R Pandharipande

A Pixton

D Zvonkine

Tautological relations via r-spin structures

Relations among tautological classes on M g,n are obtained via the study of Witten's r-spin theory for higher r. In order to calculate the quantum product, a new formula relating the r-spin correlators in genus 0 to the representation theory of sl 2 (C) is proven. The Givental-Teleman classification of CohFTs is used at two special semisimple points of the associated Frobenius manifold. At the first semisimple point, the R-matrix is exactly solved in terms of hypergeometric series. As a result, an explicit formula for Witten's r-spin class is obtained (along with tautological relations in higher degrees). As an application, the r = 4 relations are used to bound the Betti numbers of R * (M g ). At the second semisimple point, the form of the R-matrix implies a polynomiality property in r of Witten's r-spin class.

In the Appendix (with F. Janda), a conjecture relating the r = 0 limit of Witten's r-spin class to the class of the moduli space of holomorphic differentials is presented.

Introduction 0.1 Overview

Let M g,n be the moduli space of stable genus g curves with n markings. Let

RH * (M g,n ) ⊂ H * (M g,n )
be the subring of tautological classes in cohomology 1 . The subrings

RH * (M g,n ) ⊂ H * (M g,n ) g,n
are defined together as the smallest system of Q-subalgebras closed under push-forward via all boundary and forgetful maps, see [START_REF] Faber | Logarithmic series and Hodge integrals in the tautological ring[END_REF][START_REF] Faber | Tautological and non-tautological cohomology of the moduli space of curves in Handbook of moduli[END_REF][START_REF] Graber | Constructions of nontautological classes on moduli spaces of curves[END_REF]. There has been substantial progress in the understanding of RH * (M g,n ) since the study began in the 1980s [START_REF] Mumford | Towards an enumerative geometry of the moduli space of curves in Arithmetic and Geometry[END_REF]. The subject took a new turn in 2012 with the family of relations conjectured in [START_REF] Pixton | Conjectural relations in the tautological ring of M g[END_REF]. We refer the reader to [START_REF] Pandharipande | A calculus for the moduli space of curves, Proceedings of Algebraic geometry -Salt Lake City[END_REF] for a survey of recent developments.

Witten's r-spin class defines a Cohomological Field Theory (CohFT) for each integer r ≥ 2. Witten's 2-spin theory concerns only the fundamental classes of the moduli spaces of curves (and leads to no new geometry). In our previous paper [START_REF] Pandharipande | Relations on M g,n via 3-spin structures[END_REF], we used Witten's 3-spin theory to construct a family of relations among tautological classes of M g,n equivalent (in cohomology) to the relations proposed in [START_REF] Pixton | Conjectural relations in the tautological ring of M g[END_REF]. Our goal here is to extend our study of tautological relations to Witten's r-spin theory for all r ≥ 3.

Taking [START_REF] Pandharipande | Relations on M g,n via 3-spin structures[END_REF] as a starting point, Janda has completed a formal study of tautological relations obtained from CohFTs. Two results of Janda are directly relevant here:

(i) The relations for r = 3 are valid in Chow [START_REF] Janda | Comparing tautological relations from the equivariant Gromov-Witten theory of projective spaces and spin structures[END_REF][START_REF] Janda | Relations on M g,n via equivariant Gromov-Witten theory of P 1[END_REF].

(ii) The relations for r ≥ 4 are implied by the relations for r = 3 [START_REF] Janda | Comparing tautological relations from the equivariant Gromov-Witten theory of projective spaces and spin structures[END_REF][START_REF] Janda | Frobenius manifolds near the discriminant and relations in the tautological ring[END_REF]. By (i) and (ii) together, all of the r-spin relations that we find will be valid in Chow. However, since our methods here are cohomological, we will use the language of cohomology throughout the paper. Given Janda's results, why proceed with the higher r-spin analysis? There are three basic reasons:

• The r = 4 relations are simpler and easier to use than the 3-spin relations when restricted to M g . From the 4-spin relations, we derive a new bound on the rank of R d (M g ) which specializes in case d ≥ g -2 to the basic results

dim Q R g-2 (M g ) ≤ 1 , dim Q R >g-2 (M g ) = 0 (1)
of Looijenga [START_REF] Looijenga | On the tautological ring of M g[END_REF]. By (ii) above, we conclude that both statements in [START_REF] Bainbridge | Compactification of strata of abelian differentials[END_REF] follow from the restriction of the 3-spin relations to M g . The latter restriction equals the Faber-Zagier relations 2 .

The outcome is a proof that Looijenga's results [START_REF] Bainbridge | Compactification of strata of abelian differentials[END_REF] follow from the Faber-Zagier relations. Since Faber's conjectures [START_REF] Faber | A conjectural description of the tautological ring of the moduli space of curves, Moduli of curves and abelian varieties[END_REF] governing the proportionalities of κ monomials in R g-2 (M g ) are known 3 to be compatible with the Faber-Zagier relations, we can also conclude that the Faber-Zagier relations imply these proportionalities.

Relations, by themselves, cannot prove non-vanishing results. The non-

vanishing dim Q R g-2 (M g ) ≥ 1 (2)
is proven [START_REF] Faber | A conjectural description of the tautological ring of the moduli space of curves, Moduli of curves and abelian varieties[END_REF][START_REF] Faber | Logarithmic series and Hodge integrals in the tautological ring[END_REF] by Hodge integral evaluations. The results (1) and [START_REF] Belorousski | A descendent relation in genus 2[END_REF] together prove R g-2 (M g ) ∼ = Q .

• Another outcome is a much better understanding of Witten's r-spin class for higher r. We obtain an exact formula for Witten's r-spin correlators in genus 0 in terms of the representation theory of sl 2 (C). The genus 0 results and the Givental-Teleman classification of semisimple CohFTs together provide two explicit approaches to Witten's r-spin class in all genera. The first leads to a formula for all r parallel to the 3-spin formula of [START_REF] Pandharipande | Relations on M g,n via 3-spin structures[END_REF]. As an application of the second approach, we prove a new polynomiality property in r of Witten's r-spin class.

• In the Appendix with Janda, we present a new conjecture relating an appropriate limit (defined by polynomiality) of Witten's r-spin class to the class of the moduli space of holomorphic differentials with prescribed zero multiplicities.

These are unexpected developments. There is no a priori reason to believe the 4-spin relations would be algebraically simpler or that polynomiality in r holds for Witten's class. The very simple connection with the class of the moduli space of differentials leads to a much more direct calculation than in [START_REF] Farkas | The moduli space of twisted canonical divisors[END_REF]Appendix], but is available only in the holomorphic case.

Cohomological field theories

We recall here the basic definitions of a cohomological field theory by Kontsevich and Manin [START_REF] Kontsevich | Gromov-Witten classes, quantum cohomology, and enumerative geometry[END_REF], see also [START_REF] Pandharipande | Cohomological field theory calculations[END_REF] for a survey. Let V be a finite dimensional Q-vector space with a non-degenerate symmetric 2-form η and a distinguished element 1 ∈ V . The data (V, η, 1) is the starting point for defining a cohomological field theory. Given a basis {e i } of V , we write the symmetric form as a matrix η jk = η(e j , e k ) .

The inverse matrix is denoted by η jk as usual.

A cohomological field theory consists of a system Ω = (Ω g,n ) 2g-2+n>0 of elements Ω g,n ∈ H * (M g,n ) ⊗ (V * ) ⊗n .

We view Ω g,n as associating a cohomology class on M g,n to elements of V assigned to the n markings. The CohFT axioms imposed on Ω are:

(i) Each Ω g,n is S n -invariant, where the action of the symmetric group S n permutes both the marked points of M g,n and the copies of V * .

(ii) Denote the basic gluing maps by

q : M g-1,n+2 → M g,n , r : M g 1 ,n 1 +1 × M g 2 ,n 2 +1 → M g,n .
The pull-backs q * (Ω g,n ) and r * (Ω g,n ) are equal to the contractions of Ω g-1,n+2 and Ω g 1 ,n 1 +1 ⊗ Ω g 2 ,n 2 +1 by the bi-vector j,k

η jk e j ⊗ e k inserted at the two identified points.

(iii) Let v 1 , . . . , v n ∈ V be any vectors, and let p : M g,n+1 → M g,n be the forgetful map. We require

Ω g,n+1 (v 1 ⊗ • • • ⊗ v n ⊗ 1) = p * Ω g,n (v 1 ⊗ • • • ⊗ v n ) , Ω 0,3 (v 1 ⊗ v 2 ⊗ 1) = η(v 1 , v 2 ) . Definition 0.1. A system Ω = (Ω g,n ) 2g-2+n>0 of elements Ω g,n ∈ H * (M g,n ) ⊗ (V * ) ⊗n
satisfying properties (i) and (ii) is a cohomological field theory or a CohFT. If (iii) is also satisfied, Ω is a CohFT with unit.

A CohFT Ω yields a quantum product • on V via

η(v 1 • v 2 , v 3 ) = Ω 0,3 (v 1 ⊗ v 2 ⊗ v 3 ) .
Associativity of • follows from (ii). The element 1 ∈ V is the identity for • by (iii). A CohFT ω composed only of degree 0 classes,

ω g,n ∈ H 0 (M g,n ) ⊗ (V * ) ⊗n ,
is called a topological field theory. Via property (ii), ω g,n (v 1 , . . . , v n ) is determined by considering stable curves with a maximal number of nodes. Such a curve is obtained by identifying several rational curves with three marked points. The value of ω g,n (v 1 ⊗ • • • ⊗ v n ) is thus uniquely specified by the values of ω 0,3 and by the quadratic form η. In other words, given V and η, a topological field theory is uniquely determined by the associated quantum product.

Witten's r-spin class

For every integer r ≥ 2, there is a beautiful CohFT obtained from Witten's r-spin class. We review here the basic properties of the construction. The integer r is fixed once and for all. Let V r be an (r -1)-dimensional Q-vector space with basis e 0 , . . . , e r-2 , bilinear form

η ab = η(e a , e b ) = δ a+b,r-2 ,
and unit vector 1 = e 0 . Witten's r-spin theory provides a family of classes

W r g,n (a 1 , . . . , a n ) ∈ H * (M g,n ).
for a 1 , . . . , a n ∈ {0, . . . , r -2}. These define a CohFT by

W r g,n : V ⊗n → H * (M g,n ), W r g,n (e a 1 ⊗ • • • ⊗ e an ) = W r g,n (a 1 , . . . , a n ) .
Witten's class W r g,n (a 1 , . . . , a n ) has (complex) degree given by the formula

deg C W r g,n (a 1 , . . . , a n ) = D r g,n (a 1 , . . . , a n ) (3) = (r -2)(g -1) + n i=1 a i r .
If D r g,n (a 1 , . . . , a n ) is not an integer, the corresponding Witten's class vanishes.

In genus 0, the construction was first carried out by Witten [START_REF] Witten | Algebraic geometry associated with matrix models of twodimensional gravity in Topological methods in modern mathematics[END_REF] using rspin structures (r th roots of the canonical bundle) and satisfies the following initial conditions:

W r 0,3 (a 1 , a 2 , a 3 ) = 1 if a 1 + a 2 + a 3 = r -2, 0 otherwise. ( 4
) W r 0,4 (1, 1, r -2, r -2) = 1 r • [pt] ∈ H 2 (M 0,4 ) .
Uniqueness of Witten's r-spin theory in genus 0 follows from the initial conditions (4) and the axioms of a CohFT with unit. The genus 0 sector defines a quantum product • on V with unit e 0 , η(e a • e b , e c ) = W r 0,3 (a, b, c) .

The resulting algebra, even after extension to C, is not semisimple.

The existence of Witten's class in higher genus is both remarkable and highly non-trivial. An algebraic construction was first obtained by Polishchuk and Vaintrob [START_REF] Polishchuk | Algebraic construction of Witten's top Chern class in Advances in algebraic geometry motivated by physics[END_REF] defining

W r g,n (a 1 , . . . , a n ) ∈ A * (M g,n )
as a cycle class. The algebraic approach was later simplified by Chiodo [START_REF] Chiodo | The Witten top Chern class via K-theory[END_REF]. Analytic constructions have been given by Mochizuki [START_REF] Mochizuki | The virtual class of the moduli stack of stable r-spin curves[END_REF] and later by Fan, Jarvis, and Ruan [START_REF] Fan | The Witten equation, mirror symmetry and quantum singularity theory[END_REF]. As a consequence of the following result, the analytic and algebraic approaches coincide and yield tautological classes in cohomology.

Theorem 1 ([25]

). For every r ≥ 2, there is a unique CohFT which extends Witten's r-spin theory in genus 0 and has pure dimension (3). The unique extension takes values in the tautological ring

RH * (M g,n ) ⊂ H * (M g,n ).
Whether Witten's r-spin theory as an algebraic cycle takes values in

R * (M g,n ) ⊂ A * (M g,n )
is an open question.

0.4 Witten's r-spin class and representations of sl 2 (C).

Consider the Lie algebra sl 2 = sl 2 (C). Denote by ρ k the k-th symmetric power of the standard 2-dimensional representation of sl 2 ,

ρ k = Sym k (ρ 1 ) , dim ρ k = k + 1 .
The complete list of irreducible representations of sl 2 is {ρ k } k≥0 , see [START_REF] Fulton | Representation theory[END_REF]. Let

H = 1 2 1 0 0 -1 ∈ sl 2 .
The trace of the exponential is tr k e tH = e (k+1)t/2 -e -(k+1)t/2 e t/2 -e -t/2 .

The formula for the tensor product of two irreducible representations is then easily obtained:

ρ k ⊗ ρ l = ρ |k-l| ⊕ ρ |k-l|+2 ⊕ • • • ⊕ ρ k+l .
Our first result relates Witten's r-spin class in genus 0 with the representation theory of sl 2 .

Theorem 2. Let a = (a 1 , . . . , a n≥3 ) with a i ∈ {0, . . . , r -2} satisfy the degree constraint D r 0,n (a) = n -3. Then,

W r 0,n (a) = (n -3)! r n-3 dim ρ r-2-a 1 ⊗ • • • ⊗ ρ r-2-an sl 2 • [pt] ∈ H 2(n-3) (M 0,n ),
where the superscript sl 2 denotes the sl 2 -invariant subspace and the class

[pt] ∈ H 2(n-3) (M 0,n )
is Poincaré dual to a point.

The degree constraint D r 0,n (a) = n -3 in the statement of Theorem 2 can be written equivalently (using (3)) as

n i=1 a i = (n -2)r -2 .
Since a i ≤ r -2, the bound n ≤ r + 1 is a simple consequence.

Shifted Witten class

Given a vector γ ∈ V r , the shifted Witten class is defined by

W r,γ g,n (v 1 ⊗ • • • ⊗ v n ) = m≥0 1 m! p m * W r g,n+m (v 1 ⊗ • • • ⊗ v n ⊗ γ ⊗m ),
where p m : M g,n+m → M g,n is the forgetful map. The shifted Witten class W r,γ determines a CohFT, see [25, Section 1.1]. The vector space V r carries a Gromov-Witten potential F satisfying

∂ 3 F ∂t a ∂t b ∂t c (γ) = W r,γ 0,3 (e a ⊗ e b ⊗ e c )
which defines a Frobenius manifold structure on V r .

Example 0.2. For r = 3, the Gromov-Witten potential obtained from Witten's class equals

F(x, y) = 1 2 x 2 y + 1 72 y 4 ,
where x = t 0 and y = t 1 . For r = 4, the potential is

F(x, y, z) = 1 2 x 2 z + 1 2 xy 2 + 1 16 y 2 z 2 + 1 960 z 5 ,
where x = t 0 , y = t 1 , and z = t 2 .

The tangent vector space to γ ∈ V r has a natural Frobenius (or fusion) algebra structure Φ r,γ given by the structure constants

η(e a • γ e b , e c ) = ∂ 3 F ∂t a ∂t b ∂t c (γ).
Theorem 3. For (0, . . . , 0, r) ∈ V r , the algebra Φ r,(0,...,0,r) is isomorphic to the Verlinde algebra of level r for sl 2 .

Since the Verlinde algebra is semisimple, the algebra Φ r,(0,...,0,r) is semisimple as a consequence of Theorem 3. Proposition 2.3 of Section 2.4 provides a basis of idempotents for the algebras Φ (0,...,0,rφ) for all 0 = φ ∈ Q.

Theorems 2 and 3 relate Witten's r-spin class and the corresponding Frobenius manifold to the representations of sl 2 at level r. On the other hand, the Frobenius manifold associated to Witten's r-spin theory is usually constructed from the A r-1 singularity and thus is related to the Lie algebra sl r Lie algebra. Perhaps there is some form of rank-level duality for Frobenius manifolds, but we are not aware of other examples.

Euler field and Hodge grading operator.

The Frobenius manifold structure on V r includes an Euler field and a conformal dimension which determine a Hodge grading operator.

The Euler field on the Frobenius manifold V r is

E = r-2 a=0 1 - a r t a ∂ a .
The Lie derivatives with respect to E of the basis vectors fields are easily calculated:

L E (∂ a ) = [E, ∂ a ] = -1 - a r ∂ a .
The conformal dimension equals

δ = r -2 r .
Let v be a tangent vector at a point of the Frobenius manifold. We define the shifted degree operator µ(v), also called the Hodge grading operator, by

µ(v) = [E, v] + 1 - δ 2 v .
Here, the vector v is extended to a flat tangent vector field in order to compute the commutator. We have

µ(∂ a ) = 2a + 2 -r 2r ∂ a .

Tautological relations

We will construct tautological relations using Givental's R-matrix action on CohFTs. 4 The relations will be proven by studying Witten's r-spin class.

The point τ = (0, . . . , 0, rφ) ∈ V r with respect to a nonzero parameter φ ∈ Q will play a special role. Let

∂ a = φ -(2a-r+2)/4 ∂ a
be a new tangent frame on the Frobenius manifold V r at the point τ . We define a multilinear map ω r,τ g,n :

V ⊗n r → Q by the trigonometric formula ω r,τ g,n ∂ a 1 ⊗ • • • ⊗ ∂ an = r 2 g-1 φ (r-2)(2g-2+n)/4 r-1 k=1 (-1) (k-1)(g-1) n i=1 sin (a i +1)kπ r sin( kπ r ) 2g-2+n . (5)
We will prove ω r,τ is a CohFT with the right side of (5) interpreted as a multiple of the identity 1 ∈ H 0 (M g,n ). In fact, ω r,τ is the topological part of the τ -shifted r-spin CohFT W r,τ defined in Section 0.5. Our construction of tautological relations depends upon the following hypergeometric series. For every a ∈ {0, . . . , r -2}, we define

B r,a (T ) = ∞ m=0 m i=1 (2i -1)r -2(a + 1) (2i -1)r + 2(a + 1) i - T 16r 2 m .
For r even and a = r 2 -1, we have B r,a = 1. Otherwise, B r,a is a power series with all coefficients nonzero. We denote by B even r,a and B odd r,a the even and odd parts of the power series B r,a , B r,a (T ) = B even r,a (T ) + B odd r,a (T ) .

Example 0.3. For r = 3, we obtain a slight variation of the series occurring in the Faber-Zagier relations:

B 3,0 (T ) = m≥0 (6m)! (2m)! (3m)! - T 1728 m , B 3,1 (T ) = m≥0 1 + 6m 1 -6m (6m)! (2m)! (3m)! - T 1728 m .
For r = 4, we obtain:

B 4,0 (T ) = m≥0 (4m)! m! (2m)! - T 256 m , B 4,1 (T ) = 1 , B 4,2 (T ) = m≥0 1 + 4m 1 -4m (4m)! m! (2m)! - T 256 m .
Consider the matrix-valued power series R(z) ∈ End(V r )[[z]] with coefficients given by R a a = B even r,r-2-a (φ -r/2 z) , a ∈ {0, . . . , r -2} on the main diagonal,

R r-2-a a = -B odd r,a (φ -r/2 z) , a ∈ {0, . . . , r -2}
on the antidiagonal (if r is even, the coefficient at the intersection of both diagonals is 1), and 0 everywhere else.

Example 0.4. For r = 3, the R matrix is

B even 3,1 (φ -3/2 z) -B odd 3,1 (φ -3/2 z) -B odd 3,0 (φ -3/2 z) B even 3,0 (φ -3/2 z)
.

For r = 4, the R matrix is

  B even 4,2 (φ -2 z) 0 -B odd 4,2 (φ -2 z) 0 1 0 -B odd 4,0 (φ -2 z) 0 B even 4,0 (φ -2 z)   .
We will prove that the inverse matrix R -1 (z) has coefficients (R -1 ) a a = B even r,a (φ -r/2 z) , a ∈ {0, . . . , r -2} on the main diagonal, (R -1 ) r-2-a a = B odd r,a (φ -r/2 z) , a ∈ {0, . . . , r -2} on the anti-diagonal (if r is even, the coefficient at the intersection of both diagonals is 1), and 0 everywhere else.

Let Ω r,τ be the stable graph expression for the CohFT obtained by the action ω r,τ of the above R-matrix, Ω r,τ = R.ω r,τ , see [START_REF] Pandharipande | Relations on M g,n via 3-spin structures[END_REF]Definition 2.13].

Theorem 4. For every d > D r g,n (a 1 , . . . , a n ), the degree d part of Ω r,τ g,n (e

a 1 ⊗ • • • ⊗ e an ) ∈ H * (M g,n ) vanishes.
The complexity of the topological field theory ω r,τ for higher r leads to complicated relations on M g,n . However, by multiplying by ψ classes, pushing forward by forgetful maps, and then restricting to the interior, we obtain much simpler relations on M g,n . In order to write the resulting relations, we extend the definition of the power series B r,a (T ) to all a ≥ 0 satisfying a ≡ r -1 mod r by the formula B r,a+rb (T ) = T b B r,a (T ) .

The relations depend upon on a partition

5 σ = (σ 1 , σ 2 , . . . , σ (σ) )
with no part σ i congruent to r -1 mod r, and a vector of non-negative integers a = (a 1 , . . . , a n )

with no a i congruent to r -1 mod r.

rd ≡ (r -2)(g -1) + |σ| + n i=1 a i mod 2.
Then, the degree d part of

n i=1 B r,a i (ψ i )   m≥0 1 m! p (σ)+m * (σ) j=1 B r,σ j +r (ψ n+j ) m k=1 (T -T B r,0 )(ψ n+ (σ)+k )   vanishes in H 2d (M g,n ).
In the statement of Theorem 5, p (σ)+m is (as before) the forgetful map

p (σ)+m : M g,n+ (σ)+m → M g,n
forgetting the last (σ) + m points. We use the relations of Theorem 5 to bound the Betti numbers of the tautological ring of M g . Let P (n, k) denote the set of partitions of n of length at most k.

Theorem 6. For g ≥ 2 and d ≥ 0, dim Q RH d (M g ) ≤ |P (d, g -1 -d)|.
The bound of Theorem 6 implies the results

dim Q RH g-2 (M g ) ≤ 1 , dim Q RH >g-2 (M g ) = 0 . (6)
For d < g -2, Theorem 6 is new (no non-trivial bounds were known before), but is not expected to be sharp. For example, Theorem 6 yields

dim Q RH g-3 (M g ) ≤ 1 + g -3 2 ,
while the expectation based on calculations is

dim Q RH g-3 (M g ) = 1 .
Though our proofs use cohomological methods, results (i) and (ii) of Janda discussed in Section 0.1 imply Theorems 4-6 all are valid in Chow. In particular, Theorem 6 yields the bound

dim Q R d (M g ) ≤ |P (d, g -1 -d)| ,
and (6) specializes to Looijenga's result (1).

Polynomiality

Let a 1 , . . . , a n be non-negative integers satisfying the condition

n i=1 a i = 2g -2 .
If a i ≤ r-2 for all i, then Witten's r-spin class W r g,n (a 1 , . . . , a n ) is well-defined and of degree independent of the choice of r,

D r g,n (a , . . . , a n ) = (r -2)(g -1) + n i=1 a i r = g -1 .
We may reasonably ask6 here about the dependence of W r g,n (a 1 , . . . , a n ) on r.

Theorem 7. For n i=1 a i = 2g -2,

r g-1 • W r g,n (a 1 , . . . , a n ) ∈ RH g-1 (M g,n )
is a polynomial in r for all sufficiently large r.

Our proof of Theorem 7 is obtained by the analysis of the shifted CohFT W r, τ at the point τ = (0, rφ, 0, . . . , 0) ∈ V r .

The CohFTs W r,τ and W r, τ behave differently and yield different insights.

While our knowledge of the R-matrix at τ is not as explicit as at τ , the property of polynomiality is easier to see at τ . In the Appendix (with F. Janda), we conjecture the constant term of the polynomial of Theorem 7 is (-1) g times the class of the closure of the locus of holomorphic differentials with zero multiplicities given by (a 1 , . . . , a n ). In addition to a precise formulation, the evidence for the conjecture and the connection to the conjectures of [START_REF] Farkas | The moduli space of twisted canonical divisors[END_REF]Appendix] are discussed in the Appendix.

Plan of the paper

We start in Section 1 with the proof of Theorem 2. The result plays a basic role in our analysis of the r-spin theory in genus 0. The study of the shift τ = (0, . . . , 0, rφ) ∈ V r is presented in Section 2. Theorem 3 is proven in Section 2.2, and the corresponding R-matrix is solved in terms of hypergeometric series in Section 2.3. Section 3 concerns the tautological relations obtained from the shifted CohFT W r,τ . Theorems 4, 5, and 6 are proven in Sections 3.1, 3.3, and 3.4 respectively. The study of the shift τ = (0, rφ, 0, . . . , 0) ∈ V r is presented in Section 4. The polynomiality of the R-matrix is derived in Section 4.5, and Theorem 7 is proven in Section 4.6.

Two formulas for Witten's r-spin class are given: Theorem 8 of Section 3.1 via the τ -shift and Theorem 9 of Section 4.5 via the τ -shift. The Appendix (with F. Janda) conjectures a connection between Witten's r-spin class and the class of the moduli space of holomorphic differentials. 1 Representations of sl 2 (C)

Correlators

Our goal here is to prove Theorem 2 relating Witten's class and the representation theory of sl 2 = sl 2 (C).

Definition 1.1. Let a 1 , . . . , a n ∈ {0, . . . , r -2} satisfy n i=1 a i = (n -2)r -2 .
The associated genus 0 correlator is

a 1 , . . . , a n r = M 0,n W r 0,n (a 1 , . . . , a n ) ∈ Q .
Since r can be deduced from a 1 , . . . , a n , we will often drop the superscript and denote the correlator by a 1 , . . . , a n .

The WDVV equation.

Let a, b, c, d, x 1 , . . . , x k ∈ {0, . . . , r -2} satisfy

a + b + c + d + k i=1 x i = (k + 1)r -2 , so W r 0,4+k (a, b, c, d, x 1 , . . . , x k ) is a class of degree k on M 0,4+k . Since the dimension of M 0,4+k is k + 1, we can cut W r 0,k+4 (a, b, c, d, x 1 , . . . , x k
) once with boundary divisors. If we pull-back the WDVV relation from M 0,4 to M 4+k via the map forgetting the last k markings,

p k : M 4+k → M 4 ,
and apply the CohFT splitting axiom for Witten's class, we obtain

I J={1,...,k} * + * =r-2 x I , a, c, * x J , b, d, * = I J={1,...,k} * + * =r-2 x I , a, d, * x J , b, c, * , (7)
where x I and x J are the insertions

(x i : i ∈ I) and (x j : j ∈ J) ,
and * and * are non-negative integers with sum r -2.

Using the values (4) of the 3-points correlators, we can rewrite the WDVV equation [START_REF] Faber | Tautological and non-tautological cohomology of the moduli space of curves in Handbook of moduli[END_REF] as

a, c, b + d, x 1 , . . . , x k + b, d, a + c, x 1 , . . . , x k = a, d, b + c, x 1 , . . . , x k + b, c, a + d, x 1 , . . . , x k + Q,
where, by convention, a correlator vanishes if any insertion exceeds r -2. Here, Q is a sum of products of correlators with 4 to k + 2 insertions each. In particular,

Q vanishes unless k ≥ 2. Lemma 1.2. The 3-point evaluations a 1 , a 2 , a 3 = 1 if a 1 + a 2 + a 3 = r -2, 0 otherwise
together with the WDVV equation [START_REF] Faber | Tautological and non-tautological cohomology of the moduli space of curves in Handbook of moduli[END_REF] force the vanishing of all genus 0 correlators a 1 , . . . , a n with n ≥ 4 and at least one 0 insertion.

Proof. Apply the WDVV equation to the class W r 0,n+1 (a, 0, b, 0, c 1 , . . . , c n-3 ) with nonzero a and b. We obtain

a + b, 0, 0, c 1 , . . . , c n-3 + a, b, 0, c 1 , . . . , c n-3 = a, b, 0, c 1 , . . . , c n-3 + a, b, 0, c 1 , . . . , c n-3 + Q or, equivalently, a, b, 0, c 1 , . . . , c n-3 = a + b, 0, 0, c 1 , . . . , c n-3 -Q. ( 8
)
If n = 4, the quadratic term Q vanishes. Hence, we can use [START_REF] Farkas | The moduli space of twisted canonical divisors[END_REF] to increase the number of 0 insertions from 1 to 2 and then 2 to 3. The nonzero insertion of the correlator x, 0, 0, 0 must satisfy

x = 2r -2 > r -2 .
Hence x, 0, 0, 0 = 0, and the Lemma is proved for n = 4.

Assume n ≥ 5. In every term of Q in ( 8), there is a correlator with fewer than n insertions and with a 0 insertion. By induction, such correlators vanish, so Q again vanishes. We may use [START_REF] Farkas | The moduli space of twisted canonical divisors[END_REF] as before to increase the number of 0 entries until we arrive at x, 0, 0, 0 n-1 [START_REF] Fulton | Intersection theory[END_REF] Then x = (n -2)r -2 exceeds r -2, and the correlator (9) vanishes.

Lemma 1.3. The correlators a 1 , a 2 , a 3 = 1 if a 1 + a 2 + a 3 = r -2, 0 otherwise r -2, r -2, 1, 1 = 1
r and the WDVV equation [START_REF] Faber | Tautological and non-tautological cohomology of the moduli space of curves in Handbook of moduli[END_REF] uniquely determine the values of all the other correlators in genus 0.

Proof. Consider a correlator with n ≥ 4 insertions. If the correlator has at least one 0 insertion, the correlator vanishes by Lemma 1.2. If not, all insertions are positive.

Let a, b, c, d 1 , . . . , d n-3 be a correlator with positive insertions ordered in a non-increasing sequence. Assume also c ≥ 2. For n ≥ 5, the condition c ≥ 2 is automatic, while for n = 4, the condition excludes only the single correlator r -2, r -2, 1, 1 = 1 r .

By applying the WDVV equation ( 7) to Witten's r-spin class

W r 0,n+1 (a, 1, c -1, b, d 1 , . . . , d n-3 ) ,
we obtain the relation

a + c -1, b, 1, d 1 , . . . , d n-3 + a, b + 1, c -1, d 1 , . . . , d n-3 = a + b, c -1, 1, d 1 , . . . , d n-3 + a, b, c, d 1 , . . . , d n-3 + Q . ( 10
)
If we order the insertions of the correlators in non-increasing order, then our original correlator a, b, c, d 1 , . . . , d n-3 is lesser than the three others principal correlators of (10) in lexicographic order. Thus every correlator except r -2, r -2, 1, 1 can be expressed via correlators greater in lexicographic order and correlators with fewer entries.

Proof of Theorem 2.

A straightforward calculation of the sl 2 side shows that the formula of Theorem 2 is true for n ≤ 4. By Lemma 1.3, the WDVV equation ( 7) uniquely determines the correlators for n ≥ 5. To finish the proof, we need only to show that the sl 2 side of the formula of Theorem 2 satisfies the WDVV equation [START_REF] Faber | Tautological and non-tautological cohomology of the moduli space of curves in Handbook of moduli[END_REF].

The WDVV equation ( 7) starts with a choice of insertions a, b, c, d, x 1 , . . . , x k .

For convenience, we replace the above insertions by their r -2 complements:

a → r-2-a , b → r-2-b , c → r-2-c , d → r-2-d , x i → r-2-x i .
Then, we have

a + b + c + d + k i=1 x i = 3r -6 -2k .
We must prove the identity

I J={1,...,k} |I|!|J|! x I , a, c, 2r -4 -a -c -2|I| - i∈I x i sl 2 • x J , b, d, 2r -4 -b -d -2|J| - j∈J x j sl 2 = I J={1,...,k} |I|!|J|! x I , a, d, 2r -4 -a -d -2|I| - i∈I x i sl 2 • x J , b, c, 2r -4 -b -c -2|J| - j∈J x j sl 2 ,
where we define

a 1 , . . . , a k sl 2 = dim ρ a 1 ⊗ • • • ⊗ ρ a k sl 2 .
The following formula for the dimensions of the sl 2 -invariants follows easily from the multiplication rule for the ρ a :

a 1 , . . . , a n , 2s -a 1 -• • • -a n sl 2 = (1 -t -1 ) n i=1 1 -t a i +1 1 -t t s .
Applying the above formula, we can rewrite the desired identity as

I J={1,...,k} |I|!|J|!   (1 -t a+1 )(1 -t c+1 )(1 -u b+1 )(1 -u d+1 ) (1 -t)(1 -u) i∈I t -t xi+2 1 -t j∈J u -u xj +2 1 -u   t r-1 u r-1 = I J={1,...,k} |I|!|J|!   (1 -t a+1 )(1 -t d+1 )(1 -u b+1 )(1 -u c+1 ) (1 -t)(1 -u) i∈I t -t xi+2 1 -t j∈J u -u xj +2 1 -u   t r-1 u r-1
.

Subtracting one side from the other and moving factors outside the sum, we must, equivalently, show that the coefficient of

t r-1 u r-1 in (1 -t a+1 )(1 -u b+1 ) (1 -t c+1 )(1 -u d+1 ) -(1 -t d+1 )(1 -u c+1 ) (1 -t)(1 -u) • I J={1,...,k} |I|!|J|! i∈I t -t x i +2 1 -t j∈J u -u x j +2
1 -u vanishes. Interchanging t and u and adding, we can replace the latter polynomial by a symmetric one:

(1 -t a+1 )(1 -u b+1 ) -(1 -t b+1 )(1 -u a+1 ) (1 -t c+1 )(1 -u d+1 ) -(1 -t d+1 )(1 -u c+1 ) (1 -t)(1 -u) • I J={1,...,k} |I|!|J|! i∈I t -t x i +2 1 -t j∈J u -u x j +2 1 -u .
Next, we apply the change of variables

t → t v and u → u v
and multiply the outcome by v m where

m = a + b + c + d + 3 + k i=1 (x i + 2) = 3r -3 .
Then, we replace a, b, c, d by a-1, b-1, c-1, d-1 and x i by x i -2. After these transformations, the required identity is that the coefficient of t r-1 u r-1 v r-1 vanishes in the polynomial

(t a u b -u a t b -t a v b + v a t b + u a v b -v a u b )(t c u d -u d t c -t c v d + v c t d + u c v d -v c u d ) t 2 u -u 2 t -t 2 v + v 2 t + u 2 v -v 2 u •(t -u)v I J={1,...,k} |I|!|J|! i∈I t x i v -v x i t t -v j∈J u x j v -v x j u u -v .
The initial factor is invariant under cyclically permuting t, u, v. So we can instead prove the vanishing of the coefficient of t r-1 u r-1 v r-1 in the polynomial

(t a u b -u a t b -t a v b + v a t b + u a v b -v a u b )(t c u d -u d t c -t c v d + v c t d + u c v d -v c u d ) t 2 u -u 2 t -t 2 v + v 2 t + u 2 v -v 2 u • cyc (t -u)v I J={1,...,k} |I|!|J|! i∈I t x i v -v x i t t -v j∈J u x j v -v x j u u -v ,
where the sum in the second term is over the three cyclic permutations of t, u, v. In fact, we claim a stronger vanishing:

cyc (t -u)v I J={1,...,k} |I|!|J|! i∈I t x i v -v x i t t -v j∈J u x j v -v x j u u -v = 0 (11)
for any x i ∈ Z. The last step in the proof of Theorem 2 is to show the vanishing [START_REF]Gromov-Witten invariants and quantization of quadratic Hamiltonians[END_REF]. Let A B C = {1, . . . , k} be a partition of the {x i } into three sets of sizes

a = |A| , b = |B| , c = |C| .
We can compute the coefficient of i∈A t x i i∈B u x i i∈C v x i on the left side [START_REF]Gromov-Witten invariants and quantization of quadratic Hamiltonians[END_REF]:

cyc (t -u)v i+j=c c i (i + a)!(j + b)! v t -v a v u -v b -t t -v i -u u -v j ,
where now a, b, c are also cyclically permuted in correspondence with t, u, v. Multiplying by (-1)

a+b+c (t -u) a+b (u -v) b+c (v -t) c+a and dividing by a!b!c! then yields cyc i+j=c (-1) b+i a + i i b + j j t i u j v a+b+1 (t -u) a+b+1 (u -v) i (v -t) j . ( 12
) Now set z 1 = t(u -v), z 2 = u(v -t), z 3 = v(t -u)
, multiply by y a 1 y b 2 y c 3 , and sum [START_REF] Graber | Constructions of nontautological classes on moduli spaces of curves[END_REF] over all non-negative integers a, b, c. The result is

cyc z 3 (1 + z 1 y 3 -z 3 y 1 )(1 + z 3 y 2 -z 2 y 3 )
, which expands to

z 1 + z 2 + z 3 (1 + z 1 y 3 -z 3 y 1 )(1 + z 2 y 1 -z 1 y 2 )(1 + z 3 y 2 -z 2 y 3 ) . ( 13
)
Since

z 1 + z 2 + z 3 = tu -tv + uv -ut + vt -vu = 0, so (
a i = r(n -2) -2 . ( 14
) Let b i = r -2 -a i , so n i=1 b i = 2(r + 1 -n).
If one of the b i is greater than the sum of all others, then the representation ⊗ n i=1 ρ b i has no invariant vectors and the correlator [START_REF] Janda | Comparing tautological relations from the equivariant Gromov-Witten theory of projective spaces and spin structures[END_REF] vanishes by Theorem 2. Thus the greatest possible value of b i in a nonzero correlator is equal to r + 1 -n. In other words, the smallest possible value of a i is n -3.

2 The semisimple point τ = (0, . . . , 0, rφ)

Special shift

The last basis vector e r-2 ∈ V r plays a special role in Witten's r-spin theory: e r-2 corresponds, via the formula of Theorem 2, to the trivial representation of sl 2 . The shift along the last coordinate yields simpler expressions for the quantum product and the R-matrix which are calculated in Sections 2.2 and 2.3 respectively. The associated topological field theory is calculated in Sections 2.4 and 2.5.

The quantum product (proof of Theorem 3)

We study the Frobenius algebra at the point τ = (0, . . . , 0, rφ) ∈ V r and find an isomorphism with the Verlinde fusion algebra for sl 2 of level r.

Let a + b + c = r -2 + 2k. By Theorem 2, we have

W r 0,3+k (a, b, c, r -2, . . . , r -2 k ) = k! r k if min(a, b, c) ≥ k, 0 otherwise. ( 15
)
The quantum product at τ is given by

∂ a • τ ∂ b = min(a,b) k=max(0,a+b-r+2) φ k ∂ a+b-2k .
The k! in the evaluation [START_REF] Janda | Frobenius manifolds near the discriminant and relations in the tautological ring[END_REF] is cancelled by the k! in the denominator of the Gromov-Witten potential, and the r k in the evaluation is cancelled by r factor in the last coordinate of τ .

To simplify the computations, we introduce the new frame7

∂ a = φ -(2a-r+2)/4 ∂ a .
Then, the quantum product assumes a slightly simpler form:

∂ a • τ ∂ b = φ (r-2)/4 min(a,b) k=max(0,a+b-r+2) ∂ a+b-2k . ( 16
)
We have proven the following result. The symmetry under permutations of a, b, c of the structure constants of the algebra is expected. However, the tetrahedron has an extra symmetry obtained by replacing every a by r -2 -a (central symmetry) and then changing the metric from η Ver to η Frob (a vertical flip).

The R-matrix

We compute here the R-matrix of the Frobenius manifold V r at the point τ = (0, . . . , 0, rφ) .

We do not know closed form expressions for the coefficients of the R-matrix at any other point of V r .

The operator of quantum multiplication by the Euler field at τ ,

E = 2φ (r+2)/4 ∂ r-2 ,
is given in the frame { ∂ a } by the matrix

ξ = 2φ r/2                0 • • • • • • 0 1 0 0 1 0 . . . . . . . . . . . . . . . 0 1 0 0 1 0 • • • • • • 0                .
The nonvanishing coefficients8 are ξ a r-2-a = 2φ r/2 . In the same frame, the shifted degree operator is

µ = 1 2r                -(r -2) 0 • • • • • • 0 0 -(r -4) 0 0 . . . . . . . . . . . . . . . 0 0 r -4 0 0 • • • • • • 0 r -2               
.

The nonvanishing coefficients are

µ a a = 2a -r + 2 2r .
Recall the hypergeometric series defined in Section 0.7 for every integer a ∈ {0, . . . , r -2},

B r,a (T ) = ∞ m=0 m i=1 (2i -1)r -2(a + 1) (2i -1)r + 2(a + 1) i - T 16r 2 m .
We denote by B even r,a and B odd r,a the even and odd parts of the series B r,a . Proposition 2.2. The unique solution

R(z) = ∞ m=0 R m z m ∈ End(V r )[[z]] of the equations [R m+1 , ξ] = (m + µ)R m with the initial condition R 0 = 1 has coefficients R a a = B even r,r-2-a (φ -r/2 z) , a ∈ {0, . . . , r -2} on the main diagonal, R r-2-a a = -B odd r,a (φ -r/2 z) , a ∈ {0, . . . , r -2}
on the antidiagonal (if r is even, the coefficient at the intersection of both diagonals is 1), and 0 everywhere else. The inverse matrix R -1 (z) has coefficients

(R -1 ) a a = B even r,a (φ -r/2 z) , a ∈ {0, . . . , r -2}
on the main diagonal, (R -1 ) r-2-a a = B odd r,a (φ -r/2 z) , a ∈ {0, . . . , r -2} on the antidiagonal (if r is even, the coefficient at the intersection of both diagonals is 1), and 0 everywhere else.

Proof. The uniqueness of the solution follows from the semisimplicity of the Frobenius manifold V r at τ (proven in Section 2.4). We verify here that the R-matrix described in the Proposition is indeed a solution of the recursion

[R m+1 , ξ] = (m + µ)R m (17)
with initial condition9 R 0 = 1.

The coefficients of the commutator on the left side of ( 17) are given by

[R m+1 , ξ] a b = 2φ r/2 (R m+1 ) a r-2-b -(R m+1 ) r-2-a b . (18)
The right side of (18) vanishes unless a = b or a + b = r -2. The same is true of R m and therefore of (m + µ)R m , since µ is a diagonal matrix. Thus, we have two cases to consider.

Case a = b. We have

[R m+1 , ξ] a a = 2φ r/2 (R m+1 ) a r-2-a -(R m+1 ) r-2-a a = 2φ r/2 [z m+1 ] -B odd r,r-2-a (φ -r/2 z) + B odd r,a (φ -r/2 z) .
Using the definition of the B odd series, the last expression is

δ even m 2φ r/2 m+1 i=1 (2i -1)r -2(a + 1) (2i -1)r + 2(a + 1) i • (3 + 2m)r -2(a + 1) (2m + 1)r + 2(a + 1) + 1 - 1 16r 2 φ r/2 m+1 , or, equivalently, δ even m 2φ r/2 m+1 i=1 (2i -1)r -2(a + 1) (2i -1)r + 2(a + 1) i • 4(m + 1)r (2m + 1)r + 2(a + 1) - 1 16r 2 φ r/2 m+1 .
After further simplification, we conclude

[R m+1 , ξ] a a = -δ even m (2m + 1)r -2(a + 1) 2r • [z m ]B r,a .
On the other hand, we have

[(m + µ)R m ] a a = = m + 2a -r + 2 2r (R m ) a a = δ even m (2m -1)r + 2a + 2 2r • [z m ]B r,r-2-a = δ even m (2m -1)r + 2a + 2 2r 2(a + 1) -(1 + 2m)r 2(a + 1) -(1 -2m)r • [z m ]B r,a = -δ even m (2m + 1)r -2(a + 1) 2r • [z m ]B r,a .
Therefore, equation ( 17) is satisfied.

Case a + b = r -2. We have [R m+1 , ξ] r-2-a a = 2φ r/2 (R m+1 ) r-2-a r-2-a -(R m+1 ) a a = 2φ r/2 [z m+1 ] B even r,r-2-a (φ -r/2 z) -B even r,a (φ -r/2 z) .
Using the definition of the B odd series, the last expression is

δ odd m 2φ r/2 m+1 i=1 (2i -1)r -2(a + 1) (2i -1)r + 2(a + 1) i • 2(a + 1) -(3 + 2m)r 2(a + 1) + (2m + 1)r -1 - 1 16r 2 φ r/2 m+1 , or, equivalently, -δ odd m 2φ r/2 m+1 i=1 (2i -1)r -2(a + 1) (2i -1)r + 2(a + 1) i • 4(m + 1)r (2m + 1)r + 2(a + 1) - 1 16r 2 φ r/2 m+1 .
After further simplification, we conclude

[R m+1 , ξ] a a = δ odd m (2m + 1)r -2(a + 1) 2r • [z m ]B r,a .
On the other hand, we have

[(m + µ)R m ] r-2-a a = m - 2a -r + 2 2r (R m ) r-2-a a = δ odd m (2m + 1)r -2(a + 1) 2r • [z m ]B r,a .
Therefore, equation ( 17) is satisfied.

The expression for R -1 is obtained from the symplectic condition 10

R -1 (z) = R * (-z)
10 R * denotes the adjoint with respect to the metric η on V r .

29

for the R-matrix. The symplectic condition implies the identities B r,a (T )B r,r-2-a (-T ) + B r,a (-T )B r,r-2-a (T ) = 2, or, equivalently,

B even r,a (T )B even r,r-2-a (T ) -B odd r,a (T )B odd r,r-2-a (T ) = 1.
Of course, these identities can also be proved directly.

The topological field theory

We study next the topological field theory arising from the Frobenius algebra of V r at the point τ = (0, . . . , 0, rφ) .

Proposition 2.3. The basis of normalized idempotents of the quantum product ( 16) is given by

v k = 2 r r-2 a=0 sin (a + 1)kπ r ∂ a , k ∈ {1, . . . , r -1} .
More precisely, we have

η(v k , v l ) = (-1) k-1 δ k,l , v k • τ v l = φ (r-2)/4 r/2 sin( kπ r ) v k δ k,l .
Proof. The idempotents of the Verlinde algebras are known for all semisimple Lie algebras and all levels [START_REF] Verlinde | Fusion rules and modular transformations in 2D conformal field theory[END_REF]. The idempotents are automatically orthogonal to each other, so we only need to compute their scalar squares. Since the extraction of the elementary formulas for sl 2 from the general case is not so simple, we check the statements of the Proposition independently by a series of elementary computations with trigonometric functions in Section 2.5.

Proposition 2.4. For a 1 , . . . , a n ∈ {0, . . . , r -2}, we have

ω r,τ g,n ( ∂ a 1 ⊗ • • • ⊗ ∂ an ) = r 2 g-1 φ (r-2)(2g-2+n)/4 r-1 k=1 (-1) (k-1)(g-1) n i=1 sin (a i +1)kπ r sin( kπ r ) 2g-2+n
.

Proof. The topological field theory ω r,τ g,n can be computed by restricting the τ -shifted r-spin theory W r,τ g,n to

[C] ∈ M g,n ,
where C is a completely degenerate curve with 2g-2+n rational components and 3g -3 + n nodes. By Proposition 2.3, ω r,τ 0,3 (v i ⊗ v j ⊗ v k ) vanishes unless the three indices coincide. For equal indices, we have

ω r,τ 0,3 (v k ⊗ v k ⊗ v k ) = (-1) k-1 φ (r-2)/4 r/2 sin( kπ r ) . ( 19
)
After applying the splitting axioms of the CohFT to

ω r,τ g,n (v k ⊗ • • • ⊗ v k ) [C]
,

each node contributes the sign (-1) k-1 . Including the sign in ( 19) associated to each rational component, the total sign is

(-1) (k-1)(2g-2+n) • (-1) (k-1)(3g-3+n) = (-1) (k-1)(g-1)
.

Hence, for k ∈ {1, . . . , r -1},

ω r,τ g,n (v k ⊗ • • • ⊗ v k ) = r 2 2g-2+n
φ (r-2)(2g-2+n)/4 (-1) (k-1)(g-1) sin( kπ r )

2g-2+n . ( 20
)
The change of basis matrix from ∂ a to v k is self-inverse. 11 Therefore, we have

∂ a = 2 r r-1 k=1 sin (a + 1)kπ r v k .
The Proposition then follows from (20).

Proof of Proposition 2.3.

We provide here direct proofs of the claims of Proposition 2.3. The methods are via simple manipulation of trigonometric functions, but we were not able to find an elementary reference.

The scalar square η(v k , v k ). We have

η(v k , v k ) = 2 r r-2 a=0 sin (a + 1)kπ r sin (r -1 -a)kπ r = (-1) k-1 2 r r-2 a=0 sin 2 (a + 1)kπ r = (-1) k-1 1 r r-2 a=0 1 -cos 2(a + 1)kπ r = (-1) k-1 1 r (r -1) -(-1) = (-1) k-1 .
The scalar product η(v k , v l ) for k = l. We have

η(v k , v l ) = 2 r r-2 a=0 sin (a + 1)kπ r sin (r -1 -a)lπ r = (-1) l-1 2 r r-2 a=0 sin (a + 1)kπ r sin (a + 1)lπ r = (-1) l-1 1 r r-2 a=0 cos (a + 1)(k -l)π r -cos (a + 1)(k + l)π r = (-1) l-1 1 r (-1) -(-1) = 0 . The quantum square v k • τ v k . The coefficient of ∂ c in v k • τ v k is equal to 2φ (r-2)/4 r a,b sin (a + 1)kπ r sin (b + 1)kπ r ,
where the sum runs over the set shown in Figure 1. The sum can be conveniently reparameterized by the change of variables

p = a + b + 2 , q = a -b .
We obtain the following expression for the coefficient of

∂ c in v k • τ v k : 2φ (r-2)/4 r 2r-2-c p=c+2 c q=-c sin (p + q)kπ 2r sin (p -q)kπ 2r = φ (r-2)/4 r 2r-2-c p=c+2 c q=-c cos qkπ r -cos pkπ r , (21)
where the sums run over only over integers p and q which have the same parity as c. The second line of ( 21) can be rewritten as

(r -c -1) φ (r-2)/4 r c q=-c exp iqkπ r -(c + 1) φ (r-2)/4 r 2r-2-c p=c+2 exp ipkπ r ,
where denotes the real part. The first term equals (r -c -1)

φ (r-2)/4 r exp i(c+1)kπ r -exp -i(c+1)kπ r exp( ikπ r ) -exp(-ikπ r )
, and the second term equals

-(c + 1) φ (r-2)/4 r exp i(2r-1-c)kπ r -exp i(c+1)kπ r exp( ikπ r ) -exp(-ikπ r )
.

Since exp i(2r-1-c)kπ r and exp -i(c+1)kπ r are equal, the first and second terms combine as

φ (r-2)/4 exp i(c+1)kπ r -exp -i(c+1)kπ r exp( ikπ r ) -exp(-ikπ r ) = φ (r-2)/4 sin (c+1)kπ r sin( kπ r )
.

Hence, we obtain

v k • τ v k = φ (r-2)/4 sin( kπ r ) r-2 c=0 sin (c + 1)kπ r ∂ c = φ (r-2)/4 r/2 sin( kπ r ) v k . The quantum product v k • τ v l for k = l. The coefficient of ∂ c in v k • τ v l is 2φ (r-2)/4 r a,b sin (a + 1)kπ r sin (b + 1)lπ r , (22)
where the sum runs over the set shown in Figure 1. As before, we set

p = a + b + 2 , q = a -b
and write [START_REF] Pandharipande | The κ ring of the moduli of curves of compact type[END_REF]

as 2φ r-2 4 r 2r-2-c p=c+2 c q=-c sin (p + q)kπ 2r sin (p -q)lπ 2r = φ r-2 4 r 2r-2-c p=c+2 c q=-c cos q(k + l) + p(k -l) 2r π -cos p(k + l) + q(k -l) 2r π ,
where the sums run over only over integers p and q which have the same parity (mod 2) as c. The result can be rewritten as

φ r-2 4 r 2r-2-c p=c+2 c q=-c exp q(k + l) + p(k -l) 2r iπ -exp p(k + l) + q(k -l) 2r iπ ,
where denotes the real part as before. After further transformation, we obtain

φ (r-2)/4 r 2r-2-c p=c+2 c q=-c exp q(k + l) 2r iπ exp p(k -l) 2r iπ - φ (r-2)/4 r 2r-2-c p=c+2 c q=-c exp p(k + l) 2r iπ exp q(k -l) 2r iπ .
Consider the following four functions: .

A = c q=-c exp q(k + l) 2r iπ = exp (c+1)(k+l)iπ 2r -exp -(c+1)(k+l)iπ 2r exp (k+l)iπ 2r -exp -(k+l)iπ 2r , B = 2r-2-c p=c+2 exp p(k -l) 2r iπ = exp (2r-1-c)(k-l)iπ 2r -exp (c+1)(k-l)iπ 2r exp (k-l)iπ 2r -exp -(k-l)iπ 2r = - exp (c+1)(k-l)iπ 2r -exp -(c+1)(k-l)iπ 2r exp (k-l)iπ 2r -exp -(k-l)iπ 2r , C = c q=-c exp q(k -l) 2r iπ = exp (c+1)(k-l)iπ 2r -exp -(c+1)(k-l)iπ 2r exp (k-l)iπ 2r -exp -(k-l)iπ 2r , D = 2r-2-c p=c+2 exp p(k + l) 2r iπ = exp (2r-1-c)(k+l)iπ 2r -exp (c+1)(k+l)iπ
Since AB -CD = 0, we conclude

v k • τ v l = 0 for k = l.
3 Tautological relations

Proof of Theorem 4

By Teleman's classification [START_REF] Teleman | The structure of 2D semi-simple field theories[END_REF], the semisimple CohFT W r,τ for τ = (0, . . . , 0, rφ) ∈ V r defined in Section 0.5 is given by Givental's action [START_REF]Gromov-Witten invariants and quantization of quadratic Hamiltonians[END_REF] of the R-matrix computed in Section 2.3 on the topological field theory ω r,τ computed in Section 2.4. By the degree bound,

deg p m * W r g,n+m (e a 1 ⊗ • • • ⊗ e an ⊗ τ ⊗ • • • ⊗ τ ) ≤ (g -1)(r -2) + a i + m(r -2) r -m = D g,n (a 1 , . . . , a n ) - 2m r , W r,τ g,n (e a 1 ⊗ • • • ⊗ e an ) ∈ H * (M g,n ) is a cohomology class with top degree D r g,n (a 1 , . . . , a n ) = (g -1)(r -2) + a i r equal to W r g,n (a 1 , . . . , a n ) ∈ H 2D r g,n (a) (M g,n
) and lower degree terms. Hence, for any d > D r g,n (a 1 , . . . , a n ), the degree d part of the stable graph expression of Ω r,τ = R.ω r,τ vanishes. The proof of Theorem 4 is complete.

The proof of Theorem 4 also yields an explicit calculation of Witten's r-spin class.

Theorem 8. W r g,n (a 1 , . . . , a n ) equals the degree D r g,n (a 1 , . . . , a n ) part of the stable graph expression of

Ω r,τ = R.ω r,τ in H * (M g,n ).
As a consequence of Theorem 8, we see

W r g,n (a 1 , . . . , a n ) ∈ RH * (M g,n ) . ( 23
)
Since Theorem 8 and the implication [START_REF] Pandharipande | A calculus for the moduli space of curves, Proceedings of Algebraic geometry -Salt Lake City[END_REF] do not concern relations, Janda's results do not apply: we do not have a proof of the lifts to Chow of these two statements.

An example

: g = 1, n = 1 Let r ≥ 2. Let g = 1, n = 1, a 1 = a ∈ {0, . . . , r -2}, d = 1. We have D r 1,1 (a) = a r < 1 .
Thus, the degree 1 part of the stable graph expression of

Ω r,τ 1,1 (e a ) = φ (2a-r+2)/4 Ω r,τ 1,1 ( ∂ a )
is a tautological relation for every a. To write the relations, we will require the coefficient

b r,a = (2a + 2 + r)(2a + 2 -r) 16r 2 of T in B r,a .
There are exactly three stable graphs 12 which contribute to the degree 1 part of Ω r,τ 1,1 ( ∂ a ): the graph Γ 1 with one genus 1 vertex and one marked leg, the graph Γ 2 with one genus 1 vertex, one marked leg, and one κ-leg, and the graph Γ 3 with one genus 0 vertex, a leg, and a loop.

• Γ 1 contributes φ -r/2 b r,a ω r,τ 1,1 ( ∂ r-2-a ) ψ 1 , • Γ 2 contributes -φ -r/2 b r,0 ω r,τ 1,2 ( ∂ a ⊗ ∂ r-2 ) κ 1 , • Γ 3 contributes -φ -r/2 r-2 a =0 b r,r-2-a ω r,τ 0,3 ( ∂ a ⊗ ∂ a ⊗ ∂ a ) δ .
Here, δ is the class Poincaré dual to the boundary. The factor φ -r/2 comes from the series B r,a (φ -r/2 z) in the R-matrix. Using Proposition 2.4, we obtain the evaluations:

ω r,τ 1,1 ( ∂ r-2-a ) = (a + 1)φ (r-2)/4 δ even r-a , ω r,τ 1,2 ( ∂ a ⊗ ∂ r-2 ) = (a + 1)φ (r-2)/4 δ even r-a , ω r,τ 0,3 ( ∂ a ⊗ ∂ a ⊗ ∂ a ) = φ (r-2)/4 if r -a is even and r-2-a 2 ≤ a ≤ r-2+a 2 , 0 otherwise.
Hence, we have

φ -r 2 b r,a ω r,τ 1,1 ( ∂ r-2-a ) = φ -r+2 4 (2a + 2 + r)(2a + 2 -r)(a + 1) 16r 2 δ even r-a , -φ -r 2 b r,0 ω r,τ 1,2 ( ∂ a ⊗ ∂ r-2 ) = φ -r+2 4 (r + 2)(r -2)(a + 1) 16r 2 δ even r-a , -φ -r 2 r-2 a =0 b r,r-2-a ω r,τ 0,3 ( ∂ a ⊗ ∂ a ⊗ ∂ a ) = -φ -r+2 4 δ even r-a (r-2+a)/2 a =(r-2-a)/2 (2a + 2 -r)(2a + 2 -3r) 16r 2 = -φ -r+2 4 a(a + 1)(a + 2) 48r 2 δ even r-a .
After dividing by the common factor -φ -(r+2)/4 (a + 1) 16r 2 , we obtain the following statement. Proposition 3.1. For a ∈ {0, . . . , r -2} of the same parity as r, we have

(r -2a -2)(r + 2a + 2) • ψ 1 -(r -2)(r + 2) • κ 1 + a(a + 2) 3 • δ = 0 ∈ H 2 (M 1,1 ).
After regrouping the terms, we write the relation of Proposition 3.1 as

(r 2 -4)(ψ 1 -κ 1 ) + (2a + a 2 ) δ 3 -4ψ 1 = 0 ,
which can only be satisfied for all the possible choices of a and r if

δ 12 = κ 1 = ψ 1 .

Restriction to M g,n

The tautological relations of Theorem 4 become much simpler when restricted to the interior M g,n ⊂ M g,n as the graph sum is then reduced to a single term. In order to prove Theorem 5, we will alter the relations slightly before restricting to the interior. Let σ, a 1 , . . . , a n and d be as in the statement of Theorem 5, so

rd > (r -2)(g -1) + |σ| + n i=1 a i .
Since a i and σ j avoid r -1 mod r, we can write

a i = b i + rc i , σ j = b n+j + rc n+j with 0 ≤ b i ≤ r -2 for all 1 ≤ i ≤ n + (σ). For C = n+ (σ) i=1 c i , we have r(d -C) > (r -2)(g -1) + n+ (σ) i=1 b i . By Theorem 4 the degree d -C part of Ω r,τ g,n+ (σ) ( ∂ b 1 ⊗ • • • ⊗ ∂ b n+ (σ) )
yields a tautological relation on M g,n+ (σ) which we write as

X = 0 ∈ H 2(d-C) (M g,n+ (σ) ) . ( 24
)
Push-forward yields a tautological relation of degree d on M g,n ,

p (σ) *   (σ) j=1 ψ c n+j +1 n+j • X   = 0 ∈ H 2d (M g,n ) . ( 25
)
We restrict [START_REF] Pandharipande | Relations on M g,n via 3-spin structures[END_REF] to the interior to obtain a tautological relation of degree d on M g,n .

The only stable graph for M g,n+ (σ) which contributes to the relation ( 24) is the principal graph Γ • with a single vertex and no edges. All other strata classes are either annihilated by multiplying by j ψ n+j or remain supported on the boundary after push-forward by the forgetful map to M g,n .

All the factors in the contribution of Γ • match up exactly with the formula given in Theorem 5 except for the values of the topological field theory ω r,τ which are absent. A leg (or κ-leg) assigned vector ∂ a produces the series B r,a by combining the two nonzero entries in column a of the matrix R -1 . The topological field theory value which appears is

ω r,τ g,n+ (σ)+m ( ∂ b 1 , . . . , ∂ b n+ (σ) , ∂ 0 , . . . , ∂ 0 ),
where x represents either x or r -2-x depending on whether the diagonal or antidiagonal entry in R -1 was chosen. The number of times an antidiagonal 13entry is chosen is congruent mod 2 to the degree d -C. After specialization to φ = 1, the easily checked identity

ω r,τ g,s+1 ( ∂ x 1 , . . . , ∂ xs , ∂ r-2 ) = ω r,τ g,s ( ∂ x 1 , . . . , ∂ x s-1 , ∂ r-2-xs ),
then implies that the topological field theory value does not depend on any of the choices made (and can be divided out by the nonvanishing result below). The parity condition in the statement of Theorem 5,

rd ≡ (r -2)(g -1) + |σ| + n i=1 a i mod 2 ,
together with parity constraint on the number of times an antidiagonal is chosen, implies

n+ (σ) i=1 b j ≡ r(g -1) mod 2 . Lemma 3.2. Let g ≥ 1, s ≥ 0 with 2g -2 + s > 0. Let x 1 , . . . , x s ∈ {0, . . . , r -2} satisfy s i=1 x i ≡ r(g -1) mod 2.
Then, ω r,τ g,s (x 1 , . . . , x s ) = 0.

Proof. We will use the formula for ω r,τ 0,3 given by Proposition 2.1 to induct on g and n. In fact, the argument will prove ω r,τ g,s ( ∂ x 1 , . . . , ∂ xs ) > 0 for φ > 0.

Our base case is ω r,τ 1,1 (x 1 ) with x 1 even. After applying the splitting axiom of the CohFT, we obtain

ω r,τ 1,1 ( ∂ x 1 ) = r-2 x=0 ω r,τ 0,3 ( ∂ x 1 , ∂ y , ∂ r-2-y )
and all of the terms are non-negative by Proposition 2.1. For y = x 1 2 , the point (x 1 , y, y) lies in the tetrahedron described in Proposition 2.1,

ω r,τ 0,3 ∂ x 1 , ∂ x 1 2 , ∂ r-2-x 1 2 > 0 .
Next, we prove the case g = 1, n > 1 by induction on n. We have

ω r,τ 1,n ( ∂ a 1 , . . . , ∂ a n-1 , ∂ an ) = r-2 x=0 ω r,τ 1,n-1 ( ∂ a 1 , . . . , ∂ a n-2 , ∂ x )•ω 0,3 ( ∂ a n-1 , ∂ an , ∂ r-2-x ),
and we may assume a n-1 ≡ a n mod 2. As before, non-negativity means we need only find a single value of x such that ω 0,3 (a n-1 , a n , r -2-x) is nonzero. The tetrahedron constraints are satisfied for x = |a n -a n-1 |. Finally, we treat the case g > 1 by induction on g. We have

ω r,τ g,n ( ∂ a 1 , . . . , ∂ an ) = r-2 x=0 ω r,τ g-1,n+2 ( ∂ a 1 , . . . , ∂ an , ∂ x , ∂ r-2-x )
and all of the terms on the right are positive by the inductive hypothesis (since the parity condition is preserved).

Proof of Theorem 6

We will now use the relations of Theorem 5 with r = 4 to bound the Betti numbers of the tautological ring of M g . 14In the case r = 4 and n = 0, the relations of Corollary 5 are parameterized by partitions σ with no parts congruent to 3 mod 4 and positive integers d satisfying 4d > 2(g -1) + |σ| and |σ| ≡ 0 (mod 2).

We discard the relations coming from partitions σ containing an odd part and then halve all parts of σ. The remaining relations are then simply indexed by partitions σ and positive integers d satisfying 2d ≥ g + |σ|.

For D s (T ) = B 4,2s (T ), the relations of Theorem 5 are obtained by taking the degree d part of

m≥0 1 m! p m+ (σ) * (σ) j=1 (T D σ j )(ψ j ) m k=1 (T -T D 0 )(ψ (σ)+k ) . (26)
We will alter the definition of D 1 to kill the constant term:

D 1 (T ) = B 4,2 (T ) -B 4,0 (T ).
A straightforward check shows the span of the relations ( 26) is unchanged by the new definition of D 1 .

Proof of Theorem 6. The push-forward kappa polynomials

p (τ ) * (τ ) i=1 ψ τ i +1 i , p (τ ) : M g, (τ ) → M g , (27)
where τ is a partition of d, form a basis for the vector space of (formal) kappa polynomials of degree d. We will use the push-forward basis [START_REF] Pixton | The tautological ring of the moduli space of curves[END_REF] to obtain a lower bound for the rank of the relations given in [START_REF] Pixton | Conjectural relations in the tautological ring of M g[END_REF]. Given any two partitions σ and τ , let K(σ, τ ) be the coefficient of the push-forward kappa polynomial corresponding to τ in [START_REF] Pixton | Conjectural relations in the tautological ring of M g[END_REF]. Define a matrix M with rows and columns indexed by partitions of d by

M στ = K(σ -, τ ) ,
where σ -is the partition formed by reducing each part of σ by 1 and discarding the parts of size 0. By Proposition 3.3 below, M is invertible.

The invertibility of M implies Theorem 6 by the following argument. If |σ| = d and (σ) ≥ g -d, then

|σ -| ≤ d -(g -d) = 2d -g ,
so the row corresponding to σ actually contains the coefficients of a relation obtained from Theorem 5. Since M is invertible, all such relations are linearly independent, so the quotient of the space of degree d kappa polynomials by such relations has dimension at most the number of partitions of d of length at most g -1 -d, as desired.

Proposition 3.3. The matrix M is invertible.

Proof. We will show the invertibility of M by constructing another matrix A of the same size and checking that the product MA is upper-triangular (with nonvanishing diagonal entries) with respect to any ordering of the partitions of d which places partitions containing more parts of size 1 after partitions containing fewer parts of size 1.

First, we compute the coefficient K(σ, τ ) as a sum over injections from the set of parts of σ to the set of parts of τ describing which factors in [START_REF] Pixton | Conjectural relations in the tautological ring of M g[END_REF] produce which psi powers. We write such an injection as

φ : σ → τ .
The parts of τ which are not in the image of φ are produced by the factors involving D 0 . The result is

K(σ, τ ) = (-1) (τ )-(σ) |Aut(τ )| φ : σ → τ i φ →j [D i ] T j j∈(τ \φ(σ)) [D 0 ] T j . (28)
We define the matrix A as follows. For any partitions τ and µ of the same size,

A τ,µ = ψ : τ →µ refinement |Aut(τ )| k∈µ |Aut(ψ -1 (k))| k∈µ (ψ -1 (k)) + 2k + 1 ! j∈τ 1 (2j + 1)!! ,
where the sum runs over all partition refinements

ψ : τ → µ ,
functions from the set of parts of τ to the set of parts of µ such that the preimage of each part k of µ is a partition of k.

We factor the sums appearing in the entries of the product matrix MA:

τ K(σ -, τ )A τ,µ = ξ:σ -→µ k∈µ σ =ξ -1 (k) τ K(σ , τ )A τ ,(k) , (29)
where ξ : σ -→ µ is a function from the set of parts of σ -to the set of parts of µ.

In order to understand [START_REF] Polishchuk | Witten's top Chern class on the moduli space of higher spin curves in Frobenius manifolds[END_REF], we must study the sum

τ K(σ -, τ )A τ,(k) . (30)
After expanding [START_REF] Polishchuk | Moduli spaces of curves with effective r-spin structures in Gromov-Witten theory of spin curves and orbifolds[END_REF] via formula [START_REF] Polishchuk | Algebraic construction of Witten's top Chern class in Advances in algebraic geometry motivated by physics[END_REF] for K(σ -, τ ) and the definition of A τ,(k) , the result is [START_REF] Sauvaget | Cohomology classes of strata of differentials[END_REF] where the first sum is over all partitions τ of k. Next, we include formal variable t to keep track of the size of τ and factor based on the values of the images of the parts of σ -under φ via the series

(-1) (σ) τ k φ : σ -→ τ (-1) (τ ) ( (τ ) + 2k + 1)! |Aut(τ )| • j∈τ 1 (2j + 1)!! i φ →j [D i ] T j j∈(τ \φ(σ -)) [D 0 ] T j ,
D i (t) = j≥1 [D i ] T j t j+ 1 2 (2j + 1)!! .
After removing nonzero scaling factors, we rewrite [START_REF] Sauvaget | Cohomology classes of strata of differentials[END_REF] as

  D -2k-2 0 i∈σ - D i D 0   t -1 . ( 32
)
Up to a triangular change of basis in the D i , we have

D i = sin 2i + 1 2 sin -1 ( √ t) .
We define

θ = 1 2 sin -1 ( √ t)
and check the following two properties:

1 sin 4 θ t -1 = 0 , but 1 sin e θ t -1 = 0
for every even e ≥ 6. Using these facts to compute (32), we conclude

τ K(σ -, τ )A τ,(k) = 0 whenever |σ -| < k -1, and τ K((k -1), τ )A τ,(k) = 0 . (33)
We now return to the matrix MA. Suppose that σ and τ are partitions of d containing S and T parts equal to 1 respectively and MA στ = 0 .

By the identity (29) and the analysis above, there exists a function ξ : σ -→ τ for which the preimage of each part k ∈ τ is a partition of size at least k -1. Thus, we have |σ| -

(σ) = |σ -| ≥ |τ | -(τ ) . (34)
Since |σ| = |τ | = d, we see (τ ) ≥ (σ). Moreover, comparing lengths of the partitions, we obtain

(σ) -S = (σ -) ≥ (τ ) -T . ( 35
)
Adding the inequalities [START_REF] Witten | Algebraic geometry associated with matrix models of twodimensional gravity in Topological methods in modern mathematics[END_REF] and (35), we conclude S ≤ T . If S = T , then the ξ-preimage of each part k ∈ τ must be of size k -1 and have length one if k > 1, which implies τ = σ. Thus, the matrix MA is triangular. The nonvanishing of the diagonal entries follows from the nonvanishing [START_REF] Verlinde | Fusion rules and modular transformations in 2D conformal field theory[END_REF]. [START_REF] Chiodo | The Witten top Chern class via K-theory[END_REF] The semisimple point: τ = (0, rφ, 0, . . . , 0)

Another shift

The shift along the second basis vector e 1 ∈ V r also yields a semisimple CohFT with attractive properties. The associated topological field theory ω r, τ is very simple, much simpler than ω r,τ , but the R-matrix is not as explicit. A basic polynomiality property of Witten's r-spin class will be proven using τ .

The quantum product

Recall the notation for genus 0 correlators,

a 1 , . . . , a n r = M 0,n W r 0,n (a 1 , . . . , a n ) ∈ Q , discussed in Section 1.
As before, we will often drop the superscript r. All correlators involving an a i = 1 vanish whenever n ≥ 5.

Proof. The values of 3-and 4-point correlators are well-known (and were stated in (4) of Section 0.3). The only nontrivial claim here is the vanishing for n ≥ 5 which is a direct consequence of Proposition 1.4.

By Proposition 4.1, the quantum product at τ is given by

∂ a • τ ∂ b = ∂ a+b if a + b ≤ r -2 , φ ∂ a+b-r+1 if a + b ≥ r -1 .
To simplify computations, we introduce a new frame 15∂ a = φ -a/(r-1) ∂ a .

The quantum multiplication then takes the form

∂ a • τ ∂ b = ∂ a+b if a + b ≤ r -2 , ∂ a+b-r+1 if a + b ≥ r -1 .

Euler field and shifted degree

The operator of quantum multiplication by the Euler field at τ ,

E = (r -1)φ r r-1 ∂ 1 ,
is given in the frame { ∂ a } by the matrix

ξ = (r -1)φ r r-1                0 • • • • • • 0 1 1 0 0 0 1 0 0 . . . . . . . . . . . . . . . 0 • • • 0 1 0                or ξ 0 r-2 = ξ a+1 a = (r -1)φ r r-1 .
In the same frame, the shifted degree operator is

µ = 1 2r                -(r -2) 0 • • • • • • 0 0 -(r -4) 0 0 . . . . . . . . . . . . . . . 0 0 r -4 0 0 • • • • • • 0 r -2               
or µ a a = 2a-r+2 2r .

The R-matrix

Define the polynomials P m (r, a) by the following recursive procedure. Let P 0 (r, a) = 1. For m ≥ 1, let

P m (r, a) = 1 2 a b=1 (2mr -r -2b)P m-1 (r, b -1) (36) - 1 4mr(r -1) r-2 b=1 (r -1 -b)(2mr -b)(2mr -r -2b)P m-1 (r, b -1) .
The second summation can be extended to r -1 instead of r -2 because of the presence of the factor r -1 -b. The second sum is then easily seen to be divisible both by r and by r -1, so P m is indeed a polynomial. The first few values are: There appears to be no closed formula for the polynomials P m . However, we will present a closed expression for P m (0, a) in Proposition A.2 in terms of Bernoulli polynomials.

P 0 = 1, P 1 = 1 2 a(r -1 -a) - 1
Lemma 4.3. The polynomials P m satisfy the relations

P m (r, a) -P m (r, a -1) = 1 2 (2mr -r -2a)P m-1 (r, a -1) (37)
P m (r, 0) = P m (r, r -1) (38)
and are the unique solutions to these equations with initial condition P 0 = 1.

Proof. Given P m-1 , equation (37) determines P m uniquely up to a polynomial in r independent of a. Equation (38) for P m+1 then determines this polynomial in r. The uniqueness statement is therefore established.

Equation (37) follows directly from the definition of P m . To show that the second equation is satisfied by P m+1 , a calculation is required. The definition of P m implies P m+1 (r, r -1) -P m+1 (r, 0) = 1 2 In equation (39), we exchange the summation order in the first term and use the identities above. We obtain Proof. The uniqueness of the solution follows from the semisimplicity of the Frobenius manifold V r at τ (proven in Section 4.2). Since P 0 = 1, the formula for R 0 yields the identity matrix. We must check that the formula for R is indeed a solution of the recursion

(2mr + r -2a -2)(2mr -r -2b)P m-1 (r, b -1) (39) - 1 8mr(r -1) r-2 a=0 (2mr + r -2a -2) × r-2 b=1 (r -1 -b)(2mr -b)(2mr -r -2b)P m-1 (r, b -1) .

Using the evaluation

P m+1 (r, r -1) -P m+1 (r, 0) = 1 4 r-2 b=1 (r -1 -b)(2mr -b)(2mr -r -2b)P m-1 (r, b -1) - 1 8mr(r -1) 2mr(r -1) × r-2 b=1 (r -1 -b)(2mr -b)(2mr -r -2b)P m-1 (r, b -
[R m+1 , ξ] = (m + µ)R m .
Explicitly, we must show

(R m+1 ) b a+1 ξ a+1 a -ξ b b-1 (R m+1 ) b-1 a = (m + µ b b )(R m ) b a , or, equivalently, (r -1)φ r r-1 (R m+1 ) b a+1 -(R m+1 ) b-1 a = 2mr -r + 2b + 2 2r (R m ) b a ,
where both a + 1 and b -1 are understood modulo r -1.

The nonvanishing condition b + m = a (mod r -1) is simultaneously satisfied or not satisfied in all three terms of the equality. The formula for R m contains the factor -r(r -1)φ r r-1 -m . After using these two observations, we obtain the final form of the equality to be checked:

P m+1 (r, r-1-b)-P m+1 (r, r-2-b) = 1 2 (2mr-r+2b+2)P m (r, r-2-b) . ( 40
)
Given a stable graph Γ with weighting, we assign to each edge e ∈ E(Γ) the edge factor

∆(e) = 1 x a y b 1 - m, ≥0 P m (r, a)P (r, b)(xψ ) m (yψ ) ψ + ψ .
Here, a and b are the weightings of the half-edges of e, ψ and ψ are the corresponding cotangent line classes, and x, y are the vertex variables corresponding to the vertices adjacent to the edge (if the edge is a loop, then x = y).

To each leg i for 1 ≤ i ≤ n we assign the leg factor

L(i) = 1 x a i v m≥0 P m (r, a i )(x v ψ i ) m ,
where a i is the weighting of the leg, ψ i is the cotangent line associated to the leg, and x v is the vertex variable of the vertex to which the leg is attached. Finally, to each κ-leg i for n + 1 ≤ i ≤ n + k, we assign the κ-factor

K(i) = -ψ i m≥1 P m (r, 0)(x v ψ i ) m ,
where ψ i is the cotangent line class of the leg, and x v is the vertex variable of the vertex to which the leg is attached.

Proposition 4.6. The class W g;a 1 ,...,an (r) is given by the degree g -1 part of the mixed degree cohomology class

k≥0 Γ∈G g,n+k weightings a (r -1) 1-h 1 (Γ) |Aut(Γ)| p * v x gv-1 v e ∆(e) n i=1 L(i) n+k i=n+1 K(i) x ,
where p : M Γ → M g,n is the natural boundary map forgetting the last k marked points.

Proof. The formula is essentially a reformulation of the R-matrix action described in Theorem 9. To simplify the computations, we set φ = 1. The powers of x v keep track of the remainders modulo r -1. More precisely, the base vector e a corresponds to x -a . The bi-vector η ab is then encoded by the expression 1

x a y b with a+b = r -2. The matrix R -1 m takes e a to a multiple of e b with b = a-m mod r-1. Therefore the coefficients of ψ m in the formulas come with an mth power of the corresponding vertex variable. Finally, putting a factor x g-1 v on each vertex and taking the coefficient of x 0 v allows one to encode the condition g -1 -a i = 0 mod r -1 which appears in topological field theory ω r, τ g,n . In order to take into account the rescaling by r g-1 , we have removed the factor r m from the denominator of the mth term R m of the r-matrix. Thus the degree d part of the resulting mixed degree class is multiplied by r d . In particular, the degree g -1 part, which corresponds to Witten's r-spin class, is multiplied by r g-1 giving the rescaled class.

Finally, we account for all the occurrences of r -1. There is a factor of (r -1) m in the denominator or R m , leading, as above, to a global factor of (r -1) 1-g . There is also a factor (r -1) gv in the topological field theory at the vertex v. The latter yield (r -1) g-h 1 (Γ) . After multiplying the two factors, we obtain (r -1) 1-h 1 (Γ) . We will use Proposition 4.6 to prove the polynomiality assertion of Theorem 7. However, we will temporarily remove the division by ψ + ψ from the edge factor ∆. We will study the polynomiality in r of the formula of Proposition 4.6 without the division by e∈E(Γ) (ψ e + ψ e ). For each stable graph Γ, we will prove the polynomiality in r of the degree

g -1 + |E(Γ)|
part of the formula of Proposition 4.6 without denominators, where |E(Γ)| is the number of edges.

The division by e∈E(Γ) (ψ e +ψ e ) will be taken afterwards by the following argument. Consider the expression of Proposition 4.6 as an element of the strata algebra not quotiented by any tautological relations, not even the relations due to the degree of the cohomology class supported by M v being higher than the dimension of M v for some vertex v. Then, ψ e + ψ e is not a zero divisor in the strata algebra. Division by e∈E(Γ) ψ e + ψ e , when possible at all, is therefore uniquely defined and preserves the property of being a polynomial in r.

type of vertex of Γ a b effect on x midpoint of edge h-h in Γ r -2 0 x h + x h = r -2 endpoint of leg h in Γ a h 0 x h = a h vertex v of Γ g v -1 + h →v m h b v TopFT condition mod r -1 at v
The numbers b v in the table can take different values for different weightings. However, for a given graph Γ and a given choice of integers m h , there are only finitely many possible values b v . Thus, the sum S Γ,m over all weightings can be decomposed into a finite number of sums of the form of [17, Proposition A1]. Hence, by [17, Proposition A1], S Γ,m is a polynomial in r.

Lemma 4.8. The polynomial S Γ,m is divisible by (r -1) h 1 (Γ) .

Proof. Once again we follow Pixton's proof in [START_REF] Janda | Double ramification cycles on the moduli spaces of curves[END_REF]Appendix]. Let Q be a polynomial in N variables with (p-integral) Q-coefficients. According to [START_REF] Janda | Double ramification cycles on the moduli spaces of curves[END_REF]Equation 33], the sum

0≤w 1 ,...,w N ≤p Q(w 1 , . . . , w N ) (43)
over the N -tuples satisfying D integral linear equations mod p is divisible by p N -D for every large enough prime p. In our case, the sum S Γ,m has exactly the form (43) if we take p = r -1 to be prime. The number N of variables is equal to the number |E(Γ)| of edges of the graph Γ. The number of mod p linear equations is equal to |V(Γ)| -1, where V (Γ) is vertex set of Γ. Indeed, there is one mod p condition per vertex, but one condition is redundant, since the sum of the conditions is equal to 2g -2 -

a i = 0 mod p , (44)
a condition that is automatically satisfied.

We check the assertion that the sum of the vertex condition yields (44) as follows. First, we sum the vertex conditions e∈E(Γ)

w e + (r -2 -w e ) + n i=1 a i - h∈H(Γ) m h = v∈V(Γ) (g v -1) mod r -1 .
We rewrite the above as

(r -2)|E(Γ)| + n i=1 a i -(g -1) = g -1 -|E(Γ)| mod r -1 , or equivalently, (r -1)|E(Γ)| + n i=1 a i = 2g -2 mod r -1 , which is exactly (44). Thus, by [17, Appendix A.3], S Γ,m is divisible by (r -1) N -D = (r -1) |E(Γ)|-|V(Γ)|+1 = (r -1) h 1 (Γ)
for r-1 prime and large. Since we already know that S Γ,m is a polynomial in r for r large enough, we conclude the polynomial is divisible by (r -1) h 1 (Γ) .

A Holomorphic differentials is not injective, every component of H g (a 1 , . . . , a n ) has dimension at least 2g -2 + n in M g,n by degeneracy loci considerations [START_REF] Fulton | Intersection theory[END_REF]. Polishchuk [START_REF] Polishchuk | Moduli spaces of curves with effective r-spin structures in Gromov-Witten theory of spin curves and orbifolds[END_REF] has shown that H g (a 1 , . . . , a n ) is a nonsingular substack of M g,n of pure dimension 2g -2 + n. Hence, the Zariski closure H g (a 1 , . . . , a n ) ⊂ H g (a 1 , . . . , a n ) ⊂ M g,n defines a cycle class [H g (a 1 , . . . , a n )] ∈ H 2(g-1) (M g,n ) .

Our goal in the Appendix is to relate [H g (a 1 , . . . , a n )] to a certain limit of Witten's r-spin classes.

A compact moduli space of twisted canonical divisors which includes the moduli of holomorphic differentials H g (a 1 , . . . , a n ) is defined in [START_REF] Farkas | The moduli space of twisted canonical divisors[END_REF]. A detailed A.3 Evidence. Genus 1. Witten's class W r 1,n (0, . . . , 0) has degree 0. To evaluate the class, the topological field theory is enough (we do not need the R-matrix). A simple computation shows that Witten's class here is equal to r -1. Indeed, The result coincides with the well-known formula for the locus of genus 2 curves with a pair of conjugate points, see [START_REF] Belorousski | A descendent relation in genus 2[END_REF]Lemma 6].

is a power series without constant term and F = exp(f ) is the exponential, then

where p k : M g,n+k → M g,n is the forgetful map.

As before, to every vertex v of a stable graph we assign a formal variable x v satisfying x r-1 v = 1. For a polynomial Π in variables x v , we will denote by {Π} x the term of degree 0 in all variables x v .

Given a stable graph, to each edge e we assign the edge factor 18

∆(e) = Here, a and b are non-negative integers, ψ and ψ are the cotangent line classes corresponding to the half-edges of e, and x, y are the vertex variables corresponding to the vertices adjacent to the edge (if the edge is a loop, then x = y). Furthermore, to each leg i we assign the leg factor

L(i) = 1 x a i v exp - m≥1 B m+1 (a i + 1) m(m + 1) (x v ψ i ) m ,
where ψ i is the cotangent line class associated to the leg, and x v is the vertex variable of the vertex to which the leg is attached. Finally, we assign to each vertex v the vertex factor

κ(v) = x g-1 v exp m≥1 B m+1 (1) m(m + 1) x m v κ m .
Here, x v is the variable of the vertex and κ m is the κ class on the moduli space M v corresponding to v. 18 Division by ψ + ψ is only possible for the constant term in r of the edge factor.

The coefficient (-1) g W g;a 1 ,...,an (0) of the rescaled Witten class (-1) g r g-1 • W r g (a 1 , . . . , a n ) is given by the r 0 coefficient of the degree g -1 part of given in Proposition A.2. The sum over weightings a has been moved to the sum in the definition of the edge term ∆(e). The push-forward of the κ-factors associated to the legs marked by n + 1 ≤ i ≤ n + k adjacent to a vertex v is collected in the vertex factor κ(v). The factor r -1 becomes -1. Finally, we multiply the resulting expression by the global factor of (-1) g .

The advantage of Proposition A.3 is that all the inputs are explicit. Proposition 4.6 is more general (and determines the full r dependence of W g;a 1 ,...,an (r)), but involves the polynomials P (r, a) for which we know no closed formula.

Remark A.4. The class [H g (a 1 , . . . , a n )] ∈ H 2(g-1) (M g,n ) on the moduli of nonsingular curves is easily determined by the classical Thom-Porteous formula 19 to be

c g-1 R 1 π * ω - n i=1 a i p i -R 0 π * ω - n i=1 a i p i , (50)
where π : C → M g,n is the universal curve. We have checked that the restriction of Proposition A.3 to M g,n agrees with (50) calculated by Grothendieck-Riemann-Roch. As a result, the restriction of Conjecture A.1 to M g,n is correct.

Theorem 5 .

 5 Let σ and a avoid r -1 mod r, and let d satisfy rd > (r -2)(g -1) + |σ| +

0. 10 Acknowledgments

 10 We are grateful to A. Buryak, A.Chiodo, E. Clader, C. Faber, G. Farkas, J. Guéré, F. Janda, Y.-P. Lee, P. Rossi, Y. Ruan, and S. Shadrin for discussions about Witten's r-spin theory. The Workshop on Pixton's conjectures at ETH Zürich in October 2014 played in an important role in our collaboration. A. P. and D. Z. have been frequent guests of the Forschungsinstitut für Mathematik (FIM) at ETHZ. The conjecture of the Appendix was formulated at the conference Moduli spaces of holomorphic differentials at Humboldt University in Berlin in February 2016. R. P. was partially supported by SNF-200021143274, SNF-200020162928, ERC-2012-AdG-320368-MCSK, ERC-2017-AdG-786580-MACI, SwissMAP, and the Einstein Stiftung. A. P. was supported by a fellowship from the Clay Mathematics Institute. D. Z. was supported by the grant ANR-09-JCJC-0104-01. This project has received funding from the European Research council (ERC) under the European Union Horizon 2020 research and innovation program (grant agreement No. 786580).

Proposition 2 . 1 . 2 r - 2 Figure 1 :

 21221 Figure 1: Quantum product: nonzero coefficients

-

  exp -(k+l)iπ 2r

Proposition 4 . 1 .

 41 We have the evaluations a, b, c = 1 for a + b + c = r -2 , a, b, c, 1 = 1 r for a + b + c = 2r -3 .

  r -2a -2)P m (r, a) , where we have substituted a = b -1 for the summation variable. From the definition of P m , we obtain P m+1 (r, r -1) -P m+1 (r,

  r -2a -2) = (r -1 -b)(2mr -b), r -2a -2) = 2mr(r -1) .

Proposition 4 . 4 .

 44 The unique solution R(z) = ∞ m=0 R m z m ∈ End(V r )[[z]] of the equations [R m+1 , ξ] = (m + µ)R m with the initial condition R 0 = 1 has coefficients (R m ) b a = -r(rm (r, r -2 -b) , if b + m = a mod r -1and 0 otherwise. The inverse matrix R -1 (z) has coefficients(R -1 m ) b a = r(r -1)φ r r-1 -m P m (r, a) , if b + m = a mod r -1and 0 otherwise.

1

 1 by F. Janda, R. Pandharipande, A. Pixton, D. Zvonkine A.Moduli space. Let g and n be in the stable range 2g -2 + n > 0, and let (a 1 , . . . , a n ) , n i=1 a i = 2g -2 be a partition with a i ≥ 0 for all i.We define the moduli space of holomorphic differentials as the closed substackH g (a 1 , . . . , a n ) = [C, p 1 , . . . , p n ] ∈ M g,n O C n i=1 a i p i = ω C ⊂ M g,n .Since H g (a 1 , . . . , a n ) is the locus of points [C, p 1 , . . . , p n ] ∈ M g,n for which the evaluation map H 0 (C, ω C ) → H 0 C, ω C|a 1 p 1 +•••+anpn

W r 1 ,- 2 a=0 1 = r - 1 . 1 = ψ 1 1 = ψ 1 + ψ 2 1 P 1 2 P 1 ( 12 (-κ 1 +

 1211111121121121 n (0, . . . , 0) = ω 1,n (0, . . . , n+2 (0, . . . , 0, a, r -2 -a) = rThus W 1;0,...,0 (r) = r -1, W 1;0,...,0 (0) = -1. According to the conjecture we have (-1) g • (-1) = 1 = [H 1 (0, . . . , 0)] ∈ RH 0 (M 1,n ),which is, indeed, true. Genus 2, n = 1, a 1 = 2. The tautological space RH 1 (M 2,1 ) is spanned by the classes ψ 1 , δ sep , and δ nonsep , where the indices sep and nonsep refer to the boundary divisors with a separating or a nonseparating node. The class κ 1 may be expressed as κ ) is obtained by pulling back the relation on M 2 expressing κ 1 in terms of boundary divisors. Theorem 9 gives an expression for Witten's class as a linear combination The class κ 1 is expressed as κ the boundary relation on M 2 . Theorem 9 gives an expression for Witten's class as a linear combination of classes κ 1 , ψ 1 , ψ 2 , α, β, γ and δ nonsep with the following coefficients: Class R-matrix TopFT Coefficientκ 1 -P 1 (r,0) r(r-1) (r -1) 2 (r-1)(r-2)(2r-1) 24r ψ (r,1) r(r-1) (r -1) 2 -(r-1)(r-2)(2r-13) 24rψ 's class by (-1) g r g-1 = r and extracting the constant term in r, we obtain1 13ψ 1 + 13ψ 2 -37α -β -13γ -δ nonsep ) .We remove κ 1 using equation (45). Conjecture A.1 predicts [H 2 (1, 1)] = ψ 1 + ψ 2 -3α -∈ RH 1 (M 2,2 ) .

k≥0 p k * n+k i=n+1 ψ i ( 1 -

 1 F (ψ i )) = exp -m≥1 c m κ m ,

a+b=r- 2 1 x a y b 1

 21 -exp -m≥1 B m+1 (a+1)•(xψ ) m +B m+1 (b+1)•(yψ ) m m(m+1) ψ + ψ .

Proposition A. 3 .

 3 Let a 1 , . . . a n ∈ Z ≥0 satisfy n i=1 a n = 2g -2 .

,

  where p : M Γ → M g,n is the natural boundary map.Proof. The result is obtained by a mod r reduction of the formula of Proposition 4.6 for the rescaled Witten class. The polynomials P m (r, a) are replaced by the expression for Q m (a) = P m (0, a)

All cohomology and Chow groups will be taken with Q-coefficients. The tautological ring in Chow is denoted byR * (M g,n ) ⊂ A * (M g,n ) .We will use the complex grading for RH * (M g,n ), so R i (M g,n ) → RH i (M g,n ).

See[START_REF] Pandharipande | A calculus for the moduli space of curves, Proceedings of Algebraic geometry -Salt Lake City[END_REF] for a survey of the Faber-Zagier relations and related topics.

A proof (unpublished) was found by Faber and Zagier in 2002. The result is also derived in [27, Section 3].

A review of Givental's R-matrix action on CohFTS in the form we require can be found in[START_REF] Pandharipande | Relations on M g,n via 3-spin structures[END_REF] Section 2].

A partition has positive integer parts. The empty partition is permitted.

The question was posed to us by P. Rossi.

The metric is η( ∂ a , ∂ b ) = δ a+b,r+2 .

In our convention for matrix coefficients, the superscript is the row index and the subscript is the column index.

Here, R 0 = 1 denotes the identity in End(V r ).

The self-inverse property follows easily from the trigonometric identities used to calculate η(v k , v l ) in Section 2.3.

1 1 κ Γ 1 Γ

Γ

We refer the reader to [25, Section 2] for a review of Givental's action of R on ω r,τ .

The parity condition here is because the antidiagonal entries are constructed from odd functions.

The analysis here was completed by A. Pixton before our study of r-spin relations started and appears in[START_REF] Pixton | The tautological ring of the moduli space of curves[END_REF]. Since there is no published reference (and for the convenience of the reader) we have included the short argument here. Several aspects are parallel to the linear algebra required in[START_REF] Pandharipande | The κ ring of the moduli of curves of compact type[END_REF].

The metric is η( ∂ a , ∂ b ) = φ -r-2 r-1 δ a+b,r-2 .

If r -1 were smaller than a i , the weighting of the leg would be a i mod r -1, but here we assume r is large.

In fact, Sauvaget proves [H g (a 1 , . . . , a n )] is tautological in Chow.

See, for example, [3, Section 2] for the Thom-Porteous approach.

The topological field theory

Proposition 4.2. We have

where δ equals 1 if g -1 -n i=1 a i is divisible by r -1 and 0 otherwise. Proof. From Proposition 4.1 and the definition of ∂ a we get

The topological field theory ω r, τ g,n for general g and n can be computed by restricting the τ -shifted r-spin theory W r, τ g,n to

where C is a completely degenerate curve with 2g-2+n rational components and 3g -3 + n nodes. The 3g -3 + n nodes divide the C into genus 0 components with 3 special points each. By the splitting axiom, we must place insertions {0, . . . , r -2} on every branch of every node in a manner such that the following conditions are satisfied: (i) the sum of the two insertions at each node equals r -2, (ii) the sum of the three insertions on each rational component of the curve plus 1 is divisible by r -1.

Conditions (i) and (ii) are impossible to satisfy if g -1 -n i=1 a i is not divisible by r -1.

If the divisibility condition is satisfied, we can first place an arbitrary insertion on a single branch of a node of every independent cycle of the dual graph of the curve. Then, the other insertions are uniquely determined. We find exactly (r -1) g possibilities. Now each rational component contributes a factor of φ -r-2 r-1 and each node a factor φ r-2 r-1 (the inverse of the metric). Collecting all the factors we get

Here, the argument r -1 -b of the polynomials should be taken modulo r -1. In other words, when b = 0 the equality reads

We first prove (40). After replacing m with m + 1 in (37), we obtain

.

After substituting a = r -1 -b, we have

which is exactly (40). In particular, for b = 0, we find

Equation (41) now follows from the equality P m+1 (r, r -1) = P m+1 (r, 0) of equation (38).

Let W r, τ be the cohomological field theory given by the shift of Witten's r-spin class by the vector τ = (0, rφ, 0, . . . , 0). Define Ω r, τ = R.ω r, τ by the action of the R-matrix of Proposition 4.4 on the topological field theory ω r, τ g,n of Proposition 4.2. Using Teleman's classification and the dimension analysis of Section 3.1, we obtain the following result parallel to Theorem 4 and Theorem 8 for the shift by τ = (0, . . . , 0, rφ). As an outcome, we obtain a second formula for Witten's r-spin class.

The parts of Ω r, τ g,n of degree higher than D r g,n vanish. A more explicit formula for the cohomological field theory Ω r, τ is provided in Proposition 4.6 of Section 4.6.

Proof of Theorem 7

We prove here Theorem 7: for n i=1 a i = 2g -2, the rescaled Witten class

is polynomial for sufficiently large values of r. Our proof will show the polynomial ( 42) is always divisible by r -1. Theorems 8 and 9 provide two formulas for Witten's r-spin class obtained by shifting by e r-2 and e 1 respectively. The formula of Theorem 8 has an explicit R-matrix, but the topological field theory is not explicitly polynomial in r. We will therefore use the formula of Theorem 9 which is explicitly polynomial. From now on, we will always assume that r is sufficiently large.

Denote by G g,n+k the set of stable graphs with n+k legs. The legs marked by n+1 to n+k, termed the κ-legs, will correspond to marked points forgotten by a forgetful map. The ψ classes corresponding to the κ-legs push-forward to κ classes.

Theorem 9 applied when n i=1 a i = 2g -2 expresses the rescaled Witten class W g;a 1 ,...,an (r) as a sum over stable graphs with weightings. • if h and h are the two half-edges of a single edge, then a h + a h = r -2,

To every vertex v of a stable graph, we assign a formal variable x v satisfying x r-1 v = 1. For a polynomial Π in variables x v , we will denote by {Π} x the term of degree 0 in all variables x v .

Let Γ be a stable graph with n + k legs. Let m be a function

and the condition

• if h and h are the two half-edges of a single edge, then (m h , m h ) = (0, 0).

Define the sum

By Lemmas 4.7 and 4.8 below, for r large enough, S Γ,m is a polynomial in r divisible by (r -1) h 1 (Γ) . By writing the coefficients of the formula of Proposition 4.6 without denominators in terms of the S Γ,m , we obtain the polynomiality required for Theorem 7. Moreover, since the prefactor in Proposition 4.6 for the rescaled Witten class is (r-1) 1-h 1 (Γ) , we also conclude that the rescaled Witten class is a polynomial in r divisible by r -1.

Lemma 4.7. The sum S Γ,m is a polynomial in r for r large enough.

Proof. The proof here follows closely Pixton's proof of polynomiality in [START_REF] Janda | Double ramification cycles on the moduli spaces of curves[END_REF]Appendix]. We will use [17, Proposition A1], but with Pixton's r replaced with r -1 (which we assume to be large enough). Let Γ be the graph obtained from Γ by adding a vertex at the end of each leg and in the middle of each edge. Let M be the edge-vertex adjacency matrix of Γ . The matrix M satisfies the assumptions of [

imposes. In fact, the latter conditions are equivalent to x being a weighting.

study of the points of the closure H g,n (a 1 , . . . , a n ) can be found in [START_REF] Bainbridge | Compactification of strata of abelian differentials[END_REF]. In [START_REF] Farkas | The moduli space of twisted canonical divisors[END_REF]Appendix], a conjecture determining

in terms the fundamental classes of the moduli spaces of twisted canonical divisors and a formula of Pixton is presented.

The relationship of the conjecture of [START_REF] Farkas | The moduli space of twisted canonical divisors[END_REF]Appendix] to our conjecture here is a direction for future study.

A.2 The limit r = 0. Let (a 1 , . . . , a n ) be a partition of 2g -2 with nonnegative parts (as in A.1). For r -2 ≥ max{a 1 , . . . , a n } , Witten's r-spin class W r g,n (a 1 , . . . , a n ) is well-defined and of degree independent of r,

By Theorem 7, after scaling by r g-1 ,

is a polynomial in r for all sufficiently large r.

Conjecture A.1. We have

If the polynomiality of Theorem 7 were to hold in Chow (which we expect), then Conjecture A.1 could also be formulated in A g-1 (M g,n ). By Conjecture A.1, [H g (a 1 , . . . , a n )] is a tautological class -a claim which has been proven 17 by A. Sauvaget [START_REF] Sauvaget | Cohomology classes of strata of differentials[END_REF]. of classes κ 1 , ψ 1 , δ sep and δ nonsep with the following coefficients: Class R-matrix TopFT Coefficient

After multiplying Witten's class by (-1) g r g-1 = r and extracting the constant term in r, we obtain

We remove κ 1 using equation (45). Conjecture A.1 predicts

) .

The result coincides with the well-known formula for the locus of the Weierstrass points, see [START_REF] Belorousski | A descendent relation in genus 2[END_REF]Lemma 5].

Genus 2, n = 2, a 1 = a 2 = 1. The tautological space RH 1 (M 2,2 ) is spanned by six classes ψ 1 , ψ 2 , α, β, γ and δ nonsep where

• α is the locus of curves with a rational component carrying both markings and a genus 2 component,

• β is the locus of curves with two elliptic components carrying one marking each,

• γ is the locus of curves with two elliptic components one of which carries both markings and the other one no markings,

• δ nonsep is the locus of curves with a nonseparating node.

A.4 The constant term. We will now present a more explicit approach to the constant term (-1) g W g;a 1 ,...,an (0) ∈ RH g-1 (M g,n )

which, according to Conjecture A.1, equals [H g (a 1 , . . . , a n )].

We will use the shift along e 1 studied in Section 4. The corresponding R-matrix involves a sequence of polynomials P m (r, a) for which we know no closed formula. However, in Proposition A.2 below, we obtain a closed formula for the polynomials P m (0, a).

Let B m (x) be the Bernoulli polynomials defined by

Let P m (r, a) be the polynomials defined by equation (36).

Proposition A.2. We have

We will show that the polynomials Q m (a) satisfy the mod r reduction of equations ( 37) and (38) of Lemma 4.3,

Together with the initial condition Q 0 = 1, these properties determine the polynomials Q m uniquely and immediately imply Q m (a) = P m (0, a) .

• Property (48) is implied by the simple equality

for every m ≥ 1 and definition (46).

• Property (47) follows from a generating function calculation:

which is equivalent to which is a well-known property of Bernoulli polynomials.

Using Proposition A.2, we can derive a more explicit formula for the constant term (-1) g W g,(a 1 ,...,an) (0) as a sum over stable graphs. We will require here only stable graphs with exactly n legs. We will replace the push-forward of the ψ classes on the κ-legs with an equivalent vertex factor involving κ classes. The equivalence requires the following well-known equality. If

Department of Mathematics, MIT apixton@mit.edu CNRS, Institut Mathématique de Jussieu dimitri.zvonkine@ump-prg.fr