; Γ̂; ⊂; ĩ; \overline{M}_{g,n}^{G} (\overline{m})]

\overline{M}G ; Γ̃_{cut}  r̃

[figure: FΓ̂(BG); p̃r; μ̂; \overline{M}Γ̂(BG); s̃t′; ⊂; î; \overline{M}_{g,n} (BG; \overline{m}); s̃t]

[figure: \overline{M}Γ̂_{cut}; s̃t′′; r̂]

[figure: \overline{M}_{Γ_{cut}}; p̂r; μ; ŝt′′; \overline{M}_{Γ}; ŝt′; ⊂; i; \overline{M}_{g,n}; ŝt]

The above diagram has two cases. The first case corresponds to the situation where all graphs are decorated stable graphs of genus g with n tails which are trees of the form

(29)  Γ̃ =  and Γ̃_{cut} =

(30)  Γ̂ =  and Γ̂_{cut} =

and

(31)  Γ =  and Γ_{cut} =  ,

where N_{+} := {i_{1}, …, i_{n_{+}} }, is the index set of the labels for the tails on the left half of each graph above, N_{−} := {j_{1}, …, j_{n_{−}} }, is the index set of the labels for the tails on the right half of each graph above, N_{+} ⊔ N_{−} = {1, …, n}, and g_{+} + g_{−} = g.

The second case corresponds to the situation where all graphs are decorated stable graphs of genus g with n tails which are loops of the following form:

(32)  Γ̃ =  and Γ̃_{cut} =

(33)  Γ̂ =  and Γ̂_{cut} =

and

(34)  Γ =  and Γ_{cut} =  .

Note that in both cases, the graph Γ̃ has tails labeled by conjugacy classes \overline{m}_{i}, but its one edge is labeled by a specific choice of m_{+} and m_{−}.

Now we explain the various terms and morphisms. \overline{M}_{Γ} is the closure of the locus in \overline{M}_{g,n} whose dual graph is Γ, i is the inclusion morphism, and μ is the normalization morphism associated to cutting the internal edge of Γ. Similarly, \overline{M}Γ̂(BG) denotes the closure of the locus in \overline{M}_{g,n} (BG; \overline{m}) whose associated dual graph has tails decorated by \overline{m} and whose monodromies around one side of the node lie in \overline{m}_{+} and whose monodromies around the other side of the node lie in \overline{m}_{−}. The morphism îis the inclusion, and FΓ̂(BG) is the fibered product \overline{M}_{Γ_{cut}}×_{\overline{M}_{Γ}} \overline{M}Γ̂(BG). The morphisms μ̂ and p̂r are the canonical projections of the fibered product. To explain r̂, we first note that FΓ̂(BG) is the the stack of triples consisting of a cut curve C′ in \overline{M}_{Γ_{cut}}, an admissible G-cover E ) C in \overline{M}Γ̂(BG), and an isomorphism α in \overline{M}_{Γ} from the glued curve μ(C′) to C. The morphism r̂ takes such a triple to the pullback of E along the composition α ∘ μ. The morphisms ŝt, ŝt′, and ŝt′′ simply forget their respective twisted curve structures.

Similarly, \overline{M}G ; Γ̂ is the closure of the locus of pointed G-covers with dual graph Γ̂, so all tails are labeled by conjugacy classes \overline{m}_{i}; and \overline{M}G ; Γ̃_{cut} is the closure of the locus of pointed G-covers with dual graph Γ̃_{cut}, so their tails are labeled with conjugacy classes \overline{m}_{i}, but on the two sides of the node their monodromies are the specific group elements m_{+} and m_{−}. The morphisms s̃t′ and s̃t′′ simply forget the marked points in the G-cover.

The stack FG ; Γ̂ is the fibered product FΓ̂(BG) ×_{\overline{M}Γ̂(BG)} \overline{M}_{Γ}^{G} = \overline{M}_{Γ_{cut}} ×_{\overline{M}_{Γ}} \overline{M}_{Γ}^{G}, and the morphisms μ̃ and p̃r are the canonical projections. The morphism r̃ is induced by the pair of the gluing map ρ : \overline{M}G ; Γ̃_{cut} \overline{M}G ; Γ̂ and the map ŝt′′ ∘ s̃t′′ :

\overline{M}G ; Γ̃_{cut} \overline{M}_{Γ_{cut}} (actually the gluing map has as its target \overline{M}_{g,n}^{G} , but it factors through the substack \overline{M}G ; Γ̂). In particular, we can write the gluing morphism on \overline{M}^{G} as

(35)  ρ̃Γ̃ = ĩ∘ μ̃ ∘ r̃,

while the corresponding gluing morphism on \overline{M} can be written as

(36)  ρ_{Γ} = i ∘ μ.

Remark 5.3. The morphisms i, î, ĩ are regular embeddings. The remaining morphisms in the diagram are both flat and proper.

Notation 5.4. For all \overline{m} in \overline{G}, let |C(\overline{m})| denote the order of the subgroup C(m′) of G for any m′ in \overline{m}, as it is independent of the choice of m′.

Theorem 5.5. Let {Λ̂_{g,n} } be a collection of classes associated to a G-CohFT \{Λ_{g,n} \}, as in Proposition 5.2. Fix any conjugacy class \overline{m}_{+}, and let \overline{m}_{−} := \overline{m}_{+}^{−1}. Let Γ̂ be a decorated stable graph of genus g with n-tails which is either a tree, as in Equation (30), or a loop, as in Equation (33). Let v\overline{m} belong to \overline{H}\overline{m}.

When Γ̂ is a tree then (37)

r̂_{∗}μ̂^{∗} î^{∗} Λ̂_{g,n} (v\overline{m}) = \frac{deg(ŝt′)}{deg(ŝt′′)} ∑_{β[\overline{m} _{+}],β[\overline{m} _{−}]} Λ̂_{g_{+},n_{+}+1} (v_{\overline{m}_{N+}} , e_{β[\overline{m} _{+}]} )η̂^{β[\overline{m} _{+}],β[\overline{m} _{−}]} Λ̂_{g_{−},n_{−}+1} (e_{β[\overline{m} _{−}]} , v_{\overline{m}_{N−}} ),

where N_{+} ⊔ N_{−} = {1, …, n} is the partition corresponding to the tree, n_{±} = |N_{±}| and v_{\overline{m}_{N_{±}}} denotes the n_{±}-tuple ∏ _{i∈N_{±}} v_{m_{i}} , the collection \{e_{β[\overline{m} _{±}]} \} is a basis of \overline{H}_{m_{±}}, and g_{+} + g_{−} = g. And η̂^{β[\overline{m} _{+}]β[\overline{m} _{−}]} is the inverse of the metric η̂ on \overline{H} in the basis \{e_{β[\overline{m} _{±}]} \}, where

(38)  η̂(v_{\overline{m} _{+}}, v_{\overline{m} _{−}}) := |C(\overline{m}_{+})|\overline{η}(v_{\overline{m} _{+}}, v_{\overline{m} _{−}})

for all v_{\overline{m} _{±}} in \overline{H}_{\overline{m} _{±}}.

When Γ̂ is a loop then (39)

r̂_{∗}μ̂^{∗} î^{∗} Λ̂_{g,n} (v\overline{m}) = \frac{deg(ŝt′)}{deg(ŝt′′)} ∑_{β[\overline{m} _{+}],β[\overline{m} _{−}]} Λ̂_{g−1,n+2} (v\overline{m}, e_{β[\overline{m} _{+}]} , e_{β[\overline{m} _{−}]} )η̂^{β[\overline{m} _{+}],β[\overline{m} _{−}]} .

In either case, denote the right hand side of equations (37) and (39) by \frac{deg(ŝt′)}{deg(ŝt′′)} Λ̂Γ̂_{cut} .

Remark 5.6. This theorem suggests that Λ̂_{g,n} should be regarded as an analog of the virtual class c_{g,n}^{1/r} on \overline{M}_{g,n}^{1/r} , the moduli stack of r-spin curves [JKV]. Equations

(37) and (39) should be regarded as an analog of the Cutting-Edges axiom.

Proof: (of Theorem 5.5). Let m_{+} be any representative of the conjugacy class \overline{m}_{+} and let m_{−} := m_{+}^{−1}. Consider the associated commuting diagram (28) and graphs

(29) to (34).

For v\overline{m} in \overline{H}\overline{m}, let

I := s̃t′′_{∗}ρ̃∗ ; Γ̃Λ_{g,n} (v\overline{m}).

We have

ρ̃∗ ; Γ̃Λ_{g,n} (v\overline{m}) = ρ̃∗ ; Γ̃ s̃t^{∗}Λ̂_{g,n} (v\overline{m})
= (s̃t ∘ ĩ∘ μ̃ ∘ r̃)^{∗}Λ̂_{g,n} (v\overline{m})
= (î∘ μ̂ ∘ p̃r ∘ r̃)^{∗}Λ̂_{g,n} (v\overline{m}).

Therefore,

I = s̃t′′_{∗}(î∘ μ̂ ∘ p̃r ∘ r̃)^{∗}Λ̂_{g,n} (v\overline{m})
= (r̂∘ p̃r ∘ r̃)_{∗}(î∘ μ̂ ∘ p̃r ∘ r̃)^{∗}Λ̂_{g,n} (v\overline{m})
(40)  = deg(p̃r ∘ r̃)r̂_{∗}μ̂^{∗} î^{∗} Λ̂_{g,n} (v\overline{m})

because p̃r ∘ r̃ is finite and surjective.

For all m in G, let {e_{α[m]} } be a basis for H_{m} such that {e_{α[m]} } is the disjoint union of a basis {e_{μ[m]} } for H_{m}^{C(m)} and a basis {e_{ν[m]} } for H′_{m} as in Equation (23), such that for all γ in G,

(41)  ρ(γ)e_{μ[m]} = e_{μ[γ^{−1} mγ]} .

Assume that Γ̂ is a tree, then let \overline{M}_{+}^{G}(m′_{+}) := \overline{M}_{g_{+},n_{+}+1}^{G} (\overline{m}_{N_{+}} , m′_{+}) and

\overline{M}_{−}^{G}(m′_{−}) := \overline{M}_{g_{−},n_{−}+1}^{G} (m′_{−}, \overline{m}_{N_{−}} ) for all m′_{±} in \overline{m}_{±}. Let \overline{M}_{±}^{G}(\overline{m}_{±}) := ∐ _{m′_{±}∈\overline{m} _{±}} \overline{M}_{±}^{G}(m′_{±}).

We can write \overline{M}G ; Γ̃_{cut} = \overline{M}_{+}^{G}(m_{+}) × \overline{M}_{−}^{G}(m_{−}). Similarly, let

Λ_{+}(v_{m_{+}} ) := Λ_{g_{+},n_{+}+1} (v_{\overline{m}_{N+}} , v_{m_{+}} )

and

Λ_{−}(v_{m_{−}} ) := Λ_{g_{−},n_{−}+1} (v_{m_{−}} , v_{\overline{m}_{N−}} )

for all v_{m_{±}} in H_{m_{±}} . Furthermore, let Λ̂_{±} (v_{\overline{m} _{±}}) be defined by

Λ_{±}(v_{\overline{m} _{±}}) = s̃t^{∗}Λ̂_{±} (v_{\overline{m} _{±}})

for all v_{\overline{m} _{±}} in H_{\overline{m} _{±}}.

The G-CohFT axioms imply that

I = ∑_{α[m_{±}]} s̃t′′_{∗}(Λ_{+}(e_{α[m_{+}]} )η^{α[m_{+}]α[m_{−}]} Λ_{−}(e_{α[m_{−}]} ))
= ∑_{α[m_{±}]} s̃t_{\overline{m} _{+}∗}Λ_{+}(e_{α[m_{+}]} )η^{α[m_{+}]α[m_{−}]} s̃t_{\overline{m} _{−}∗}Λ_{−}(e_{α[m_{−}]} ),

where we have the natural forgetful morphisms s̃t_{\overline{m} _{+}} : \overline{M}_{+}^{G}(\overline{m}_{+}) \overline{M}_{g_{+},n_{+}+1} (BG; \overline{m}_{N_{+}} , \overline{m}_{+}) and s̃t_{\overline{m} _{−}} : \overline{M}_{−}^{G}(\overline{m}_{−}) \overline{M}_{g_{−},n_{−}+1} (BG; \overline{m}_{−}, \overline{m}_{N_{−}} ).

Therefore, since Λ is G-equivariant and the fibers of s̃t_{\overline{m} _{±}} are G-orbits, we have

I = ∑_{α[m_{±}]} s̃t_{\overline{m} _{+}∗}Λ_{+}(π_{G}(e_{α[m_{+}]} ))η^{α[m_{+}]α[m_{−}]} s̃t_{\overline{m} _{−}∗}Λ_{−}(π_{G}(e_{α[m_{−}]} ))
= ∑_{μ[m_{±}]} s̃t_{\overline{m} _{+}∗}Λ_{+}(π_{G}(e_{μ[m_{+}]} ))η^{μ[m_{+}]μ[m_{−}]} s̃t_{\overline{m} _{−}∗}Λ_{−}(π_{G}(e_{μ[m_{−}]} ))
= ∑_{μ[m_{±}]} s̃t_{\overline{m} _{+}∗}s̃t_{\overline{m} _{+}}^{∗} Λ̂_{+} (π_{G}(e_{μ[m_{+}]} ))η^{μ[m_{+}]μ[m_{−}]} s̃t_{\overline{m} _{−}∗}s̃t_{\overline{m} _{−}}^{∗} Λ̂_{−} (π_{G}(e_{μ[m_{−}]} ))
= ∑_{μ[m_{±}]} deg(s̃t_{\overline{m} _{+}}) deg(s̃t_{\overline{m} _{−}})Λ̂_{+} (π_{G}(e_{μ[m_{+}]} ))η^{μ[m_{+}]μ[m_{−}]} Λ̂_{−} (π_{G}(e_{μ[m_{−}]} ))
= ∑_{μ[m_{±}]} deg(s̃t_{\overline{m} _{+}} × s̃t_{\overline{m} _{−}})Λ̂_{+} (π_{G}(e_{μ[m_{+}]} ))η^{μ[m_{+}]μ[m_{−}]} Λ̂_{−} (π_{G}(e_{μ[m_{−}]} )),

where the first equality holds because s̃t_{∗}Λ_{g,n} belongs to H^{•}(\overline{M}_{g,n} (BG)) ⊗ \overline{H}^{∗⊗n}, and the second follows from the choice of basis and Proposition 3.7(1).

Furthermore, let s̃t′′_{(m′_{+},m′_{−})} denote the forgetful morphism

\overline{M}_{+}^{G}(m′_{+})×\overline{M}_{−}^{G}(m′_{−}) \overline{M}_{g_{+},n_{+}+1} (BG; \overline{m}_{N_{+}} , \overline{m}_{+})×\overline{M}_{g_{−},n_{−}+1} (BG; \overline{m}_{−}, \overline{m}_{N_{−}} )

for all m′_{±} in \overline{m}_{±}, then

deg(s̃t_{\overline{m} _{+}} × s̃t_{\overline{m} _{−}}) = ∑_{m′_{±}∈\overline{m} _{±}} deg(s̃t′′_{(m′_{+},m′_{−})} ) = \frac{|G|^{2}}{|C(\overline{m}_{+})|^{2}} deg(s̃t′′_{(m_{+},m_{−})} ),

where in the second equality, we have used that deg(s̃t′′_{(m′_{+},m′_{−})} ) is independent

of the choice m′_{±} in \overline{m}_{±}, the fact that \overline{m}′_{±} contains \frac{|G|}{|C(\overline{m} _{±})|} elements, and that |C(\overline{m})| = |C(\overline{m}^{−1})| for all conjugacy classes \overline{m} in G. Thus,

(42) I = ∑_{μ[m_{±}]} deg(s̃t′′)\frac{|G|^{2}}{|C(\overline{m}_{+})|^{2}} Λ̂_{+} (π_{G}(e_{μ[m_{+}]} ))η^{μ[m_{+}]μ[m_{−}]} Λ̂_{−} (π_{G}(e_{μ[m_{−}]} )),

but s̃t′′ = r̂∘ p̃r ∘ r̃, hence, (43)

I = ∑_{μ[m_{±}]} deg(r̂) deg(p̃r ∘ r̃)\frac{|G|^{2}}{|C(\overline{m}_{+})|^{2}} Λ̂_{+} (π_{G}(e_{μ[m_{+}]} ))η^{μ[m_{+}]μ[m_{−}]} Λ̂_{−} (π_{G}(e_{μ[m_{−}]} )).

Equating Equations (40) and (43) and canceling factors of deg(p̃r ∘ r̃), we obtain (44)

r̂_{∗}μ̂^{∗} î^{∗} Λ̂_{g,n} (v\overline{m}) = \frac{|G|^{2}}{|C(\overline{m}_{+})|^{2}} deg(r̂) ∑_{μ[\overline{m} _{±}]} Λ̂_{+} (π_{G}(e_{μ[m_{+}]} ))η^{μ[\overline{m} _{+}]μ[\overline{m} _{−}]} Λ̂_{−} (π_{G}(e_{μ[m_{−}]} )).

Let ϵ_{μ[\overline{m} _{±}]} := π_{G}(e_{μ[m_{±}]} ). Notice that the left hand side only depends upon \overline{m}_{±} because of Equation (41). Since {e_{μ[m_{±}]} } is a basis for H_{m_{±}}^{C(m_{±})} , then by Proposition 3.7(2), \{ϵ_{μ[\overline{m} _{±}]} \} is a basis for \overline{H}_{\overline{m} _{±}}. Let

\overline{η}_{μ[\overline{m} _{+}]μ[\overline{m} _{−}]} := \overline{η}(ϵ_{μ[\overline{m} _{+}]} , ϵ_{μ[\overline{m} _{−}]} ) = \frac{1}{|G|}η(ϵ_{μ[\overline{m} _{+}]} , ϵ_{μ[\overline{m} _{−}]} ) = \frac{|C(m_{+})|}{|G|^{2}}η_{μ[m_{+}]μ[m_{−}]}

where η_{μ[m_{+}]μ[m_{−}]} = η(e_{μ[m_{+}]} , e_{μ[m_{−}]} ). Therefore, taking inverses,

(45)  \overline{η}^{μ[\overline{m} _{+}]μ[\overline{m} _{−}]} = \frac{|G|^{2}}{|C(\overline{m}_{+})|}η^{μ[m_{+}]μ[m_{−}]}

and

η̂^{μ[\overline{m} _{+}]μ[\overline{m} _{−}]} = \frac{|G|^{2}}{|C(\overline{m}_{+})|^{2}} η^{μ[m_{+}]μ[m_{−}]}

by Equation (38), so (46)

r̂_{∗}μ̂^{∗} î^{∗} Λ̂_{g,n} (v\overline{m}) = deg(r̂) ∑_{μ[\overline{m} _{±}]} Λ̂_{g_{+},n_{+}+1} (v_{\overline{m}_{N+}} , ϵ_{μ[\overline{m} _{+}]} ))η̂^{μ[\overline{m} _{+}]μ[\overline{m} _{−}]} Λ̂_{g_{−},n_{−}+1} (ϵ_{μ[\overline{m} _{−}]} , v_{\overline{m}_{N−}} ).

To conclude, note that p̂r = ŝt′′ ∘ r̂ and deg(p̂r) = deg(ŝt′), so

deg(r̂) = \frac{deg(p̂r)}{deg(ŝt′′)} = \frac{deg(ŝt′)}{deg(ŝt′′)}.

This finishes the tree case.

Suppose now that Γ̂ is a loop and that \overline{M}G ; Γ̃_{cut} = \overline{M}_{g−1,n+2}^{G} (\overline{m}, m_{+}, m_{−}). Following the analogous steps to the case of the tree, we obtain the counterpart of Equation (43): (47)

I = ∑_{μ[m_{±}]} deg(r̂) deg(p̃r∘r̃)\frac{|G|^{2}}{|C(\overline{m}_{+})|^{2}} Λ̂_{g−1,n+2} (v\overline{m}, π_{G}(e_{μ[m_{+}]} ), π_{G}(e_{μ[m_{−}]} ))η^{μ[m_{+}]μ[m_{−}]} .

Proceeding further, the counterpart of Equation (46) is

(48) r̂_{∗}μ̂^{∗} î^{∗} Λ̂_{g,n} (v\overline{m}) = deg(r̂) ∑_{μ[\overline{m} _{±}]} Λ̂_{g−1,n+2} (v\overline{m}, ϵ_{μ[\overline{m} _{+}]} ), ϵ_{μ[\overline{m} _{−}]} )η̂^{μ[\overline{m} _{+}]μ[\overline{m} _{−}]} .

The rest of the proof is essentially the same as in the case of the tree. □

5.2. From \overline{M}_{g,n} (BG) to \overline{M}_{g,n} .

Definition 5.7. Let (H, η, Λ_{g,n} , 1) be a G-CohFT. Define \overline{Λ} _{g,n} := ŝt_{∗}Λ̂_{g,n} in H^{•}(\overline{M}_{g,n} ) ⊗ H^{∗⊗n}.

Theorem 5.8. If (H, η, Λ_{g,n} , 1) is a G-CohFT, then (\overline{H}, \overline{η}, \overline{Λ} _{g,n} , 1) forms a CohFT.

Proof. We begin by observing that

(49)  \frac{1}{deg ŝt}i^{∗}ŝt_{∗} = \frac{1}{deg s̃t′} ŝt′_{∗}î^{∗},

since the lower right square is not Cartesian, due to ramification over \overline{M}_{Γ}. Next, we observe that

(50)  μ^{∗}ŝt′_{∗} = p̂r_{∗}μ̂^{∗},

since the lower left square is Cartesian by definition.

Therefore, for all v\overline{m} in \overline{H}\overline{m},

ρ_{Γ}^{∗} \overline{Λ} _{g,n} (v\overline{m}) = ρ_{Γ}^{∗} ŝt_{∗}Λ̂_{g,n} (v\overline{m})
= μ^{∗}i^{∗}ŝt_{∗}Λ̂_{g,n} (v\overline{m})
= μ^{∗} (\frac{deg(ŝt)}{deg(ŝt′)}ŝt′_{∗}î^{∗}) Λ̂_{g,n} (v\overline{m})

= \frac{deg(ŝt)}{deg(ŝt′)}p̂r_{∗}μ̂^{∗} î^{∗} Λ̂_{g,n} (v\overline{m})

= \frac{deg(ŝt)}{deg(ŝt′)}ŝt′′_{∗}(r̂_{∗}μ̂^{∗} î^{∗})Λ̂_{g,n} (v\overline{m})

= \frac{deg(ŝt)}{deg(ŝt′)}ŝt′′_{∗} (\frac{deg(ŝt′)}{deg(ŝt′′)}Λ̂Γ̂_{cut} )

= \frac{deg(ŝt)}{deg(ŝt′′)}ŝt′′_{∗}Λ̂Γ̂_{cut} ,

where Equations (37) and (39) have been used in the sixth equality.

Assume that Γ̂ is a tree. Adopting the notation from the proof of Theorem 5.5, we obtain

ρ_{Γ}^{∗} \overline{Λ} _{g,n} (v\overline{m}) = \frac{deg(ŝt)}{deg(ŝt′′)} ∑_{β[\overline{m} _{+}],β[\overline{m} _{−}]} ŝt′′_{∗}(Λ̂_{+} (e_{β[\overline{m} _{+}]} )η̂^{β[\overline{m} _{+}]β[\overline{m} _{−}]} Λ̂_{−} (e_{β[\overline{m} _{−}]} ))

= \frac{deg(ŝt)}{deg(ŝt′′)} ∑_{β[\overline{m} _{+}],β[\overline{m} _{−}]} \overline{Λ} _{+}(e_{β[\overline{m} _{+}]} )η̂^{β[\overline{m} _{+}]β[\overline{m} _{−}]} \overline{Λ} _{−}(e_{β[\overline{m} _{−}]} ),

where \overline{Λ} _{±} := st_{\overline{m} _{±}∗}Λ̂_{±} . This can be rewritten as (51)

deg(ŝt′′)ρ_{Γ}^{∗} \overline{Λ} _{g,n} (v\overline{m}) = deg(ŝt) ∑_{β[\overline{m} _{+}],β[\overline{m} _{−}]} \overline{Λ} _{+}(e_{β[\overline{m} _{+}]} )η̂^{β[\overline{m} _{+}]β[\overline{m} _{−}]} \overline{Λ} _{−}(e_{β[\overline{m} _{−}]} ).

Following [JK], let

(52)  Ω_{g,n} (\overline{m}) := deg(ŝt)

for all \overline{m} in \overline{G}^{n}. We have

deg(ŝt′′) = Ω_{g_{+},n_{+}+1} (\overline{m}_{N_{+}} , \overline{m}_{+})Ω_{g_{−},n_{−}+1} (\overline{m}_{−}, \overline{m}_{N_{−}} ),

and Equation (51) becomes, after multiplying both sides by |C(\overline{m}_{+})|, using the definition of η̂, and summing over all conjugacy classes \overline{m}_{±} such that \overline{m}_{−} = \overline{m}_{+}^{−1},

∑_{\overline{m} _{±}:\overline{m} _{−}=\overline{m} _{+}^{−1}} |C(\overline{m}_{+})| Ω_{g_{+},n_{+}+1} (\overline{m}_{N_{+}} , \overline{m}_{+})Ω_{g_{−},n_{−}+1} (\overline{m}_{−}, \overline{m}_{N_{−}} )ρ_{Γ}^{∗} \overline{Λ} _{g,n} (v\overline{m}) =

Ω_{g,n} (\overline{m}) ∑_{\overline{m} _{±}:\overline{m} _{−}=\overline{m} _{+}^{−1}} ∑_{β[\overline{m} _{+}],β[\overline{m} _{−}]} \overline{Λ} _{+}(e_{β[\overline{m} _{+}]} )\overline{η}^{β[\overline{m} _{+}]β[\overline{m} _{−}]} \overline{Λ} _{−}(e_{β[\overline{m} _{−}]} ).

But Lemma 3.5(1) from [JK] states that

∑_{\overline{m} _{±}:\overline{m} _{−}=\overline{m} _{+}^{−1}} |C(\overline{m}_{+})|Ω_{g_{+},n_{+}+1} (\overline{m}_{N_{+}} , \overline{m}_{+})Ω_{g_{−},n_{−}+1} (\overline{m}_{−}, \overline{m}_{N_{−}} ) = Ω_{g,n} (\overline{m}).

Therefore, by canceling Ω_{g,n} (\overline{m}) from both sides, we obtain the desired result. In the case of the loop, we have

ρ_{Γ}^{∗} \overline{Λ} _{g,n} (v\overline{m}) = \frac{deg(ŝt)}{deg(ŝt′′)} ∑_{β[\overline{m} _{+}],β[\overline{m} _{−}]} ŝt′′_{∗}(Λ̂_{g−1,n+2} (v\overline{m}, e_{β[\overline{m} _{+}]} , e_{β[\overline{m} _{−}]} ))η̂^{β[\overline{m} _{+}]β[\overline{m} _{−}]}

= \frac{deg(ŝt)}{deg(ŝt′′)} ∑_{β[\overline{m} _{+}],β[\overline{m} _{−}]} \overline{Λ} _{g−1,n+2} (v\overline{m}, e_{β[\overline{m} _{+}]} , e_{β[\overline{m} _{−}]} )η̂^{β[\overline{m} _{+}]β[\overline{m} _{−}]} .

Multiplying both sides by deg(ŝt′′)|C(\overline{m}_{+})|, plugging in deg(ŝt′′) = Ω_{g−1,n+2} (\overline{m}, \overline{m}_{+}, \overline{m}_{−}), deg(ŝt) = Ω_{g,n} (\overline{m}), and then summing over all conjugacy classes \overline{m}_{±} such that \overline{m}_{−} = \overline{m}_{+}^{−1}, we obtain

∑_{\overline{m} _{±}:\overline{m} _{−}=\overline{m} _{+}^{−1}} |C(\overline{m}_{+})| Ω_{g−1,n+2} (\overline{m}, \overline{m}_{+}, \overline{m}_{−})ρ_{Γ}^{∗} \overline{Λ} _{g,n} (v\overline{m}) =

Ω_{g,n} (\overline{m}) ∑_{\overline{m} _{±}:\overline{m} _{−}=\overline{m} _{+}^{−1}} ∑_{β[\overline{m} _{+}],β[\overline{m} _{−}]} \overline{Λ} _{g−1,n+2} (v\overline{m}, e_{β[\overline{m} _{+}]} , e_{β[\overline{m} _{−}]} )\overline{η}^{β[\overline{m} _{+}]β[\overline{m} _{−}]} .

Since Lemma 3.5(2) from [JK] states that

∑_{\overline{m} _{±}:\overline{m} _{−}=\overline{m} _{+}^{−1}} |C(\overline{m}_{+})|Ω_{g−1,n+2} (\overline{m}, \overline{m}_{+}, \overline{m}_{−}) = Ω_{g,n} (\overline{m}),

we may cancel Ω_{g,n} (\overline{m}) from both sides to obtain the desired result.

This completes the proof of the factorization axiom of the CohFT.

The invariance under the symmetric group is manifest.

The flat identity axiom follows from considering the following commuting diagram:

\overline{M}_{g,n+1}^{G} (\overline{m}, 1) \overline{τ̃} \overline{M}_{g,n}^{G} (\overline{m})

[figure: \overline{M}_{g,n+1} (BG; \overline{m}, 1); s̃t_{1}; τ̂; \overline{M}_{g,n} (BG; \overline{m}); s̃t]

[figure: \overline{M}_{g,n+1}; ŝt_{1}; τ; \overline{M}_{g,n}; ŝt]

The horizontal morphisms are forgetting-tails morphisms and are both flat and proper. The vertical morphisms are forgetful morphisms and are all quasi-finite, flat, and proper.

By Lemma 5.1 We have

s̃t_{1}^{∗}Λ̂_{g,n+1} (v\overline{m}, 1) = Λ_{g,n+1} (v\overline{m}, 1)
= τ̃^{∗}Λ_{g,n} (v\overline{m})
= τ̃^{∗}s̃t^{∗}Λ̂_{g,n} (v\overline{m})
= s̃t_{1}^{∗}τ̂^{∗}Λ̂_{g,n} (v\overline{m})

By the uniqueness of the classes Λ̂ (again, see Lemma 5.1) we conclude that

(53)  τ̂^{∗}Λ̂_{g,n} (v\overline{m}) = Λ̂_{g,n+1} (v\overline{m}, 1).

On the other hand, while the bottom square of this diagram is not Cartesian, it is almost so—the stack \overline{M}_{g,n+1}^{G} (BG; \overline{m}, 1) is the universal orbicurve over

\overline{M}_{g,n}^{G} (BG; \overline{m}), and it is birational to its coarse moduli space, the universal curve

over \overline{M}_{g,n}^{G} (BG; \overline{m}). Thus, we have

τ^{∗}\overline{Λ} _{g,n} (v\overline{m}) = τ^{∗}ŝt_{∗}Λ̂_{g,n} (v\overline{m}) = st_{1∗} τ̂^{∗}Λ̂_{g,n} (v\overline{m}) = ŝt_{1∗} Λ̂_{g,n+1} (v\overline{m}, 1) = \overline{Λ} _{g,n+1} (v\overline{m}, 1).

The last property that must be verified is

(54)  \overline{η}(v_{\overline{m} _{+}}, v_{\overline{m} _{−}}) = \overline{Λ} _{0,3} (v_{\overline{m} _{+}}, v_{\overline{m} _{−}}, 1)

for all v_{\overline{m} _{±}} in \overline{H}_{±}, where we have identified H^{•}(\overline{M}_{0,3} ) with the ground ring ℂ. Since this identity holds trivially if \overline{m}_{−} ≠ \overline{m}_{+}^{−1}, let us assume that \overline{m}_{−} = \overline{m}_{+}^{−1}.

We have the morphisms

∐_{m′_{+}∈\overline{m}_{+}} ξ_{0,3} (m′_{+}, m_{+}^{′−1} , 1) ^{s̃t_{ξ}} \overline{M}_{0,3} (BG; \overline{m}_{+}, \overline{m}_{+}^{−1} , 1) ŝt \overline{M}_{0,3} ,

and we let st_{ξ} := ŝt ∘ s̃t_{ξ}. Since η is defined by

η(v_{m_{+}} , v_{m_{−}} ) = ∫_{[[ξ(m_{+},m_{−},1) ]]} Λ_{0,3} (v_{m_{+}} , v_{m_{−}} , 1)st_{ξ∗}Λ_{0,3} (v_{m_{+}} , v_{m_{−}} , 1),

we have

\overline{Λ} _{0,3} (v_{\overline{m} _{+}}, v_{\overline{m} _{−}}, 1) = ŝt_{∗}Λ̂_{0,3} (v_{\overline{m} _{+}}, v_{\overline{m} _{−}}, 1)
= ŝt_{∗}(s̃t_{ξ∗} s̃t_{ξ}^{∗} \frac{1}{deg s̃t_{ξ}} )Λ̂_{0,3} (v_{\overline{m} _{+}}, v_{\overline{m} _{−}}, 1)
= \frac{1}{deg s̃t_{ξ}} ŝt_{∗}s̃t_{ξ∗}Λ_{0,3} (v_{\overline{m} _{+}}, v_{\overline{m} _{−}}, 1)
= \frac{1}{deg s̃t_{ξ}} st_{ξ∗}Λ_{0,3} (v_{\overline{m} _{+}}, v_{\overline{m} _{−}}, 1)
= \frac{1}{deg s̃t_{ξ}} η(v_{\overline{m} _{+}}, v_{\overline{m} _{−}}),

but

deg(s̃t_{ξ}) = |\overline{m}_{+}||C(\overline{m}_{+})| = |G|,

since a generic point of \overline{M}_{g,n} (BG; \overline{m}_{+}, \overline{m}_{−}, 1) has automorphism group isomorphic to C(m_{+}). Therefore, Equation (54) is satisfied. □

Remark 5.9. The CohFT (\overline{H}, \overline{η}, \{\overline{Λ} _{g,n} \}, 1) constructed above has more structure than a generic CohFT, as it is \overline{G}-graded; that is, (\overline{H}, \overline{η}) is a \overline{G}-graded vector space with metric, and for all v\overline{m} in \overline{H}\overline{m}, the class \overline{Λ} _{g,n} (v\overline{m}) vanishes unless there exist representatives m′_{i} in \overline{m}_{i} for all i = 1, …, n such that ∏ _{i=1}^{n} m′_{i} belongs to the subgroup [G, G]^{g}. This is follows from the fact that \overline{M}_{g,n} (BG; \overline{m}) is empty unless this holonomy condition holds.

Proposition 5.10. Let (H, η, {Λ_{g,n} }, 1) be a G-CohFT. For all nonzero λ in ℂ, (H, λ^{−2}η, {λ^{2g−2} Λ_{g,n} }, 1) is a G-CohFT.

The proof is immediate from the definition.

Remark 5.11. One can eliminate the annoying factor of \frac{1}{|G|} in the definition of \overline{η} by choosing λ such that λ^{2} = \frac{1}{|G|} . In this case, the associated “quotient” by G of the G-CohFT (H, |G|η, \{|G|^{1−g} Λ_{g,n} \}, 1) is the CohFT (\overline{H}, |G|\overline{η}, \{|G|^{1−g} Λ_{g,n} \}, 1), but |G|\overline{η} is equal to the restriction of η to \overline{H}.

5.3. Coinvariants of G-Frobenius algebras.

Let ((H, ρ), ·, 1, η) be a G-Frobenius algebra. We now have two ways to endow its space of coinvariants \overline{H} with the structure of a Frobenius algebra. The first is purely algebraic. The tuple (\overline{H}, ·, 1, η) is a Frobenius algebra where the multiplication on \overline{H} is inherited by restriction from the multiplication on H and the metric η is the restriction of the metric on H.

The second is to apply the geometric procedure described in the previous subsection to H, regarded as a G-CohFT, to induce the structure of a Frobenius algebra on \overline{H}. It turns out that these two Frobenius structures are identical after a rescaling.

In order to simplify the proof, we note that the structure of the G-Frobenius algebra ((H, ρ), ·, 1, η) can also be described as the tuple ((H, ρ), μ, 1), where μ belongs to H^{∗⊗3} and is defined by

(55)  μ(v_{m_{1}} , v_{m_{2}} , v_{m_{3}} ) := η(v_{m_{1}} , v_{m_{2}} · v_{m_{3}} ),

since it follows that η(v_{m_{1}} , v_{m_{2}} ) = η(v_{m_{1}} , v_{m_{2}} ·1). If μ̃ denotes the restriction of μ to \overline{H}, then the data (\overline{H}, μ̃, 1) is an equivalent description of the Frobenius algebra structure on \overline{H} induced by restriction.

Proposition 5.12. Let ((H, ρ), μ, 1) be a G-Frobenius algebra arising from a G-CohFT (H, η, \{Λ_{g,n} \}, 1). The Frobenius algebra structure on \overline{H} arising from the CohFT (H, \overline{η}, \{\overline{Λ} _{g,n} \}, 1) is (\overline{H}, \overline{μ}, 1), where

\overline{μ} = \frac{1}{|G|}μ̃,

and μ̃ is the restriction of μ to \overline{H}.

Remark 5.13. The Frobenius algebra (\overline{H}, \overline{μ}, 1) can also be described as the tuple (\overline{H}, ·, \overline{η}, 1), where the multiplication · on \overline{H} is inherited from the multiplication on H, but where \overline{η} is the restriction of \frac{1}{|G|} η to \overline{H}.

Proof. (of Proposition 5.12)

Since \overline{μ}(v\overline{m}) = \overline{Λ} _{0,3} (v\overline{m}), after identifying H^{•}(\overline{M}_{0,3} ) with ℂ, we need only prove that

\overline{Λ} _{0,3} (v\overline{m}) = \frac{1}{|G|}μ(v\overline{m})

for all v\overline{m} in \overline{H}\overline{m}. In order to proceed, let us introduce some notation.

For all m := (m_{1}, m_{2}, m_{3}) belonging to G^{3} such that m_{1}m_{2}m_{3} = 1, we have the following forgetful morphisms

\overline{M}_{0,3}^{G} (\overline{m}) s̃t \overline{M}_{0,3} (BG; \overline{m}) ŝt \overline{M}_{0,3} ,

and we let st := ŝt ∘ s̃t.

Furthermore, if Q is a substack of \overline{M}_{0,3}^{G} (\overline{m}) then we let s̃t_{Q} denote the restriction of s̃t to Q. Let

ξ := ∐_{m′∈χ(\overline{m})} ξ(m′),

where

χ(\overline{m}) := \{(m′_{1}, m′_{2}, m′_{3}) ∈ \overline{m}|m′_{1}m′_{2}m′_{3} = 1\}.

Henceforth, fix an element m in χ(\overline{m}) once and for all.

Let us adopt the notation that for any v\overline{m} in \overline{H}\overline{m}, and for any m′ ∈ \overline{m}, the vector v_{m′} denotes the m′-graded component of v\overline{m}; that is,

v\overline{m} =: ∑_{m′∈\overline{m}} v_{m′}.

Note that v_{m′} belongs to the subspace of C(m′)-invariant vectors in H_{m′}, where C(m′) := C(m′_{1}) × C(m′_{2}) × C(m′_{3}).

For all m′ in χ(\overline{m}), we have

μ(v′_{m}) = \frac{1}{deg(st_{ξ(m′)})}st_{ξ(m′)∗}Λ_{0,3} (v′_{m}).

Otherwise, μ(v′_{m}) = 0. Since μ̃ is the restriction of μ to \overline{H},

μ̃(v\overline{m}) = ∑_{m′∈χ(\overline{m})} \frac{1}{deg(st_{ξ(m′)})}st_{ξ(m′)∗}Λ_{0,3} (v_{m′})

= \frac{1}{deg(st_{ξ(m)})} ∑_{m′∈χ(\overline{m})} st_{ξ(m′)∗}Λ_{0,3} (v_{m′})

= \frac{1}{deg(st_{ξ(m)})}st_{ξ∗}Λ_{0,3} (v\overline{m}),

where the second equality comes from the fact that the degree of st restricted to any connected component of \overline{M}_{0,3}^{G} (\overline{m}) is independent of the choice of connected component. This statement follows from the fact that every connected component of \overline{M}_{0,3}^{G} (\overline{m}) is ρ(γ)ξ(m) for some γ in G^{3}, but ρ(γ) is an isomorphism.

However, we have

st_{ξ∗}Λ_{0,3} (v\overline{m}) = ∑_{m′∈χ(\overline{m})} st_{ξ(m′)∗}Λ_{0,3} (v_{m′})
= |χ(\overline{m})|st_{ξ(m)∗}Λ_{0,3} (v_{m})
= |G|Ω_{0,3} (\overline{m})st_{ξ(m)∗}Λ_{0,3} (v_{m}),

where the second equality follows from the observation that every connected component of ξ can be obtained by the action of some element of G^{3}, and the fact that v\overline{m} and Λ_{0,3} are G^{3}-invariant. The third equality is from Proposition 3.4 of [JK], where Ω_{0,3} is defined in Equation (52). Therefore, we obtain

(56)  μ̃(v\overline{m}) = \frac{|G|Ω_{0,3} (\overline{m})}{deg(st_{ξ(m)})}st_{ξ(m)∗}Λ_{0,3} (v_{m}).

On the other hand, the definition of Λ̂_{0,3} implies that

Λ̂_{0,3} (v\overline{m}) = \frac{1}{deg(s̃t)}s̃t_{∗}Λ_{0,3} (v\overline{m}),

hence

\overline{Λ} _{0,3} (v\overline{m}) = ŝt_{∗}Λ̂_{0,3} (v\overline{m})
= \frac{1}{deg(s̃t)}ŝt_{∗}s̃t_{∗}Λ_{0,3} (v\overline{m}),

and we obtain

(57)  \overline{Λ} _{0,3} (v\overline{m}) = \frac{1}{deg(s̃t)}st_{∗}Λ_{0,3} (v\overline{m}).

Using the fact that deg(st_{Q} ) is independent of the choice of connected component Q of \overline{M}_{0,3}^{G} (\overline{m}), we can write

(58)  deg(s̃t) = A(\overline{m}) deg(s̃t_{ξ(m)}),

where A(\overline{m}) is the number of connected components of \overline{M}_{0,3}^{G} (\overline{m}). Similarly, let I(\overline{m}) consist of all elements γ in G^{3} such that the collection \{ρ(γ)ξ(m)\} is in oneto-one correspondence with the connected components of \overline{M}_{0,3}^{G} (\overline{m}), then

st_{∗}Λ_{0,3} (v\overline{m}) = ∑_{γ ∈I(\overline{m})} st_{ρ(γ )ξ(m)∗} Λ_{0,3} (v_{ρ(γ )m})
= ∑_{γ ∈I(\overline{m})} st_{ξ(m)∗}ρ(γ^{−1})_{∗}Λ_{0,3} (v_{ρ(γ )m})
= ∑_{γ ∈I(\overline{m})} st_{ξ(m)∗}ρ(γ)^{∗}Λ_{0,3} (v_{ρ(γ )m})
= ∑_{γ ∈I(\overline{m})} st_{ξ(m)∗}Λ_{0,3} (ρ(γ^{−1})v_{ρ(γ )m})
= ∑_{γ ∈I(\overline{m})} st_{ξ(m)∗}Λ_{0,3} (v_{m})
(59)  = A(\overline{m})st_{ξ(m)∗}Λ_{0,3} (v_{m}),

where the first equality is the sum over contributions from each connected components of \overline{M}_{0,3}^{G} (\overline{m}), and the second is from the fact that, for all γ in G^{3}, we have

(60)  st_{ξ(m)} = st_{ρ(γ )ξ(m)} ∘ ρ(γ).

The fourth equality is from the G^{3}-invariance of Λ_{0,3} and the fifth is from the G^{3}-invariance of v\overline{m}. Putting together Equations (57), (58), and (59), we obtain

\overline{Λ} _{0,3} (v\overline{m}) = \frac{1}{deg(s̃t_{ξ(m)})}st_{ξ(m)∗}Λ_{0,3} (v_{m})
= \frac{Ω_{0,3} (\overline{m})}{deg(st_{ξ(m)})}st_{ξ(m)∗}Λ_{0,3} (v_{m}),

since

deg(st_{ξ(m)}) = deg(s̃t_{ξ(m)}) deg(ŝt) = deg(s̃t_{ξ(m)})Ω_{0,3} (\overline{m}).

However,

st_{∗}Λ_{0,3} (v\overline{m}) = ∑_{m′∈χ(\overline{m})} st_{ξ(m′)∗}Λ_{0,3} (v_{m′})
= |χ(\overline{m})|st_{ξ(m)∗}Λ_{0,3} (v_{m})
= |G|Ω_{0,3} (\overline{m})st_{ξ(m)∗}Λ_{0,3} (v_{m}).

Putting this all together, we obtain the desired result

\overline{Λ} _{0,3} (v\overline{m}) = \frac{Ω_{0,3} (\overline{m})}{deg(st_{ξ(m)})} \frac{1}{|G|Ω_{0,3} (\overline{m})} st_{ξ∗}Λ_{0,3} (v\overline{m})
= \frac{1}{|G|}μ̃(v\overline{m}).
□

The results of this section can be applied to the example of the group algebra G-CohFT and its associated G-Frobenius algebra to yield the (stringy) orbifold cohomology of a point with trivial G-action.

Proposition 5.14. The Frobenius algebra \overline{H} induced from the G-Frobenius algebra H = ℂ[G] is the Frobenius algebra Z(ℂ[G]), the center of the group ring, with its induced multiplication, identity, and the metric \overline{η}.

The resulting Frobenius algebra is isomorphic to the orbifold (stringy) quantum cohomology of BG, the classifying stack of G.

We refer the reader to [JK] where the calculation is worked through in detail.

5.4. The quotient stack [\overline{M}_{g,n}^{G} /G^{n} ].

The process of obtaining a CohFT from a G-CohFT involved the stack \overline{M}_{g,n} (BG). However, there is another stack that one could have used instead, namely, the quotient stack Q_{g,n} := [\overline{M}_{g,n}^{G} /G^{n} ] and its substacks Q_{g,n} (\overline{m}) := [\overline{M}_{g,n}^{G} (\overline{m})/G^{n} ]. We will show that one can construct a CohFT by replacing \overline{M}_{g,n} (BG) by Q_{g,n} (\overline{m}) := [\overline{M}_{g,n}^{G} (\overline{m})/G^{n} ], but that the resulting CohFT is isomorphic to the original one.

We have the following sequence of forgetful morphisms

(61)  \overline{M}_{g,n}^{G} (\overline{m}) ^{št′} Q_{g,n} (\overline{m}) ^{št} \overline{M}_{g,n} (BG; \overline{m}) ŝt \overline{M}_{g,n} ,

where s̃t := št ∘ št′. The stack Q is a smooth, Deligne-Mumford stack, and all of these morphisms are proper and flat. Observe that while the morphism št induces an isomorphism at the level of the corresponding coarse moduli spaces, they are not isomorphic as stacks, since an object in Q_{g,n} (\overline{m}) has a larger automorphism group than the corresponding object in \overline{M}_{g,n} (BG; \overline{m}).

Definition 5.15. Let ((H, ρ), η, {Λ_{g,n} }, 1) be a G-CohFT. Define the elements

Λ̌_{g,n} in ⨁_{\overline{m}∈\overline{G}^{n}} H^{•}(Q_{g,n} (\overline{m})) ⊗ \overline{H}∗ ; \overline{m} via

(62)  Λ̌_{g,n} (v\overline{m}) := št^{∗}Λ̂_{g,n} (v\overline{m})

for all v\overline{m} in \overline{H}\overline{m} and \overline{m} in \overline{G}^{n}. Define \overline{Λ} ′_{g,n} in H^{•}(\overline{M}_{g,n} ) ⊗ \overline{H}∗ ; \overline{m} via

\overline{Λ} ′_{g,n} (v\overline{m}) := (ŝt ∘ št)_{∗}Λ̌_{g,n} (v\overline{m}).

Let

\overline{η}′(v_{\overline{m} _{+}}, v_{\overline{m} _{−}}) := \overline{Λ} ′_{0,3} (v_{\overline{m} _{+}}, v_{\overline{m} _{−}}, 1)

for all v_{\overline{m} _{±}} in \overline{H}_{\overline{m} _{±}}.

Proposition 5.16. Let ((H, ρ), η, {Λ_{g,n} }, 1) be a G-CohFT.

(1) We have the identity

Λ_{g,n} (v\overline{m}) = št^{′∗}Λ̌_{g,n} (v\overline{m}).

(2) We also have

\overline{Λ} ′_{g,n} (v\overline{m}) := (∏_{i=1}^{n} \frac{1}{k_{\overline{m} _{i}}} ) Λ_{g,n} (v\overline{m}),

where k\overline{m} is the order of the cyclic subgroup generated by any representative of \overline{m} in \overline{G}.

(3) (\overline{H}, \overline{η}′, \{\overline{Λ} ′_{g,n} \}, 1) is a CohFT.
(4) The linear map ϕ : \overline{H} \overline{H}, where
ϕ(v\overline{m} ) := k\overline{m} v\overline{m}

for all v\overline{m} in \overline{H}\overline{m} and \overline{m} in \overline{G}, is an isomorphism between the CohFTs (\overline{H}, \overline{η}, \{\overline{Λ} _{g,n} \}, 1) and (\overline{H}, \overline{η}′, \{\overline{Λ} ′_{g,n} \}, 1).

Proof. Since št^{′∗}Λ̌_{g,n} (v\overline{m}) = s̃t^{∗}Λ̂_{g,n} (v\overline{m}) = št^{′∗}št^{∗}\overline{Λ} _{g,n} (v\overline{m}), we obtain

Λ_{g,n} (v\overline{m}) = št^{′∗}Λ̌_{g,n} (v\overline{m}).

For the second part, apply št_{∗} to both sides of equation (62) and use the fact that

deg(št) = ∏_{i=1}^{n} \frac{1}{k_{\overline{m} _{i}}}

to get

št_{∗}Λ̌_{g,n} (v\overline{m}) = (∏_{i=1}^{n} \frac{1}{k_{\overline{m} _{i}}} ) Λ̂_{g,n} (v\overline{m}).

Thus,

Λ̂_{g,n} (v\overline{m}) = (∏_{i=1}^{n} k_{\overline{m} _{i}} ) št_{∗}Λ̌_{g,n} (v\overline{m}).

However,

\overline{Λ} _{g,n} (v\overline{m}) = ŝt_{∗}Λ̂_{g,n} (v\overline{m})
= (∏_{i=1}^{n} k_{\overline{m} _{i}} ) ŝt_{∗}št_{∗}Λ̌_{g,n} (v\overline{m})

= (∏_{i=1}^{n} k_{\overline{m} _{i}} ) \overline{Λ} ′_{g,n} (v\overline{m}).

This establishes the second part of the proposition.

Clearly, ϕ^{∗}\overline{Λ} ′_{g,n} = \overline{Λ} _{g,n} , ϕ^{∗}\overline{η}′ = \overline{η}, and ϕ(1) = 1. Since (\overline{H}, \overline{η}, \{\overline{Λ} _{g,n} \}, 1) is a CohFT, so is (\overline{H}, \overline{η}′, \{\overline{Λ} ′_{g,n} \}, 1), and ϕ is an isomorphism. □

Remark 5.17. A similar rescaling was observed in [AGV], and the previous proposition could be regarded as its origin in the framework of G-CohFTs.

6. G-stable maps

In this section we briefly describe the main source of examples of G-CohFTs; namely, Gromov-Witten style classes on the moduli space of G-stable maps.

Definition 6.1. A genus g, n-pointed G-stable map over a base T into a global quotient [X/G] is a G-equivariant morphism f : E X from an admissible Gcover π : E C of a genus g prestable curve C/T with n sections p̃_{i} : T E such that the induced morphism of stacks f̄: [E/G] [X/G] with marked points p_{i} := π ∘ p̃_{i} is an n-pointed orbifold (a.k.a. twisted) stable map of genus g (as defined in [CR1, AGV]).

We denote the stack of genus g, n-pointed G-stable maps by \overline{M}_{g,n}^{G} (X), and if β ∈ H_{2}(X/G, ℤ), then we denote the substack of maps whose image lies in the homology class β by \overline{M}_{g,n}^{G} (X, β).

Theorem 6.2. If the quotient [X/G] admits a projective coarse moduli space X/G, then the stack \overline{M}_{g,n}^{G} ([X/G], β) is a proper Deligne-Mumford stack, which itself admits a projective coarse moduli space.

The proof follows from the results of [AGV] in essentially the same way that Theorem 2.4 follows from the results of [ACV].

There is a natural forgetful morphism st_{(X,β)} : \overline{M}_{g,n}^{G} ([X/G], β) \overline{M}_{g,n}^{G} obtained by forgetting the morphism f and contracting components in a manner similar to that described in the definition of the forgetting tails morphism of Section 2. There are also natural evaluation morphisms ev_{i} from \overline{M}_{g,n}^{G} ([X/G], β) to the space

X̂:= {(x, g)|x ∈ X, g ∈ stab(x)} = ∐_{g∈G} X^{g} ⊆ X × G,

with ev_{i}((f : E X, p̃_{i} )) = (f(p̃_{i} ), m_{i}), where m_{i} is the monodromy of E around p̃_{i} and X^{g} is the fixed point locus in X of the subgroup ⟨g⟩ ⊆ G. These are compatible in the sense that the following diagram commutes

[figure: \overline{M}_{g,n}^{G} (X, β); ev_{i}; X̂; \overline{M}_{g,n}^{G}; st_{(X,β)}; ev_{i}; G; pr_{2}],

where the map pr_{2} is the projection onto the second factor and the lower map ev_{i} is the ith component of the map e of Definition 2.3.

Definition 6.3. We denote by \overline{M}_{g,n}^{G} (X, β, m) the component st_{(X,β)} ^{−1}(\overline{M}_{g,n}^{G} (m)) that maps to m ∈ G^{n} via e ∘ st_{(X,β)} .

Definition 6.4. Let H(X) := H^{2•}(X̂; Θ) = ⨁_{m∈G} H(X)_{m}, where H(X)_{m} := H^{2•}(X^{m} ; Θ), and Θ is the usual ring (see [Ma]) associated to X with generators {q^{β} } over ℂ, satisfying q^{β+β′} = q^{β} q^{β′} .

Remark 6.5. Of course, one could allow odd-dimensional cohomology classes as well, after inserting the necessary signs for skew-symmetry, but for simplicity we will work only with even-dimensional classes.

In a subsequent paper [IP], we will describe the details of how the classes

Λ_{g,n}^{G,X} (v_{1}, …, v_{n} ) := ∑_{β} st_{(X,β)} _{∗}(∏_{i=1}^{n} ev_{i}^{∗}(v_{i}) ∩ [\overline{M}_{g,n}^{G} (X)]^{vir} )q^{β}

form a G-CohFT, and how the CohFT of coinvariants of {Λ_{g,n}^{G,X} } agrees with the orbifold Gromov-Witten classes of Chen-Ruan [CR2].

In the remainder of this section we will briefly treat two special cases. In Subsection 6.1 we describe the case of β = 0, and show that it gives the ring H^{•}(X, G) of Fantechi and Göttsche—and therefore the stringy orbifold cohomology of Chen and Ruan—as special cases. In Subsection 6.2 we describe the G-CohFT {Λ_{g,n}^{G,X} } for all β in the case that G acts trivially on X.

6.1. The degree zero case, the Fantechi-Göttsche ring, and Chen-Ruan orbifold cohomology.

We will now study the case of degree-zero G-stable maps in more detail. We will explicitly prove that the degree-zero G-stable maps endow H(X) with the structure of a G-Frobenius algebra, the genus-zero part of which agrees with the ring H^{•}(X, G) in [FG].

Throughout this section, we will use the ground ring ℂ instead of Θ in the definition of H(X), since we are restricting to degree-zero maps. We will also assume that X is a smooth variety with projective coarse moduli space, unless otherwise stated.

Definition 6.6. Let s̃t_{X} : \overline{M}_{g,n}^{G} (X, 0, m) \overline{M}_{g,n}^{G} (m) denote the morphism st_{(X,β=0)} . We define

ξ(X, 0, m) := s̃t_{X}^{−1}(ξ(m)).

Similarly, if m, a, b ∈ G are chosen such that m ∈ [a, b], we let

ξ_{1,1} (X, 0, (m, a, b)) := s̃t_{X}^{−1}(ξ_{1,1} (m, a, b)).

We also define X^{⟨m⟩} to be the locus in X of points fixed by the subgroup ⟨m⟩ ≤ G generated by all of the elements m_{1}, …, m_{n} in m.

Since the marked points p̃_{i} in the universal G-cover E over ξ(m) all lie in the same connected component of E, it is straightforward to see that any G-stable map f into X of degree 0 that maps by s̃t_{X} to ξ(m) is determined only by the underlying G-cover (the point s̃t_{X}([E ) C] ∈ ξ(m)) and by the point f(p̃_{1}) = ⋯ = f(p̃_{n} ). Moreover, the point f(p̃_{i} ) must have a stabilizer that includes the monodromy element m_{i}, so the following proposition is now easy to see.

Lemma 6.7. The substack ξ(X, 0, m) of \overline{M}_{0,3}^{G} (X, 0, m) is canonically isomorphic to the product

ξ(X, 0, m) = ξ(m) × X^{⟨m⟩},

and the substack ξ_{1,1} (X, 0, (m, a, b)) is canonically isomorphic to the product

ξ_{1,1} (X, 0, (m, a, b)) = ξ_{1,1} (m, a, b) × X^{m} .

Proof. For an object in \overline{M}_{0,3}^{G} (X, 0, m), the isomorphism is given by the morphism (E ^{f} X^{⟨m⟩}; p̃_{1}, …, p̃_{n} ) ) (E; p̃_{1}, …, p̃_{n} ) × f(p̃_{1}), where E is the G-cover (we have suppressed the underlying curve C since it is determined by E), and its inverse is given by (E; p̃_{1}, …, p̃_{n} ) × q ) (E ^{f} X^{⟨m⟩}; p̃_{1}, …, p̃_{n} ), where if E′ is the connected component of E containing the marked points p̃_{1}, …, p̃_{n} , then f(E′) := q and f(ρ(γ)p̃′) := ρ(γ)f(p̃′) for all γ in G and p̃′ in E′. The maps for ξ_{1,1} (X, ), (m, a, b)) are similar. □

6.1.1. The minimal cover ξ′(m).

Definition 6.8. Let G be a finite group and fix m in G^{n} such that ∏ _{i=1}^{3} m_{i} = 1, and let G′ := ⟨m⟩ denote the subgroup of G generated by the components of m.

Let ξ′(m) denote the connected component of \overline{M}_{0,3}^{G′} (m) which is defined in the same way as ξ(m) but with the group G replaced by G′.

Lemma 6.9. Let G be a finite group m ∈ G^{3} with ∏ _{i=1}^{3} m_{i} = 1, and G′ = ⟨m⟩. Consider the morphism I : ξ(m) ξ′(m) taking the object (E C; p̃_{1}, …, p̃_{n} ) to the object (E′ C; p̃_{1}, …, p̃_{n} ), where E′ is the connected component of E which contains p̃_{i} for all i = 1, …, n. The morphism I is an isomorphism.

Proof. Since E′ is a G′-cover (see Appendix of [FG]), I is a morphism.

The inverse morphism takes (E′ C; p̃′_{1}, …, p̃′_{n} ) to (E C; p̃_{1}, …, p̃_{n} ), where E = E′ ×_{G′} G and G′ acts on E from the right in the usual way, G′ acts on G by left multiplication, and p̃_{i} := [p̃′_{i}, 1] for all i = 1, …, n. □

Consider the following commutative diagram

(63)

X

[figure: E′ × X^{⟨m⟩}; Ĩ; f′; E × X^{⟨m⟩}; f]

[figure: C′ × X^{m}; C × X^{m}]

[figure: ξ′(m) × X^{⟨m⟩}; π̃′; I; ξ(m) × X^{⟨m⟩}; π̃]

[figure: ξ′(m); pr_{ξ′}; I; ξ(m); pr_{ξ}]

where Ĩ and I are the isomorphisms induced by I, C′ and C are the universal curves, E′ and E are the universal G′ and G covers, respectively, and f, f′ are the universal stable maps.

Proposition 6.10. I^{∗}Rπ̃_{∗}^{G} (f^{∗}TX) is canonically isomorphic to Rπ̃_{∗}^{′G′} (f^{′∗}(TX)) in the K-theory of ξ′(m) × X^{⟨m⟩}, where Rπ̃_{∗}^{G} denotes the G-invariant derived pushforward, and Rπ̃_{∗}^{′G′} denotes the G′-invariant derived push-forward.

Proof. The fiber of I^{∗}Rπ̃_{∗}^{G} (f^{∗}TX) over ξ′(m) × q for all q in X^{⟨m⟩} is H^{•}(E × q, T ), where the sheaf T over E × q is f^{∗}(TX). Since E′ is the connected component of E containing p̃_{i} for all i = 1, …, n we have T |_{p̃×q} = T_{q} V for all p̃ in E′. Henceforth, let us regard T as a bundle over E to avoid notational clutter.

Observe that T is a G-equivariant trivial bundle on E. Denote the restriction of T to E′ by T ′ and observe that it is a G′-equivariant bundle. We will now construct a bundle from T ′ on E′ which is isomorphic as a G-equivariant bundle to T on E as follows.

Consider the bundle T ′ ⊗ O_{G} on E′ × G. We observe that E′ is a right G′-space and G is a left G-space by left multiplication. Similarly, there is a right G′ action on T ′ and a left G′-action on O_{G}. Therefore, T ′ ⊗ O_{G} over E′ × G is a G′-equivariant bundle with respect to the diagonal G′ action. Quotienting by G′ and using the identification of O_{G} with ℂ[G], we obtain T ′ ⊗_{ℂ[G′]} ℂ[G] over E′ ×_{G′} G, which is a G-equivariant bundle, where an element γ̃ in G acts upon an element of the base as [e′, γ] ) [e′, γγ̃], and similarly in the bundle. We now have the isomorphism of G-equivariant vector bundles

[figure: T ′ ⊗_{ℂ[G′]} ℂ[G]; λ̃; T; E′ ×_{G′} G; λ; E,]

where λ̃([v′, γ]) := ρ(γ)v′, and λ([e′, γ]) := ρ(γ)v′, and where ρ(γ) indicates the right G action.

Therefore,

H^{•}(E, T ) = H^{•}(E′ ×_{G′} G, T ′ ⊗_{ℂ[G′]} O_{G})
= H^{•}(E′ × G, T ′ ⊗ ℂ[G])^{G′}
= (H^{•}(E′, T ′) ⊗ H^{•}(G, O_{G}))^{G′}
= (H^{•}(E′, T ′) ⊗ ℂ[G])^{G′}
= H^{•}(E′, T ′) ⊗_{ℂ[G′]} ℂ[G].

Taking G-invariants, we have

H^{•}(E, T )^{G} = (H^{•}(E′, T ′) ⊗_{ℂ[G′]} ℂ[G])^{G} ≅ H^{•}(E′, T ′)^{G′}.

The latter is precisely the fiber of Rπ̃_{∗}^{′G′} (f^{′∗}(TX)) over ξ′(m) × q. □

Proposition 6.11. When β = 0, the sheaf R^{1} π̃_{∗}^{G} (f^{∗}TX) is locally free on ξ(m) × X^{⟨m⟩} = ξ(X, 0, m) and the virtual fundamental class of ξ(X, 0, m) is simply the top Chern class c_{top} (R^{1} π̃_{∗}^{G} (f^{∗}TX)).

Proof. This follows immediately from the construction of \overline{M}_{g,n}^{G} (X) as a fibered product of sections over \overline{M}_{g,n} (X), the stack of orbifold stable maps to X, and the fact that the proposition holds there (see e.g., [AGV]). □

Definition 6.12. Let c(m) := c_{top}(R^{1} π̃_{∗}^{G} (f^{∗}TX)) and c′(m) := c_{top}(R^{1} π̃_{∗}^{′G′} (f^{′∗}TX)), where c_{top} denotes the top Chern class.

Corollary 6.13. For all m in G^{3} such that ∏ _{i=1}^{3} m_{i} = 1, we have

(64)  I^{∗}c(m) = c′(m).

We now prove that the 3-point correlator responsible for the multiplication in the G-Frobenius algebra can be identified by the isomorphism in Lemma 6.7.

Proposition 6.14. For all m in G^{3} such that ∏ _{i=1}^{3} m_{i} = 1 and α_{m_{i}} in H^{•}(X^{m_{i}} ), let Λ_{0,3}^{ξ′} (α_{m}) in H^{•}(ξ′(m)) and Λ_{0,3}^{ξ} (α_{m}) in H^{•}(ξ′(m)) be given by

Λ_{0,3}^{ξ′} (α_{m}) := pr_{ξ′∗}(∏_{i=1}^{3} (ev_{m_{i}}^{′∗} α_{m_{i}} )) ∪ c′(m))

and

Λ_{0,3}^{ξ} (α_{m}) := pr_{ξ∗}((∏_{i=1}^{3} (ev_{m_{i}}^{∗} α_{m_{i}} )) ∪ c(m)).

We have

{ \atop I^{∗}}Λ_{0,3}^{ξ} (α_{m}) = Λ_{0,3}^{ξ′} (α_{m}),

where ev_{m_{i}} : ξ(V, 0, m)  X^{m_{i}} and ev′_{m_{i}} : ξ′(m) × X^{⟨m⟩}  X^{m_{i}} are the

evaluation morphisms, and pr_{ξ′} : ξ′(m) × X^{⟨m⟩} ξ′(m) and pr_{ξ} : ξ(m) × X^{⟨m⟩} ξ(m) are the projections, which can be identified with the morphism forgetting the G-stable maps.

Proof.

{ \atop I^{∗}}Λ_{0,3}^{ξ} (α_{m}) = { \atop I^{∗}}pr_{ξ∗}((∏_{i=1}^{3} (ev_{m_{i}}^{∗} α_{m_{i}} )) ∪ c(m))

= { \atop I_{∗}^{−1}}pr_{ξ∗}((∏_{i=1}^{3} (ev_{m_{i}}^{∗} α_{m_{i}} )) ∪ c(m))

= (({ \atop I^{−1}} ∘ pr_{ξ})_{∗}((∏_{i=1}^{3} (ev_{m_{i}}^{∗} α_{m_{i}} )) ∪ c(m))

= (pr_{ξ′} ∘ I^{−1})_{∗}((∏_{i=1}^{3} (ev_{m_{i}}^{∗} α_{m_{i}} )) ∪ c(m))

= pr_{ξ′∗}I^{∗}((∏_{i=1}^{3} (ev_{m_{i}}^{∗} α_{m_{i}} )) ∪ c(m))

= pr_{ξ′∗}((∏_{i=1}^{3} ((ev_{m_{i}} ∘ I)^{∗}α_{m_{i}} )) ∪ I^{∗}c(m))

= pr_{ξ′∗}((∏_{i=1}^{3} ((ev_{m_{i}} ∘ I)^{∗}α_{m_{i}} )) ∪ I^{∗}c(m))

= pr_{ξ′∗}((∏_{i=1}^{3} (ev_{m_{i}}^{′∗} α_{m_{i}} )) ∪ c′(m))
= Λ_{0,3}^{ξ′} (α_{m}),

where we have used Equation (64) in the penultimate equality. □

Corollary 6.15. For all m in G^{3} such that ∏ _{i=1}^{3} m_{i} = 1, and for α_{m_{i}} in H(X)_{m_{i}} = H^{•}(X^{m_{i}} ), we have

μ(α_{m}) = ∫_{[[ξ(m)]]} Λ_{0,3}^{ξ} (α_{m}) = ∫_{[[ξ′(m)]]} Λ_{0,3}^{ξ′} (α_{m}),

where μ is defined as in Equation (55).

μ completely determines the multiplication and metric. We will now prove that it yields a G-Frobenius algebra.

6.1.2. The genus-zero part of the G-Frobenius algebra.

For this subsection, we can assume, without loss of generality, that G′ = G in light of the results of the previous section.

Definition 6.16. Since the virtual fundamental class c(m) belongs to H^{•}(ξ(m) × X^{⟨m⟩}) ≅ H^{•}(ξ(m)) ⊗ H^{•}(X^{⟨m⟩}), define c̃(m) in H^{•}(X^{⟨m⟩}) to be the unique class such that

c(m) = 1_{ξ(m)} ⊗ c̃(m),

where 1_{ξ(m)} is the unit in H^{•}(ξ(m)).

We will now write μ(v_{m}) as an integral over X^{⟨m⟩}.

Proposition 6.17. For all v_{m} in H(X)_{m} where ∏ _{i=1}^{3} m_{i} = 1, we have

(65)  μ(v_{m}) = ∫_{[X^{⟨m⟩}]}(∏_{i=1}^{3} j_{m_{i}}^{∗} v_{m_{i}} ) ∪ c̃(m),

where j_{m_{i}} : X^{⟨m⟩} ^{⊂} X^{m_{i}} is the inclusion and [X^{⟨m⟩}] is the fundamental class of the variety X^{⟨m⟩}. In particular, when m_{i} = 1 for all i = 1, 2, 3, then c̃(1, 1, 1) = 1, the unit in H^{•}(X). The restriction of the multiplication and metric to the untwisted sector H(X)_{1} agree with the usual cup product and metric from H^{•}(X). Furthermore, (\overline{H}(X), \overline{μ}, 1) is isomorphic as a \overline{G}-graded Frobenius algebra to the Chen-Ruan orbifold cohomology of [X/G].

Proof. The first statement is a straightforward calculation. The second follows from the observation that the appropriate obstruction bundle vanishes when m_{1} = m_{2} = m_{3} = 1. The third follows from the following remark and Section 2 of [FG]. □

Remark 6.18. The vector bundle R^{1} π̃_{∗}^{G} (f^{∗}TX) ξ(m) × X^{⟨m⟩} is not the pullback of a vector bundle via the projection ξ(m) × X^{⟨m⟩} X^{⟨m⟩} because the the automorphism group of a G-cover (which is isomorphic to H(m)) in ξ(m) acts non-trivially on R^{1} π̃_{∗}^{G} (f^{∗}TX) ξ(m) as the action of the automorphism group commutes with the action of G. Nevertheless, one can interpret the bundle R^{1} π̃_{∗}^{G} (f^{∗}TX) ξ(m) × X^{⟨m⟩} as an H(m)-equivariant vector bundle R^{1} π̃_{∗}^{G} (f^{∗}TX)

X^{⟨m⟩}. This bundle can be identified with the bundle F(m_{1}, m_{2}) X^{⟨m⟩} introduced in [FG]. Therefore, their cohomology class c(m_{1}, m_{2}) can be identified with c̃(m), which is a class on X^{⟨m⟩}, so Equation (65) is consistent with their multiplication.

They also prove that the vector bundle F(m_{1}, m_{2}) restricted to a connected component U of X^{⟨m⟩} has rank a(m_{1}, U) + a(m_{2}, U) − a(m_{1}m_{2}, U) − codim(U ⊆ X^{m_{1} m_{2}} ). To explain this notation, let X have dimension D, q belong to X, and m belong to the isotropy subgroup of G at q. Denote the set of eigenvalues of the action of m on T_{q} X by {exp(2πir_{1}), …, exp(2πir_{D} )} for all j = 1, …, D where r_{j} belongs to the interval [0, 1). The age of m in q, a(m, q), is defined to be ∑ _{j=1}^{D} r_{j}. Since a(m, q) depends only upon the connected component containing q, a(m, U) is defined to be a(m, q) for any q in U.

Proposition 6.19. The triple (H(X), μ, 1) satisfies all of the axioms of a G-Frobenius algebra except, perhaps, for the trace axiom. Our multiplication, metric, and identity agrees with that [FG] on the ring H^{•}(X, G). Furthermore, η on H(X) has nonzero homogeneous components H(X)_{m_{+}} ⊗ H(X)_{m_{−}} ℂ only if m_{+}m_{−} = 1, in which case H(X)_{m_{+}} = H(X)_{m_{−}} = H^{•}(X^{m_{+}} , ℂ), and η agrees with the usual Poincaré pairing.

Proof. This result follows from the previous remark and [FG]. □

Remark 6.20. Proposition 5.12 explains the origin of the factor of \frac{1}{|G|} in the definition of \overline{η} from the viewpoint of intersection theory. This factor may be removed, if desired, as per Remark 5.11.

6.1.3. The trace axiom.

We will now prove that the trace axiom, which is a genus-one condition, holds for (H(X), μ, 1) = H^{•}(X, G).

Proposition 6.21. The Trace Axiom (Definition 4.13 (11)) holds for the triple (H(X), μ, 1).

Proof. The proof of the trace axiom in Theorem 4.15 shows that it suffices for us to check that the cutting loops property 4.1(5a) holds in the special cases of ρ_{a} : ξ(m_{1}, bab^{−1}, a^{−1}) ξ_{1,1} (m_{1}, a, b) and ρ_{b} : ξ(m_{1}, b, ab^{−1}a^{−1}) ξ_{1,1} (m_{1}, a, b) for the virtual class. So it suffices to check that ρ_{a}^{∗}c_{top} R^{1}π_{∗}^{G} f^{∗}TX = c_{top} R^{1} π̃_{∗}^{G} f̃TX (and the same for ρ_{b}) where the morphisms are those of the following diagram

(66)  \begin{aligned}& X \\ E′ × X^{⟨m,a,b⟩} & \overline{ρ̃_{a}} & ^{f̃} & E × X^{m} & \frac{}{f} \\ ξ(m, bab^{−1}a^{−1}) × X^{⟨m,a,b⟩} & ^{π̃} & \overline{ρ_{a} × i} & ξ_{1,1} (m, a, b) × X^{m} & π \\ ξ(m, bab^{−1}, a^{−1}) & ^{p̃r^{ξ}} & \overline{ρ_{a}} & ξ_{1,1} (m, a, b) & ^{pr^{ξ}} \end{aligned}

where f and f′ are the universal stable maps from the universal admissible covers E× X^{m} and E′ × X^{⟨m,a,b⟩} , respectively, i is the obvious inclusion i : X^{⟨m,a,b⟩} X^{m}, and ρ̃_{a} is the induced map making the top square Cartesian. Since ρ_{a} is the composition of a regular embedding and a flat morphism, we have

(ρ_{a} × i)^{∗}R^{1}π_{∗}^{G} f^{∗}TX = R^{1} π̃_{∗}^{G} ρ̃_{a}^{∗} f^{∗}TX
= R^{1}π_{∗}^{G} f̃^{∗}TX,

which gives the desired result. □

Remark 6.22. Finally, we note that the G-Frobenius algebra (H(X), μ, 1) enjoys some functoriality properties, as Fantechi-Göttsche have showed that it pulls back along étale maps [FG, pg. 11].

6.1.4. Tensor products.

We now work out the tensor products of the equivariant CohFTs described above and show that they reduce to the obvious notions of tensor products for G-Frobenius algebras.

Proposition 6.23. Let X′ be a smooth, projective variety with a G′-action and let ((H(X′), ρ′), μ′, 1′) be the G′-Frobenius algebra associated to contributions from maps of degree zero where μ′ is defined by Equation (55). Let X′′ be a smooth, projective variety with a G′′-action and let ((H(X′′), ρ′′), μ′′, 1′′) be its similarly associated G′′-Frobenius algebra.

(1) Consider X′×X′′ with its G′×G′′ action. The associated G′×G′′-Frobenius algebra ((H(X′ × X′′), ρ), μ, 1) is canonically isomorphic to the external tensor product of ((H(X′), ρ′), μ′, 1′) and ((H(X′′), ρ′′), μ′′, 1′′).

(2) Suppose that G′ = G′′ = G, and consider X′ × X′′ with its diagonal G action. Its associated G-Frobenius algebra ((H(X′ × X′′), ρ), μ, 1) is canonically isomorphic to the tensor product of ((H(X′), ρ′), μ′, 1′) and ((H(X′′), ρ′′), μ′′, 1′′).

Proof. To prove the first part, we need to understand the behavior of the obstruction bundle when g = 0, n = 3, and β = 0.

For all m′ in G^{′n} and m′′ in G^{′′n}, consider the substack ξ^{G′×G′′} (m′ × m′′) of

\overline{M}_{0,3}^{G′×G′′} (m′ × m′′), where we adopt the notation m′ × m′′ from Proposition 4.6. Since \overline{M}_{0,3} is a point, ξ^{G′×G′′} (m′ × m′′) is canonically isomorphic to ξ^{G′}(m′) × ξ^{G′′}(m′′). Similarly, ξ^{G′×G′′} (X′ × X′′, 0, m′ × m′′) is canonically isomorphic to ξ^{G′}(m′) × X^{′⟨m′⟩} × ξ^{G′′}(m′′) × X^{′′⟨m′′⟩}. Thus, we have

X′×X′′ ^{f′×f′′} E′×X^{′⟨m′⟩}×E′′×X^{′′⟨m′′⟩} ^{π̃′ ×π̃′′} ξ^{G′}(m′)×X^{′⟨m′⟩}×ξ^{G′′}(m′′)×X^{′′⟨m′′⟩},

where E′ and E′′ are the universal curves, and f′ and f′′ are universal evaluation morphisms. It is easy to see that there is a canonical isomorphism

R^{•}(π̃′ ×π̃′′)_{∗}^{G′×G′′} (f′ ×f′′)^{∗}(T(X′ ×X′′)) ≅ R^{•}π̃_{∗}^{′′G′} (f^{′∗}(TX′))⊕R^{•}π̃_{∗}^{′′G′′} (f^{′′∗}(TX′′)).

By multiplicativity of the top Chern class, we obtain

(67)  c(m′ × m′′) = c′(m′) ⊗ c′′(m′′),

where c(m′ × m′′) is the virtual fundamental class of ξ^{G′×G′′} (X′ × X′′, 0, m′ × m′′) ≅ ξ^{G′}(X′, 0, m′) × ξ^{G′′}(X′′, 0, m′′), c′(m′) is the virtual fundamental class of ξ^{G′}(X′, 0, m′), and c′′(m′′) is the virtual fundamental class of ξ^{G′′}(X′′, 0, m′′).

It is a straightforward exercise using Equation (67) and the identification H(X′ × X′′) ≅ H(X′) ⊗ H(X′′) to show that

μ = μ′ ⊗ μ′′.

The trace axiom then follows immediately from the fact that trace of a tensor product is the product of the corresponding traces over each tensor factor. This finishes the proof of the first part of the proposition.

The second part of the proposition follows from Remark 4.11 and the first part of this proposition.

□

6.2. Trivial G-actions.

In the special case that the action of G on X is trivial, the data of a G-stable map to X is the same as a stable map from the underlying curve C to X and the data of a positive admissible G-cover, that is

\overline{M}_{g,n}^{G} (X) = \overline{M}_{g,n}^{G} ×_{\overline{M}_{g,n}} \overline{M}_{g,n} (X).

Moreover, since \overline{M}_{g,n}^{G} is smooth, it is evident that the virtual fundamental class

on \overline{M}_{g,n}^{G} (X) is simply the pullback of the virtual fundamental class of \overline{M}_{g,n} (X)

[\overline{M}_{g,n}^{G} (X)]^{vir} = pr_{2}^{∗}[\overline{M}_{g,n} (X)]^{vir} ,

and the evaluation map \overline{M}_{g,n}^{G} (X) (X̂)^{n} = (X × G)^{n} is simply the product of

the evaluation maps e : \overline{M}_{g,n}^{G}  G^{n} and ev : \overline{M}_{g,n} (X)  X^{n} .

Thus in this special case, we have

Λ_{g,n}^{G,X} = st^{∗}Λ_{g,n}^{X} ,

where \{Λ_{g,n}^{X} \} is the usual Gromov-Witten CohFT for X, and st : \overline{M}_{g,n}^{G} \overline{M}_{g,n} is the forgetful map (st := ŝt ∘ s̃t).

Since G acts trivially, we have

H(X)_{m_{i}} = H^{•}(X^{m_{i}} ; Θ) ≅ H^{•}(X; Θ)

for every m_{i} ∈ G. So the state space H(X) is just

H^{•}(X; Θ) ⊗ ℂ[G]

and Λ_{g,n}^{G,X} ((v_{1} ⊗ m_{1}), …, (v_{n} ⊗ m_{n} )) = st^{∗}Λ_{g,n}^{X} (v_{1}, …, v_{n} ) ∪ e^{∗}(1),

which is clearly just the external tensor product of Λ_{g,n}^{X} with ℂ[G]. Thus we have proved the following.

Proposition 6.24. If X is a smooth, projective variety with a trivial G action, then its associated G-CohFT is isomorphic to the external tensor product of the CohFT of stable maps associated to X (regarded as an equivariant CohFT for the trivial group) with ℂ[G], the group ring G-CohFT (see Example 4.3).

Remark 6.25. In the previous example, the induced CohFT on the space of Gcoinvariants agrees with Proposition 3.7 in [JK].

References

ACV. D. Abramovich, A Corti, and A. Vistoli, Twisted bundles and admissible covers.

math.AG/0106211.

AGV. D. Abramovich, T. Graber, and A. Vistoli, Algebraic orbifold quantum products. In A. Adem, J. Morava, and Y. Ruan (eds.), Orbifolds in Mathematics and Physics, Contemp. Math., Amer. Math. Soc., Providence, RI. 310, (2002), 1–25. math.AG/0112004.

BaKi. B. Bakalov and A. Kirillov, Lectures on tensor categories and modular functors. Univ.

Lecture Series 21, Amer. Math. Soc., Providence, RI, 2001. CoKat. D. Cox and S. Katz, Mirror symmetry and algebraic geometry. Math. Surv. 68, Amer.

Math. Soc., Providence, RI, 1999. CR1. W. Chen and Y. Ruan, A new cohomology theory for orbifold. math.AG/0004129. CR2. , Orbifold Gromov-Witten theory. In A. Adem, J. Morava, and Y. Ruan (eds.),

Orbifolds in Mathematics and Physics, Contemp. Math., Amer. Math. Soc., Providence, RI. 310, (2002), 25–85. math.AG/0103156.

FG. B. Fantechi and L. Göttsche, Orbifold cohomology for global quotients. math.AG/0104207. Fu. W. Fulton, Algebraic topology. Springer, NY, 1995. IP. In preparation. JK. T. Jarvis and T. Kimura, The orbifold quantum cohomology of the classifying space of

a finite group. In A. Adem, J. Morava, and Y. Ruan (eds.), Orbifolds in Mathematics and Physics, Contemp. Math., Amer. Math. Soc., Providence, RI. 310, (2002), 123–134. math.AG/0112037.

JKV. T. Jarvis, T. Kimura, and A. Vaintrob, Moduli spaces of higher spin curves and integrable

hierarchies. Compositio Math. 126 (2001), no. 2, 157–212. math.AG/9905034. Kas. C. Kassel, Quantum groups. Springer-Verlag, New York, NY, 1995. Kau. R. Kaufmann, Orbifold Frobenius algebras, cobordisms, and monodromies. In A. Adem,

J. Morava, and Y. Ruan (eds.), Orbifolds in Mathematics and Physics, Contemp. Math., Amer. Math. Soc., Providence, RI. 310, (2002), 135–162. And Orbifolding Frobenius algebras. Int. J. of Math. To appear math.AG/0107163.

Kir. A. Kirillov, Jr. Modular categories and orbifold models. Comm. Math. Phys. 229 (2002),

no. 2, 309–335. KoMa. M. Kontsevich and Yu. Manin, Gromov-Witten classes, quantum cohomology, and enu-

merative geometry. Commun. Math. Phys. 164 (1994), 525–562. Ma. Yu. Manin, Frobenius manifolds, quantum cohomology, and moduli spaces. Colloquium

Publ., 47, Amer. Math. Soc., Providence, RI, 1999. Mo. G. Moore, Lectures on D-branes, RR-fields, and K-theory. Miniprogram (June 18 - July

13, 2001) The Duality Workshop, Institute for Theoretical Physics, Santa Barbara, CA.

       http://online.itp.ucsb.edu/online/mp01/moore2/

PoVa. A. Polishchuk and A. Vaintrob, Algebraic construction of Witten’s top Chern class. In E. Previato (ed.), Contemp. Math. 276, Amer. Math. Soc., (2001), 229–249.

       math.AG/0011032.

Po2. A. Polishchuk, Witten’s top Chern class on the moduli space of higher spin curves.

       math.AG/0208112.

Tu. V. Turaev, Homotopy field theory in dimension 2 and group-algebras. math.QA/99100110.

Department of Mathematics, Brigham Young University, Provo, UT 84602, USA

   E-mail address: jarvis@math.byu.edu

Department of Mathematics, Oklahoma State University; Stillwater, OK and the Max Planck Institut für Mathematik; Bonn, Germany

   E-mail address: ralphk@mpim-bonn.mpg.de

Department of Mathematics and Statistics; 111 Cummington Street, Boston University; Boston, MA 02215, USA and School of Mathematics; Institute for Advanced Study; 1 Einstein Dr.; Princeton, NJ 08540, USA

   E-mail address: kimura@math.bu.edu