Contemporary Mathematics

Algebraic orbifold quantum products

Dan Abramovich, Tom Graber, and Angelo Vistoli

1. Introduction

The purpose of this note is to give an overview of our work on defining algebraic counterparts for W. Chen and Y. Ruan’s Gromov-Witten theory of orbifolds. This work will be described in detail in a subsequent paper.

The presentation here is generally based on lectures given by two of us at the Orbifold Workshop in Madison, Wisconsin. Following the spirit of the workshop, our presentation is intended to be understandable not only to algebraic geometers, but also practitioners of the other disciplines represented there, including differential geometers and mathematical physicists.

We make a special effort to make our constructions as canonical as we can, systematically using the language of algebraic stacks. Our constructions are based on the theory of twisted stable maps developed in [ℵ-V3], but require making explicit some details which were not studied in the paper [ℵ-V3]. Apart from the pleasure we take in understanding these details, our efforts bear some concrete fruits; in particular, we are able to define the Chen-Ruan stringy product in degree 0 (the so called stringy cohomology) with integer coefficients.

We work over the field ℂ of complex numbers (although our discussion works just as well over any field of characteristic 0).

2. Stacks and their moduli spaces

There are two, quite different, ways in which “orbifolds” or “stacks” arise.

2.1. Groupoids in schemes. This is the way most differential geometers, as well as many algebraic geometers, are introduced to the subject, since it is in some sense concrete and geometric: one thinks about an object which is locally modeled on a quotient of a variety by the action of an algebraic group. Then one needs to define a good notion of maps between such objects - this is the difficult part of the picture.

Concretely, one is given a “relation”, namely a morphism R → U ×U, where each of the projections R → U is étale (or, more generally, smooth), with the extra data of an “inverse” R → R and “composition” R×_{U} R → R satisfying natural assumptions which the reader may guess. This is what one calls a groupoid. One then defines morphisms of groupoids, Morita equivalences, and finally one defines the category of orbifolds using groupoids up to Morita equivalence.

In the theory of stacks, there is a natural algebraic stack [U/R] associated to a groupoid. Two groupoids are Morita equivalent if and only if the associated stacks are isomorphic; therefore the groupoid appears as a presentation of the stack. The stack is a more intrinsic object.

For instance, the quotient stack [U/G] of a scheme U by the action of a finite group G is the stack associated to the groupoid given by R = G × U. A particularly simple and useful case is the classifying stack BG = [pt/G] of the group G (where pt stands for the point Spec ℂ).

\begin{matrix}c \\ ◯ \end{matrix}0000 (copyright holder)

Research of D.A. partially supported by NSF grant DMS-0070970.

Research of T.G. partially supported by an NSF post-doctoral research fellowship.

Research of A.V. partially supported by the University of Bologna, funds for selected research topics.

2.2. Moduli problems. This is the way stacks arise in a more abstract, categorical incarnation, but still extremely natural.

One can encode the data of a “moduli problem” in the category of families of objects which we want to parametrize. For instance, the data of “moduli of curves of genus g” is encoded in the category M_{g} whose objects are

Ob(M_{g}) = \{ \begin{matrix}C \\ ↓ \\ S \end{matrix} |  S is a scheme, and C → S is a smooth family of curves of genus g  \} ,

and the arrows from one object C → S to another C′ → S′ are the fiber diagrams.

\xymatrix{C \ar[r] \ar[d] & C′ \ar[d] \\ S \ar[r] & S′}

There is also a structural functor M_{g} → Sch which sends a family of curves C → S to the underlying scheme S.

Formally, an algebraic stack is a category M with a functor M → Sch, satisfying certain assumptions which guarantee that it is “somewhat close to being representable by a scheme”. These assumptions imply, in particular, that an algebraic stack is always the stack associated to a groupoid in schemes [U/R]. If the automorphism group of every object in the category is finite, then in fact M is locally given as the quotient [U/G] of a scheme by a finite group. Such a stack is called a Deligne–Mumford stack.

If M is a stack of a moduli problem, then exactly as in the example, the objects are families C → S, and the arrows are fiber diagrams.

\xymatrix{C \ar[d] \ar[r] & C′ \ar[d] \\ S \ar[r] & S′}

The functor M → Sch simply sends a family C → S to its base scheme S. Simple examples include: M_{g}, the moduli stack of smooth curves of genus g, and BG, the moduli stack of principal homogeneous G-spaces, say for a finite group G.

2.3. Coarse moduli spaces. Every Deligne–Mumford stack M has an associated algebraic space called the coarse moduli space M (for a general proof see [K-M]):

• In case M is given via a groupoid as [U/R], the coarse moduli space is the orbit space U/R.

• In case M is the moduli stack of a moduli problem, then M is the usual coarse moduli space parametrizing objects modulo isomorphisms.

3. Twisted stable maps

3.1. Stable maps into a projective variety. Recall that, when X is a projective variety, one has a projective moduli space \overline{M}_{g,n}(X, β) of n-pointed stable maps of genus g and image class β. This is always the coarse moduli space of a Deligne–Mumford stack \overline{M}_{g,n}(X, β), whose objects over a scheme T are triples:

(C → T, {s_{1}, … s_{n} : T → C}, f : C → X) .

Here

(1) C → T is a family of prestable curves of arithmetic genus g,

(2) s_{i} : T → C are sections of C → T with disjoint images lying in the smooth locus, and

(3) f : C → X is a morphism with f_{∗}([C_{t}]) = β for all points t ∈ T, such that the group of automorphisms of fibers of C → T commuting with f and fixing all the sections s_{i} is finite.

It is no secret in this workshop that the spaces (and stacks) of stable maps have been useful in symplectic geometry, enumerative geometry and mathematical physics through Gromov–Witten invariants. They have also served as a stepping stone for constructing objects of interest in algebraic geometry (such as other moduli spaces). For a recent application in higher dimensional geometry see [G-H-S].

3.2. Stable maps into a stack. Since here we are in the business of enlarging our vocabulary of geometric objects from “varieties” to “orbifolds” (or “stacks”), one may wonder if it is worthwhile to replace the projective variety X in the definition of a stable map by a stack X.

Let us consider two examples of stacks which were mentioned above:

(1) Fix an integer h > 1, and say M = \overline{M}_{h}. Then a map of a curve C to M is the same as an object of M over C, in other words a family X → C of stable curves of genus h over C. Thus the moduli space of all such things is in particular a moduli space of a certain type of surfaces with extra structure (given by a map to a curve). For an algebraic geometer, it is evident that moduli of surfaces are of interest. So having a space of stable maps into \overline{M}_{g} is useful at least as a construction tool. See [ℵ-V1].

(2) Fix a finite group G, and say M = BG. Then a map of a curve C to M is a family P → C of principal homogeneous G-spaces, in other words, a principal G-bundle. The moduli of these objects are closely related to Hurwitz schemes, level structures and admissible covers, which have been a subject of interest in algebraic geometry for years. See [ℵ-C-V].

It is from this point of view, of using stable maps as a basic construction tool, that two of us approached the problem of constructing a suitable space of stable maps into a stack, see [ℵ-V3]. Admittedly, Gromov–Witten theory for stacks seemed to us a distant possibility until the work of W. Chen and Y. Ruan on the subject was made public.

3.3. The non-compactness problem. One still needs to give a definition of what one means by a “stable map into a stack”. As a first attempt, one can define stable maps into a stack X just as stable maps into a projective variety were defined. The only problem with this is that the stack of usual stable maps into X is not necessarily compact.

This phenomenon is already manifest with the simplest series of examples where X = BG, so that maps to X correspond to principal G-bundles, see [ℵ-V3]. Say G = (ℤ/mℤ)^{4}, m ≥ 2, and g = 2. A smooth curve C of genus 2 carries a connected principal G-bundle P → C, since its first homology group is ℤ^{4}. But if one takes a one-parameter family P_{t} → C_{t}, t ≠ 0 of such bundles where the underlying curve C_{t} degenerates, as t approaches 0, to a nodal curve C_{0} of geometric genus 1, then there is nothing to which the one-parameter family of bundles can degenerate. This is simply because the first homology of the limiting curve C_{0} is ℤ^{3}, and thus C_{0} carries no connected principal G-bundle.

An example of a similar phenomenon with X = \overline{M}_{1,1} is described in [ℵ-V1].

3.4. Adding orbispace structure on curves. A simple analysis, detailed in [ℵ-V2], [ℵ-V3], of what goes wrong in the example above, reveals that the problem is centered above the node of the degenerate curve C_{0}, and moreover, at the node there is a natural orbispace structure C_{0} on C_{0} which does afford a limiting connected principal bundle P_{0} → C_{0}.

One then realizes that, in order to have a compact moduli space (or stack) of maps into a stack X, one needs to allow the source curves to acquire an orbispace structure as well. This may seem as a surprise at first, but we have come to believe that this is completely natural and beautiful. In a way, nature imposes this solution upon us, so why fight the elements?

3.5. Twisted curves. Let us describe in some detail the orbispace curves - which we call twisted curves, underlying these new maps. We restrict ourselves here to balanced twisted curves, the only type that is relevant to our discussion in this paper.

3.5.1. Nodes. First, consider the structure at a node. The orbispace curve here is locally the quotient [U/μ_{r}] of a nodal curve U = {xy = f(t)} by the following action of the cyclic group μ_{r} of r-th roots of 1:

(x, y) ↦ (ζx, ζ^{−1}y).

The coarse curve C underlying the orbispace curve C is, locally, the schematic quotient U/μ_{r} defined by uv = f(t)^{r}, where u = x^{r} and v = y^{r}.

This kind of orbispace structure by itself would suffice for solving the compactification problem we discussed above. But we also know that it is useful to be able to describe the “boundary” of moduli by gluing marked curves into nodal ones. Reversing this line of thought, it is useful to understand what kind of structure one obtains when separating a node into two marked points on a maked curve.

3.5.2. Markings. We are led to consider an orbispace structure along a marking. Here the orbispace curve C → T is locally the quotient [U/μ_{r}] of a smooth curve U, with local coordinate z defining the marking, by the following action of μ_{r}:

z ↦ ζz.

The coarse curve C is locally a smooth curve with coordinate w = z^{r}.

The integer r in these local descriptions is called the index of the orbifold curve at the point in question. 3.5.3. Global structure of a marking. Before imposing on the reader the formal definition of a twisted curve,

we need to give a warning. Having given the local description of a family of twisted curves C → T at a marking, one might be misled to believe that there is automatically globally a section T → C. This is not the case in general. This fact is going to be important later when we consider evaluation maps.

Take the coarse family of curves C → T, with its section T → C, having image Σ^{C} ⊂ C. The reduction Σ^{C} ⊂ C of the inverse image of Σ^{C} in the twisted curve C is canonically a gerbe over T banded by the group μ_{r}. This means, in particular, that locally Σ^{C} is isomorphic to the stack-theoretic quotient of Σ^{C} by the trivial action of μ_{r}, but globally there is an obstruction for such a quotient description. The obstruction, which naturally lies in H^{2}(T, μ_{r}), comes in our case from H^{1}(T, 𝔾_{m}) = Pic(T) in the following manner:

Let L = s^{∗}N be the pullback to T of the normal bundle of Σ^{C} ⊂ C. Then Σ^{C} is the moduli stack of r-th roots of L, whose objects over a scheme S are triples

(g : S → T, M, σ : M^{⊗r} ˜ ⟶g^{∗}L),

where

• g : S → T is a morphism,

• M is a line bundle on S, and

• σ is an isomorphism of line bundles.

The group μ_{r} acts on every object here by fiberwise multiplication on M. All automorphisms arise this way. It is clear from this description that Σ^{C} → T has a section if and only if the line bundle L has an r-th root over T.

Incidentally, there is also a simple prescription for obtaining C from the coarse curve C, at least away from the nodes. We will not need that description in the present exposition.

3.5.4. Finally, here is a formal definition of twisted curves, as in [ℵ-V3]:

Definition 3.5.5. A twisted nodal n-pointed curve over a scheme T is a diagram

\xymatrix{Σ^{C} \ar@{^{(}->}[r] \ar[ddr] & C \ar[d] \\  & C \ar[d] \\  & T}

where

(1) C is a Deligne-Mumford stack, proper over T, which étale locally is a nodal curve over T;

(2) Σ^{C} = ∪_{i=1}^{n}Σ_{i}^{C}, where Σ_{i}^{C} ⊂ C are disjoint closed substacks in the smooth locus of C → S;

(3) Σ_{i}^{C} → T are étale gerbes;

(4) the morphism C → C exhibits C as the coarse moduli scheme of C; and

(5) C → C is an isomorphism over C_{gen}.

The notation C_{gen} in the definition above stands for the generic locus, namely the complement of the nodes and markings on C.

One categorical issue we have to mention is the following: the collection of all stacks naturally forms a 2-category, and therefore families of twisted curves also form a 2-category. However, the fact that each twisted curve has a dense open subset which is a scheme, can be shown to imply that the 2-category of twisted curves is equivalent to a category (so it has a chance of being a stack).

3.6. Twisted stable maps. We can now define a twisted stable map as follows (see [ℵ-V3]):

Definition 3.6.1. Let X be a Deligne–Mumford stack, X → X ⊂ ℙ^{N} a projectively embedded coarse moduli scheme.

An n-pointed twisted stable map f : C → X to X of genus g, degree d, is a diagram

\xymatrix{C \ar[d] \ar[r]^{f} & X \ar[d] \\ C \ar[r]^{\overline{f}} & X}

where

(1) C is a twisted marked curve, with coarse moduli space C → C,

(2) C → X a stable map of genus g and degree d, and

(3) C → X representable

We recall that a morphism of algebraic stacks is representable if and only if for any geometric object ξ ∈ X (Spec ℂ), the map f : Autξ → Aut f(ξ) is a monomorphism.

With this definition, we have the following theorem:

Theorem 3.6.2. The category of n-pointed twisted stable maps of genus g and degree d to X is a proper Deligne–Mumford stack K_{g,n}(X , d) admitting a projective coarse moduli space K_{g,n}(X , d).

Of course, the stacks K_{g,n}(X , d) can be further decomposed using a numerical equivalence class, or a homology class β ∈ H_{2}(X, ℤ) instead of the degree.

The notation using the letter K rather than \overline{M} was chosen originally to avoid confusion when inserting something like \overline{M}_{γ} as an argument in place of X. In this note we will introduce the notation \overline{M}_{g,n}(X , d) for a certain variant of K_{g,n}(X , d) (namely the stack of twisted stable maps with trivialized markings).

The stacks K_{g,n}(X , d) can also be decomposed in terms of the type of the markings. To avoid cumbersome notation, we will not do this explicitly. Note, however, that the index r_{i} of the i-th marking is a locally constant function K_{g,n}(X , d) → ℤ.

Also, as in the case of usual stable maps, there is a stabilization morphism K_{g,n}(X , ∗) → K_{g,n}(Y, ∗) associated to a morphism of stacks X → Y. This is going to be useful in this paper in the special case where Y is a point.

Theorem 3.6.2 is closely related to Proposition 2.3.8 in Chen and Ruan’s paper [C-R2]. There are several differences: Chen and Ruan work with pseudo-holomorphic maps into smooth symplectic orbifolds in general. In [ℵ-V3] we work only with algebraic Deligne–Mumford stacks. On the other hand we allow the target stack X to be singular (and allow tame stacks in positive characteristics). Chen and Ruan’s moduli space is a differential orbifold, whereas we work systematically with stacks.

Some words on the proof of the theorem in [ℵ-V3]: often, the construction of a moduli stack (or space) is a standard process based on the existence of Hilbert and Quot schemes. Unfortunately, there is no well-developed machinery of universal constructions analogous to Hilbert and Quot schemes for stacks. The proof in [ℵ-V3] constructs the stack K_{g,n}(X , d) “with bare hands” using deformation theory and algebraization, following the list of axioms in M. Artin’s paper [Ar]. Some of the methods developed in [ℵ-V3] might serve as building blocks for a future theory of universal constructions for stacks.

4. Gromov-Witten theory of varieties and stacks

4.1. Quantum intersection theory. In standard treatments of Gromov-Witten theory in symplectic geometry, as well as [C-R2], one defines the quantum product in terms of “n-point Gromov-Witten numbers”, using the inverse matrix of the intersection pairing on cohomology, both for defining the product and proving associativity. In the context of algebraic cycles on varieties, let alone stacks, this cannot be done in general, since there is no analogue of Poincaré duality. A formalism which circumvents this issue was described in the paper [G-P]:

Let X be a smooth projective variety. Consider the correspondence

\xymatrix{\overline{M}_{g,n+1}(X, β) \ar[r]^{e_{n+1}} \ar[d]_{e_{1,... ,n}} & X \\ X^{n}}

where e_{1,... ,n} : \overline{M}_{g,n+1}(X, β) → X^{n} is the evaluation map at the first n points, and e_{n+1} is the last evaluation map.

Define a map

A^{∗}(X)^{n} ⟶ A^{∗}(X)
γ_{1} × ⋯ × γ_{n} ↦ ⟨γ_{1}, …, γ_{n}, ∗⟩_{g,β} ,

where

⟨γ_{1}, …, γ_{n}, ∗⟩_{g,β} = (e_{n+1})_{∗} ( e_{1,... ,n}^{∗}(γ_{1} × ⋯ × γ_{n}) ∩ [\overline{M}_{g,n+1}(X, β)]^{v} )

Here we used the notation [\overline{M}_{g,n+1}(X, β)]^{v} for the virtual fundamental class of \overline{M}_{g,n+1}(X, β), defined using the cotangent complex and deformation theory, see, e.g., [B-F], [L-T].

Gromov–Witten invariants of smooth projective varieties satisfy a number of relations which are coded by a number of well known axioms. In particular, the genus 0 invariants can be combined to give the multiplication law of the associative quantum cohomology ring. For instance, the so-called “small” quantum product is defined by

γ_{1} ∗ γ_{2} = ∑_{β∈A^{1}(X)} ⟨γ_{1}, γ_{2}, ∗⟩_{0,β} q^{β}.

To prove the axioms of Gromov–Witten invariants and, in particular, to prove associativity of the quantum product, one needs to compare the fiber product

\overline{M}_{g_{1},n_{1}+1}(X, β_{1}) ×_{X} \overline{M}_{g_{2},n_{2}+1}(X, β_{2}),

along with its virtual fundamental class, with a boundary divisor in

\overline{M}_{g_{1}+g_{2},n_{1}+n_{2}}(X, β_{1} + β_{2}),

as we will illustrate later.

In order to generalize this picture, we will need

(1) an analogue of A^{∗}(X) for a smooth Deligne–Mumford stack,

(2) an analogue of the evaluation maps e_{1,... ,n} and e_{n+1},

(3) an analogue of the virtual fundamental class, and

(4) an analogue of the fiber product description of the boundary divisors.

4.2. Intersection theory of a stack. The first ingredient in the “non-orbifold” case which needs a revision in the stack situation is the intersection theory A^{∗}(X).

Two important ingredients in intersection theory of smooth varieties are

(1) a ring A^{∗}(X), functorial under pull-back along a morphism f : X → Y, and

(2) push-forward maps f : X → Y for f proper, satisfying the projection formula.

It has been observed long ago that such a theory with integer coefficients cannot exist for Deligne–Mumford stacks. The problem lies with the push-forward. Let G be a finite group of r elements, and let p = Spec k be a point. Consider the fiber diagram.

\xymatrix{G \ar[r] \ar[d]_{ϕ} & p \ar[d]^{ψ} \\ p \ar[r] & BG}

Since the degree of G → p is the order r of G (i.e. ϕ_{∗}[G] = r[p]), and since p → BG is proper and flat, it follows by the projection formula that the degree of ψ : p → BG is r as well. But we also have the map π : BG → p, since p is the coarse moduli space of BG, and of course π ∘ ψ = id : p → p. So the degree of π must be 1/r, so rπ_{∗}[BG] = [p], which is impossible since the generator [p] is not divisible by r.

In [Gi] and [V], Gillet and Vistoli defined an analogous intersection theory with rational coefficients for smooth Deligne-Mumford stacks. Basically

(1) as a group, A^{∗}(X)_{ℚ} is isomorphic to A_{∗}(X)_{ℚ}, where X is the coarse moduli space, and it has the structure of a commutative ring, but

(2) push forward is a bit more subtle, and takes into account the degrees of stabilizers.

This construction can be generalized in the sense of [F], Chapter 17 (see also [B-F]) to a bivariant intersection theory of Deligne–Mumford stacks - we will use this when proving associativity of the quantum product.

In [K], A. Kresch defined an intersection theory A^{∗}(X) with integer coefficients but where push-forward maps are defined only for representable morphisms f : X → Y. We will use Kresch’s theory when we define an integer-coefficients version of Chen and Ruan’s stringy cohomology ring of X.

The constructions of Kresch and Vistoli are related by the fact that A^{∗}(X) ⊗ ℚ = A^{∗}(X)_{ℚ}.

4.3. A^{∗}(X) is too small. Both constructions discussed above are not quite sufficient for what we need. Even for global quotients, it has long been observed that the K theory of X = [V/G], namely the equivariant K-theory of V, is bigger than the K theory of V/G, even with rational coefficients. Indeed, the decomposition theorem says that the equivariant K-ring K_{0}(V//G) is a product of K_{0}(V^{g}/C(g)), suitably twisted by cyclotomic rings. Thus any reasonable Riemann-Roch formula must take into account contributions from all the fixed-point sets. These contributions are lost in A_{∗}(X)_{ℚ}. A detailed theory of Riemann-Roch type theorems for stacks, in which such contributions are introduced, was developed in Toen’s work [T].

In enumerative geometry, T. Graber studied in [Gr] the number of hyperelliptic curves of genus g and degree d in ℙ^{2} through 3d + 1 general points. It is explained there that the cohomology of Sym^{2}ℙ^{2} does not carry enough information to encode all these invariants, and therefore the Hilbert scheme Hilb_{2}(ℙ^{2}) was used instead. The extra information in Hilb_{2}(ℙ^{2}) is completely accounted for by contributions of fixed-point sets. It is now known (see [B-K-R]) that the derived categories of Hilb_{2}(ℙ^{2}) and the stack symmetric square [(ℙ^{2})^{2}/(ℤ/2ℤ)] are equivalent, and therefore their K groups are isomorphic.

In [C-R2], Chen and Ruan were motivated by string theory to introduce similar contributions of fixed-point sets. We follow their approach, with a slight modification which avoids any choice of roots of 1.

4.4. The inertia stack and variants. The stringy cohomology of X according to Chen-Ruan is, as a vector space, the cohomology of another smooth stack, namely the inertia stack

X Chen-Ruan_{1} = I_{X} = { (x, g) | x ∈ Ob(X), g ∈ Aut x}
= {(x, H, g) | x ∈ Ob(X), H ⊂ Aut x, g a generator of H}

A slightly more canonical Gromov-Witten formalism arises if instead one takes a Galois twist of this:

X_{1} = {(x, H, χ) | x ∈ Ob(X), H ⊂ Aut x, χ : H ˜→μ_{r} for some r}
= ⋃_{r} HomRep(Bμ_{r}, X)

i.e., the stack of representable morphism from a constant cyclotomic gerbe to X.

Of course, over ℂ there is a natural isomorphism of μ_{r} with ℤ/rℤ, which gives an isomorphism

X Chen-Ruan_{1} ≃ X_{1}.

But over other fields the group-schemes are different, in which case one has to use μ_{r}: with ℤ/rℤ one cannot even define the Gromov-Witten invariants of X.

There is an interesting variant of this stack, which we denote by \overline{X}_{1}. It arises as a rigidification of X_{1}. The point is that the group μ_{r} acts on every object of X_{1} in the component corresponding to μ_{r}, and the rigidification process (see [ℵ-C-V]) allows one to remove this action from the picture (just as the Picard functor is obtained by removing the ℂ^{∗}-action from the stack of line bundles).

\overline{X}_{1} = \{(x, H, χ) up to μ_{r}\}
= stack of cyclotomic gerbes in X

The geometric objects of X_{1} and \overline{X}_{1} are the same (x, H, χ), but the automorphisms are different: in the first case an automorphism is an element of the centralizer C(H), in the second it is an element of C(H)/H.

We note that

• there are obvious inclusions X ⊂ X_{1} and X ⊂ \overline{X}_{1}.

• There is also a “forgetful map” X_{1} → X, but there is in general no map from \overline{X}_{1} to X.

• Finally the rigidification map π_{1} : X_{1} → \overline{X}_{1} exhibits X_{1} as the universal gerbe over \overline{X}_{1}. This implies that the coarse moduli space X_{1} of X_{1} is also the moduli space of \overline{X}_{1}.

The total stringy Chow group of X is defined as the total Chow group A^{∗}(X_{1})_{ℚ}. We can also consider the integral version A^{∗}(X_{1}) defined by Kresch.

There is a useful endomorphism

r : A^{∗}(X_{1}) → A^{∗}(X_{1}),

induced by the locally constant function

r : X_{1} → ℤ

with value r on the piece corresponding to HomRep(Bμ_{r}, X).

4.5. Evaluation maps. Given a family of n-pointed twisted stable maps parametrized by a scheme S:

(C → S, f : C → X)

we have, for each i in the range 1 ≤ i ≤ n, a map of the marking (Σ_{i}) into X. This means that S parametrizes a family of cyclotomic gerbes in X. By definition this gives a morphism (ē_{i})_{S} : S → \overline{X}_{1}, and since this is true for every twisted stable map, we get a map of moduli stacks

ē_{i} : K_{g,n}(X , β) → \overline{X}_{1}.

Passing to moduli spaces we get

ei^{coarse} : K_{g,n}(X , β) → X_{1}.

Chen and Ruan [C-R2] use this latter map ei^{coarse} of coarse moduli spaces. It so happens that the formalism of Gromov-Witten theory basically goes through (with an important twist to be introduced soon) if one works with this map as if it came from a map ei^{orb} : K_{g,n}(X , β) → X_{1}. It is a fact of life that such a “virtual map” does not exist in general. However, as long as one works with cohomology or intersection theory with rational coefficients, then one can define a “cohomological pull-back”:

(ei^{orb})^{∗} = r^{−1} · ē_{i}^{∗} (π_{1})_{∗}.

(here again r denotes the operator which multiplies a class on a component of index r by the integer r). A similar trick works also for pushforward

(ei^{orb})_{∗} = r · π_{1}^{∗} (ē_{i})_{∗}.

In order to avoid the confusion of working with such “weighted maps” we can simply work with the map ē_{i} : K_{g,n}(X , β) → \overline{X}_{1}, changing the products by a suitable factor. For the sake of exposition, it may be a bit more transparent to use X_{1} and work with the following formalism:

Define \overline{M}_{g,n}(X , β) to be the stack of twisted stable maps with sections of all the gerbes. This is simply the fibered product of the n universal gerbes over K_{g,n}(X , β). Now we have evaluation maps

\xymatrix{\overline{M}_{g,n}(X , β) \ar[r]^{e_{i}} \ar[d] & X_{1} \ar[d] \\ K_{g,n}(X , β) \ar[r]^{ē_{i}} & _{\overline{X}_{1}}}

There is an obvious involution ι on \overline{X}_{1} and X_{1} via χ ↦ χ^{−1}. We define the twisted evaluation map ě_{i} = ι∘e_{i}. The twisted evaluation map comes about when one glues two twisted stable maps along two markings to form a balanced twisted stable map.

We can work with \overline{M}_{g,n}(X , β), as long as we remember to account for the degree 1/(r_{1} ⋯ r_{n}) of the map \overline{M}_{g,n}(X , β) → K_{g,n}(X , β). Note that this degree is only locally constant, and varies from one connected component to another.

4.6. Deformation and obstruction. We want to define an appropriate virtual fundamental class for \overline{M}_{g,n}(X , β), generalizing the situation of the space of maps into a smooth projective variety.

The first obstacle one needs to overcome is mostly psychological - does deformation theory as we know it work for stacks?

The answer is yes - and already in the literature. The point is that Illusie’s work on deformation theory [I] works by deforming a ringed topos (essentially the category of sheaves on a site), as soon as it is reasonable enough to allow the definition of a cotangent complex. The fact that a stack has such a topos is discussed in [L-MB], Chapter 12, and the existence of a cotangent complex is discussed in [L-MB], Chapter 17 (in fact the case of Deligne–Mumford stacks can be more easily deduced from the case of schemes).

In particular, as in the case of schemes, the infinitesimal deformations of a twisted curve C are measured by Ext^{1}(Ω_{C}^{1}, O_{C}), and if one wants to deform some additional non-twisted marking, then infinitesimal deformations are in Ext^{1}(Ω_{C}^{1}(D), O_{C}), where D is the divisor given by these markings. Obstructions would lie in the corresponding Ext^{2} group, which turns out to be zero for twisted pointed curves (see [ℵ-C-V]).

Similarly, infinitesimal deformations of a map f : C → X fixing the structure of C (as well as of X) are measured by Hom(f^{∗}Ω_{X}^{1} , O_{C}) = H^{0}(C, f^{∗}T X). Obstructions lie in H^{1}. More precisely, R^{•}π_{∗}(f^{∗}T X) gives a relative obstruction theory for this moduli problem over the stack of marked twisted prestable curves, where the symbols refer to the diagram:

\xymatrix{C \ar[r]^{f} \ar[d]_{π} & X \\ \overline{M}_{g,n}(X , β)}

Now, the formalism of the virtual fundamental classes (see [L-T], [B-F], [K]) automatically allows one to define a virtual fundamental class [K_{g,n}(X , β)]^{v}. There is similarly a class [\overline{M}_{g,n}(X , β)]^{v}, which is just the pullback of [K_{g,n}(X , β)]^{v}. As we indicated before, to get the degrees right we need to replace the latter by a multiple: [\overline{M}_{g,n}(X , β)]^{w} = r_{1} ⋯ r_{n}[\overline{M}_{g,n}(X , β)]^{v}.

We remark that in the special case where the moduli space is smooth and the morphism from the deformation space of the map surjects onto the deformation space of the underlying twisted curve for all points in the moduli space, there is a simple expression for the virtual fundamental class. The formula in this case is

[\overline{M}_{g,n}(X , β)]^{v} = c_{top}(E)

where E = R^{1}π_{∗}(f^{∗}T X) is the obstruction bundle. (The hypotheses above ensure that this sheaf is locally free.) There are two important cases where these conditions are satisfied. One is when g = 0 and the target space is a homogeneous variety. In this case the obstruction bundle is trivial, and the virtual class can be identified with the ordinary fundamental class. The other case is of special interest in this paper. It is when d = 0 – the case of constant maps. When the map is constant, it is clear that the underlying curve can be deformed freely, which explains why we get the surjectivity on deformation spaces. It is by studying the spaces of constant maps that we define the product structure on the stringy Chow rings in Section 6. As a result, we see that this product can be understood without recourse to the somewhat technical theory of virtual classes. In order to make this exposition more accessible, we will assume that we are in one of these nice cases and identify the virtual class with this chern class in some of our arguments below. Those familiar with the arguments of [B] and [B-F] should have no trouble adapting them in general. Those who are not can still take this presentation as a complete proof in the case of constant maps which is in some sense the most interesting from our point of view.

4.7. The product. We can now define the Gromov-Witten invariants and the quantum product just as it is done for smooth projective varieties. Consider the diagram

\xymatrix{\overline{M}_{g,n+1}(X , ·) \ar[d]_{e_{1,... ,n}} \ar[r]^{ě_{n+1}} & X_{1} \\ X_{1}^{n}}

and define

A^{∗}(X_{1})_{ℚ}^{n} ⟶ A^{∗}(X_{1})_{ℚ}
γ_{1} × ⋯ × γ_{n} ↦ ⟨γ_{1}, …, γ_{n}, ∗⟩_{g,β} ,

where

⟨γ_{1}, …, γ_{n}, ∗⟩_{g,β} = ě_{n+1 ∗} ( e_{1,... ,n}^{∗}(γ_{1} × ⋯ × γ_{n}) ∩ [\overline{M}_{g,n+1}(X , β)]^{w} ) .

Again - the reason we are using the twisted evaluation map ě_{n+1} has to do with the fact that the twisted curves are balanced, and will become more explicit below.

Having defined the Gromov-Witten invariants, we immediately get a new product structure on A_{∗}(X_{1})_{ℚ}[[q]] defined by taking

γ_{1} ∗ γ_{2} = ∑_{β}⟨γ_{1}, γ_{2}, ∗⟩_{0,β}q^{β}.

5. Associativity

As in the now “classical” case of smooth varieties, the most subtle property of the quantum product is its associativity. Here we go through the main steps in the proof. We revisit these steps in the next section when discussing the stringy cohomology ring of X.

5.1. Associativity: the product diagram. For simplicity of the discussion, we restrict from now on to the case where g = 0 and n = 3.

We want to prove that the “3-point” quantum product is associative. Since it is commutative, it suffices to show

Theorem 5.1.1.

∑_{β_{1}+β_{2}=β} ⟨ ⟨γ_{1}, γ_{2}, ∗⟩_{0,β_{1}}, γ_{3}, ∗ ⟩_{0,β_{2}} = ∑_{β_{1}+β_{2}=β} ⟨ ⟨γ_{1}, γ_{3}, ∗⟩_{0,β_{1}}, γ_{2}, ∗ ⟩_{0,β_{2}} .

Fix two homology classes β_{i} ∈ H_{1}(X, ℤ), and for simplicity write M_{i} = \overline{M}_{0,3}(X , β_{i}) for the corresponding stacks of twisted stable maps. The corresponding evaluation maps are denoted e_{i}^{1} : M_{1} → X_{1} and e_{i}^{2} : M_{2} → X_{1}.

The relevant diagram is the following.

\xymatrix{ &  & M_{2} \ar[r]^{ě_{3}^{2}} \ar[d]^{e_{2}^{2}} \ar[dl]_{e_{1}^{2}} & X_{1} \\ M_{1} \ar[r]^{ě_{3}^{1}} \ar[d]^{e_{2}^{1}} \ar[d]_{e_{1}^{1}} & X_{1} & X_{1} \\ X_{1}}

As a first step we complete the top left corner of the diagram.

\xymatrix{M_{1} ×_{X_{1}} M_{2} \ar[r]^{p_{2}} \ar[d]_{p_{1}} & M_{2} \ar[d]^{e_{1}^{2}} & equivalently & M_{1} ×_{X_{1}} M_{2} \ar[r]^{p_{1}×p_{2}} \ar[d] & M_{1} × M_{2} \ar[d]^{ě_{3}^{1}×e_{1}^{2}} \\ M_{1} \ar[r]^{ě_{3}^{1}} & X_{1} &  & X_{1} \ar[r]^{δ} & X_{1}^{2}}

Consider the following morphisms from M_{1} ×_{X_{1}} M_{2} to X_{1} :

e_{1}^{×} = e_{1}^{1} ∘ p_{1},
e_{2}^{×} = e_{2}^{1} ∘ p_{1},
e_{3}^{×} = e_{1}^{2} ∘ p_{2},
e_{4}^{×} = e_{1}^{2} ∘ p_{2}.

Starting with the definition of the Gromov-Witten invariants, a straightforward push-pull argument detailed in [G-P] gives

⟨⟨γ_{1}, γ_{2}, ∗⟩_{0,β_{1}}, γ_{3}, ∗⟩_{0,β_{2}} = ě_{4∗}^{×} ( ( e_{1}^{×∗}γ_{1} · e_{2}^{×∗}γ_{2} · e_{3}^{×∗}γ_{3}) ∩ δ^{!}( [M_{1}]^{w} × [M_{2}]^{w} ) ) . (1)

As we will see, the class δ^{!}( [M_{1}]^{w} × [M_{2}]^{w} ) has an interpretation as a weighted virtual fundamental class of M_{1} ×_{X_{1}} M_{2} with respect to a natural obstruction theory.

5.2. Associativity: the divisor diagram. The stack M_{1} ×_{X_{1}} M_{2} can be viewed as the stack of pairs of 3-pointed genus 0 twisted stable maps with balanced gluing data along ě_{3}^{1} and e_{1}^{2}, and trivialization of all the gerbes. Writing β = β_{1} + β_{2}, this means that there is a gluing morphism

gl : ⋃_{β_{1}+β_{2}=β} M_{1} ×_{X_{1}} M_{2} → \overline{M}_{0,4}(X, β),

just as in the case of usual stable maps. It is essential here, however, that the fiber product be taken with respect to the twisted evaluation on the second map. Otherwise the curve that we would obtain by gluing would not be balanced.

Consider the stabilization morphism st : \overline{M}_{0,4}(X, β) → \overline{M}_{0,4} and the divisor D := (12|34) ⊂ \overline{M}_{0,4} (corresponding to the reducible stable 4-pointed curve of genus 0 where the two first points are on the same component). We have the top and bottom of the following fiber diagram.

\xymatrix{⋃_{β_{1}+β_{2}=β} M_{1} ×_{X_{1}} M_{2} \ar[d]^{g} \\ D(X) \ar[r] \ar[d] & D ×_{\overline{M}_{0,4}} \overline{M}_{0,4}(X) \ar[d] \ar@{^{(}->}[r] & \overline{M}_{0,4}(X , β) \ar[d] \\ D^{orb} \ar[r]^{j} & D ×_{\overline{M}_{0,4}} \overline{M}_{0,4}^{orb} \ar[d] \ar@{^{(}->}[r] & _{\overline{M}{orb \atop 0,4}} \ar[d] \\ & D \ar@{^{(}->}[r] & _{\overline{M}_{0,4}}} (2)

The notation D(X) stands for the substack of reducible twisted stable maps to X with a specified node separating the first two markings from the last two. We also denote by e_{i} : \overline{M}_{0,4}(X , β) → X_{1} the evaluation maps. Note that by construction, we have

e_{i}^{×} = e_{i} ∘ gl. (3)

On the second row we inserted the Artin stack \overline{M}_{0,4}^{orb} of quasistable twisted curves, i.e. connected proper twisted curves without a stability requirement. This is a smooth stack which is neither separated nor of finite type, yet it comes in handy just as the usual stack of quasistable curves was useful in [B]. We let D^{orb} denote the smooth stack of quasistable curves with a specified node separating the first two markings from the last two, and let

j : D^{orb} → \overline{M}_{0,4}^{orb}

denote the natural morphism from D^{orb} to the boundary of \overline{M}_{0,4}^{orb}. This morphism is analogous to the standard gluing maps between moduli spaces of ordinary stable curves. In particular, it is birational onto its image, although not an embedding. We will use [D^{orb}] to denote the class of this boundary divisor in \overline{M}_{0,4}^{orb}.

A local analysis establishes the following two equations.

Proposition 5.2.1.

(1) The map g : ⋃ M_{1} ×_{X_{1}} M_{2} → D(X) has locally constant degree 1/r_{×}. In fact ⋃ M_{1} ×_{X_{1}} M_{2} is the universal gerbe over the distinguished node of the universal curve of D(X).

(2) We have an equality of divisors on \overline{M}_{0,4}^{orb}:

[D ×_{\overline{M}_{0,4}} \overline{M}_{0,4}^{orb}] = r_{×}[D^{orb}].

Here we are using r_{×} to denote the locally constant function of D^{orb} giving the index of the distinguished node. By abuse of notation, we will sometimes use the same symbol to refer to the pullback of this function to M_{1} ×_{X_{1}} M_{2} where it is also equal to p_{1}^{∗}r_{3} or p_{2}^{∗}r_{1}.

5.3. Associativity: compatibility of virtual fundamental classes. A more subtle issue is the following:

Proposition 5.3.1.

δ^{!}( [M_{1}]^{w} × [M_{2}]^{w} ) = r_{×}^{2} · g^{∗}(j^{!}[\overline{M}_{0,4}(X)]^{w})

Proof. It is for the proof of this statement that we will make the simplifying assumptions mentioned at the end of Section 4.6. It is true in complete generality, however. By definition of [M_{1}]^{w}, [M_{2}]^{w}, [\overline{M}_{0,4}(X)]^{w}, and r_{×} we need to show

δ^{!}(c_{top}(E_{1}) × c_{top}(E_{2})) = g^{∗}(j^{!}c_{top}(E)) (4)

where E_{1}, E_{2}, and E are the relevant obstruction bundles.

The right hand side of this equation is easy to understand. Since we have assumed that the morphism from \overline{M}_{0,4}(X) to \overline{M}_{0,4}^{orb} is smooth, the j^{!} just means restriction to D(X), and the right hand side of Equation (4) is equal to c_{top}(gl^{∗}E). Denote

Y = M_{1} ×_{X_{1}} M_{2}.

By the theorem on cohomology and base change, we can compute gl^{∗}E as a higher direct image sheaf on Y. If we take C_{Y} to be the pull-back under gl of the universal family of twisted curves over \overline{M}_{0,4}(X), then gl^{∗}E = R^{1}π_{∗}(f^{∗}T X) where by abuse of notation we let π be the map from C_{Y} to Y. Essentially by the definition of Y, C_{Y} is naturally a union of two families of curves, which we will denote C_{1} and C_{2}, which are pulled back from the projections from Y to M_{1} and M_{2} respectively. If we let C′_{Y} be the disjoint union of these two families, then there is a normalization map ν : C′_{Y} → C_{Y} which glues the two curves together along the gerbe Σ associated to the distinguished node of C_{Y}. We consider the normalization sequence

0 → f^{∗}T X → ν_{∗}f′^{∗}T X → (f^{∗}T X)|_{Σ} → 0.

Pushing forward to Y gives

0 → π_{∗}f^{∗}T X → π′_{∗}f′^{∗}T X → π_{∗}(f^{∗}T X)|_{Σ} → gl^{∗}E → p_{1}^{∗}E_{1} ⊕ p_{2}^{∗}E_{2} → 0.

We break this into two shorter exact sequences.

0 → π_{∗}f^{∗}T X → π′_{∗}f′^{∗}T X → π_{∗}(f^{∗}T X)|_{Σ} → Q → 0 (5)

and

0 → Q → gl^{∗}E → p_{1}^{∗}E_{1} ⊕ p_{2}^{∗}E_{2} → 0.

At this point we conclude that

c_{top}(gl^{∗}E) = c_{top}(Q) · p_{1}^{∗} c_{top}(E_{1}) · p_{2}^{∗} c_{top}(E_{2}).

Comparing to the equation we are trying to prove, and considering the diagram

\xymatrix{M_{1} ×_{X_{1}} M_{2} \ar[r]^{p_{1}×p_{2}} \ar[d] & M_{1} × M_{2} \ar[d]^{ě_{3}^{1}×e_{1}^{2}} \\ X_{1} \ar[r]^{δ} & X_{1}^{2}}

we see that what we need to prove is that Q is the excess bundle for the intersection of M_{1} and M_{2} over X_{1}. The result would then follow immediately from the excess intersection formula. Using our identification of the relative tangent space to the space of stable maps as H^{0}(C, f^{∗}T X), and of the curve C′_{Y} as the union of C_{1} and C_{2}, we can rewrite sequence (5) as

0 → T Y → p_{1}^{∗}T M_{1} ⊕ p_{2}^{∗}T M_{2} → π_{∗}(f^{∗}T X)|_{Σ} → Q → 0.

Now it is evident that the result will follow from the following lemma.

Lemma 5.3.2. π_{∗} ((f^{∗}T X)|_{Σ}) ≃ p_{1}^{∗} (ě_{3}^{1})^{∗} T X_{1}

Proof of the Lemma. Over a geometric point y of Y, we can identify the fiber of Σ with Bμ_{r}. The point y maps to a geometric point x of X, with stabilizer G, and we can locally describe X around x as [U/G]. The pulback T of the tangent space of X to Y has a natural action of μ_{r}, and the fiber of π_{∗}(f^{∗}T X)|_{Σ} is naturally the space of invariants T^{μ^{r}}. This is naturally isomorphic to the tangent space of X_{1} at the point ě_{3}^{1}(p_{1}(y)), which is what we needed. ♣

5.4. Associativity: end of proof. The associativity now follows formally from our calculations. Consider the following commutative diagram.

\xymatrix{⋃ M_{1} ×_{X_{1}} M_{2} \ar[d]^{g} \ar[dr]^{gl} \\ D(X) \ar[r]^{ι} \ar[d] & \overline{M}_{0,4}(X , β) \ar[d]^{s} \ar[dd]^{st} \\ D^{orb} \ar[d] \ar[r]^{j} & _{\overline{M}{orb \atop 0,4}} \ar[d] \\ D \ar@{^{(}->}[r] & _{\overline{M}_{0,4}}}

Note that this is a simplified version of diagram (2). In particular, the bottom square of this diagram is not cartesian, although the middle square is. Denote

c = (e_{1}^{×})^{∗}γ_{1} · (e_{2}^{×})^{∗}γ_{2} · (e_{3}^{×})^{∗}γ_{3}

and

[\overline{M}]^{w} = [\overline{M}_{0,4}(X , β)]^{w}.

Combining Equation (1) with Proposition 5.3.1 we obtain

S := ∑ ⟨⟨γ_{1}, γ_{2}, ∗⟩_{0,β_{1}}, γ_{3}, ∗⟩_{0,β_{2}} = ě_{4∗}^{×}(gl^{∗}(c) ∩ (r_{×}^{2} · g^{∗}j^{!}[\overline{M}]^{w})).

Now we use the fact that ě_{4}^{×} = ě_{4} ∘ gl to simplify to

S = ě_{4∗} gl_{∗} (gl^{∗}c ∩ (r_{×}^{2} · g^{∗}j^{!}[\overline{M}]^{w})).

The projection formula gives

S = ě_{4∗} (c ∩ gl_{∗}(r_{×}^{2} · g^{∗}j^{!}[\overline{M}]^{w})).

Factoring gl = ι ∘ g gives

S = ě_{4∗} (c ∩ ι_{∗}g_{∗}(g^{∗}r_{×}^{2} · j^{!}[\overline{M}]^{w}))

and since g has degree 1/r^{×} we get

S = ě_{4∗} (c ∩ ι_{∗}(r_{×} · j^{!}[\overline{M}]^{w})).

Now, standard intersection theory gives us

S = ě_{4∗} (c · r_{×} · s^{∗}[D^{orb}] ∩ [\overline{M}]^{w}).

Finally, by the second part of Proposition 5.2.1, we can conclude

∑ ⟨⟨γ_{1}, γ_{2}, ∗⟩_{0,β_{1}}, γ_{3}, ∗⟩_{0,β_{2}} = ě_{4∗} (c · st^{∗}[D] ∩ [\overline{M}]^{w}).

The final expression is clearly independent of the way the markings are grouped (since ě_{4} is invariant under the S_{3} action permuting the first three points), which is what we need for the associativity theorem:

∑ ⟨⟨γ_{1}, γ_{2}, ∗⟩_{0,β_{1}}, γ_{3}, ∗⟩_{0,β_{2}} = ∑ ⟨⟨γ_{1}, γ_{3}, ∗⟩_{0,β_{1}}, γ_{2}, ∗⟩_{0,β_{2}} .

♣

6. Stringy Chow rings with integral coefficients

We now concentrate on the part of quantum Chow ring of X involving only g = 0, n = 3 and β = 0. Following a suggestion of Y. Ruan during a discussion at the conference, we call it the stringy Chow ring of X.

6.1. The main result. A-priori, one gets a new ring structure on the group A_{st}^{∗}(X)_{ℚ} := A^{∗}(X_{1})_{ℚ}. It is easy to see that this is well defined whenever X is smooth and separated, without any properness assumptions. Our main claim is the following:

Theorem 6.1.1. Let X be a smooth and separated Deligne–Mumford stack, not necessarily proper. There is a commutative, associative ring structure on Kresch’s Chow group A_{st}^{∗}(X) := A^{∗}(X_{1}) lifting the stringy chow ring structure (with rational coefficients) on A^{∗}(X_{1})_{ℚ}.

Moreover, let j : X ↪ X_{1} be the embedding. Then the group homomorphism j_{∗} : A^{∗}(X) → A^{∗}(X_{1}) = A_{st}^{∗}(X) is a ring homomorphism, in particular j_{∗}1 is the identity in A_{st}^{∗}(X).

The proof of this theorem is more subtle than the rational case.

6.2. Refined evaluation and the product law. First, we need to define the product. The crucial point is that in the case g = 0, n = 3 and β = 0 we have refined evaluation maps:

Lemma 6.2.1. Each evaluation map ē_{i} : K_{0,3}(X , 0) → X̄_{1} admits a lifting e_{i} : K_{0,3}(X , 0) → X_{1}. The morphism e_{i} is representable and finite.

Proof. The existence of e_{i} stems from the following: given a family of twisted stable maps (C → S, C → X) in K_{0,3}(X , 0) (S), the coarse curve is just ℙ^{1} × S, and the markings can be identified as {0} × S, {1} × S and {∞} × S. Constructing a lifting e_{i} is equivalent to constructing an r_{i}-th root of the normal bundle of the i-th marking, functorially in S, which is the same as constructing an r_{i}-th root of the tangent space of ℙ^{1} restricted to the i-th marking. But a line bundle over a point obviously has an r_{i}-th root.

The morphism e_{i} is easily seen to be proper and with finite fibers. Representability is a straightforward calculation of stabilizers. ♣

We note that K_{0,3}(X , 0) is always smooth. The obstruction class O_{0,3} defining [K_{0,3}(X , 0)]^{v} is given by the top Chern class of the bundle

E_{0,3} = R^{1}(π_{0,3})_{∗}f_{0,3}^{∗}T X .

We can now define the product as follows: Definition 6.2.2. Let γ_{1}, γ_{2} ∈ A^{∗}(X_{1}). Define

γ_{1} ⌣ γ_{2} = ě_{3∗} ( e_{1}^{∗}(γ_{1}) · e_{2}^{∗}(γ_{2}) · O_{0,3} )

As things stand, this definition depends on the choice of the liftings e_{i}. However it is not difficult to show that the group homomorphisms e_{i∗} and e_{i}^{∗} are independent of the choices, and therefore the product γ_{1} ⌣ γ_{2} is independent of the choices. Note that after tensoring with ℚ this definition agrees with the definition of the quantum product in degree 0.

6.3. Associativity. Next we need to prove associativity. The discussion in section 5.1 goes through word for word and one obtains (with notation analogous to that in section 5.1)

(γ_{1} ⌣ γ_{2}) ⌣ γ_{3} = (ě_{4}^{×})_{∗} ( ( (e_{1}^{×})^{∗}γ_{1} (e_{2}^{×})^{∗}γ_{2} (e_{3}^{×})^{∗}γ_{3} ) δ^{!}( [K_{1}]^{v} × [K_{2}]^{v} ) ) .

We can now follow the arguments of section 5.2. As before, we have a gluing morphism

gl : K_{1} ×_{X_{1}} K_{2} →^{g} D(X) ↪^{ι} K_{0,4}(X , 0).

This time, the morphism g is a μ_{r_{×}}-bundle.

There is a little problem with Equation (3) in section 5.2: the morphism ē_{i} : K_{0,4}(X , 0) → X̄_{1} does not lift to X_{1}, since the normal bundle to the i-th section of the universal curve over \overline{M}_{0,4} has degree 1, and thus has no r_{i}-th root unless r_{i} = 1. However, if we restrict to the inverse image K̃_{0,4}(X , 0) of the open set 𝔸^{1} = \overline{M}_{0,4} (14|23) ⊂ \overline{M}_{0,4}, then a section does exist. We denote by gl : K_{1} ×_{X_{1}} K_{2} → K̃_{0,4}(X , 0) the gluing map. So we can define

e_{i} : K̃_{0,4}(X , 0) → X_{1} , i = 1, …, 4

and if we do so carefully then

e_{i} ∘ gl = e_{i}^{×}.

Continuing with the arguments, the proof of Proposition 5.3.1 goes through and gives

δ^{!}( [K_{1}]^{v} × [K_{2}]^{v} ) = g^{∗}(j^{!}[K_{0,4}(X , 0)]^{v}).

We also have that K_{0,4}(X , 0) → K_{0,4}^{orb} is flat, therefore

g^{∗}(j^{!}[K_{0,4}(X , 0)]^{v}) = gl^{∗}[K_{0,4}(X , 0)]^{v}.

This implies that

(γ_{1} ⌣ γ_{2}) ⌣ γ_{3} = ě_{4∗}^{×} gl^{∗} ( e_{1}^{∗}γ_{1} e_{2}^{∗}γ_{2} e_{3}^{∗}γ_{3} [K̃_{0,4}(X , 0)]^{v} ) .

Unfortunately, e_{4} is neither proper nor representable, so we cannot replace ě_{4∗}^{×} by ě_{4∗}gl_{∗}. We remark that Chen and Ruan utilize a certain identification of the coarse moduli space K_{0,4}(X , 0) = \overline{M}_{0,4} × X_{3}, which does not lift to the level of stacks.

Let τ : K̃_{0,4}(X , 0) → K̃_{0,4}(X , 0) be the automorphism induced by the involution (2, 3) on the markings, and denote gl_{τ} = τ ∘ gl. One immediately sees that associativity follows from the following:

Claim.

ě_{4∗}^{×} gl^{∗} = ě_{4∗}^{×} (τ ∘ gl)^{∗} : A^{∗}(K̃_{0,4}(X , 0)) → A^{∗}X_{1}.

To show this equality it is useful to refine the divisor diagram. We deal with one connected component of K̃_{0,4}(X , 0) at a time, so let K̃ be such a component. We denote Z := gl^{−1}K̃ and similarly Z_{τ} := gl_{τ}^{−1}K̃. One can show that the locally constant function r_{×} is constant on Z and Z_{τ}, and we denote these constants by r_{Z} and r_{Z_{τ}} respectively.

We have a fiber diagram

\xymatrix{Z \ar[r] \ar[d] & D̃(X) \ar@{^{(}->}[r] \ar[d] & K̃ \ar[d] \\ Spec ℂ \ar[r] & Bμ_{r_{Z}} \ar@{^{(}->}[r] & M̃_{0,4}}

and a similar diagram for the τ version.

Here M̃_{0,4} denotes the open stack of twisted 4-pointed curves over 𝔸^{1} with stable 4-pointed coarse curve, with markings of indices r_{1}, …, r_{4}, and nodes of index r_{Z} over {0} ⊂ 𝔸^{1} and index r_{Z_{τ}} over {1} ⊂ 𝔸^{1}. It is simply obtained by endowing 𝔸^{1} with orbifold structure of index r_{Z} at {0} ⊂ 𝔸^{1} and index r_{Z_{τ}} at {1} ⊂ 𝔸^{1}. It thus contains two divisors G_{0} and G_{1} over {0} and {1} in 𝔸^{1}, isomorphic to Bμ_{r_{Z}} and Bμ_{r_{Zτ}}, respectively.

The square on the left is cartesian since Spec ℂ → Bμ_{r_{Z}} is the universal principal bundle. Ignoring the middle column, we can extend the fiber diagram as follows:

\xymatrix{Z \ar[r]^{gl} \ar[d]_{ě_{4}^{×}} & K̃ \ar[d]^{s×ě_{4}} \\ X_{1} \ar[r]^{p×id} \ar[d] & M̃_{0,4} × X_{1} \ar[d] \\ Spec ℂ \ar[r]^{p} & M̃_{0,4}}

(and a similar diagram with τ).

Considering the projection formula for the upper square, associativity follows if we prove that

(p × id)^{∗} = (p_{τ} × id)^{∗} : A^{∗}(M̃_{0,4} × X_{1}) → A^{∗}(X_{1}).

We can apply the projection formula here, because s×ě_{4} is proper and representable. Given ξ ∈ A^{∗}(M̃_{0,4} ×X_{1}), we can restrict it to the open set over M̃_{0,4} (G_{0} ∪ G_{1}) ≃ 𝔸^{1} {0, 1}.

\xymatrix{ & M̃_{0,4} \ar[d] \\ 𝔸^{1} \{0, 1\} \ar@{-->}[ur] \ar@{^{(}->}[r] & 𝔸^{1}}

The usual exact sequence says that this restriction extends to a class on 𝔸^{1} × X_{1}, which can be pulled back to a class ξ′ on M̃_{0,4} × X_{1}. On the one hand, A^{∗}(𝔸^{1} × X_{1}) = A^{∗}(X_{1}), and therefore (p × id)^{∗}ξ′ is independent of the choice of p. On the other hand, ξ and ξ′ differ by a class supported on G_{0} ∪ G_{1}. It thus suffices to show that

(p × id)^{∗}ξ = 0

whenever ξ is supported on (G_{0} ∪ G_{1}) × X_{1}. It suffices to treat one of the two.

Say i : G_{0} × X_{1} ↪ M̃_{0,4} × X_{1} is the embedding of the gerbe over 0, with normal sheaf N_{0}. The point p : Spec ℂ → M̃_{0,4} lifts to p̃: Spec ℂ → G_{0}. We may assume the class ξ = i_{∗}ξ̃. Then

i^{∗}i_{∗}ξ̃ = c_{1}(N_{0})ξ̃.

Pulling back to X_{1} = Spec ℂ × X_{1}, we have

(p × id)^{∗}i_{∗}ξ̃ = (p̃× id)^{∗}c_{1}(N_{0})ξ̃.

The fact that p^{∗}N( G ⊂ M̃_{0,4} ) is trivial implies that

(p̃× id)^{∗}c_{1}(N_{0}) = 0,

therefore (p × id)^{∗}ξ = (p × id)^{∗}i_{∗}ξ̃ = 0. ♣

7. Remarks on grading, expected dimension, divisors

7.1. Degree shifting. When we read about virtual fundamental classes on moduli spaces of “usual” stable maps, we often find it stressed that the virtual fundamental class has the “expected dimension”. In our discussion here we (somewhat perversely) avoided any discussion of the expected dimension, and evidently it is not at all necessary for proving the associativity of the quantum product. On the other hand, the existence of such a dimension and its properties are of interest.

An interesting feature of the expected dimension of twisted stable maps is that it depends on the component of the moduli space in a subtle way, determined by the type of the evaluation maps, using the so called “degree shifting number” or “age” of the image component in X_{1}.

The complex Rπ_{g,n,β∗}f^{∗}T X was used for defining the relative obstruction theory, and thus the class [K_{g,n}(X , β)]^{v}. There is a naïve approximation for the degree of the top Chern class, obtained by using the usual Grothendieck–Riemann–Roch formula. However, it is well known that the usual Grothendieck–Riemann– Roch formula does not hold for Deligne–Mumford stacks; it is explained in Toen’s work [T] that one needs to introduce terms corresponding to pullback of the sheaf to the twisted sectors to correct the formula.

The issue can be understood if one considers a smooth marked twisted curve C with a representable map f : C → X. Let V = f^{∗}T X. If π : C → C is the map to the coarse curve, then χ(C, V) = χ(C, π_{∗}V), since π_{∗} is an exact functor. Also, deg_{C} π_{∗}V = deg_{C} π^{∗}π_{∗}V. Thus the failure of Riemann–Roch is exactly the degree of the torsion sheaf V/π^{∗}π_{∗}V supported at the twisted markings.

A neighborhood of a twisted marking locally looks like [Spec ℂ[[t]]/μ_{r}]. We describe the action as follows: we let μ_{r} act on the tangent space of C via the fundamental character. We can choose the parameter t to be an eigenvector, and since it generates the cotangent space at x, the group μ_{r} acts on t via the inverse of the fundamental character:

t ↦ ζ_{r}^{−1} t.

The group μ_{r} is identified via f with a subgroup H of the stabilizer G_{x} of a point of X. Let the tangent space V_{x} of X at x have basis (v_{1}, …, v_{n}) consisting of μ_{r}-eigenvectors. The degree of V/π^{∗}π_{∗}V is then exactly \frac{1}{r} ∑_{i}^{n} k_{i} where

k_{i} = min{l|v_{i} · t^{l} is invariant }.

In other words, μ_{r} acts on v_{i} via

v_{i} ↦ ζ_{r}^{k_{i}}v_{i},

with 0 ≤ k_{i} < r. Thus the contribution of a marking to the degree depends only on the component of X_{1} where it evaluates. This contribution \frac{1}{r} ∑_{i}^{n} k_{i} is known as the degree shifting number ([C-R2]) or the age (Miles Reid’s terminology). If the connected components of X_{1} are X_{1}^{i}, we denote this value by a(X_{1}^{i}).

As in [C-R1], one defines a grading on the group A_{st}^{∗}(X) as follows:

A_{st}^{m}(X) = ⊕_{i}A^{m−a(X_{1}^{i})}(X_{1}^{i}).

An argument similar to that of Chen and Ruan shows that this makes the stringy Chow ring with integer coefficients A_{st}^{∗}(X) into a graded ring.

7.2. Forgetting a marking. The Gromov–Witten invariants of projective varieties satisfy the so-called “divisor axiom”, which comes from the fact that \overline{M}_{g,n+1}(X, β) is the universal curve over \overline{M}_{g,n}(X, β). This still works for stacks when we restrict to the case where the (n + 1)-st marking is untwisted. In our setup there is no “forgetful map” for a twisted marking. There is, however, a way to change the formalism which allows for such forgetful maps. One way to do this entails

(1) systematically breaking up \overline{M}_{g,n}(X , β) according to the types of the markings, with components \overline{M}_{g,n}(X , β, τ), where τ ∈ π_{0}(X_{1})^{n} is an n-tuple of connected components of X_{1} indicating where each evaluation map e_{i} lands;

(2) for each index set I ⊂ \{1, …, n\} of twisted markings in \overline{M}_{g,n}(X , β, τ) of size |I| = l introducing a new moduli stack

\overline{M}_{g,n−l}(X , β, τ′) = [\overline{M}_{g,n}(X , β, τ)/S_{l}].

The right hand side is the quotient of \overline{M}_{g,n}(X , β, τ) by the action of the symmetric group on l letters S_{l} acting on the indices of I. The point is that the only thing we can “forget” about a collection of twisted markings is their order. The notation τ′ on the left denotes the image of τ in π_{0}(X_{1})^{n}/S_{l} obtained by forgetting the order of the indices in I.

At this point we do not know which formalism carries more computational or theoretical advantages.

8. Examples

We include a few examples of integral stringy Chow rings, both to give the reader a sense of what these rings look like, and to illustrate the fact that stacks tend to have a lot of torsion in their Chow rings. The most basic example is Bμ_{r}. We recall that the non-stringy Chow ring, A^{∗}(Bμ_{r}) is ℤ[t]/⟨rt⟩, where t is the first Chern class of the standard line bundle over Bμ_{r}, see [To]. The inertia stack of Bμ_{r} is simply r disjoint copies of Bμ_{r}, so additively we know exactly what the stringy Chow ring is. It is easy to compute the product structure in this case. If we let s be the element of A_{st}^{∗}(Bμ_{r}) corresponding to the fundamental class of the component of the inertia stack given by the identity map from Bμ_{r} to itself, then its powers give the fundamental classes of the other components of the inertia stack. We conclude that

A_{st}^{∗}(Bμ_{r}) = ℤ[s, t]/⟨rt, s^{r} − 1⟩.

Another simple example is the stack quotient X = [𝔸^{1}/μ_{r}]. This stack is a vector bundle over Bμ_{r}, so its ordinary Chow ring is isomorphic to the Chow ring of Bμ_{r}. Under this isomorphism, the generator t corresponds to the substack of X isomorphic to Bμ_{r} supported at the origin. The inertia stack X_{1} is a copy of X together with r − 1 disjoint copies of Bμ_{r}. In this case, if we let s denote the fundamental class of the component of X_{1} corresponding to the natural inclusion of Bμ_{r} into X_{1}, we again find that powers of s correspond to the fundamental classes of the other components of X_{1}, except for the r^{th} power, which now gives the generator t of the Chow ring of the identity component of X_{1}. We conclude that

A_{st}^{∗}(X) = ℤ[s]/⟨rs^{r}⟩.

Note that even after tensoring with ℚ, this ring does not agree with A_{st}^{∗}(Bμ_{r}), showing that the stringy Chow ring does not enjoy the functorial properties of the standard Chow ring.

We conclude with a description of the stringy Chow ring of \overline{M}_{1,1}. The ordinary Chow ring of \overline{M}_{1,1} was computed in [E-G] to be ℤ[λ]/⟨24λ^{2}⟩, where λ is the first Chern class of the Hodge bundle. The inertia stack consists of two copies of \overline{M}_{1,1}, two copies of Bμ_{4}, and four copies of Bμ_{6}. We let u denote the fundamental class of the second copy of \overline{M}_{1,1}, a denote the fundamental class of the copy of Bμ_{4} corresponding to an elliptic curve with four automorphisms and an isomorphism of Aut(E) with μ_{4} such that μ_{4} acts on H^{1}(O_{E}) via the standard representation, and let b denote the analogous class for the elliptic curve with six automorphisms. Then considerations similar to those above (although more involved) lead to

A_{st}^{∗}(\overline{M}_{1,1}) = ℤ[u, a, b, λ]/⟨u^{2} − 1, 4aλ, 6bλ, a^{2} − 6λu, b^{3} − 4λu, ab⟩.

As in the nonstringy case, we get the stringy Chow ring for the open stack M_{1,1} by adding the additional relation 12λ = 0.

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Department of Mathematics, Boston University, 111 Cummington Street, Boston, MA 02215, U.S.A.

    E-mail address: abrmovic@math.bu.edu

Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge MA 02138, U.S.A.

    E-mail address: graber@math.harvard.edu

Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato 5, 40127 Bologna, Italy

    E-mail address: vistoli@dm.unibo.it