Topology of Moduli Spaces and Operads

Home , Dan Abramovich, Maxim Kontsevich, Mikio Sato, Ravi Vakil

Topology of moduli spaces and operads

DAN PETERSEN

Doctoral Thesis Stockholm, Sweden 2013 TRITA-MAT-13-MA-02 KTH ISSN 1401-2278 Institutionen för Matematik ISRN KTH/MAT/DA 13/03-SE 100 44 Stockholm ISBN 978-91-7501-759-4 SWEDEN

Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan fram- lägges till oﬀentlig granskning för avläggande av teknologie doktorsexamen i matematik fredagen den 31 maj 2013 kl 14.00 i sal E2, Kungl Tekniska högskolan, Lindstedtsvägen 3, Stockholm.

c Dan Petersen, 2013

Tryck: Universitetsservice US AB iii

Abstract

This thesis contains five articles related to the moduli space of curves and to the theory of operads. Paper A contains proofs of two results on the even rational cohomology groups of M1,n: that these groups are generated additively by cycle classes of strata, and that all relations between these generators follow from the WDVV relation on M0,4 and Getzler’s relation on M1,4. These results were announced by Ezra Getzler in the mid-90’s, but this is the first complete proof. Paper B proves the existence of an integer n such that the tautological ring of M2,n is not a Gorenstein algebra. Thus the Faber conjecture fails for this space. In fact n is the smallest integer for which there is a non- tautological cohomology class of even degree on M2,n; we prove that n ∈ {8, 12, 16, 20} and conjecture that n = 20. Paper C explains how to compute the cohomology groups of natural symplectic local systems on certain Humbert surfaces in A2 in the categories of mixed Hodge structures or `-adic Galois representations. Paper D considers moduli spaces of curves equipped with an admissible G-cover, where G is a fixed finite group, together with trivializations of the covering over each marked point. It is explained in what sense these spaces form an operad, and it is proven in detail that the Gromov–Witten theory of global quotient orbifolds provides examples of algebras over this operad. Paper E uses the action of the Grothendieck–Teichmüller group and a version of a formality criterion due to Sullivan to gives a new short proof of a result of Tamarkin, that the operad of little disks is formal. iv

Sammanfattning

Denna avhandling består av fem artiklar om modulirum för kurvor och operadteori. Artikel A ger bevis av två resultat om de jämna ra- tionella kohomologigrupperna till M1,n: att dessa grupper genereras additivt av cykelklasser av strata, och att alla relationer mellan dessa generatorer följer från WDVV-relationen på M0,4 och Getzlers relation på M1,4. Dessa resultat tillkännagavs av Ezra Getzler på mitten av 90-talet, men detta är det första fullständiga beviset. Artikel B bevisar existensen av ett heltal n sådant att den tautologiska ringen till M2,n in- te är en Gorensteinalgebra. Alltså ger detta ett motexempel mot Fabers förmodan för detta rum. Mer speciﬁkt är n det minsta heltalet för vilket det existerar en icke-tautologisk kohomologiklass av jämn grad på M2,n; vi visar att n ∈ {8, 12, 16, 20} och förmodar att n = 20. Artikel C förkla- rar hur kohomologigrupperna av naturliga symplektiska lokala system på vissa Humbertytor i A2 kan beräknas, antingen som `-adiska Galois- representationer eller som (orena) Hodgestrukturer. Artikel D behandlar modulirum för kurvor med en godtagbar G-övertäckning, där vi ﬁxerat en ändlig grupp G, tillsammans med en trivialisering av sagda övertäck- ning över varje markerad punkt. Artikeln förklarar i vilken mening dessa modulirum utgör en operad, och bevisar i detalj att Gromov–Witten- teori för globala kvot-orbifolder ger exempel på algebror över denna operad. Artikel E använder verkan av Grothendieck–Teichmüller-gruppen och en version av ett formalitetskriterium som formulerades av Sullivan för att ge ett nytt kort bevis av en sats av Tamarkin, att operaden av små diskar är formell. Contents

Contents v

Acknowledgements vii

Part I: Introduction and summary

1 Introduction 3

2 Summary of results 29

References 35

Part II: Scientiﬁc papers

Paper A The structure of the tautological ring in genus one Submitted. 16 pages.

Paper B The Gorenstein conjecture fails for the tautological ring of M2,n (joint with Orsola Tommasi) Inventiones Mathematicae, to appear. 21 pages.

Paper C Cohomology of local systems on loci of d-elliptic abelian surfaces Michigan Mathematical Journal, to appear. 14 pages.

Paper D On the operad structure of admissible G-covers Algebra & Number Theory, to appear. 26 pages.

v vi CONTENTS

Paper E Minimal models, GT-action and formality of the little disk operad Submitted. 6 pages. Acknowledgements

I wish to express my gratitude to many different people. My scientific advisor Carel Faber made this thesis possible in the first place and has given me plenty of guidance and inspiration. I have enjoyed a nice collaboration with Orsola Tommasi, which became a significant part of this thesis. My colleagues and fellow graduate students at the mathematics depart- ments of KTH and SU have provided a stimulating environment, where it’s fun to go to work in the morning and you can knock on anyone’s door with a question. My friends inside and outside of the mathematics department have given me valuable distractions these last few years and kept me grounded in reality. Finally I am deeply grateful to my wonderful family: Erik, Lisa, Amanda and Oskar; Mariette; and Lukas, my son.

vii

Part I Introduction and summary

1 Introduction

In this introduction I will begin by trying to explain some of the words in the title of this thesis: the first parts of the introduction are introductions to the moduli space of curves and to the theory of operads, and I will try to explain in particular why operads can shed light on the combinatorial structure of the moduli spaces. This part is somewhat informal, and for more authoritative treatments we refer the reader to [ACG11, HM98] and [MSS02, LV12], respectively. After this I give an introduction to the tautological ring of the moduli space of curves and to the Faber conjectures, which play a large role in the first two papers of this thesis. This is a long story involving the work of a large amount of different people, and this section is more technical. Other references for the tautological rings and the Faber conjectures are [Fab13, Vak08].

Moduli of curves The starting point for the moduli theory of curves is thinking about n-pointed algebraic curves and how they vary in families. By an n-pointed algebraic curve, we mean a projective nodal geometrically connected curve X → S equipped with n ordered disjoint sections σi : S → X contained in the smooth locus of X. Over the complex numbers, an equivalent deﬁnition is that we are considering compact connected Riemann surfaces X which are allowed to have simple nodes, and which are equipped with an embedding [n] = {1, 2, . . . , n} ,→ X \ Xsing. We say that an n-pointed algebraic curve X is stable if its automorphism group is ﬁnite. If X is smooth, i.e. Xsing = ∅, then X is stable if and only if 2g − 2 + n > 0, or equivalently, the Euler characteristic of the complement of the markings is negative.

The spaces Mg,n.

Let 2g − 2 + n > 0. We denote by Mg,n the moduli space which parametrizes smooth n-pointed algebraic curves of genus g. Over C, Mg,n can be thought of as the space parametrizing isomorphism classes of complex structures on a compact oriented surface S with an embedding [n] ,→ S. One can also think of Mg,n in terms of hyperbolic geometry: since the Euler characteristic of

3 4 CHAPTER 1. INTRODUCTION the surface minus the punctures is negative, by uniformization it is a quotient Γ \ H of the upper half plane by a discrete group of orientation-preserving isometries. This gives a hyperbolic metric on S \ [n], and one can thus think of Mg,n as the space parametrizing ﬁnite area hyperbolic metrics on S\[n]. In this picture, the n marked points are actually cusps of the hyperbolic surface and more naturally thought of as punctures. A priori it is not even clear that Mg,n should be a ‘space’, and not just a set (or groupoid). Indeed, why should there even be a topology on the set Mg,n(C), i.e. the set

{compact Riemann surfaces of genus g with n markings}/isomorphisms, in a natural way? One needs to start thinking of curves varying in families. One can for instance consider the family of curves

y2 = x(x − 1)(x − t) as t varies in C; when t 6∈ {0, 1} we get an elliptic curve, i.e. a smooth curve of genus 1 with a marked point. We would like to say that the topology on M1,1(C) should be such that elliptic curves deﬁned by nearby values of t should be close to each other in moduli space. If one contemplates how to formalize this vague idea in terms of algebraic geometry, one thinks perhaps after a while of the condition that the map

C \{0, 1} → M1,1 that assigns to a number t the isomorphism class of the curve y2 = x(x − 1)(x−t) is a morphism of algebraic varieties. In fact, if we impose the stronger requirement that morphisms S → M1,1 should be in bijective correspondence with isomorphism classes of families of elliptic curves over S, for any base space S, then this determines the complex structure on M1,1 completely (via Yoneda). This is essentially the definition of the spaces Mg,n. It is probably worth pointing out that the above discussion sweeps lots of stacky issues under the rug. In particular, according to the definition above, spaces Mg,n do not even exist. To get a correct definition, one needs to replace isomorphism classes with groupoids everywhere, in which case the moduli spaces are not varieties or schemes but rather stacks [DM69, Vis05]. In fact, the stability condition implies that these stacks are even Deligne– Mumford, which is the kind of stack closest to an ordinary algebraic variety. Over the complex numbers, a Deligne–Mumford stack corresponds to a not necessarily effective orbifold.

Mapping class groups of surfaces Yet another way to think about the moduli space of curves is via its fundamental group. Let us consider a loop γ in Mg,n(C), and let us for simplicity assume that it does not meet the locus of curves with nontrivial automorphism group. Fix an n-pointed smooth surface S of genus g. We identify the starting point of γ with a choice of a complex structure on S. We can represent 5 moving along γ by a continuous deformation of the chosen complex structure. As we have reached the end of the loop, we have a new complex structure on the same underlying surface S. But we know moreover that the resulting two Riemann surfaces are uniquely isomorphic, and this isomorphism is realized by a diffeomorphism φγ : S → S (which fixes the marked points). Now (con- trary to what the notation suggests) φγ does not only depend on the loop γ, since we needed a choice of a lift of γ to the space of all complex structures on S. But it is true that φγ is uniquely determined up to isotopy, and if we modify γ by a homotopy, then the isotopy class remains invariant. Hence we obtain a map from π1(Mg,n(C)) to the mapping class group Γg,n of S, i.e. the group of all diffeomorphisms of S moduli isotopy. This group is of central importance in low-dimensional topology and geometric group theory. In the preceding paragraph we assumed for simplicity that γ does not meet the locus of curves with automorphisms, so that there was a single well defined isomorphism between the two complex structures on S. All we said remains true without this assumption if we take π1(Mg,n(C)) to be the orbifold fundamental group, i.e. when we think of Mg,n(C) as a topological stack rather than a topological space; for this one needs to be a bit more careful in defining precisely what it means to have a loop in an orbifold. From now on we always mean the orbifold fundamental group when we write π1. In fact, the map π1(Mg,n(C)) → Γg,n is an isomorphism. This amounts to saying that the space of complex structures on S, considered up to diffeomorphisms isotopic to the identity, is simply connected. The latter space is called the Teichmüller space Tg,n of the pointed surface S, and in fact more is true: Tg,n is a complex manifold which is homeomorphic to a ball in R6g−6+2n [Ahl54, Ahl60], so it is in fact contractible. This shows that 1 Mg,n(C) is homotopy equivalent to the classifying space BΓg,n. Thus all homotopic information of Mg,n is contained in the mapping class group. See e.g. [FM12] as well as [Pap07, Pap09] for this viewpoint on moduli spaces.

Compactiﬁcation

A basic fact of life is that the space Mg,n is not compact. What this means concretely is that one can ﬁnd 1-parameter families of Riemann surfaces which ‘diverge’; one cannot assign to them a limiting complex structure. In the hyperbolic picture, one can visualize this phenomenon e.g. by considering the family obtained by letting the length of a bounding curve shrink to zero. In the limit, the surface acquires a cusp, as in Figure 1.1. In the algebraic picture, this means that one needs to allow singular algebraic curves in order to assign a limit to this family. By now, there is a well established practice that whenever a moduli space is not compact, one should

1One must be slightly careful here. Either one should work with coarse moduli spaces, in which case the correct statement is that Mg,n(C) is only rationally homotopic to BΓg,n, as Γg,n acts on Teichmüller space with ﬁnite stabilizers. Alternatively, one may work with ﬁne moduli spaces and say that Mg,n(C) and BΓg,n are homotopy equivalent as topological stacks. 6 CHAPTER 1. INTRODUCTION

Figure 1.1: The length of a curve shrinks to zero and a cusp appears in the limit.

try to construct a compactification by enlarging the scope of the moduli problem to encompass also some appropriately ‘mild’ degenerations. In general, allowing ‘too few’ kinds of degenerations will result in a moduli space which is not compact, whereas allowing ‘too many’ results in a space which is not Hausdorff. A miracle is that the stability condition introduced earlier gives the nicest possible compactification of Mg,n. 1 If X is a stable n-pointed curve, we define its genus as dim H (X, OX ); when X is smooth, this coincides with the usual definition of the genus of a surface. Over C, there is a topological definition of the genus of a stable curve: if one removes a neighborhood of each node and instead glues in a cylinder, then the genus of the nodal curve can be defined as the genus of the resulting smooth surface. Pictorially, this definition is what you get when you run the process of Figure 1.1 backwards. Cf. also Figure 1.2, where each stable curve in the picture has genus two.

Let Mg,n be deﬁned as the moduli space parametrizing stable n-pointed curves of genus g. Then clearly Mg,n ⊂ Mg,n, and the fact is that Mg,n is smooth, projective, irreducible, and the complement Mg,n \Mg,n is a normal crossing divisor. This is part of what was alluded to earlier — that the compactiﬁcation Mg,n is the nicest possible — but there is also a rich combinatorial structure in the structure of the boundary, as we shall see now.

The combinatorics of the boundary

The spaces Mg,n admit a stratiﬁcation, the so-called stratiﬁcation by topological type. The name comes from the picture over the complex numbers: two stable curves in Mg,n(C) lie in the same stratum if and only if their underlying topological spaces are homeomorphic. The nice feature is that each stratum looks roughly like a product of lower-dimensional moduli spaces of smooth pointed curves.

Figure 1.2: A marked Riemann surface degenerates, acquiring more and more nodes. 7

As an example, consider the sequence of degenerations illustrated in Figure 1.2. All four pictures define strata in the moduli space M2,3 of 3-pointed curves of genus 2. In the picture, each stratum lies in the closure of the strata to its left. This expresses that any curve of a certain topological type can be written as a limit of curves whose topological type is defined by smoothing a subset of its nodes. For instance, the leftmost picture corresponds to the stratum M2,3 of smooth curves, whose closure is the entire moduli space. Now consider the second of the four pictures. Putting a complex structure on this surface is very nearly the same thing as putting a complex structure on a genus one surface with five marked points, as illustrated in Figure 1.3.

Figure 1.3: ‘Pulling apart’ a singularity.

Algebraically, the process illustrated in Figure 1.3 is that the smooth curve to the right is the normalization of the singular curve to the left. And it is easy to see that any complex structure on the surface to the right defines a complex structure on the nodal curve to the left by identifying two points; conversely, a complex structure on the surface on the left almost defines a 5-pointed genus 1 curve via normalization, except for the fact that the two markings obtained by resolving the node are not ordered. Indeed, the two extra markings correspond to the two branches at the node, and there is no natural ordering on these branches. What the discussion in the above paragraph shows is that the stratum defined by this topological type is isomorphic to the moduli space

M1,5/S2, where the symmetric group S2 acts by permuting two of the marked points. Similarly the remaining two strata illustrated in Figure 1.2 are isomorphic to

(M0,3 × M1,4)/S2 and (M0,3 × M0,6)/(S2 × S2).

Dual graphs These strata are often encoded combinatorially in terms of so-called dual graphs. These are graphs whose vertices are labeled by nonnegative integers, and which are allowed to have loops, multiple edges, and half-edges which are only connected to a vertex on one end. To each topological type one assigns a dual graph as follows: for each irreducible component of the normalization, one draws a vertex labeled by 8 CHAPTER 1. INTRODUCTION

2 1 0 1 0 0

Figure 1.4: The same sequence of degenerations, drawn in terms of stable graphs.

the genus of that component. For each node, one draws an edge connecting the two components connected by that node (if they are different) or a loop at the vertex corresponding to that component (if both branches of the node lie in the same component). Finally, one draws a half-edge for each marked point on the curve. These half-edges corresponding to markings are usually called legs. A dual graph is said to be stable if each vertex labeled 0 has at least three incoming edges,2 and each vertex labeled 1 has at least one incoming edge. Then a nodal curve is stable if and only if the corresponding dual graph is stable in this sense. Figure 1.4 illustrates the same sequence of degenerations as before in terms of dual graphs. When the notion of an automorphism of a dual graph has been defined appropriately, one can define the stratum corresponding to a stable graph Γ as   Y .  Mg(v),n(v) Aut(Γ) (1.1) v∈Vert(Γ) where g(v) is the genus of a vertex and n(v) is the number of incoming edges at a vertex. In fact, this defines for each stable graph Γ a morphism of moduli spaces   Y .  Mg(v),n(v) Aut(Γ) → Mg,n v∈Vert(Γ) where (g, n) is the genus and number of markings of the graph Γ. The image of the open subspace of products of smooth curves is exactly the stratum corresponding to Γ.

A∞-spaces. We now switch tracks and introduce the notion of an operad. The deﬁnition itself is not complicated but can be intimidating, and we begin by considering a motivating example going back to [Sta63]. A minor hiccup is that our example is most naturally thought of as a nonsymmetric operad, rather than as an operad proper; we will get back to this point later. Recall that a topological monoid is a space X (in some convenient category of topological spaces) with a multiplication map X × X → X which is

2which must be counted with the appropriate multiplicity. For instance, a loop at a vertex should count as two incoming edges. 9 associative and has a unit. We would like to consider a version of this notion where the associativity constraint has been suitably weakened. One could, for instance, remove the associativity condition completely: this give the notion of an H-space.3 An intermediate condition one could impose is that the diagram µ × id X × X × X - X × X

id × µ µ (1.2) ? ? X × X - X µ does not commute strictly but only up to homotopy; this is a homotopy associative H-space. But we would like to define the notion of a multiplication which is associative ‘up to coherent homotopy’, that is, we do not want only that all maps Xn → X obtained by compositions of µ are homotopic, but we want there to be a specific homotopy between any two of them in a coherent way. Our goal is to define the notion of an A∞-space, which accomplishes exactly this. Two motivating examples are the following. 1. Let Y be a pointed space, and X = ΩY its loop space. If loops are parametrized by [0, 1], then we can define the following multiplication on X: α (2t) t ∈ [0, 1/2) (α · β)(t) = β(2t − 1) t ∈ [1/2, 1]. Clearly this multiplication is not literally associative: the compositions α · (β · γ) and (α · β) · γ are not equal, but they only differ by a reparametrization. So it is homotopy associative, but there is in fact far more structure present: this structure is precisely that of an A∞-space. 2. Suppose X is a topological monoid, and X0 is homotopy equivalent to X. What is the induced structure on X0? As it turns out, X0 does not in general support a structure of topological monoid. But there is 0 always an A∞-structure on X .

The associahedra

As we said, an A∞-space is based on a strengthened notion of homotopy associativity, where associativity holds up to a coherent system of higher homotopies. In particular there should be a canonical homotopy between the two compositions in the diagram (1.2). We write this as a map

I → map(X3,X), (1.3) where I is the unit interval, and the images of 0 and 1 are µ ◦ (µ × id) and µ ◦ (id × µ), respectively. Pictorially, this homotopy is illustrated by Figure 1.5.

3But note that most authors only require H-spaces to have units up to homotopy. 10 CHAPTER 1. INTRODUCTION

Figure 1.5: The homotopy between µ ◦ (µ × id) and µ ◦ (id × µ).

Figure 1.6: Homotopies between maps with four inputs.

Now we consider maps X4 → X obtained by composing µ: there are exactly five distinct such maps (corresponding to the five distinct rooted planar trees with four leaves). Applying the homotopy (1.3) gives exactly five homotopies between these maps. The situation is illustrated in Figure 1.6. Thus we have a map → map(X4,X), (1.4) where is the hollow pentagonD in Figure 1.6. The ‘higher coherence’ condition weD impose now is that we are given an extension of this map from to the solid pentagon. D We consider maps with five inputs. The situation is now shown in Figure 1.7. In the figure, every vertex of the polyhedron corresponds to a map X5 → X obtained by composing µ. Each edge connects two such maps that are related by the homotopy (1.3). There are two kinds of 2-dimensional faces: firstly, we glue in solid pentagons according to (1.4) whenever we have five edges which fit together; secondly, we observe that some of the homotopies corresponding to edges clearly commute with each other, and in this case we may fill in a solid square I2 connecting said homotopies. In the end we obtain the boundary of a polyhedron, and the condition is that we should be given an extension of the map

(boundary of polyhedron) → map(X5,X) to a map (solid polyhedron) → map(X5,X).

Let us deﬁne K2 = (point), K3 = I, K4 to be the solid pentagon, and K5 the solid polyhedron considered above. Iterating the constructions illustrated 11

Figure 1.7: The polytope K5. We no longer draw the tree corresponding to each face of the polytope, but three examples are shown.

here indefinitely produces an infinite sequence Kn of cell complexes which are topologically solid balls (and which actually can be realized as convex polytopes [Lee89], as suggested by the pictures). We define an A∞-space to be a space X and a collection of maps

n Kn → map(X ,X) for all n, such that the gluing maps Kn × Km → Kn+m−1 corresponding to grafting of trees make the diagram n m Kn × Km - map(X ,X) × map(X ,X)

(1.5) ? ? n+m−1 Kn+m−1 - map(X ,X) commute, where the vertical arrow on the right hand side is the obvious one given by composition of maps.

The definition of an operad The correct formalism for keeping track of all the maps in the definition of an n A∞-space is the following. The collections of spaces {Kn} and {map(X ,X)} are both examples of nonsymmetric operads, and an A∞-structure on X is n exactly a morphism of nonsymmetric operads from {Kn} to {map(X ,X)}. Recall that if V is a vector space, then Hom(V,V ) is an associative algebra, and an action of an associative algebra A on V is an algebra homomorphism A → Hom(V,V ). In this way the notion of a (nonsymmetric) operad is analogous to that of an associative algebra, except we allow operations taking an arbitrarily large number of inputs. Let us now define our terms. We begin by defining ordinary operads. An operad P is a functor which assigns to a rooted tree Γ an object P(Γ) of a symmetric monoidal category E (in our previous example we had E = (Top, ×), making P a topological operad), such that: 12 CHAPTER 1. INTRODUCTION

1. P is functorial with respect to graph automorphisms. For instance, the group Sn acts by permuting the ‘input edges’ of the tree γn with one vertex and n inputs (cf. Figure 1.8), so P(γn) is a representation of Sn. 2. The value of P on any graph is the tensor product of the value of P on all vertices in the graph, as illustrated in Figure 1.9. Thus the values of P are determined by the values P(γn). 3. P is functorial with respect to edge contractions: if Γ0 is obtained from Γ by contracting some internal edges, then there is an induced map P(Γ) → P(Γ0). This must in particular be associative, in the sense that if several edges are contracted, then the order of these contractions does not matter.

γ5 = γ3 =

Figure 1.8: Two one-vertex trees.

  =     P   P ⊗P                           Figure 1.9: The value of P on a tree is the tensor product of the values at each vertex.

Thus an operad is deﬁned by a collection of Sn-representations P(n) = P(γn) and a collection of composition rules

◦i : P(n) ⊗ P(m) → P(n + m − 1)

(given by ‘gluing γm to the ith input of γn’ satisfying an associativity condition. One may or may not wish to include as an axiom the existence of a unit; we silently pass over this point. The deﬁnition of a nonsymmetric operad is identical, except the functor P is required to take values on planar rooted trees. This has the eﬀect of remov- ing the action of Sn from the picture: for instance, the two one-vertex trees in Figure 1.8 have no automorphisms compatible with their planar structure. A bewildering number of variations of the notion of operad can be obtained like this, i.e. by modifying what conditions are imposed on the kind of graphs are being glued together. See for instance [Mar08] for more on this perspective on

1 13 operads. We will return to this point shortly. In any case, the associahedra form a nonsymmetric operad K such that

K(γn) = Kn; if Γ is a rooted planar tree with N input legs, then the product of associahedra K(Γ) embeds into the boundary of KN , which deﬁnes the structure maps in the nonsymmetric operad K. This should be visible already in Figures 1.5–1.7. We call K the A∞-operad. If E has an internal Hom functor, then for any object X of E the collection Hom(Xn,X) is an operad, which we denote End(X). We say that a morphism P → End(X) gives X the structure of a P-algebra. The notion of a P-algebra makes sense also more generally (that is, without demanding that E has an internal Hom), by considering instead maps

P(n) ⊗ X⊗n → X satisfying a natural list of axioms. So an A∞-space is an algebra over a particular (nonsymmetric) operad K. We remark that strictly speaking there was no need to introduce nonsymmetric operads for this motivating example. We could instead have considered the collection of spaces a Kn

σ∈Sn

(with their natural Sn-action) — these form an operad whose algebras are exactly the same as algebras over the nonsymmetric operad K, i.e. A∞-spaces. The introduction of the notion of an operad doesn’t just allow a concise definition of an A∞-space, but it also explains in a sense why this is the correct notion to consider. The point is that a topological monoid is also an algebra over a certain nonsymmetric operad Ass, with Ass(n) = {point} for all n. Then the fact that ‘being a topological monoid’ is not a homotopy invariant notion amounts to saying that Ass is not cofibrant in the natural model structure on topological (nonsymmetric) operads. But the obvious map K → Ass is a cofibrant replacement [BV73, BM03], which is why A∞-spaces are homotopically better behaved — as mentioned earlier, A∞-structures can be transferred along homotopy equivalences. The structure of an A∞-space is the smallest one which contains topological monoids as a special case and which has this homotopy transfer property.

Little disk operad Let us give one more important example of an operad, which will appear in this thesis. Fix an integer k ≥ 1, and let Dk(n) be the space parametrizing collections of n disjoint disks inside of the k-dimensional unit disk. Then Dk is a topological operad, where Sn acts on Dk(n) by permuting the disks and the composition laws are obtained by placing a given conﬁguration of disks inside each other. The composition is most easily understood from a picture, 14 CHAPTER 1. INTRODUCTION

3 3 4 2 2 2 1 5 1 1 4

3 6

Figure 1.10: Left: two conﬁgurations of small disks, corresponding to points in D2(3) and D2(4), respectively. Right: The point of D2(6) obtained by composing the other two conﬁgurations. In this composition, the output leg of the point of D2(4) has been glued to the second input leg of the point of D2(3).

rather than writing down formulas: see Figure 1.10. We have already seen 4 one case of this operad, in a sense: the operad D1 is equivalent to the A∞- operad. The little disk operad was introduced by Boardman and Vogt, see [BV68, May72]. Examples of algebras over Dk are given by iterated loop spaces. Let Y be a pointed space, and set X = ΩkY . Points of X can be identiﬁed with maps from the k-dimensional unit disk D to Y which map the boundary of the disk to the basepoint. Then there are maps

n Dk(n) × X → X, defined as follows: given a configuration of n smaller disks D1,..., Dn in D, and n maps φ1, . . . , φn : D → Y as above, we define their composition by taking the map which is given by a rescaling of φi on the ith smaller disk Di, and which is constant at the basepoint outside of each of the smaller disks. Up to homotopy, these are in fact all of the Dk-operads: a space X is homotopy equivalent to a k-fold loop space if and only if it admits the structure of a Dk-algebra. This characterization of the homotopy types of iterated loop spaces was the original motivation for their introduction. If we are given an operad in a symmetric monoidal category E and a symmetric monoidal functor F : E → E 0, then applying F yields a new operad in E 0. This can be used to provide geometric models for algebraic structures. The little disk operad is a prime example of this: taking homology gives an operad H•(Dk) in dg vector spaces (with zero differential). Cohen [CLM76] proved that algebras over H•(D2) are exactly the Gerstenhaber algebras of deformation theory [Ger63], and algebras over H•(Dk) for k > 2 are Gersten- haber or Poisson algebras (depending on the parity of k) with a degree shift for the bracket. The formality of the little disk operad, a purely topological result, can then be applied to the algebraic study of Gerstenhaber algebras up to homotopy, see the discussion of Paper E in the summery of results in this thesis for references.

4Both operads are coﬁbrant replacements of the associative operad Ass. 15

Back to moduli spaces So far, our examples of operads have been classical ones from the original advent of operads, relevant to homotopy theory. In the 90’s the theory of operads enjoyed a renaissance, partially because of a realization that operads played a role in a web of string-theoretically inspired mathematical theories, e.g. that BRST cohomology and various notions of field theories admit operadic descriptions. We mention only as starting points [Sta97, Sta99]. This is also where moduli of curves enter the picture. The associahedra Kn have some properties in common with the spaces N Mg,n. Any polytope P ⊂ R may be thought of as a stratified space, where the strata are the interiors of the faces of the polytope. For the polytopes Pn this gives a stratification where the strata are indexed by graphs, as one sees in Figures 1.5–1.7. This stratification has the following properties: 1. The codimension of a stratum equals the number of the internal edges in the graph. If Γ0 is obtained by contracting internal edges in Γ, then the closed stratum corresponding to Γ0 contains the stratum corresponding to Γ — so the combinatorics of the graphs determines the inclusions between strata.

2. The strata of positive codimension are precisely the part of Kn which is the image of gluing maps of lower-dimensional associahedra. 3. The stratum of an arbitrary tree Γ is isomorphic to

Y ◦ Kn(v), (1.6) v∈Vert(Γ)

◦ where Kn is the interior of Kn, and n(v) denotes the number of inputs adjacent to the vertex v.

This is exactly analogous to the stratification of the spaces Mg,n by topological type. Compare Equations 1.6 and 1.1. (The automorphism group of Γ does not appear in the stratification of the associahedra, as a rooted tree with labeled inputs has no automorphisms.) The only difference is the type of graph involved: the associahedra are stratified by rooted planar trees, whereas the spaces Mg,n are stratified by dual graphs (which are undirected, the vertices are decorated by genus, and multiple edges and loops are allowed). But this difference is not essential: the definition of an operad, as explained earlier, makes sense in exactly the same manner for various other choices of categories of graphs with suitable notions of grafting of graphs and an associative contraction of edges. For an example, if we still only consider trees, but they are not rooted (so the edges do not carry a direction), then one obtains the definition of a cyclic operad [GK95]. In this case P(γn) (where γn still denotes a one-vertex graph as in Figure 1.8, but its half-edges are now unoriented) is a representation of Sn+1. By forgetting the choice of root, we see that every cyclic operad has an underlying non-cyclic operad. The main example of a cyclic operad is for us the collection of spaces M0,n+1. Again, if we want to consider also curves 16 CHAPTER 1. INTRODUCTION of higher genus then we should consider operads modelled on dual graphs, i.e. which moreover have genus decorations at the vertices and where loops are allowed: this is precisely the notion of a modular operad, introduced in [GK98]. When we defined algebras over ordinary operads we considered the operad given by the collection of spaces Hom(X⊗n,X). This also made pictorial sense, since the graph γn looks like it has n inputs and one output. For operad-like structures modeled on undirected graphs the distinction between input and output seems to have disappeared, making this definition much less natural. In fact the most natural algebras over cyclic or modular operads are given by vector spaces with an inner product, i.e. an isomorphism V =∼ V ∨. ⊗n ∼ ⊗n+1 Then since Hom(V ,V ) = V , the Sn-action extends to an Sn+1-action as desired, giving the collection V ⊗n+1 the structure of a cyclic (or modular) operad in vector spaces, and one can directly define as above what it means for V to be an algebra over such an operad.

Cohomological ﬁeld theories

So far, it would seem that the assertion that the spaces Mg,n form a modular operad is little more than a highly abstract way of encoding the combinatorial structure of the moduli spaces. Moreover, our presentation above may seem slightly idiosyncratic to an ‘operadchik’: we have so far emphasized the combinatorial structure of the operad itself, whereas usually the primary object of interest is the collection of algebras over the operad. All this raises the question: what are the algebras over the operad Mg,n? The answer is that such algebras are obtained from Gromov–Witten theory [KM94, Man99]. For this one needs to consider the spaces of stable maps. Fix a smooth projective variety X and a class β ∈ H2(X, Z). Deﬁne

Mg,n(X, β) to be the moduli space parametrizing pairs consisting of an n-pointed nodal curve C of genus g and a map f : C → X for which f∗[C] = β. These pairs are moreover required to be stable, which means that the group of automorphisms of C commuting with f is ﬁnite. Concretely, this means that a component of C which is contracted to a point can not be a rational curve with less than three ‘special’ points (markings or nodes), or a curve of genus one without any special points at all. The space Mg,n(X, β) carries a forgetful map to Mg,n, given by forgetting the map f and possibly contracting any components which become unstable upon doing so. It also carries n evaluation maps to X, mapping (C, f) to the value of f at each of the marked points on C. We get in this way a correspondence

ev n Mg,n(X, β) - X

π ?

Mg,n 17

n which can be thought of as a morphism from Mg,n to X in a symmetric monoidal category whose objects are smooth projective varieties (or rather, DM-stacks) and whose morphisms are correspondences. More formally, a morphism X → Y in this category is an element in the Chow ring of X × Y , and in Gromov–Witten theory this morphism is given by the cycle class

(π × ev)∗)[Mg,n(X, β)], where [Mg,n(X, β)] is the fundamental class of the space of stable maps (or rather, the so called virtual fundamental class of [BF97], since the moduli space is in general singular and of the wrong dimension). Finally one should sum over all β, either formally or by proving convergence of this infinite series in an appropriate sense. This makes X an algebra over the operad Mg,n. In particular, since correspondences as above induce pullback-pushforward maps in cohomology, the cohomology of X becomes an algebra over the cohomology of Mg,n. But now we should more carefully speak of co-operads rather than operads; these are the gadgets obtained by reversing the direction of all arrows in the definition of an operad. This structure is called a Coho- mological Field Theory, or CohFT. Note that H•(X) carries an inner product (by Poincaré duality), as we wanted from an algebra over a modular operad. The simplest example is when g = 0, n = 3 and β = 0. Since M0,3 is just a point, the corresponding Gromov–Witten invariant is a class in H•(X)⊗3. With Poincaré duality this can be interpreted as an element of Hom(H•(X)⊗2,H•(X)); this homomorphism is simply the cup product. Keeping (g, n) = (0, 3) but allowing β to vary gives an infinite family of products on the cohomology of X, which may be assembled to a power series which can be thought of as a deformation of the usual cup product. This is called the small quantum product. By allowing also n ≥ 3 one can write down an even larger deformation, the big quantum product, from which all genus zero Gromov–Witten invariants may be recovered. This structure on H•(X) is called quantum cohomology. The notion of a CohFT is not only abstraction for abstraction’s sake, but it turns out that many features of Gromov–Witten theory are formal conse- quences of the CohFT structure present in the background. Teleman [Tel12] has proven a very general and powerful classification theorem for CohFT’s, and used this to confirm a prediction of Givental [Giv01b]: for any target X with semi-simple quantum cohomology, the higher genus Gromov–Witten invariants can be reconstructed from genus zero invariants. Other striking applications of Teleman’s classification result are [FSZ10, PPZ13]. Related to Teleman’s result is the large group of universal symmetries acting on any CohFT, the Givental group [Giv01a, Giv01b]. This symmetry group appears to be of operadic origin; it is “explained” by the homotopical algebra of the CohFT operad [DSV13b, DSV13a].

The definition of tautological ring The study of the tautological rings of the moduli spaces of curves originates in David Mumford’s seminal paper [Mum83]. His motivation was enumerative 18 CHAPTER 1. INTRODUCTION geometry: in the same way as enumerative problems regarding configurations of lines, planes etc. can be translated into intersection-theoretic computations in the Chow ring of a Grassmannian or flag variety, it seems plausible that enumerative problems regarding curves of given genus can be studied by means of the Chow ring of an appropriate moduli space of curves. After the ‘Gromov–Witten revolution’ in enumerative geometry, this is even more true. However, the Chow rings of Grassmannians are vastly simpler than those of moduli spaces of curves. Grassmannian varieties are rational and have an algebraic cell decomposition, and one can write down a basis for the whole Chow ring and a combinatorial description of the structure constants for the multiplication. This is ‘Schubert calculus’ and ‘Littlewood–Richardson coefficients’ [KL72]. But the moduli spaces of curves are of general type unless g and n are small [HM82, EH84], and their Chow rings are in general infinite dimensional. It is a hopeless task in general to describe the entire Chow ring. But if our intended application is enumerative geometry, then there is in any case no real need to understand the whole Chow ring — in a concrete enumerative problem, one should expect to meet only ‘natural’ classes, defined canonically in terms of the moduli problem. Mumford reasons by analogy to the case of a Grassmannian, where one obtains geometrically significant classes from the Chern classes of the universal vector bundle. On the moduli space of curves, there are similar classes that can be written down only in terms of the geometry of the universal curve; the only difference is that on the moduli space of curves, these do not exhaust the Chow/cohomology ring. It is these classes that Mumford refers to as tautological. Specifically, let π : C → Mg be the universal family over the space of smooth genus g curves, and ω its relative dualizing sheaf. One can consider def K = c1(ω), and for each i the natural class i+1 i κi = π∗(K ) ∈ A (M). These are called the kappa classes, or MMM-classes (after Mumford, Morita [Mor87] and Miller [Mil86]). Note that the definition would make sense verba- tim for any family of smooth curves whatsoever. One can also define natural classes via λi = ci(π∗ω), the Chern classes of the so called Hodge bundle. Mumford proves, however, that no new classes are obtained this way: the even λi are polynomials in the odd ones, and the odd λi are polynomials in the κ-classes. The tautolog- • ical ring R (Mg) is then defined to be the Q-subalgebra of the Chow ring generated by the κ-classes. Another motivation for introducing the tautological ring, and/or another argument for the naturality of the κ-classes, is Mumford’s conjecture (the Madsen–Weiss theorem [MW07]): without making this precise, we have • lim H (Mg) = Q[κ1, κ2, κ3,...], g→∞ where the limit should be interpreted as saying that for each fixed i, the i sequence {H (Mg)}g≥2 is eventually constant. This says that any characteristic class of surface bundles — i.e. a rule which assigns to any surface bundle 19

Σ → B a functorial class in Hi(B) — that one can write down without specifying the genus of the ﬁbers is a polynomial in the κ-classes. Later, the notion of tautological ring was extended also to moduli spaces of pointed curves and to (partial) compactiﬁcations. In this case, one should consider as tautological also the classes

ψi = c1(Li), where Li is the line bundle on the moduli space given by the cotangent line at the ith marking, and also the fundamental classes of all strata in the stratification by topological type. It turns out that in this more general setting one can give a less ad hoc definition of the tautological rings, by defining the • entire collection of all tautological rings R (Mg,n) simultaneously. Namely, the system of tautological rings is the smallest collection of Q-subalgebras • of A (Mg,n) which is closed under pushforward along all gluing and point- forgetting morphisms. This definition of tautological ring was formulated in [FP05]. The naturality of this definition reflects that the ‘tower’ of all spaces Mg,n and their natural morphisms may be a more fundamental object of study than the spaces Mg themselves, cf. [Gro97]. The tautological ring of Mg,n is also natural in Gromov–Witten theory. Every relation between tautological classes gives rise to a universal partial differential equation for generating series of Gromov–Witten invariants for an arbitrary target space X. For instance, the fact that all three boundary points of M0,4 are linearly equivalent gives rise to the Witten–Dijkgraaf–Verlinde– Verlinde equation, which expresses associativity of the quantum product. If U ⊂ Mg,n is any Zariski open subset, one can then define its tautological • • • ring R (U) as the image of the restriction morphism R (Mg,n) → A (U). Two cases in particular deserve mention. We define a stable curve C of genus g to be of compact type if its dual graph is a tree, or equivalently, if the geometric genera of all irreducible components add up to g. This is equivalent to saying that its Jacobian is a compact variety, which explains the name. We say that C has rational tails if one of its irreducible components has geometric genus g, so that the rest of the curve consists of trees (‘tails’) of rational curves attached at markings of this nonsingular genus g curve. We denote by

rt ct Mg,n ⊂ Mg,n ⊂ Mg,n the open substacks consisting of curves with rational tails and curves of compact type, respectively.

Conjectures on tautological classes I: Witten’s conjecture The ﬁrst hint of the richness of the combinatorial structure of the tautological classes came unexpectedly from string theory via the Witten conjecture. Deﬁne Z k1 k2 kn ψ1 ψ2 ··· ψn Mg,n 20 CHAPTER 1. INTRODUCTION

k1 k2 kn to be the degree of the 0-cycle ψ1 ψ2 ··· ψn if k1 +k2 +...+kn = 3g −3+n, and zero otherwise. Introduce the generating function Z ! X 1 X k1 k2 kn Fg = ψ1 ψ2 ··· ψn tk1 tk2 ··· tkn n! M n≥0 k1,k2,...,kn g,n in inﬁnitely many variables. Each Fg encodes all genus g intersection numbers, and we may form the combination

X 2g−2 F = λ Fg. g≥0

Witten [Wit91] conjectured that exp F , the generating function counting con- tributions from potentially disconnected surfaces, is a Sato τ-function for the KdV hierarchy. Concretely this means that exp F satisfies an infinite sequence of partial differential equations, which we do not write down, but let us just say that exp F is annihilated by a particular representation of the Virasoro algebra. Witten’s conjecture implies in particular that all intersection numbers may be recursively computed from the initial data provided by intersection numbers on M0,3, a single point! I will try to say something about Witten’s physical motivation for this conjecture, but the reader should keep in mind that I know nothing about physics. The basic idea is an analogy between two different theories of two-dimensional quantum gravity: the one-matrix model, based on random triangulations of surfaces, and topological gravity, based on integrals over moduli spaces of curves. The partition function in the matrix model was known to be such a Sato τ-function, which motivated Witten’s conjecture. Our treatment below is very brief and we refer to [DFGZJ95, Dij92] for more information. See also Witten’s ICM talk [Wit87] which in particular gives an idea of how integrals on moduli spaces appear in quantum gravity. The idea in the matrix model is that one wants to compute a sum over surfaces of all genera, and for each such surface an integral over the space of all Riemannian metrics on the surface modulo diffeomorphism. But the latter space is infinite dimensional and the integral is not well defined mathematically. Instead one considers a sequence of random metrics obtained from triangulations of the surfaces, for which all the curvature is concentrated at the vertices of the triangulations. This makes sense mathematically for any finite number of triangles, and one expects the limit as the number of triangles approaches infinity and the areas of the triangles shrink to zero to converge to the previous ill-defined integrals. Using the ‘Wick expansion’ [BIZ80, LZ04] one can write down a generating series for triangulations of surfaces of all genera simultaneously, based on simple graphical rules. This generating series admits an interpretation as a certain integral over the space of N × N hermitian matrices. It can be shown that these integrals satisfy the Toda lattice equations and, in the large N limit, that the KdV hierarchy arises. It is in fact a general phenomenon that matrix integrals of a certain form give rise to τ-functions of integrable hierarchies. 21

In the approach of topological gravity each integral over the space of Rie- mannian metrics is instead reduced to the finite dimensional space of con- formal structures on the surface, i.e. to the moduli space of curves. Now the integrals make sense mathematically and can be identified with the ψ-integrals above, and the resulting generating series is the series F defined earlier. The Witten conjecture was proven in a tour-de-force paper of Maxim Kont- sevich [Kon92]. Kontsevich showed that the series F also admitted an interpretation as a matrix integral, but in a completely different way. The series in the matrix model admit an expansion into Feynman diagrams, where each term describes a particular triangulation of a surface. In Kontsevich’s approach there is also such a Feynman diagram expansion, except each term now describes an integral over a cell in a particular triangulation of the moduli space. This triangulation of the moduli space is obtained using Jenkins– Strebel quadratic differentials on Riemann surfaces and is attributed by Kont- sevich to Penner–Harer–Mumford–Thurston, see [MP98, LZ04] for clear ex- positions. In the end both approaches give rise to certain sums over ribbon graphs which are interpreted as matrix integrals. Since then, several more proofs of the Witten conjecture have appeared, all with distinctly different flavors: Okounkov and Pandharipande [OP09] gave a proof using Hurwitz theory and random trees, Mirzakhani [Mir07] gave a proof using hyperbolic geometry and asymptotics for Weil–Petersson volumes, Kazarian–Lando [KL07] gave the first algebro-geometric proof using the ELSV formula [ELSV01], and Kim–Liu [KL08] gave a second algebro- geometric proof using the technique of localization.

Conjectures on tautological classes II: Faber’s conjectures Although Faber’s original conjectures concerned only intersection theory on the spaces Mg, it seems today more natural to formulate a ‘trinity’ of con- rt ct jectures for the spaces Mg,n, Mg,n and Mg,n. This perspective on the Faber conjectures was formulated by Faber and Pandharipande, see [Fab01, Pan02, FP00b]. The original conjectures are specializations of these, since rt Mg,0 = Mg. In what follows we take this slightly ahistorical point of view on the conjectures.

Vanishing/Socle conjecture The ﬁrst part of the conjecture is the assertion that the tautological ring is one-dimensional in its top degree, and a simple expression for what this top degree is. Namely, the claim is that Ri(M) vanishes for i > N and is one-dimensional for i = N, where:

rt 1. N = g − 2 + n − δ0,g if M = Mg,n

ct 2. N = 2g − 3 + n if M = Mg,n

3. N = 3g − 3 + n if M = Mg,n. 22 CHAPTER 1. INTRODUCTION

The term δ0,g is the Kronecker delta, that is, it denotes 1 if g = 0 and 0 otherwise. We recognize N = 3g − 3 + n as the complex dimension of the moduli space. It is natural to wonder what the other two numbers ‘mean’. One way to arrive at the formulas for the top degree N is via integration against the classes λgλg−1 and λg, respectively. (This also explains the correction term δ0,g — it comes about because the class λ−1 does not exist.) i rt g−2+n−i rt Suppose that α ∈ R (Mg,n) and β ∈ R (Mg,n), and choose liftings of these classes α, β in the tautological ring of Mg,n. Then the intersection number Z def hα, βi = α ∪ β ∪ λgλg−1 Mg,n is independent of the choice of liftings α and β, because λgλg−1 annihilates rt every class supported on Mg,n \Mg,n, as explained in [Fab99b]. The vanishing and socle conjectures then say that h−, −i is a pairing inducing an ∼ g−2+n rt isomorphism Q = R (Mg,n). In exactly the same way the integral R ct α ∪ β ∪ λg defines a pairing on the tautological ring of Mg,n, and Mg,n ∼ 2g−3+n ct an isomorphism Q = R (Mg,n). Conjecturally, a different way to think about these numbers is that they describe the largest compact subvariety of the respective moduli spaces. This connection comes about because the tautological ring contains an ample class, N κ1. If X ⊂ M were a compact subvariety of dimension N, then κ1 would N necessarily be nonzero (because κ1 would have nonzero intersection number with the class of X). Hence the vanishing part of the conjecture immediately implies that the integer N above gives an upper bound on the dimension on rt ct a compact subvariety of Mg,n and Mg,n respectively. The fact that g − 2 is an upper bound of the dimension of a compact subvariety of Mg is the celebrated Diaz’s theorem [Dia84]; whether or not this upper bound is sharp is a major open problem. (Even the existence of a compact surface in M4 is notoriously unknown.) Thus the vanishing part of the Faber conjectures can be seen as a generalization of the Diaz bound. For curves of compact type more is known: Faber and Van der Geer [FvdG04a] proved, using work of Oort [Oor74] and Koblitz [Kob75], that the largest ct dimension of a compact subvariety of Mg is exactly 2g − 3 in characteristic p > 0, namely, one considers the locus of curves whose Jacobian has vanishing p-rank. (The condition that the p-rank is zero defines a closed subvariety of Mg, and it does not intersect δ0 since a torus has positive p-rank.) Keel and Sadun [KS03] prove that there is no such subvariety in characteristic zero, so the dimension of a largest compact subvariety depends on the characteristic! See also Krichever [Kri12] for a sharpened statement. Since the forgetful maps rt ct down to Mg resp. Mg are proper, the statements in this paragraph remain true with marked points added, with the obvious modifications. All parts of the vanishing/socle conjectures are now theorems. Looi- rt g−2+n rt jenga [Loo95] proved the vanishing part for Mg,n, and that R (Mg,n) is at most one-dimensional. Faber [Fab99b] then proved nonvanishing of 23

g−2 R (Mg) by proving the identity

Z |B |(g − 1)! κ λ λ = 2g (1.7) g−2 g g−1 2g(2g)! Mg from the Witten conjecture, see also [FP00b]. Here B2g is a Bernoulli number; these are known never to be zero. The socle statement for Mg,n was proved in [GV01]. More uniform proofs of the vanishing and socle statements on rt ct Mg,n, Mg,n and Mg,n were given in [GV05] and in [FP05].

Intersection number conjecture The second part of the conjecture describes explicitly the pairing h−, −i introduced in the previous section. If we choose a generator for the top degree RN (M) of the tautological ring, then the pairing between Ri(M) and RN−i(M) can be described by an intersection matrix; the intersection number conjecture describes the entries of this matrix. On Mg,n, it was proved by Faber [Fab99a] that the intersection numbers involving monomials in the ψ-classes determine completely all intersection numbers in the tautological rings — this was the basis for Faber’s computer program KaLaPs calculating such intersection numbers. The intersection numbers involving only ψ-classes are of course exactly those which are determined by Witten’s conjecture. rt ct On Mg,n and Mg,n it turns out similarly that all pairings are determined by the pairings of monomials in the ψ-classes. But here this pairing means integrating against λgλg−1 and λg, respectively. So the ‘intersection number conjecture’ takes the form of an explicit formula for the integrals of ψ-classes against these classes, viz. Z Z k1 kn (2g − 3 + n)!(2g − 1)!! g−1 ψ1 ··· ψn λgλg−1 = Qn ψ1 λgλg−1 (2g − 1)! (2ki − 1)!! Mg,n i=1 Mg,1

(the λgλg−1-conjecture), and Z Z k1 kn 2g − 3 + n 2g−2 ψ1 ··· ψn λg = ψ1 λg k1, . . . , kn Mg,n Mg,1

(the strikingly simple λg-conjecture). The integrals on the right hand side were computed by Faber and Pandharipande: the former follows from (1.7) and the projection formula, and the latter was calculated in [FP00a]. These conjectures, too, are theorems now. In [GP98] it was proved that the λgλg−1-conjecture follows from the degree zero case of the Virasoro conjecture 2 of Eguchi–Hori–Xiong [EHX97] applied to P , and that the λg-conjecture similarly follows from the degree zero Virasoro conjecture applied to P1. After Givental’s proof [Giv01a] of the Virasoro conjecture for projective spaces, both of these are thus known. But this is a rather roundabout way of proving 24 CHAPTER 1. INTRODUCTION

the formulas, and in fact the λg-conjecture had been proved already before Givental’s work in [FP03]. Since the λ-classes are tautological, and we have already mentioned that all intersection numbers in the tautological ring are determined by integrals of ψ-classes and hence by the Witten conjecture, it might seem natural to try to prove this result by expressing λ-classes in terms of boundary-, κ- and ψ-classes in order to eventually obtain integrals containing only ψ-classes, and applying Witten’s conjecture. A proof of the λgλg−1-formula along these lines was in fact given by Liu–Xu [LX09] (it turns out not to be so easy!). A different proof of the λgλg−1-conjecture was given by Buryak–Shadrin [BS11] using double ramification cycles. Two different further proofs of the λg-conjecture also exist: one by Goulden–Jackson–Vakil [GJV09], using the ELSV formula [ELSV01], and one by Liu–Liu–Zhou [LLZ06], using their proof of the Mariño– Vafa formula [LLZ03] via the cut-and-join equation. The reader may be puzzled by the fact that our formulation of the intersection number conjecture involved only integrals of ψ-classes, whereas we also said that Faber’s original formulation was on Mg, without any marked points. For completeness let us state Faber’s original version of the intersection number conjecture. Fix any n-tuple of non-negative integers (d1, . . . , dn) summing to g − 2. For any permutation σ ∈ Sn, define Y κσ = κd +d +...+d . i1 i2 ik (i1i2···ik) a cycle of σ Then the conjecture asserts the equality X (2g − 3 + n)!(2g − 1)!! κσ = Qn κg−2. (2g − 1)! i=1(2di + 1)!! σ∈Sn

Gorenstein conjecture Finally we come to the Gorenstein part of the conjectures. This is the the part which has turned out most resilient to proof, and the part which will play a role in this thesis. This conjecture asserts that the pairings introduced in our description of the vanishing/socle conjecture are perfect, i.e. that

h−, −i: Ri(M) × RN−i(M) → RN (M) has the property that the functional hα, −i vanishes if and only if α vanishes, for all i. (This is equivalent to that the zero-dimensional local ring R•(M) is Gorenstein, hence the name.) Thus the tautological rings satisfy a form of Poincaré duality. An equivalent formulation of the Gorenstein conjecture is that any poten- tial relation between tautological classes which is consistent with the pairings — which are completely determined by the previous parts of the conjectures — is actually a true relation. Formulated this way, it is clear that the Goren- stein conjecture can be veriﬁed for any given g and n by writing down ﬁnitely many relations between tautological classes. 25

Faber’s original paper on the conjectures [Fab99b] describes an algorithm that will, in principle, generate an infinite amount of relations between tautological classes. If we heuristically assume that this algorithm will produce a randomly chosen point in the vector space of all relations (and except for small exceptional cases, there is no reason not to believe this), then such an algorithm should with high probability produce a complete list of tautological relations in short order. Unfortunately the computations required are massive and a great amount of trickery is required in order to carry out the calculations as the dimensions of the spaces involved grow. At the time of [Fab99b] the Gorenstein conjecture had been verified in this way on Mg for all g ≤ 15. This may seem as very strong evidence, but we know from the study of the Kodaira dimension of Mg that g needs to be surprisingly large for the large-genus behaviour of these spaces to become visible. Since then, Faber has verified the Gorenstein conjecture on Mg for all g ≤ 23. However, on M24 a problem appears. Even after generating hundreds of relations using Faber’s algorithm, the end result is still that the quotient of the polynomial ring in the κ-classes by these relations is 36-dimensional in degree 10 and 37-dimensional in degree 12, so there can be no perfect pairing. Thus either the Gorenstein conjecture fails in this case, or the heuristic that the relations obtained from Faber’s algorithm are random is false: surely one can not pick hundreds of random points from a finite dimensional vector space (in fact, 41-dimensional) and have them all land in a codimension 1 subspace. Different methods for generating relations between tautological classes have been constructed by Randal-Williams [RW12] and Yin [Yin12], but neither of their methods will produce a further relation. Yin also studies the tautological rings of Mg,1 and proves that they are Gorenstein for g ≤ 19, but his algorithm refuses to find sufficiently many relations when g = 20. Hence one might start to doubt the validity of the Gorenstein conjecture. But although it was possible to algorithmically prove the Gorenstein conjecture for a given genus, it appears harder to disprove the conjecture. A counterexample to the Gorenstein conjecture would require exhibiting a tautological class which pairs trivially with all classes of complementary degree, but which does not vanish. Proving that a given expression does not vanish in the Chow ring is not easy, given that its product with any ‘natural’ thing we could write down should vanish (given that any ‘natural’ class is tautological). But see [GG03, Yin13] for successful work in this direction, namely a counterexample to a similar Gorenstein conjecture for the tautological ring of a single curve [Fab01].

FZ relations After extensive computer calculations and inspired guesswork, Faber and Za- gier wrote down in the early 00’s a highly complicated power series whose coefficients are polynomials in the κ-classes. We do not recall the definition of this power series, but see [Fab13]. They conjectured that all coefficients satisfying a certain parity condition should correspond to polynomials which actually vanish in the tautological ring of Mg. The relations obtained in this way have since been christened FZ-relations. For g ≤ 24 they computed 26 CHAPTER 1. INTRODUCTION that all known relations between tautological classes can be obtained from the FZ-relations, but the ‘missing’ relation in g = 24 is not an FZ-relation, either. For g > 24 the situation seems only to get worse, in the sense that the FZ-relations yield rings further and further from being Gorenstein. This situation opens up the (admittedly unlikely) possibility that the FZ- relations are a proper subset consisting of relations which are ‘special’ in some unknown sense, and that known algorithms for constructing relations suffer from the defect that they only produce ‘special’ relations. On the other hand, Pandharipande has conjectured that the FZ-relations are in fact all relations (and thus that the Gorenstein conjecture fails). The conjecture of Faber and Zagier was proved by Pixton and Pandhari- pande [PP13], that is, they proved that the FZ-relations are actually true relations. Moreover, Pixton has found a version of the FZ power series which produces tautological relations on Mg,n for all g and n. These relations, too, are true relations, as proved very recently [PPZ13].

The Gorenstein conjecture in low genus Let us now consider the opposite case of small g and large n. When g = 0 rt ct one has M0,n = M0,n = M0,n. From Keel [Kee92] we know that in this case the tautological, Chow and cohomology rings coincide, which proves the Gorenstein conjecture by the usual Poincaré duality. rt ct For g = 1 one can first of all consider M1,n = M1,n. In this case the tautological ring was proved to be Gorenstein for all n by Tavakol [Tav10]. • rt Extending his methods to genus 2, Tavakol proved also that R (M2,n) is Gorenstein for all n [Tav11]. Ezra Getzler announced without proof in [Get97] that all even cohomology classes on M1,n are tautological, and moreover a complete description of all relations between tautological classes in cohomology; his claim was that all relations follow from Keel’s relation in genus zero (the WDVV relation) and Getzler’s relation on M1,4, proven in said paper. After [Pan99] it is known that Getzler’s relation holds also in the Chow ring, so that Getzler’s claims would imply that the tautological ring of M1,n is Gorenstein (since it coincides with the even cohomology ring of M1,n). Getzler’s claims were proved in Paper A in this thesis. The case of M2,n is interesting and is the subject of one of the papers in this thesis. When n is small, all even cohomology of M2,n is tautological, and hence the tautological cohomology ring is Gorenstein. However, in Paper B in this thesis we prove that as soon as an even non-tautological cohomology class appears on M2,n, so that the tautological cohomology ring isn’t Gorenstein for trivial reasons, the Gorenstein property fails immediately. It is easy to see that failure of the Gorenstein conjecture for the tautological cohomology ring implies the same for the full tautological ring. Graber and Pandharipande [GP03] construct such an even non-tautological cohomology class on M2,20, so the Gorenstein conjecture fails on M2,n for some n ≤ 20. It is notable that in all the low genus cases where the Gorenstein conjecture holds, the tautological ring can be identified with (some subring of) the coho- 27 mology ring of a compact manifold, so that the Gorenstein conjecture is true • rt for ‘trivial reasons’. For M0,n and M1,n this is clear. For R (M2,n), which Tavakol proved Gorenstein, we let F denote the fiber of the forgetful map rt M2,n → M2; that is, F is the Fulton–MacPherson compactification [FM94] of n points on a genus two curve. Then it turns out that there is an isomor- • rt • phism between R (M2,n) and the subalgebra of H (F ) consisting of classes which are invariant under the monodromy of the Gauss–Manin connection, see Section 4 of Paper B in this thesis. Of course both H•(F ) and this subalgebra satisfy Poincaré duality, since F is smooth and proper. The same result rt holds also for M1,n, with basically the same proof. Finally, as per the preceding paragraph, it seems highly likely that the tautological ring of M2,n is Gorenstein precisely in those cases when the tautological ring coincides with the even cohomology ring of M2,n. On the other hand, there is no ‘reason’ whatsoever for the Gorenstein conjecture to be true for Mg when g ≤ 23; this is a phenomenon that deserves an explanation (assuming that the Gorenstein conjecture on Mg fails).

Odds and ends The original Faber conjectures included further statements, which do not rt ct generalize readily to the spaces Mg,n ⊂ Mg,n ⊂ Mg,n. The ﬁrst of these is • that R (Mg) is generated by the classes

κ1, κ2, . . . , κbg/3c.

This was proved by Morita [Mor03] for the image of the tautological ring in cohomology, and by Ionel [Ion05] on the level of Chow rings. Morita uses the representation theory of the symplectic group, and Ionel uses a careful study of the relations obtained from Faber’s algorithm, mentioned in our above discussion on the Gorenstein conjecture. Secondly, Faber conjectured that there are furthermore no relations between the κ-classes in degrees at most g/3, so that in low degrees the tautological ring is just a polynomial algebra. It suffices to prove that there are no relations in cohomology, and therefore the result would follow from the Mumford conjecture [MW07] and a sharp enough version of Harer’s stability theorem [Har85]. Harer’s theorem has successively been improved by the work of many different people, see [Wah13], and a sufficiently strong statement was proved by Boldsen [Bol12].

2 Summary of results

Paper A

The ﬁrst paper is concerned with the even cohomology groups of M1,n. Specif- ically, the paper proves two unproven claims made by Ezra Getzler in [Get97]. These claims are:

2k (i) for every k, the group H (M1,n, Q) is spanned (additively) by cycle classes of strata in the stratiﬁcation by topological type;

(ii) all (additive) relations between the above generators follow from the WDVV relations on M0,4 and the Getzler relation on M1,4.

One can compare this to the situation in genus zero. Keel [Kee92] proved that the cohomology ring of M0,n is generated as an algebra by boundary divisors, and that the ideal of relations is generated by the pullbacks of the WDVV relations. For the purposes of Gromov–Witten theory, additive relations are more interesting, and Kontsevich and Manin [KM94] proved similarly, using Keel’s theorem, that boundary strata span the cohomology of M0,n and that all additive relations follow from WDVV. Thus the results of this paper are genus one analogues of Kontsevich’s and Manin’s calculations. Keel proved moreover that the Chow ring coincides with the cohomology ring in genus zero. For g = 1 this is not true anymore: firstly because of the existence of odd cohomology, secondly because the Chow ring becomes infinite dimensional when n ≥ 11 (for n ≤ 10 there is such an isomorphism, though, see [Bel98]). What is true, as a consequence of our result, is that the even cohomology ring coincides with the tautological ring. Namely, claim (i) above implies that the map from the tautological ring to the even cohomology ring is surjective, and claim (ii) implies injectivity since both relations are known to hold on the level of Chow rings [Pan99]. Thus the Gorenstein part of the Faber conjectures is verified for these spaces, by Poincaré duality. The main technical tools are Deligne’s theory of weights in cohomology [Del75] via mixed Hodge theory, the Hodge-theoretic interpretation of the Eichler–Shimura isomorphism [Del69, Zuc85], and the Cohen–Taylor–Totaro spectral sequence for the cohomology of configuration spaces [Tot96].

29 30 CHAPTER 2. SUMMARY OF RESULTS

Paper B

The second paper is joint work with Orsola Tommasi. Our starting point was the paper [GP03], which proves (among other things) the existence of an even cohomology class on M2,20 (in fact, the class of an algebraic cycle) which is non-tautological. The results of Paper A imply that this class is even non-tautological on M2,20. Thus the analogue of the results of Paper A can not be true when g > 1, and our ﬁrst goal was to see at what point the genus one proof breaks down. Already at the outset it was clear that the problem would be related to the cohomology of symplectic local systems on M2. On M1,1, the aforementioned Eichler–Shimura theory allows the computation of the cohomology of any such local system, together with its mixed Hodge structure. For higher genus, the problem of computing these cohomology groups becomes signiﬁcantly harder. Yet our knowledge is not incomplete: Faber and Van der Geer [FvdG04b, FvdG04c] have a conjectural formula for the “motivic” Euler characteristic of each such local system, and while we were working on this project Harder [Har12] determined completely the Eisenstein cohomology of all these local systems (on A2), which turned out to be very helpful in our investigations. What we ended up showing was that non-tautological even cohomology classes on M2,n (which are not supported on the boundary) necessarily come from 2 cohomology groups of the form H (M2, V2a,2a), where V2a,2a is a symplectic local system of weight 4a. As a consequence we were able to deduce the following main theorem: Let N be the smallest integer for which there is a non-tautological cohomology class of even degree on M2,N . Then N ∈ {8, 12, 16, 20}, and the degree of this cohomology class is 2 + N. In particular, there are no non-tautological classes in other degrees. The relation between the theorem and the above results on local systems rt is that N = 4a, and that if π : M2,N → M2 is the forgetful map, then V2a,2a N occurs in R π∗Q. So under the Leray spectral sequence for π, a class in 2 rt H (M2, V2a,2a) will contribute to cohomological degree 2 + N of M2,N .

This theorem easily implies that the tautological ring of M2,N can not be Gorenstein. Thus a counterexample is found to the Faber conjecture for this space. In brief, the point is that if the tautological ring is Gorenstein, then so is its image in the cohomology ring. But the theorem implies that the tautological cohomology ring cannot be Gorenstein, because it has diﬀerent dimensions in cohomological degrees 2 + N and 4 + N, which are supposed to pair perfectly with each other. The reasons these dimensions are not equal is that the whole cohomology ring has the same dimension in degrees 2 + N and 4 + N (by Poincaré duality), but in degree 4 + N every class is tautological which is not true in degree 2 + N.

Along the way we also need to show that every class on M2,20 which is pushed forward from the boundary is tautological. This answers a question posed in [GP03] and revisited in [FP13]. 31

Paper C The third paper explains how one may compute the cohomology of symplectic local systems on loci in A2 of d-elliptic abelian surfaces, considered as mixed Hodge structures or `-adic Galois representations. We should explain the name “d-elliptic’, which is not really standard. We say that a curve C of genus at least 2 is d-elliptic if it admits a degree d map to an elliptic curve which does not factor through an isogeny. Then the locus of d-elliptic genus 2 curves is an irreducible surface in M2 for every d, and the loci of d-elliptic abelian surfaces are their closures in A2. (Recall that there is an open immersion M2 ⊂ A2.) One can give an intrinsic description of what it means for an abelian surface to be d-elliptic. Let us take two elliptic curves E and E0, and suppose we are given an isomorphism of d-torsion subgroups φ: E[d] → E0[d] which inverts the Weil pairing. Then the quotient of E × E0 by the graph of φ is a d- elliptic abelian surface, and every d-elliptic abelian surface arises this way. From this description one also sees that these loci are quotients of products of modular curves: the loci of d-elliptic abelian surfaces can be written as (Y (d)×Y (d))/SL(2, Z/d), where SL(2, Z/d) acts by a twisted diagonal action. This part is folklore, see e.g. [FK91]. This description also shows that the cohomology groups we are after may be expressed completely in terms of the cohomology of local systems on Y (d) (which are determined by the Eichler–Shimura theory) and their decomposi- tions into irreducible representations of SL(2, Z/d). We also need to determine certain branching formulas (to see how local systems decompose when restricted to the above loci) and to study how representations of SL(2, Z/d) behave under conjugation by an element of GL(2, Z/d). We give special attention to the case d = 2, and show how our results allow one to algorithmically calculate the ‘motivic’ Sn-equivariant Euler characteristic of the space of n-pointed bi-elliptic curves of genus two. This part uses results of Getzler [Get99].

Paper D The fourth paper aims to describe in what sense the moduli spaces of admissible G-covers are modular operads. These spaces form a modular operad G in a subtler sense than the spaces Mg,n. We define Mg,n to be the moduli space parametrizing stable n-pointed genus g curves equipped with an admissible G-cover which is allowed to ramify over the marked points, and which is equipped with a trivialization of the covering over each marking. (The latter condition is necessary in order for it to be possible to glue admissible covers together.) These spaces of admissible covers were constructed in [AV02, ACV03]. The admissibility condition implies that one can not glue any two points together: one needs to assume a compatibility condition between the mon- odromies around the two marked points that are glued. Moreover, when gluing two marked points together, one should forget the trivializations over the markings. This extra structure is summarized by stating that these spaces 32 CHAPTER 2. SUMMARY OF RESULTS form a colored modular operad, where the colors are a groupoid (namely the action groupoid of G acting on itself by conjugation). The notion of an operad colored by something more general than a set does not appear to have been explicitly considered in the literature; the closest thing is probably the semidirect product operads of Salvatore–Wahl [SW03]. We define first what it means for an operad to be colored by the objects of a category C. For cyclic or modular operads, which are modeled on undirected graphs, one should require the colors to form a category with duality, i.e. a category with a coherent isomorphism C =∼ Cop. With the operadic formalism in place, we prove that coalgebras over the • G cohomology co-operad H (Mg,n) are the same thing as G-Cohomological Field Theories in the sense of Jarvis–Kaufmann–Kimura [JKK05]. We also prove that the Gromov–Witten invariants of global quotient orbifolds [X/G] give examples of G-CohFTs, a result left open in loc. cit.

Paper E The fifth paper contains a new short proof of a theorem due to Tamarkin, that the operad of little disks (in two dimensions) is formal. This terminology comes by analogy from rational homotopy theory, where a topological space X is said to be formal if the dg algebra of cochains on X is connected by a zig-zag of quasi-isomorphisms to the cohomology algebra of X; in this case, the rational homotopy type of X is a ‘formal consequence’ of its cohomology ring. Specifically, a dg operad P is called formal if there is a zig-zag of quasi- isomorphisms of operads connecting P and H•(P). This theorem of Tamarkin has some history. The first claimed proof of the result was given in [GJ94]. Then in 1997, a preprint of Kontsevich [Kon97, Kon03] appeared with a breakthrough result in mathematical physics: that every Poisson manifold admits a deformation quantization via an explicit universal formula for a star product. A key ingredient in the proof was his formality theorem, which upgrades the Hochschild–Kostant–Rosenberg theorem [HKR62] to a L∞-quasi-isomorphism of certain dg Lie algebras. Soon after, Tamarkin gave an alternative proof of the formality theorem [Tam98, Hin03], in which formality of the little disks was an intermediate result. At this point he realized that Getzler’s and Jones’s proof was incorrect; a certain cellular decomposition which they used turned out not to be cellular. See [Vor00] for more details. Tamarkin gave his own proof of formality of the little disk operad, which however also turned out to be incorrect. But the first correct proof was shortly after given by Tamarkin in [Tam03]. See also [Kon99] for more about this whole story, and a sketch of an alternative proof of formality, valid for disks of arbitrary dimension. The proof given in Paper E starts out similarly as Tamarkin’s. Since all spaces D2(n) are K(π, 1) spaces, one may replace the little disk operad with a certain combinatorially defined operad of groupoids, whose nerve is quasi-isomorphic to D2. Then one may take the pro-unipotent completion of this operad of groupoids, since this does not change the rational homotopy type. At this point we take a different path than Tamarkin. We use the fact 33 that the Grothendieck–Teichmüller group acts on this operad of pro-unipotent completions [BN98]. It follows from well known results in Grothendieck– Teichmüller theory [Dri90] that every grading automorphism of the homology operad H•(D2) lifts to a chain automorphism. By an operadic version of a formality criterion for commutative dg algebras due to Sullivan [Sul77], one then deduces formality of the little disk operad. As we explain, this proof is strongly related to the idea of weights in the cohomology of algebraic varieties [Del75].

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