Half-Spin Tautological Relations and Faber's Proportionalities of Kappa Classes
We employ the 1/2-spin tautological relations to provide a particular combinatorial identity. We show that this identity is a statement equivalent to Faber's formula for proportionalities of kappa-classes on Mg, g≥2. We then prove several cases of the combinatorial identity, providing a new pro...
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| Цитувати: | Half-Spin Tautological Relations and Faber's Proportionalities of Kappa Classes / E. Garcia-Failde, R. Kramer, D. Lewański, S. Shadrin // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 22 назв. — англ. |
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| author | Garcia-Failde, E. Kramer, R. Lewański, D. Shadrin, S. |
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| citation_txt | Half-Spin Tautological Relations and Faber's Proportionalities of Kappa Classes / E. Garcia-Failde, R. Kramer, D. Lewański, S. Shadrin // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 22 назв. — англ. |
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| description | We employ the 1/2-spin tautological relations to provide a particular combinatorial identity. We show that this identity is a statement equivalent to Faber's formula for proportionalities of kappa-classes on Mg, g≥2. We then prove several cases of the combinatorial identity, providing a new proof of Faber's formula for those cases.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 15 (2019), 080, 27 pages
Half-Spin Tautological Relations
and Faber’s Proportionalities of Kappa Classes
Elba GARCIA-FAILDE †, Reinier KRAMER ‡, Danilo LEWAŃSKI ‡ and Sergey SHADRIN §
† Institute de Physique Théorique, CEA Paris-Saclay, Orme des Merisiers,
91191 Gif-sur-Yvette, France
E-mail: elba.garcia-failde@ipht.fr
‡ Max Planck Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany
E-mail: rkramer@mpim-bonn.mpg.de, ilgrillodani@mpim-bonn.mpg.de
§ Korteweg-de Vries Instituut voor Wiskunde, Universiteit van Amsterdam,
Postbus 94248, 1090GE Amsterdam, The Netherlands
E-mail: s.shadrin@uva.nl
Received June 19, 2019, in final form October 14, 2019; Published online October 18, 2019
https://doi.org/10.3842/SIGMA.2019.080
Abstract. We employ the 1/2-spin tautological relations to provide a particular combi-
natorial identity. We show that this identity is a statement equivalent to Faber’s formula
for proportionalities of kappa-classes on Mg, g ≥ 2. We then prove several cases of the
combinatorial identity, providing a new proof of Faber’s formula for those cases.
Key words: tautological ring; tautological relations; moduli spaces of curves; Faber intersec-
tion number conjecture; odd-even binomial coefficients
2010 Mathematics Subject Classification: 14H10; 05A10
1 Introduction
The moduli spaces of curves Mg,n and their Deligne–Mumford compactifications Mg,n are
central objects in modern mathematics. Although in general their Chow rings are inifinite-
dimensional, there are finite-dimensional subrings, the tautological rings R∗, that contain most
‘naturally occuring’ classes. These rings have been studied since the foundational work of
Mumford [13] and Faber [4]. Overviews of the main results on these rings can be found in
[14, 19, 20, 22].
The system of tautological rings {R∗(Mg,n)}g,n can be defined succinctly as the smallest
system of subalgebras of the Chow rings closed under pushforwards along the three tautological
maps
π : Mg,n+1 →Mg,n,
ρ : Mg,n+1 ×Mh,m+1 →Mg+h,n+m,
σ : Mg,n+2 ×Mg+1,n,
where the first map forgets the last marked point and the other two glue two marked points
together, see [5]. This system of rings is also closed under pullbacks along the above-mentioned
maps, and it contains the natural tautological ψ-, κ-, and λ-classes, after which the rings are
named.
This paper is a contribution to the Special Issue on Integrability, Geometry, Moduli in honor of Motohico Mu-
lase for his 65th birthday. The full collection is available at https://www.emis.de/journals/SIGMA/Mulase.html
elba.garcia-failde@ipht.fr
rkramer@mpim-bonn.mpg.de
ilgrillodani@mpim-bonn.mpg.de
mailto:s.shadrin@uva.nl
https://doi.org/10.3842/SIGMA.2019.080
https://www.emis.de/journals/SIGMA/Mulase.html
2 E. Garcia-Failde, R. Kramer, D. Lewański and S. Shadrin
In fact, a set of additive generators of the tautological rings can be given by dual graphs,
which are graphs with n leaves (or labelled half-edges) decorated as follows: to each vertex v
we attach a genus g(v) and a product of κ-classes, and to each half-edge we attach a power
of a ψ-class. Vertices represent stable components of algebraic curves, half-edges represent
special points (i.e., can be either nodes of the curve or leaves), among which labelled half-edges
represent the n leaves. We interpret the dual graphs as follows: we attach to vertices of genus g′
and valency n′ a copy of Mg′,n′ , we form the product of all κ- and ψ-classes attached to the
vertex or to its half-edges, and we push it forward along the gluing tautological maps given by
the edges.
The tautological rings of the open space Mg,n and its partial compactifications are defined
via restriction from Mg,n. As Mg,n corresponds to all smooth curves, all graphs with at least
one edge restrict to zero on this space; the only tautological classes are polynomials in κ- and
ψ-classes. We denote by Mct
g,n and Mrt
g,n the partial compactifications of Mg,n by stable nodal
curves of compact type and with rational tails, respectively.
In fact there are many relations between dual graphs in the tautological ring. These relations
are called tautological relations and they encode the structure of the tautological rings. There-
fore, the understanding of tautological rings boils down to the understanding of their tautological
relations.
1.1 Half-spin relations
One way of approaching tautological relations is via cohomological field theories (CohFTs).
One particular CohFT has played a distinguished role in this context. It is a shifted version
of Witten’s r-spin class [18, 21], and has been thoroughly studied by Pandharipande–Pixton–
Zvonkine in two different ways [15, 16]. On the one hand, Witten’s class is quasi-homogeneous,
and this gives a degree bound for its shifted version. On the other hand, any semi-simple
CohFT can be constructed via Givental’s action from its degree zero part, and this gives an
explicit description for the shifted Witten’s class, that seemingly has non-trivial contributions
in high degrees. As both approaches should lead to the same result, this gives tautological
relations in degrees above the bound, called r-spin relations.
In [16], it was also proved that the Witten r-spin class is polynomial in r for r large. This
makes it possible to choose r, which a priori should be an integer greater or equal to two, to be
any number. In [9], the authors observed that taking the value r = 1
2 results in much simplified
relations compared to the case of general r. These relations are called half-spin relations.
The coefficients of the half-spin relations are proportional to expressions of the type
(
2a+ 1
2d
)
· (2d− 1)!!, a, d ∈ Z≥0 (1.1)
(cf. [9, Lemma 2.1]). It turns out that further applications of half-spin relations require a
better understanding the combinatorial structure of these numbers. We propose some purely
combinatorial questions about them, cf. Question 5.2 and Conjecture 5.7 that arose naturally
from our analysis of Faber’s conjecture.
1.2 Faber’s intersection numbers conjecture
The top tautological group Rg−2(Mg) is one-dimensional, spanned by the class κg−2, g ≥ 2
[3, 12]. All other monomials of kappa-classes, κa1 · · ·κa` , ` ≥ 1, a1, . . . , a` ≥ 1, a1 + · · · + a` =
g− 2, are proportional to κg−2 with some coefficients of proportionality. These coefficients were
conjectured by Faber in [4, Conjecture 1c], and he also observed in op. cit. that the class λgλg−1
vanishes onMg,n\Mrt
g,n. An equivalent form of his conjecture (now theorem) can be represented
as follows:
Half-Spin Tautological Relations and Faber’s Proportionalities of Kappa Classes 3
Theorem 1.1 (Faber’s intersection numbers conjecture). Let n ≥ 2 and g ≥ 2. For any
d1, . . . , dn ≥ 1, d1 + · · ·+ dn = g− 2 + n, there exists a constant Cg that only depends on g such
that
1
(2g − 3 + n)!
∫
Mg,n
λgλg−1
n∏
i=1
ψdii (2di − 1)!! = Cg. (1.2)
Remark 1.2. In particular,
∫
Mg,1
λgλg−1ψ
g−1
1 = (2g−2)!
(2g−3)!!Cg. This integral is computed in [4,
Theorem 2], so it is known that Cg =
|B2g |
22g−1(2g)!
, where B2g is the Bernoulli number.
This theorem has several proofs: Getzler and Pandharipande [6] derived it from the Virasoro
constrains for P2 proved by Givental [7]. Liu and Xu [11] derived it from an identity for the
n-point functions of the intersection numbers of ψ-classes that comes from the KdV equation.
Goulden, Jackson, and Vakil proved it for n ≤ 3 using the reductions of Faber–Hurwitz classes [8].
Buryak and the fourth author proved it using relations for double ramification cycles [1]. Finally,
Pixton showed the compatibility of this theorem with Faber–Zagier relations in [17], also proved
by Faber and Zagier (unpublished, see a remark in [16]). Together with a result of [16], this
shows that Faber–Zagier relations imply this theorem.
In fact, all these independent proofs are inspired by quite different ideas and they all lead
to a deeper understanding of the geometry of the moduli spaces of curves. In this paper, we
use the half-spin relations to transform Faber’s conjecture into a combinatorial identity. This
gives insight into the use of half-spin relations and the related combinatorics of expressions of
the form of (1.1). On the other hand, it gives insight into Faber’s formula itself, as we extend it
to formal negative powers of ψ-classes.
We then prove several cases of the combinatorial identity, providing a new proof of Faber’s
conjecture for n less than or equal to five.
1.3 Organization of the paper
In Section 2, we give the definition of the half-spin relations. In Section 3, we reduce Faber’s con-
jecture (Theorem 1.1) to a combinatorial identity using the half-spin relations. In Section 4, we
introduce formal negative powers of ψ-classes to reduce the combinatorial identity to a simpler
one, which we refer to as the main combinatorial identity of the paper. In Section 5, we investi-
gate this identity from a combinatorial viewpoint and conjecture a refinement. In Appendix A,
we give a combinatorial proof of the identity in low-degree cases.
2 Definition of half-spin relations
We will define two specific cases of the half-spin relations in R≥g(Mct
g,n), as this is all we need
for the rest of the paper. For a more general version and the construction, see [9].
First we need to define stable graphs.
Definition 2.1. A stable graph is the data Γ = (V,H,L,E, g : V → Z≥0, v : H → V, ι : H → H)
such that
1) V is the vertex set with genus function g;
2) ι is an involution of H, the set of half-edges;
3) the set L of legs or leaves is given by the fixed points of ι;
4) the set E of edges is given by the two-point orbits of ι;
5) v sends a half-edge to the vertex it is attached to;
4 E. Garcia-Failde, R. Kramer, D. Lewański and S. Shadrin
6) the graph given by (V,E) is connected;
7) for each vertex w ∈ V , the stability condition holds: 2g(w) − 2 + n(w) > 0, where
n(w) = |v−1(w)| is the valence of w.
For such a stable graph, its genus is given by g(Γ) =
∑
v∈V g(v) +h1(Γ), where h1(Γ) is the first
Betti number of the graph. The type of a stable graph Γ is given by (g(Γ), |L|).
We recall that the r-spin relations are proved in [16] by taking the Cohomological Field
Theory given by Witten’s r-spin class, and showing that it is polynomial in r in a certain way.
This is a subtle argument, hinging on the primary fields a1, . . . , an attached to the leaves. For
the r-spin theory, these are numbers between 0 and r − 2, such that A :=
∑
ai ≡ g − 1 + D
mod r − 1, where D is the degree of the relation. To show polynomiality in r, this congruence
is lifted to an equality A = g − 1 + D + x(r − 1) for some x ∈ Z≥0. If x = 0, all the primary
fields can be taken constant in r, and polynomiality follows from the argument of [16].
For x ≥ 1, however, the argument is more complicated, as the polynomiality does not hold
over all of Mg,n. However, under certain conditions, it still holds on certain subspaces, where
we can then use it to get half-spin relations on these subspaces. As taking r = 1
2 is in effect
taking a linear combination of relations for integer graphs, we do get relations on all of Mg,n,
but their description is not explicit outside the given subspace. For more details, see [9]. We
will only give the half-spin relations needed for this paper; they only use trees.
Definition 2.2. Define the polynomials
Qm(a) :=
(−1)m
2mm!
2m∏
k=1
(
a+ 1− k
2
)
. (2.1)
Let n ≥ 2, D ≥ g and a1, . . . , an be non-negative integers, called primary fields, with sum
A :=
∑n
i=1 ai = g − 1 + D. Consider all stable trees Γ = (V,H,L,E, g, v, ι) of type (g, n) and
decorate them in the following way:
• On each leg labeled by i, place the sum
∑ai
di=0Qdi(ai)ψ
di
i , and place the integer ai− di on
the corresponding half-edge fixed by ι.
• On each vertex v, we use the tree structure to work inwards from the leaves. If we
have determined all half-integers bi at its incident half-edges except one, say b0, then
b0 := g(v)− 1−∑i bi if this is at least zero. Otherwise, set b0 := g(v)− 3
2 −
∑
i bi.
• On each edge with half-integers a and b on its two half-edges, place the sum−∑m>0Qn(a+
m)(ψ+ψ′)m−1δa+b+m,− 3
2
, where ψ and ψ′ are the ψ-classes corresponding to the two half-
edges.
The half-spin relation for x = 0, ΩD
g,n(a1, . . . , an) = 0 ∈ RD(Mrt
g,n), is given by the sum of these
decorated stable graphs with these coefficients being zero in degree D.
Remark 2.3. Although the coefficient on the edge does not seem to be symmetric in a and b,
a simple calculation shows it actually is.
In fact, the coefficient on an edge with a and b on its two half-edges coming from the r-spin
relations is
1
ψ + ψ′
(
δa+b,− 3
2
−
∞∑
m,m′=0
∑
c,d∈ 1
2
Z
Qm(c)Qm′(d)δa,c−mδb,d−m′δc+d,− 3
2
ψm(ψ′)m
′
)
. (2.2)
This is equal to the coefficient given in the definition, but we give this equation as well, as it is
closer to the form of the r-spin relations in [16], and because it is useful for the rest of the paper.
Half-Spin Tautological Relations and Faber’s Proportionalities of Kappa Classes 5
In this formula, the numbers c and d should be interpreted as being placed near the middle of
the edge, or at the end of the half-edges. In this way, they are similar to the ai on the leaves,
and they will also be called primary fields. Meanwhile, the ai − di are similar to the a and b on
the edges. This analogy will be used in the proof of Proposition 3.1. The equality can be seen
from the relation
Qm(a+m)Qm′(b+m′) =
(
m+m′
m
)
Qm+m′(a+m+m′) if a+ b+m+m′ = −3
2
.
Remark 2.4. These relations have been proved in [9], by specialization of the r-spin relations
proved in [16]. The proof in [9] uses the polynomiality of the r-spin relations in r, which is
also proved in [16]. The half-spin relations can be extended to all of Mg,n, but this extension
is not unique, much less explicit, and unnecessary for our purpose. However, their extension is
of principal importance for applications in Gromov–Witten theory, since it allows to prove the
following statement (a reformulation of [9, Lemma 5.2]):
Proposition 2.5. Any monomial of ψ-classes of degree at least max(g, 1) on Mg,n can be
expressed in terms of the boundary classes that involve no κ-classes, that is, in terms of the dual
graphs with at least one edge, decorated only by ψ-classes.
This reformulation of [9, Lemma 5.2] is noted in [2] (where an alternative approach to the
same statement is developed), and in this reformulation Proposition 2.5 immediately resolves
Conjecture 3.14 in [10] and Conjecture 3 in [5].
We will also need the half-spin relation on M0,n for x = 1. We give them here on Mct
g,n for
general g, which reduces to M0,n for g = 0.
Definition 2.6. Now, let n ≥ 2, D ≥ g+ 1, and the primary fields a1, . . . , an−1 be non-negative
integers, and an ≤ −3
2 with sum A = g + D − 3
2 . Then the half-spin relation for x = 1,
ΩD
g,n(a1, . . . , an) = 0 ∈ RD(Mct
g,n), is given by a sum over decorated stable trees with the same
conditions as the ones for x = 0.
Remark 2.7. Although the (local) conditions are the same, the (global) relations are different,
because the sum of the primary fields is different.
3 A combinatorial identity from half-spin relations
In this section, we employ the half-spin relations to prove the following proposition. We shall
denote by JnK the set {1, . . . , n}.
Proposition 3.1. For any g ≥ 2 and n ≥ 2, for any a1, . . . , an ∈ Z≥0, a1+ · · ·+an = 2g−3+n,
we have the following equation:
0 =
n∑
k=1
(−1)k
k!
∑
I1t···tIk=JnK
Ij 6=∅,∀ j∈JkK
∑
d1,...,dk∈Z≥0
d1+···+dk=g−2+k
〈τd1 · · · τdk〉g ·
k∏
j=1
Qdj+|Ij |−1(a[Ij ]). (3.1)
Here we denote
∑
`∈Ij a` by a[Ij ] and
〈τd1 · · · τdk〉g :=
1
Cg
∫
Mg,k
λgλg−1
k∏
i=1
ψdii , (3.2)
where Cg is an arbitrary constant depending only on g (for instance, it is convenient to assume
that Cg is the constant given in Remark 1.2).
Moreover, for a fixed g ≥ 0, the whole system of equations (3.1) (we can vary parameters
n ≥ 2 and a1, . . . , an) determines all integrals 〈τd1 · · · τdk〉g, k ≥ 2, in terms of 〈τg−1〉g.
6 E. Garcia-Failde, R. Kramer, D. Lewański and S. Shadrin
Remark 3.2. Note that equation (3.1) can also be considered as an equation for the classes
in Rg−2(Mg), once we replace the symbols 〈τd1 · · · τdk〉g by the restrictions to Rg−2(Mg) of the
classes π∗(ψ
d1
1 · · ·ψdkk ), where π : Mg,k →Mg.
Proof. We will use relations in Rg−2+n(Mrt
g,n) given by half-spin relations for A = 2g − 3 + n.
Note that to produce relations D := g − 2 + n must be at least g, and hence we have n ≥ 2.
The restriction toMrt
g,n means that all allowed stable trees must have one vertex vg of genus g,
and all other vertices have genus 0. If we cut vg from such a stable tree, it falls apart in several
connected components, which are called rational tails.
The leaves are then distributed among these rational tails, and this gives a decomposition
JnK =
⊔k
i=1 Ii. If |Ii| = 1, this corresponds to a leaf attached to vg. We will therefore consider all
graphs where the points with indices in Ii lie on a separate rational tail, for every i = 1, . . . , k.
We want to simplify these relations by applying half-spin relations in genus zero to each of
the tails. Hence, we will now consider a particular rational tail that contains points with indices
in I ⊂ JnK, with |I| ≥ 2. Consider the edge that attaches this rational tail to the genus g
component, and assume that it is decorated by ψd at the node on the genus g component. We
call this edge the root edge, er, for this tail.
The total (cohomological) degree of the rest of this tail is given by the number of edges,
excluding this one, together with the total number of ψ-classes, excluding this one. We will call
this degree DI . It cannot be larger than |I| − 2, since dimCM0,|I|+1 = |I| − 2 and the graph is
constructed via pushforward along a map from this space. This means that the end of the root
edge which connects to the rational tail is decorated with ψ` for some 0 ≤ ` ≤ |I| − 2.
6 E. Garcia-Failde, R. Kramer, D. Lewański and S. Shadrin
Remark 3.2. Note that equation (3.1) can also be considered as an equation for the classes
in Rg−2(Mg), once we replace the symbols 〈τd1 · · · τdk〉g by the restrictions to Rg−2(Mg) of the
classes π∗(ψ
d1
1 · · ·ψdkk ), where π : Mg,k →Mg.
Proof. We will use relations in Rg−2+n(Mrt
g,n) given by half-spin relations for A = 2g − 3 + n.
Note that to produce relations D := g − 2 + n must be at least g, and hence we have n ≥ 2.
The restriction toMrt
g,n means that all allowed stable trees must have one vertex vg of genus g,
and all other vertices have genus 0. If we cut vg from such a stable tree, it falls apart in several
connected components, which are called rational tails.
The leaves are then distributed among these rational tails, and this gives a decomposition
JnK =
⊔k
i=1 Ii. If |Ii| = 1, this corresponds to a leaf attached to vg. We will therefore consider all
graphs where the points with indices in Ii lie on a separate rational tail, for every i = 1, . . . , k.
We want to simplify these relations by applying half-spin relations in genus zero to each of
the tails. Hence, we will now consider a particular rational tail that contains points with indices
in I ⊂ JnK, with |I| ≥ 2. Consider the edge that attaches this rational tail to the genus g
component, and assume that it is decorated by ψd at the node on the genus g component. We
call this edge the root edge, er, for this tail.
The total (cohomological) degree of the rest of this tail is given by the number of edges,
excluding this one, together with the total number of ψ-classes, excluding this one. We will call
this degree DI . It cannot be larger than |I| − 2, since dimCM0,|I|+1 = |I| − 2 and the graph is
constructed via pushforward along a map from this space. This means that the end of the root
edge which connects to the rational tail is decorated with ψ` for some 0 ≤ ` ≤ |I| − 2.
g
RTI1
RTIi
RTIk
ψd[g] ψ`[0]
b[g] b[0]
Figure 1. A dual graph of genus g with rational tails and n leaves. The marked points with indices in
Ii ⊂ JnK are attached to the rational tail denoted by RTIi , for all i = 1, . . . , k.
Let us now discuss the coefficient corresponding to the root edge. Using the congruences
for the primary fields for the leaf contributions and the vertex contributions to be non-zero,
together with the fact that all vertices in the rational tail correspond to genus 0 components
and the total number of remaining ψ classes and edges is equal to DI − `, the primary field at
the genus 0 end of the root edge must be equal to b[0] := −3
2 − a[I] + (DI − `). The primary field
at the genus g end of the root edge must be equal to b[g] := a[I] − (DI + d+ 1).
The coefficient of the contribution of the root edge reads:
−
DI∑
`=0
ψd[g]ψ
`
[0]
[
Qd+1(
b[g]+d+1︷ ︸︸ ︷
a[I] −DI)Q`
(
b[0]+`︷ ︸︸ ︷
−3
2
− a[I] +DI
)
−Qd+2(a[I] −DI + 1)Q`−1
(
− 3
2
− a[I] +DI − 1
)
+Qd+3(a[I] −DI + 2)Q`−2
(
− 3
2
− a[I] +DI − 2
)
Figure 1. A dual graph of genus g with rational tails and n leaves. The marked points with indices in
Ii ⊂ JnK are attached to the rational tail denoted by RTIi , for all i = 1, . . . , k.
Let us now discuss the coefficient corresponding to the root edge. Using the congruences
for the primary fields for the leaf contributions and the vertex contributions to be non-zero,
together with the fact that all vertices in the rational tail correspond to genus 0 components
and the total number of remaining ψ classes and edges is equal to DI − `, the primary field at
the genus 0 end of the root edge must be equal to b[0] := −3
2 − a[I] + (DI − `). The primary field
at the genus g end of the root edge must be equal to b[g] := a[I] − (DI + d+ 1).
The coefficient of the contribution of the root edge reads
−
DI∑
`=0
ψd[g]ψ
`
[0]
[
Qd+1(
b[g]+d+1︷ ︸︸ ︷
a[I] −DI)Q`
(
b[0]+`︷ ︸︸ ︷
−3
2
− a[I] +DI
)
−Qd+2(a[I] −DI + 1)Q`−1
(
−3
2
− a[I] +DI − 1
)
Half-Spin Tautological Relations and Faber’s Proportionalities of Kappa Classes 7
+Qd+3(a[I] −DI + 2)Q`−2
(
−3
2
− a[I] +DI − 2
)
...
+ (−1)`Qd+`+1(a[I] −DI + `)Q0
(
−3
2
− a[I] +DI − `
)]
,
where the alternating sum comes from the division by ψ[g] + ψ[0], following equation (2.2). Let
us take this sum in a bit different way, with respect to the argument of the second factor
a0 = −3
2 − a[I] + DI − j, where j runs from DI to 0, and decompose the exponent of ψ[0] as
` = j + k. We have
−
−3/2−a[I]+DI∑
a0=−3/2−a[I]
ψd[g](−1)−3/2−a0−a[I]+DIQd+1−3/2−a[I]+DI−a0
(
−3
2
− a0
)
ψ[0]
j︷ ︸︸ ︷
−3/2−a[I]+DI−a0
×
a[I]−(−3/2−a0)∑
k=0
Qk(a0)ψ
k
[0]
.
The sum over a0 here is over half-integers, with integer steps.
Let us analyse the sum
∑a[I]+3/2+a0
k=0 Qk(a0)ψ
k
[0]. We cut the root edge and assign a0 as primary
field for the new leaf on the rest of the tail, which is decorated with ψk[0]. The total dimension
of the class on the rest of the tail is D0 = DI − j = DI −
(
−3
2 − a[I] +DI − a0
)
= 3
2 + a[I] + a0.
Thus a0 + a[I] = D0 − 3
2 . Therefore, if D0 ≥ 1, then with this sum on the root edge the total
sum of all graphs in the tail (for a fixed a0) is the half-spin relation for x = 1, with primary
field a0 at the root edge and ai, i ∈ I, for the marked points on the tail.
Thus the only nontrivial contribution of the tail comes from the case D0 = 0 which produces
no relation for the tail, with a0 = −3
2−a[I]. In this case there is the unique non-trivial summand
in the sum above that is equal to
(−1)DI+1ψd[g]ψ
DI
[0] Qd+DI+1(a[I]).
Moreover, the only non-trivial ψ-classes are on the root edge and there are no more internal
edges on the tail.
In the end, modulo the relations in genus 0 on the tails, the only graphs that remain in the
relation in degree D = g − 2 + n are the following. The marked points are split in k non-empty
sets I1, . . . , Ik, corresponding to different rational tails. If Ii is a set of one element, then the
tail is just a leaf decorated with ψdi and the coefficient is Qdi(a[Ii]). If Ii is a set of two or more
points, then this tail is just one rational vertex with all leaves from Ii on it, attached by an edge
to the genus g vertex. The ψ-classes are only on this edge, ψdi on the genus g side and ψDI on
the genus 0 side, with the coefficient (−1)DI+1Qdi+DI+1(a[Ii]).
Up to now, everything we described was done in Rg−2+n(Mrt
g,n). Hence, we still need to
pushforward to Mg,k, along the map forgetting some of the marked points. For each decorated
graph we constructed, we will pushforward until each tail has exactly one marked point left, and
hence must be a leaf.
We can do this on each tail individually, first using the string equation, which in this case
reads
∫
M0,|I|+1
ψDI
[0] =
∫
M0,|I|
ψDI−1
[0] . Therefore, pushing forward along a map forgetting a point
in I decreases the exponent of ψ[0] by one. As this can be done until the rational tail has two
marked points, we must get DI = |I| − 2.
Finally, the pushforward of a rational tail with two marked points along the map forgetting
one of those marked points just collapsed the tail and moves the remaining marked point to the
collapsed node.
8 E. Garcia-Failde, R. Kramer, D. Lewański and S. Shadrin
Summarising, the only surviving terms are the terms where all the marked points are parti-
tioned as
⊔
k Ik = JnK over rational tails consisting of a leaf or a single rational curve with all
marked points attached to it, with coefficient
(−1)|I|−1ψd[g]ψ
|I|−2
[0] Qd+|I|−1(a[I]).
These terms pushforward to terms on Mg,k given by
(−1)|I|−1ψdIQd+|I|−1(a[I]).
Taking the product over all the tails and taking into account that the linear function
1
Cg
∫
Mg
λgλg−1· : Rg−2(Mg)→ Q
is an isomorphism, the half-spin relations we found for D = g − 2 + n imply the combinatorial
identity (3.1).
On the other hand, it is easy to see that these relations determine the intersections of all
possible monomials in ψ-classes in terms of
∫
Mg,1
λgλg−1ψ
g−1
1 (using the natural lexicographic
order). �
We relate Proposition 3.1 to Faber’s conjecture and refine it using the string equation, which
turns the result into a combinatorial identity.
Corollary 3.3. Let g ≥ 2 and n ≥ 2. The following two statements are equivalent:
i) Faber’s Conjecture 1.1: there exists a constant Cg that only depends on g such that
〈τd1 · · · τdk〉g :=
1
Cg
∫
Mg,n
λgλg−1
n∏
i=1
ψdii =
(2g − 3 + n)!
n∏
i=1
(2di − 1)!!
(3.3)
for any d1, . . . , dn ≥ 1.
ii) For any a1, . . . , an ∈ Z≥0 such that a1 + · · ·+ an = 2g − 3 + n, we have
0 =
n∑
k=1
(−1)k
k!
∑
I1t···tIk=JnK
Ij 6=∅,∀ j∈JkK
∑
d1,...,dk∈Z≥0
d1+···+dk=g−2+k
〈τd1 · · · τdk〉?g ·
k∏
j=1
Qdj+|Ij |−1(a[Ij ]), (3.4)
where
〈τd1 · · · τdk〉?g =
(2g − 3 + k)!
k∏
i=1
(2di − 1)!!
in case d1, . . . , dk ≥ 1, (3.5)
and determined by the string equation
〈τd1 · · · τdkτ0〉?g =
k∑
j=1
dj≥1
〈τd1−δ1j · · · τdk−δkj 〉?g (3.6)
otherwise. Here a[Ij ] denote
∑
`∈Ij a`.
Proof. By Proposition 3.1, the left-hand side of equation (3.3) satisfies equation (3.4), and the
integrals can be recovered from this equation. Therefore, both sides of equation (3.3) are equal
if and only if the right-hand side also satisfies equation (3.4). �
Half-Spin Tautological Relations and Faber’s Proportionalities of Kappa Classes 9
4 Psi-classes of negative degree
In the previous section, we showed that Theorem 1.1 is equivalent to a system of combinatorial
identities. The goal of this section is to reduce this system to a much nicer system of identities. In
order to do this, we need to consider formal systems of correlators satisfying the string equation.
4.1 Formal negative degrees of psi-classes
Definition 4.1. Let g ≥ 2. Consider a system of numbers
〈∏k
i=1 τdi
〉•
g
that depends on
d1, . . . , dk ∈ Z, d1 + · · ·+ dk = g − 2 + k, and is symmetric in these variables. We say that this
system of numbers satisfies the string equation if
〈τd1 · · · τdkτ0〉•g =
k∑
j=1
〈τd1−δ1j · · · τdk−δkj 〉•g.
Example 4.2. The system of numbers
〈 k∏
i=1
τdi
〉•
g
:=
〈 k∏
i=1
τdi
〉?
g
, d1, . . . , dk ≥ 0,
0, at least one di is negative
satisfies the string equation, as follows, by definition, from equation (3.6).
Example 4.3. The system of numbers
〈∏k
i=1 τdi
〉•
g
:= (2g−3+k)!/
∏k
i=1(2di−1)!! also satisfies
the string equation (this can be checked by direct inspection).
Remark 4.4. These two examples coincide in the case when all di’s are positive and also in the
case when all di’s except for one are positive and the remaining one is equal to zero. For other
values of (d1, . . . , dk) the numbers in these two examples are generally different.
The string equation allows to choose the values of all numbers
〈∏k
i=1 τdi
〉•
g
,
∏k
i=1 di 6= 0,
k ≥ 1 in an arbitrary way, and the rest of the numbers (where at least one index di is equal to
zero) are linear combinations of these initial values with non-negative integer coefficients.
4.2 Q-polynomials and a refined string equation
Fix g ≥ 2 and n ≥ 2 and let a1, . . . , an be formal variables. Define Qi(a) ≡ 0 for i < 0. Fix an
arbitrary system of numbers
〈∏k
i=1 τdi
〉•
g
, d1, . . . , dk ∈ Z, d1 + · · · + dk = g − 2 + k, symmetric
in these variables and satisfying the string equation.
Consider the following expression
Eg,n(~a) :=
n∑
k=1
(−1)k
k!
∑
I1t···tIk=JnK
Ij 6=∅,∀ j∈JkK
∑
d1,...,dk∈Z
d1+···+dk=g−2+k
〈τd1 · · · τdk〉•g ·
k∏
j=1
Qdj+|Ij |−1(a[Ij ]) (4.1)
as a polynomial in a1, . . . , an and a linear function in 〈τd1 · · · τdk〉•g, d1 · · · dk 6= 0.
Proposition 4.5. For any d1, . . . , dk ∈ Z, d1 + · · ·+dk = g− 2 +k, d1 · · · dk 6= 0, where at least
one index di is negative, we have
∂Eg,n(~a)
∂〈τd1 · · · τdk〉•g
≡ 0.
10 E. Garcia-Failde, R. Kramer, D. Lewański and S. Shadrin
Proof. Assume d1, . . . , d` are negative, and the rest of the indices di are positive. Let us fix
I1 t · · · t Ik ⊂ JnK that satisfy the condition |Ii|+ di − 1 ≥ 0 for any i = 1, . . . , `.
Consider all terms in the expression E satisfying the following conditions:
• The correlator factor is a coefficient
〈∏k
i=1
∏mi
j=1 τdij
〉•
g
such that
– for each i = 1, . . . , k, we have
∑mi
j=1 dij = di +mi − 1;
– for each i = 1, . . . , `, at most one of dij , j = 1, . . . ,mi is negative. It is at least di,
and at least di + 1 if mi ≥ 2 (this ensures there exists a zero index to use the string
equation);
– for each i = `+ 1, . . . , k, all dij , j = 1, . . . ,mi are non-negative. Moreover, one index
must be at least di, and at least di + 1 if mi ≥ 2.
This list of conditions is equivalent to
∂
〈∏k
i=1
∏mi
j=1 τdij
〉•
g
∂〈τd1 ···τdk 〉•g
6= 0. In other words, the corre-
lator in the denominator can be deduced from the one in the numerator via successive
applications of the string equation.
• The sets Iij satisfy tmi
j=1Iij = Ii for each i = 1, . . . , `.
• For each i = 1, . . . , k the sets Iij are arranged in such a way that min(Iij) < min(Iij′) if
and only if j < j′ (this condition is necessary to have control on the combinatorial factor).
We can refine (4.1) as follows: we define “refined correlators”
〈∏k
i=1 τdi(Ji)
〉•
g
, now depending
formally on subsets Ji ⊂ JnK, and subject to a natural refinement of the string equation
〈τ0(Jk+1)
k∏
i=1
τdi(Ji)〉•g =
k∑
j=1
〈τdj−1(Jj t Jk+1)
k∏
i=1
i 6=j
τdi(Ji)〉•g.
We then define Erefg,n(~a) to be
Erefg,n(~a) :=
n∑
k=1
(−1)k
k!
∑
I1t···tIk=JnK
Ij 6=∅,∀ j∈JkK
∑
d1,...,dk∈Z
d1+···+dk=g−2+k
〈τd1(I1) · · · τdk(Ik)〉•g ·
k∏
j=1
Qdj+|Ij |−1(a[Ij ]).
Clearly, Erefg,n(~a) reduces to Eg,n(~a) after setting τd(I)→ τd.
Using this notation, if we fix mi, dij and Iij for i > 1, and let m1, d1j , and I1j vary
in all possible ways such that the conditions above are satisfied, we can split the derivative
∂〈∏k
i=1
∏mi
j=1 τdij 〉•g/∂〈τd1 · · · τdk〉•g into the sum of “refined derivatives”
∂
〈 k∏
i=1
mi∏
j=1
τdij (Iij)
〉•
g
∂
〈 k∏
i=1
τdi(Ii)
〉•
g
=
∂
〈 k∏
i=1
mi∏
j=1
τdij (Iij)
〉•
g
∂
〈
τd1(I1)
k∏
i=2
mi∏
j=1
τdij (Iij)
〉•
g
∂
〈
τd1(I1)
k∏
i=2
mi∏
j=1
τdij (Iij)
〉•
g
∂
〈 k∏
i=1
τdi(Ii)
〉•
g
.
The derivative is clearly zero if the partition {Iij} is not a refinement of the partition {Ii}.
Thus we obtain the following expression for the derivative of Eg,n(~a):
∂Eg,n(~a)
∂〈τd1 · · · τdk〉•g
=
∑
I1t···tIk=JnK
∂Erefg,n(~a)
∂〈τd1(I1) · · · τdk(Ik)〉•g
∣∣∣∣∣
τd(I)=τd
Half-Spin Tautological Relations and Faber’s Proportionalities of Kappa Classes 11
=
(−1)k
k!
∑
I1t···tIk=JnK
Ij 6=∅,∀ j∈JkK
mi,dij ,Iij for i≥2
(−1)
k∑
i=2
(mi−1) k∏
i=2
mi∏
j=1
Qdij+|Iij |−1(a[Iij ])
∂
〈
τd1(I1)
k∏
i=2
mi∏
j=1
τdij (Iij)
〉•
g
∂
〈 k∏
i=1
τdi(Ii)
〉•
g
×
∑
m1,d1j ,I1j
(−1)m1−1
m1∏
j=1
Qd1j+|I1j |−1(a[I1j ])
∂
〈 k∏
i=1
mi∏
j=1
τdij (Iij)
〉•
g
∂
〈
τd1(I1)
k∏
i=2
mi∏
j=1
τdij (Iij)
〉•
g
∣∣∣∣∣
τd(I)=τd
. (4.2)
In order to prove the proposition, it is sufficient to show that the factor in the third line of this
expression is always equal to zero. Note that this factor is a polynomial in the variables ap,
p ∈ I1, of degree 2 (d1 +m1 − 1 + |I1| −m1). The degree of this polynomial is less than twice
the number of its variables (since d1 < 0). Therefore, in order to show the constant vanishing
of this polynomial, it is sufficient to show that it constantly vanishes for two specific values
of each of its variables, namely, at the points ap = 0 and ap = −1/2 for each p ∈ I1. Since
this polynomial is symmetric in its variables, it is sufficient to prove this vanishing for just one
variable.
We assume, for simplicity, that 1 ∈ I1, and prove the vanishing for a1 = 0,−1/2. In order to
use the string equation, we split the terms in the third line of (4.2) in two parts: those where
I1,1 = {1}, and those where I1,1 ) {1} (as 1 ∈ I1,1 by the third bullet of conditions). The
first part is parametrised by partitions I1,2 t · · · t I1,m1 = I1 \ {1}, and the second part can
be reparametrised by the same partitions, plus a choice of one of these sets which should also
contain 1. Hence, up to a common sign factor, we can split the third line of (4.2) as terms of
the form
Qd11+1−1(a1)
m1∏
j=2
Qd1j+|I1j |−1(a[I1j ])
∂
〈 k∏
i=1
mi∏
j=1
τdij (Iij)
〉•
g
∂
〈
τd1(I1)
k∏
i=2
mi∏
j=1
τdij (Iij)
〉•
g
(4.3)
−
m1∑
r=2
m1∏
j=2
Qd1j+|I1j |−1(a[I1j ] + a1δj,r)
∂
〈
τd1r−1(I1r t {1})
m1∏
j=2
j 6=r
τd1j (I1j)
k∏
i=2
mi∏
j=1
τdij (Iij)
〉•
g
∂
〈
τd1(I1)
k∏
i=2
mi∏
j=1
τdij (Iij)
〉•
g
.
In both the cases a1 = 0 and a1 = −1/2, d11 must be equal to zero here (otherwise Qd11(a1) = 0
in the first term and the indices dij in the other terms do not add up to di + mi − 1). Then,
for a1 = 0 all Q-coefficients in (4.3) are literally the same, so it vanishes using the following
derivative of the refined string equation
∂
〈
τ0({1})
m1∏
j=2
τd1j (I1j)
k∏
i=2
mi∏
j=1
τdij (Iij)
〉•
g
∂
〈
τd1(I1)
k∏
i=2
mi∏
j=1
τdij (Iij)
〉•
g
=
m1∑
r=2
∂
〈
τd1r−1(I1r t {1})
m1∏
j=2
j 6=r
τd1j (I1j)
k∏
i=2
mi∏
j=1
τdij (Iij)
〉•
g
∂
〈
τd1(I1)
k∏
i=2
mi∏
j=1
τdij (Iij)
〉•
g
. (4.4)
12 E. Garcia-Failde, R. Kramer, D. Lewański and S. Shadrin
The case a1 = −1/2 is more subtle. We use induction on |I1|. Using the identity Qp(a) −
Qp(a − 1/2) = −(a/2) · Qp−1(a − 1) and the derivative of the refined string equation (4.4), we
can rewrite expression (4.3) as
−1
2
m1∑
k=2
a[I1k] ·
m1∏
j=2
Qd̃1j+|I1j |−1(ã[I1j ])
∂
〈 m1∏
j=2
τd̃1j (I1j)
k∏
i=2
mi∏
j=1
τdij (Iij)
〉•
g
∂
〈
τd1(I1 \ {1})
k∏
i=2
mi∏
j=1
τdij (Iij)
〉•
g
, (4.5)
where in the coefficient of ak we use the notation ã[I1p] := a[I1p] − δpk, p = 2, . . . ,m1, and
d̃1q := d1q − δkq, q = 2, . . . ,m1.
By our assumptions, d1 < 0 and |I1| + d1 − 1 ≥ 0, so |I1| ≥ 2. For the base case of the
induction, |I1| = 2, we then have d1 = −1, so d11 + d12 = d1 + m1 − 1 = 0, and therefore
d12 = 0. In this case, the k-sum and j-product in equation (4.5) collapse, yielding a coefficient
Qd̃12+|I12|−1(ãI[12]) = Q−1(aI[12] − 1) = 0, proving the basis step.
For the induction step, we see that for each k the coefficient of a[I1k] in (4.5) is the polynomial
in one fewer variables (namely, taking out the variable a1) and size of |I1| one less (by removing 1)
of exactly the same form as the summands in the third line of (4.2). Resumming over m1, d1j ,
and I1,j with the sign coming from the third line of (4.2), this becomes equal to the third line
of (4.2), which was zero by induction (where we use already that this third line vanishes for all
ai = 0,−1/2).
Thus, the third line in (4.2) is equal to zero for a1 = −1/2 as well as a1 = 0, which implies
it vanishes constantly. This implies the proposition. �
4.3 Applying formal negative degrees of ψ-classes
to the combinatorial identity
We use the result of the previous section to reduce the system of identities (3.4) to a simpler
one.
Proposition 4.6. Faber’s conjecture (Theorem 1.1) is equivalent to the following system of
combinatorial identities.
For any g ≥ 2 and n ≥ 2, for any a1, . . . , an ∈ Z≥0, a1 + · · ·+ an = 2g − 3 + n,
0 =
n∑
k=1
(−1)k
k!
∑
I1t···tIk=JnK
Ij 6=∅,∀ j∈JkK
∑
d1,...,dk∈Z
d1+···+dk=g−2+k
(2g − 3 + k)!
k∏
i=1
(2di − 1)!!
·
k∏
j=1
Qdj+|Ij |−1(a[Ij ]). (4.6)
Proof. We have already shown that Faber’s conjecture is equivalent to the system of identi-
ties (3.4), where the correlators are replaced by the predicted value from the conjecture. So, it is
sufficient to show that the system of identities (3.4) is equivalent to the system of identities (4.6).
Note that the right hand side of (3.4) is a specialization of the expression Eg,n(~a) for the values
of 〈τd1 · · · τdk〉•g given in Example 4.2. The right hand side of (4.6) is a specialization of the
expression Eg,n(~a) for the values of 〈τd1 · · · τdk〉•g given in Example 4.3.
As observed before, a system of numbers 〈τd1 · · · τdk〉•g satisfying the string equation, see
Definition 4.1, is fixed by choosing all numbers for
∏k
i=1 di 6= 0 arbitrarily and inferring the
other cases from the string equation. We call the chosen numbers initial values.
The initial values of the system of numbers 〈τd1 · · · τdk〉•g given in Examples 4.2 and 4.3 coincide
for all d1, . . . , dk > 0 and differ when at least one of di’s is negative. Proposition 4.5 implies
that the initial values 〈τd1 · · · τdk〉•g with at least one di negative have no impact on the value of
Eg,n(~a). Therefore, a specialization of the expression Eg,n(~a) for the values of 〈τd1 · · · τdk〉•g given
Half-Spin Tautological Relations and Faber’s Proportionalities of Kappa Classes 13
in Example 4.2 is equal to zero if and only if the same specialization of the expression Eg,n(~a)
for the values of 〈τd1 · · · τdk〉•g given in Example 4.3 is equal to zero. Therefore, the system of
identities (3.4) is equivalent to the system of identities (4.6). �
5 The main combinatorial identity and its structure
In the previous section we used formal negative degree psi-classes in order to simplify the system
of combinatorial identities to which Faber’s conjecture is equivalent (Proposition 4.6). We now
want to substitute the Q-polynomials by their definition and rearrange the terms to obtain the
following statement.
Corollary 5.1 (of Proposition 4.6). Faber’s conjecture (Theorem 1.1) is equivalent to the fol-
lowing system of combinatorial identities.
For any g ≥ 2 and n ≥ 2, for any a1, . . . , an ∈ Z≥0, a1 + · · ·+ an = 2g − 3 + n, we have
0 =
n∑
k=1
(−1)k(2g − 3 + k)!
k!
×
∑
I1t···tIk=JnK
Ij 6=∅,∀ j∈JkK
∑
d1,...,dk∈Z≥0
d1+···+dk=g−2+n
k∏
j=1
(
2a[Ij ] + 1
2dj
)
(2dj − 1)!!
(2dj + 1− 2|Ij |)!!
. (5.1)
Here by a[Ij ] we denote
∑
`∈Ij a` and by |Ij | we denote the cardinality of the set Ij ⊂
{1, . . . , n}, j = 1, . . . , k.
Proof. Recall that, for m ≥ 0, Qm(a) = ((−1)m/23mm!) ·∏2m−1
i=0 (2a + 1 − i). If its argument
is a non-negative integer, we can rewrite Qm(a) as
(
2a+1
2m
)
· (2m− 1)!! · (−1)m/22m. For m < 0,
Qm(a) ≡ 0. Then it is easy to see that equation (5.1) is obtained from equation (4.6) by the
relabelling dj + |Ij | − 1 dj and dividing by a common factor of (−1)g−2+n/22(g−2+n). �
In the rest of the paper we refer to equation (5.1) as the main combinatorial identity. This
section is devoted to a purely combinatorial analysis of this identity. Clearly, since Faber’s
conjecture is proved, the main combinatorial identity holds true. However, we are interested in
an independent proof of it, in order to obtain a new proof of Faber’s conjecture. We produce
such a proof for n ≤ 5 in the next section.
Question 5.2. Is there a purely combinatorial way to prove the combinatorial identity (5.1)
for all n?
In fact, one of the purposes of the present article is to pose the question to the combinatorial
community about a possible enumerative interpretation of our identity.
5.1 Polynomials vanishing in the integer points of some simplices
We denote the right hand side of (5.1) by P (a1, . . . , an). It can be considered as a polynomial of
degree 2g−4 + 2n in a1, . . . , an (since the binomial coefficient
(2a[I]+1
2d
)
is naturally a polynomial
of degree 2d in a[I]). We can also rewrite it as
n∑
k=1
(−1)k(2g − 3 + k)!
k!
∑
I1t···tIk=JnK
Ij 6=∅, ∀ j∈JkK
∑
f1,...,fk∈Z≥0
f1+···+fk=g−1
k∏
j=1
(
2a[Ij ] + 1
2fj + 1
)
(2a[Ij ] − 2fj − 1)!!
(2a[Ij ] − 2fj + 1− 2|Ij |)!!
,
14 E. Garcia-Failde, R. Kramer, D. Lewański and S. Shadrin
and refer to it as R(a1, . . . , an). The function R(a1, . . . , an) is also a polynomial in a1, . . . , an,
where each term in the sum over k = 1, . . . , n has degree
∑k
j=1 2fj + |Ij | = 2g − 2 + n, so the
total degree of R is 2g − 2 + n. Note that P 6= R (they even have different degrees); from the
construction they coincide only on the simplex a1, . . . , an ∈ Z≥0, a1 + · · ·+ an = 2g − 3 + n.
Proposition 5.3. We have P |ai=0 ≡ 0, i = 1, . . . , n.
Proof. Since P is symmetric in its variables, it is enough to prove this proposition for an = 0.
Consider an arbitrary splitting I1 t · · · t Ik = {1, . . . , n− 1}. We want to append this splitting
with the element n: either we add {n} as one of the sets (there are k + 1 ways to do this, since
we can choose the number of this set from 1 to k + 1 shifting the indices of Ij accordingly), or
we append n to one of the existing sets I`, ` = 1, . . . k. Consider the sum of all terms in P that
correspond to these choices of splitting of {1, . . . , n}. Since the first k+ 1 terms are all equal to
each other, we can assume that we have k+ 1 copies of the case Ik+1 = {n} instead. Therefore,
if we split the terms of P in this way, we get summands of the following form:
(k + 1)
(−1)k+1(2g − 3 + k + 1)!
(k + 1)!
∑
d1+···+dk+1
=g−2+n
k∏
j=1
(
2a[Ij ] + 1
2dj
)
(2dj − 1)!!
(2dj + 1− 2|Ij |)!!
·
(
2an + 1
2dk+1
)
+
(−1)k(2g − 3 + k)!
k!
∑
d1+···+dk
=g−2+n
k∑
`=1
k∏
j=1
(
2a[Ij ] + 2anδj,` + 1
2dj
)
(2dj − 1)!!
(2dj + 1− 2|Ij |)!!
.
If an = 0, then the first summand is nontrivial only for dk+1 = 0. So, if we substitute an = 0,
then this expression is equal to
(−1)k(2g − 3 + k)!
k!
∑
d1+···+dk
=g−2+n
k∏
j=1
(
2a[Ij ] + 1
2dj
)
(2dj − 1)!!
(2dj + 1− 2|Ij |)!!
×
(
−(2g − 2 + k) +
k∑
`=1
(2d` + 1− 2|I`|)
)
.
Note that the last factor is equal to zero. Since the definition of P reduces to the sum over
I1 t · · · t Ik = {1, . . . , n − 1} of the terms that we considered here, and we never used the
restriction of P to the simplex a1 + · · ·+ an = 2g − 3 + n, we have P (a1, . . . , an−1, 0) ≡ 0. �
Corollary 5.4. The function P̃ (a1, . . . , an) := P (a1, . . . , an)/
n∏
i=1
ai is a polynomial in a1, . . . , an
of degree 2g − 4 + n.
So, we have a collection of symmetric polynomials of quite small degree (that is, smaller
than what one expects trying to construct such non-trivial polynomials using the Lagrange
interpolation, for instance) vanishing in all integer points of the certain simplices:
• P (a1, . . . , an) is a polynomial of degree 2g − 4 + 2n that vanishes at all integer points of
the simplex a1, . . . , an ≥ 0, a1 + · · ·+ an = 2g − 3 + n.
• R(a1, . . . , an) is a polynomial of degree 2g− 2 +n that vanishes at all integer points of the
simplex a1, . . . , an ≥ 0, a1 + · · ·+ an = 2g − 3 + n.
• P̃ (a1 +1, . . . , an+1) is a polynomial of degree 2g−4+n that vanishes at all integer points
of the simplex a1, . . . , an ≥ 0, a1 + · · ·+ an = 2g − 3.
Half-Spin Tautological Relations and Faber’s Proportionalities of Kappa Classes 15
5.2 Combinatorial reduction of the identity for ai = 1
In this section we give a combinatorial reduction of the identity (5.1) in the case one of the
arguments ai is equal to 1.
Proposition 5.5. For any g ≥ 2 and n ≥ 1, for any a1, . . . , an ∈ Z≥0, we have
P (a1, . . . , an, 1)− P (a1, . . . , an, 0) = P (a1, . . . , an) ·
(
4
n∑
i=1
ai − 8g + 10− 2n
)
. (5.2)
Proof. Let {n + 1} t J ⊂ {1, . . . , n + 1}. Consider the corresponding factor in a summand in
P (a1, . . . , an, 1) assuming Ij := {n+1}tJ , for some j, and denoting the corresponding index dj
by d. We have the following decomposition:
(
2(a[J ] + 1) + 1
2d
)
(2d− 1)!!
(2d− 1− 2|J |)!! =
(
2(a[J ] + 0) + 1
2d
)
(2d− 1)!!
(2d− 1− 2|J |)!!
+
(
2a[J ] + 1
2d̃
)
(2d̃− 1)!!
(2d̃+ 1− 2|J |)!!
·
(
4a[J ] + 3− 2d̃
)
, (5.3)
where d̃ = d − 1. The first term on the right hand side is equal to the corresponding factor in
the same summand in P (a1, . . . , an, 0). The second term gives a summand in P (a1, . . . , an) with
a coefficient.
There are (k + 1) ways to obtain the summand
(−1)k(2g − 3 + k)! ·
k∏
j=1
(
2a[Ij ] + 1
2dj
)
(2dj − 1)!!
(2dj + 1− 2|Ij |)!!
in P (a1, . . . , an) (here I1t· · ·tIk = JnK, d1+· · ·+dk = g−2+n, and we omit the factor 1/k! that
controls the permutations of the sets I1, . . . , Ik) from the second term of the decomposition (5.3):
either J = Ij , j = 1, . . . , k, or J = ∅. In the latter case, the extra coefficient that we get is
equal to −3(2g − 2 + k). Thus, the total coefficient of this summand is equal to
k∑
j=1
(4a[J ] + 3− 2dj)− 3(2g − 2 + k) = 4
n∑
i=1
ai − 8g + 10− 2n,
which does not depend on the choice of I1 t · · · t Ik = JnK and d1 + · · ·+ dk = g − 2 + n. This
implies equation (5.2). �
If we restrict equation (5.2) to the simplex a1 + · · · + an = 2g − 3 + n and use that
P (a1, . . . , an, 0) ≡ 0 (without any assumptions on a1, . . . , an), we obtain the following corol-
lary:
Corollary 5.6. For any g ≥ 2 and n ≥ 1, for any a1, . . . , an ∈ Z≥0, a1 + · · ·+ an = 2g− 3 + n,
we have
P (a1, . . . , an, 1) = (2n− 2)P (a1, . . . , an).
In particular, P (2g−2, 1) = 0, and for n ≥ 2 the vanishing of P (a1, . . . , an) implies the vanishing
of P (a1, . . . , an, 1).
16 E. Garcia-Failde, R. Kramer, D. Lewański and S. Shadrin
5.3 A conjectural refinement of the identity
In this section we formulate a conjectural refinement of the identity (5.1), which gives a natural
strategy for its combinatorial proof. In particular, it allows to prove it for n ≤ 5 for all g.
We replace each factor
(2a[I]+1
2d
)
in each summand of the identity by the sum
(2a[I]
2d
)
+
(2a[I]
2d−1
)
.
Then we collect all terms with the fixed number of factors where we have chosen to decrease 2d
to 2d− 1. Let us define Pn,t as
n∑
k=1
(−1)k(2g − 3 + k)!
k!
×
∑
I1t···tIk=JnK
Ij 6=∅, ∀ j∈JkK
∑
d1,...,dk∈Z≥0
d1+···+dk=g−2+n
∑
A⊂JkK
|A|=n−t
k∏
j=1
(
2a[Ij ]
2dj − δj∈A
)
(2dj − 1)!!
(2dj + 1− 2|Ij |)!!
for t = 0, . . . , n. Here δj∈A is equal to 1 for j ∈ A and to 0 otherwise. For instance,
Pn,0 = (−1)n(2g − 3 + n)!
∑
o1,...,on∈(2Z+1)>0
o1+···+on=2g−4+n
n∏
j=1
(
2aj
oj
)
,
Pn,1 = (−1)n(2g − 3 + n)!
n∑
i=1
∑
o1,...,ôi,...on∈(2Z+1)>0
ei∈(2Z)≥0
o1+···ôi···+on+ei=2g−3+n
n∏
j=1
j 6=i
(
2aj
oj
)(
2ai
ei
)
+ (−1)n−1(2g − 4 + n)!
n∑
i,`=1
i<`
∑
o1,...,ôi,...,ô`,...on∈(2Z+1)>0
oi`∈(2Z+1)>0
o1+···ôi,ô`···+on+oi`=2g−3+n
n∏
j=1
j 6=i,`
(
2aj
oj
)(
2ai + 2a`
oi`
)
· oi`,
Pn,n =
n∑
k=1
(−1)k(2g − 3 + k)!
k!
∑
I1t···tIk
={1,...,n}
∑
e1,...,ek∈(2Z)≥0
e1+···+ek=2g−4+2n
k∏
j=1
(
2a[Ij ]
ej
)
(ej − 1)!!
(ej + 1− 2|Ij |)!!
.
Here we use the notation o• (resp., e•) to stress that these are odd (resp., even) non-negative
numbers, and ôi means that this particular index is skipped.
Denote by An the sum (−1)n(2g − 4 + n)!
∑
o1,...,on∈(2Z+1)>0
o1+···+on=2g−4+n
∏n
j=1
(
2aj
oj
)
.
Conjecture 5.7. For any n ≥ 2 and t = 0, . . . , n, a1, . . . , an ∈ Z≥0, a1 + · · ·+ an = 2g− 3 + n,
we have
Pn,t = (−1)t
[((
n− 1
t
)
−
(
n− 1
t− 1
))
(2g − 3 + n+ t) + 2(t− 1)
(
n− 1
t− 1
)]
An. (5.4)
Observe that the right hand side is equal to
(−1)t
[(
n− 1
t
)
(2g − 2 + n+ (t− 1))−
(
n− 1
t− 1
)
(2g − 2 + n− (t− 1))
]
An.
Remark 5.8. This conjecture does not follow from identity (5.1), so the equivalence of Faber’s
conjecture and identity (5.1) does not prove equation (5.4). On the other hand, let us prove
that equation (5.4) implies the main combinatorial identity (5.1). Indeed,
n∑
t=0
(−1)t
[(
n− 1
t
)
(2g − 2 + n+ (t− 1))−
(
n− 1
t− 1
)
(2g − 2 + n− (t− 1))
]
Half-Spin Tautological Relations and Faber’s Proportionalities of Kappa Classes 17
= (2g − 3 + n)
n∑
t=0
(−1)t
(
n− 1
t
)
+ (n− 1)
n∑
t=0
(−1)t
(
n− 2
t− 1
)
− (2g − 2 + n)
n∑
t=0
(−1)t
(
n− 1
t− 1
)
+ (n− 1)
n∑
t=0
(−1)t
(
n− 2
t− 2
)
= 0.
Therefore, a combinatorial proof of Conjecture 5.7 would immediately give a new proof of Faber’s
conjecture.
Proposition 5.9. Conjecture 5.7 is true for n ≤ 5, any t, and for t = 0, 1, 2, 3, any n.
We prove this proposition in Appendix A.
Remark 5.10. Note that surprisingly this proposition is also true for n = 1, though in this
case we have no identity (5.1). Indeed, for n = 1 we have
A1 = −(2g − 3)!
(
4g − 4
2g − 3
)
,
P1,0 = −(2g − 2)!
(
4g − 4
2g − 3
)
= (2g − 2)A1,
P1,1 = −(2g − 2)!
(
4g − 4
2g − 2
)
= (2g − 1)A1,
which matches exactly equation (5.4) for n = 1 and t = 0, 1.
5.4 An equivalent formulation of the conjecture
In this section we reformulate Conjecture 5.7 via a 3-term recursion in the Pn,t. Let P̃n,t be
P̃n,t := (−1)t
[((
n− 1
t
)
−
(
n− 1
t− 1
))
(2g − 3 + n+ t) + 2(t− 1)
(
n− 1
t− 1
)]
An.
Proposition 5.11. Let n ≥ 2 and a1, . . . , an ∈ Z≥0, a1 + · · ·+ an = 2g − 3 + n. The following
three statements, are equivalent:
i) Conjecture 5.7 holds:
Pn,t = P̃n,t for all t = 0, 1, . . . , n.
ii) The Pn,t obey the following 3-term recursion for all t = 0, . . . , n:
(t+ 1)Pn,t+1 + (n− (t+ 1))Pn,t =
(−1)t(2g − 1)
(2g − 3 + n)
(
n
t
)
Pn,0.
iii) The following expression does not depend on t:
t!(n− t)!(−1)t [(t+ 1)Pn,t+1 + (n− (t+ 1))Pn,t] .
Proof. Let Sg,n(x) :=
∑n
t=0 Pn,tx
t, for which we already know the values
Sg,n(0) = Pn,0,
Sg,n(1) =
n∑
t=0
Pn,t
18 E. Garcia-Failde, R. Kramer, D. Lewański and S. Shadrin
=
n∑
k=1
(−1)k(2g − 3 + k)!
k!
∑
I1t···tIk=JnK
Ij 6=∅, ∀ j∈JkK
∑
d1,...,dk∈Z≥0∑
i di=g−2+n
k∏
j=1
(
2a[Ij ] + 1
2dj
)
(2dj − 1)!!
(2dj + 1− 2|Ij |)!!
.
Let us now compute the generating polynomial for the P̃n,t
S̃g,n(x) :=
n∑
t=0
P̃n,tx
t
=
n∑
t=0
(−x)t
[((
n− 1
t
)
−
(
n− 1
t− 1
))
(2g − 3 + n+ t) + 2(t− 1)
(
n− 1
t− 1
)]
An.
First substitute An = Pn,0/(2g − 3 + n) and expand the whole expression in terms of the type
m∑
t=0
(
m
t
)
p(t)(−x)txa,
where p(t) is a polynomial in t and a is an integer. For each such summand substitute p(t) with
p
(
x d
dx
)
, apply Newton binomial theorem to
∑m
t=0
(
m
t
)
(−x)t = (1 − x)n, and finally apply the
operator p
(
x d
dx
)
to the summand. Collecting the summands’ resulting contributions together
gives
S̃g,n(x) =
(1− x)n−1
(2g − 3 + n)
((2g − 1)(x+ 1) + (n− 2))Pn,0. (5.5)
Observe that Sg,n(0) = S̃g,n(0), and S̃(x) satisfies the non-homogeneous first order ODE
(2g − 3 + n)
[
(1− x)f ′(x) + (n− 1)f(x)
]
= (1− x)n(2g − 1)Pn,0, (5.6)
which Sg,n(x) also satisfies if and only if
(t+ 1)Pn,t+1 + (n− (t+ 1))Pn,t =
(−1)t(2g − 1)
(2g − 3 + n)
(
n
t
)
Pn,0, (5.7)
for t = 0, 1, . . . , n. This proves the equivalence between (i) and (ii). Clearly (ii) implies (iii).
Let us see that (iii) also implies (ii). Assuming (iii), we can evaluate (ii) at any t. Let us pick
t = 0 for simplicity. Then (ii) reads
Pn,1 =
(2g − 1)
(2g − 3 + n)
Pn,0 − (n− 1)Pn,0 =
(2g − 1)− (n− 1)(2g − 3 + n)
(2g − 3 + n)
Pn,0
=
(2− n)(2g − 2 + n)
(2g − 3 + n)
Pn,0
= −
[((
n− 1
1
)
−
(
n− 1
0
))
(2g − 2 + n) + 2 · 0 ·
(
n− 1
0
)]
An,
which holds true from the case t = 1 in Proposition 5.9. This concludes the proof of the
proposition. �
A Proof of the main combinatorial identity for several cases
In this section we prove Proposition 5.9. By Remark 5.8, this implies a proof of the main
combinatorial identity (5.1) for n ≤ 5 and g ≥ 2. By Corollary 5.1, this implies a new proof of
Faber’s conjecture for n ≤ 5 and g ≥ 2.
We would like to emphasize that we present in this proof a brief exposition of a quite intricate
computation involving an interplay between binomials with even and odd denominators that we
could not find deeply analyzed in the literature on combinatorics.
Half-Spin Tautological Relations and Faber’s Proportionalities of Kappa Classes 19
Proof of Proposition 5.9. The case t = 0 is obvious from the definition. For the other cases,
we perform quite intricate computations based on the following lemma.
Lemma A.1. For any non-negative integers a1, . . . , an, a1 + · · · + an = A, and t1, . . . , tn,
t1 + · · ·+ tn = T , and for an arbitrary vector of parities (p1, . . . , pn), pi ∈ Z2, we have
∑
f1+···+fn=B
f̃i=pi, i=1,...,n
n∏
i=1
(
2ai
fi
)
(fi)ti =
∑
f1+···+fn=2A−B+T
f̃i=pi+t̃i, i=1,...,n
n∏
i=1
(
2ai
fi
)
(fi)ti .
Here by f̃ ∈ Z2 we denote the parity of f ∈ Z, and by (f)t we denote the Pochhammer symbol,
(f)t := f(f − 1) · · · (f + 1− t).
Proof. It follows from the following identity
(
2a
f
)
(f)t =
(
2a− t
f − t
)
(2a)t =
(
2a− t
2a− f
)
(2a)t =
(
2a
2a− f + t
)
(2a− f + t)t. �
Example A.2. If a1 + · · ·+ an = 2g − 3 + n, then
∑
o1,...,on∈(2Z+1)>0
o1+···+on=2g−4+n
n∏
j=1
(
2aj
oj
)
=
∑
o1,...,on∈(2Z+1)>0
o1+···+on=2g−2+n
n∏
j=1
(
2aj
oj
)
.
Thus we have an alternative definition of An as the sum
(−1)n(2g − 4 + n)!
∑
o1,...,on∈(2Z+1)>0
o1+···+on=2g−2+n
n∏
j=1
(
2aj
oj
)
.
Below in all arguments we apply Lemma A.1 assuming the condition a1+· · ·+an = 2g−3+n.
We will also make use several times of the Chu–Vandermonde identity
∑
k+`=n
(
r
k
)(
s
`
)
=
(
r+s
n
)
,
which follows from the expansion of the identity (1 + x)r(1 + x)s = (1 + x)r+s.
A.1 Case t = 1
We have
Pn,1 = (−1)n(2g − 3 + n)!
n∑
i=1
∑
o1,...,ôi,...on∈(2Z+1)>0
ei∈(2Z)≥0
o1+···ôi···+on+ei=2g−3+n
n∏
j=1
j 6=i
(
2aj
oj
)(
2ai
ei
)
+ (−1)n−1(2g − 4 + n)!
n∑
i,`=1
i<`
∑
o1,...,ôi,...,ô`,...on∈(2Z+1)>0
oi`∈(2Z+1)>0
o1+···ôi,ô`···+on+oi`=2g−3+n
n∏
j=1
j 6=i,`
(
2aj
oj
)(
2ai + 2a`
oi`
)
· oi`.
In the first term we can replace the factor (2g−3+n) by the sum of the indices o1+ · · · ôi · · ·+
on + ei. In the second term we can apply the Chu-Vandermonde identity
(
2ai + 2a`
oi`
)
· oi` =
∑
o∈(2Z+1)≥0
e∈(2Z)>0
o+e=oil
((
2ai
o
)(
2a`
e
)
+
(
2ai
e
)(
2a`
o
))
(e+ o).
20 E. Garcia-Failde, R. Kramer, D. Lewański and S. Shadrin
On the right-hand side of this equation, we always have one even bottom argument in the
binomial coefficients and all other bottom arguments are odd in both terms of this expression.
The expression is totally symmetric with respect to the choice of the place of the even bottom
argument. The coefficient in each summand of
(−1)n(2g − 4 + n)!
∑
o2...,on∈(2Z+1)>0
e1∈(2Z)≥0
e1+o2+···+on=2g−3+n
(
2a1
e1
) n∏
j=2
(
2aj
oj
)
(that is, we collect the terms where the even bottom argument is below 2a1) is equal to
(
e1 +
n∑
i=2
oi
)
−
n∑
i=2
(e1 + oi) = −e1 ·
((
n− 1
1
)
−
(
n− 1
0
))
.
Applying Lemma A.1 to this term, we obtain
(−1)n(2g − 4 + n)!
∑
o2...,on∈(2Z+1)>0
e1∈(2Z)≥0
e1+o2+···+on=2g−3+n
(
2a1
e1
) n∏
j=2
(
2aj
oj
)
· (−1) · e1 ·
((
n− 1
1
)
−
(
n− 1
0
))
= (−1)n(2g − 4 + n)!
∑
o1,...,on∈(2Z+1)>0
o1+···+on=2g−2+n
n∏
j=1
(
2aj
oj
)
· o1 · (−1) ·
((
n− 1
1
)
−
(
n− 1
0
))
.
Now, since the even bottom argument could be at any place, not only at the first one, we have
to replace in the full computation of Pn,1 the factor o1 above by o1 + · · ·+on = 2g−2 +n. Thus
we have
Pn,1 = An · (2g − 2 + n) · (−1) ·
((
n− 1
1
)
−
(
n− 1
0
))
,
which is exactly the desired result for t = 1.
A.2 Case t = 2
Let us describe Pn,2. All terms there have a common factor of (2g−5 +n)!. The sum of bottom
arguments of all binomial coefficients is always equal to S := 2g−2+n. Taking into account the
total symmetry with respect to the permutations of a1, . . . , an, we see that (−1)nPn,2/(2g−5+n)!
has
(
n
3
)
terms of the type
(
2a1 + 2a2 + 2a3
o123
) n∏
i=4
(
2ai
oi
)
· o123(o123 − 2),
3
(
n
4
)
terms of the type
(
2a1 + 2a2
o12
)(
2a3 + 2a4
o34
) n∏
i=5
(
2ai
oi
)
· o12o34,
(
n
2
)(
n− 2
1
)
terms of the type −
(
2a1 + 2a2
o12
)(
2a3
e3
) n∏
i=4
(
2ai
oi
)
· o12(S − 2),
(
n
2
)
terms of the type −
(
2a1 + 2a2
e12
) n∏
i=3
(
2ai
oi
)
· (e12 − 1)(S − 2),
(
n
2
)
terms of the type
(
2a1
e1
)(
2a2
e2
) n∏
i=3
(
2ai
oi
)
· (S − 1)(S − 2).
Half-Spin Tautological Relations and Faber’s Proportionalities of Kappa Classes 21
For instance, in the first line we mean that we have the following sum of
(
n
3
)
summands
∑
i<j<k
∑
oijk∈(2Z+1)>0
o`∈(2Z+1)>0, `∈{1,...,n}\{i,j,k}
oijk+
∑
`∈{1,...,n}\{i,j,k} o`=2g−2+n
(
2ai + 2aj + 2ak
oijk
) n∏
`=1
` 6=i,j,k
(
2a`
o`
)
· oijk(oijk − 2).
We assume that the parity of the bottom arguments denoted by o (resp., e) is odd (resp., even).
Let us expand all binomial coefficients using the Chu–Vandermonde identity, that is, in such
a way that we have exactly n factors of the type
(
2ai
fi
)
, where we also keep track of the possible
parity of the bottom arguments. For instance,
(
2a1 + 2a2 + 2a3
o123
)
=
∑
e1+e2+o3=o123
(
2a1
e1
)(
2a2
e2
)(
2a3
o3
)
+
∑
e1+o2+e3=o123
(
2a1
e1
)(
2a2
o2
)(
2a3
e3
)
+
∑
o1+e2+e3=o123
(
2a1
o1
)(
2a2
e2
)(
2a3
e3
)
+
∑
o1+o2+o3=o123
(
2a1
o1
)(
2a2
o2
)(
2a3
o3
)
.
We compute all the coefficients to obtain
(
n
2
)
terms of the type
(
2a1
e1
)(
2a2
e2
) n∏
i=3
(
2ai
oi
)
· 2e1e2
(
n− 2
2
)
,
(
n
1
)
terms of the type
n∏
i=1
(
2ai
oi
)
· o1(o1 − 1)
((
n− 1
2
)
−
(
n− 1
1
))
,
(
n
2
)
terms of the type
n∏
i=1
(
2ai
oi
)
· 2o1o2
((
n− 2
1
)
−
(
n− 1
1
))
,
(
n
1
)
terms of the type
n∏
i=1
(
2ai
oi
)
· o1
(
−
(
n− 1
2
)
+
(
n− 1
1
)
+
(
n
2
))
,
1 term of the type −
n∏
i=1
(
2ai
oi
)
· 2
(
n
2
)
.
Applying Lemma A.1 to the sum of all terms in the first line, we obtain the same terms as in the
third line, with the coefficient 2o1o2
(
n−2
2
)
. This, together with all terms in the second line and
the third line, gives us the term in the fifth line with the coefficient
((
n−1
2
)
−
(
n−1
1
))
S(S − 1).
The sum of all terms in the fourth line gives us also the term in the fifth line with the coefficient
S
(
−
(
n−1
2
)
+
(
n−1
1
)
+
(
n
2
))
. The observation that
((
n− 1
2
)
−
(
n− 1
1
))
S(S − 1) + S
(
−
(
n− 1
2
)
+
(
n− 1
1
)
+
(
n
2
))
− 2
(
n
2
)
= (S − 2) ·
[((
n− 1
2
)
−
(
n− 1
1
))
(S + 1) + 2
(
n− 1
1
)]
is exactly the product of (2g− 4 + n) and the desired coefficient of Pn,2/An completes the proof
of this case.
A.3 Case t = 3
Let us describe Pn,3. All terms there have a common factor of (2g−6 +n)!. The sum of bottom
arguments of all binomial coefficients is always equal to S := 2g−1+n. Taking into account the
22 E. Garcia-Failde, R. Kramer, D. Lewański and S. Shadrin
total symmetry with respect to the permutations of a1, . . . , an, we see that (−1)nPn,3/(2g−6+n)!
has terms of the following type:
−
(
2a1 + 2a2 + 2a3 + 2a4
o1234
) n∏
i=5
(
2ai
oi
)
· o1234(o1234 − 2)(o1234 − 4),
−
(
2a1 + 2a2 + 2a3
o123
)(
2a4 + 2a5
o45
) n∏
i=6
(
2ai
oi
)
· o123(o123 − 2)o45,
−
(
2a1 + 2a2
o12
)(
2a3 + 2a4
o34
)(
2a5 + 2a6
o56
) n∏
i=7
(
2ai
oi
)
· o12o34o56,
(
2a1 + 2a2 + 2a3
o123
)(
2a4
e4
) n∏
i=5
(
2ai
oi
)
· o123(o1234 − 2)(S − 4),
(
2a1 + 2a2 + 2a3
e123
) n∏
i=4
(
2ai
oi
)
· (e123 − 1)(e123 − 3)(S − 4),
(
2a1 + 2a2
o12
)(
2a3 + 2a4
o34
)(
2a5
e5
) n∏
i=6
(
2ai
oi
)
· o12o34(S − 4),
(
2a1 + 2a2
e12
)(
2a3 + 2a4
o34
) n∏
i=5
(
2ai
oi
)
· (e12 − 1)o34(S − 4),
−
(
2a1 + 2a2
o12
)(
2a3
e3
)(
2a4
e4
) n∏
i=5
(
2ai
oi
)
· o12(S − 3)(S − 4),
−
(
2a1 + 2a2
e12
)(
2a3
e3
) n∏
i=4
(
2ai
oi
)
· (e12 − 1)(S − 3)(S − 4),
(
2a1
e1
)(
2a2
e2
)(
2a3
e3
) n∏
i=4
(
2ai
oi
)
· (S − 2)(S − 3)(S − 4).
Let us expand all binomial coefficients using the Chu–Vandermonde identity, that is, in
such a way that we have exactly n factors of the type
(
2ai
fi
)
, where we also keep track of the
possible parity of the bottom arguments. Computing the coefficients, we obtain terms of the
type
(
2a1
e1
)(
2a2
e2
)(
2a3
e3
)∏n
i=4
(
2ai
oi
)
and
(
2a1
e1
)∏n
i=2
(
2ai
oi
)
with some complicated coefficients that we
want to collect in several disjoint groups.
A.3.1 First group of terms
Denote −
((
n−1
3
)
−
(
n−1
2
))
by C1. With this coefficient we have terms of the following type:
(
2a1
e1
) n∏
i=2
(
2ai
oi
)
· e1(e1 − 1)(e1 − 2) · C1,
(
2a1
e1
)(
2a2
e2
)(
2a3
e3
) n∏
i=4
(
2ai
oi
)
· e1e2e3 · 6C1,
(
2a1
e1
)(
2a2
e2
) n∏
i=3
(
2ai
oi
)
· e1o2(o2 − 1) · 3C1.
Applying Lemma A.1 to these terms, we obtain
∑
o1+···+on
=2g−2+n
n∏
i=1
(
2ai
oi
)
· (2g − 2 + n)(2g − 3 + n)(2g − 4 + n)C1.
Half-Spin Tautological Relations and Faber’s Proportionalities of Kappa Classes 23
A.3.2 Second group of terms
Denote −4
(
n−1
2
)
by C2. With this coefficient we have terms of the following type:
(
2a1
e1
) n∏
i=2
(
2ai
oi
)
· e1 · e1C2,
(
2a1
e1
)(
2a2
o2
) n∏
i=3
(
2ai
oi
)
· e1 · o2C2,
(
2a1
e1
) n∏
i=2
(
2ai
oi
)
· e1 · (−4)C2.
We collect these terms into
(
2a1
e1
)∏n
i=2
(
2ai
oi
)
· e1 · (2g − 5 + n)C2. Applying Lemma A.1 to all
these terms, we obtain
∑
o1+···+on
=2g−4+n
n∏
i=1
(
2ai
oi
)
· (o1 + · · ·+ on)(2g − 5 + n)C2
=
∑
o1+···+on
=2g−4+n
n∏
i=1
(
2ai
oi
)
· (2g − 4 + n)(2g − 5 + n)C2.
Applying Lemma A.1 again, we obtain
∑
o1+···+on
=2g−2+n
n∏
i=1
(
2ai
oi
)
· (2g − 4 + n)(2g − 5 + n)C2.
A.3.3 Third group of terms
Denote 6
((
n−1
3
)
−
(
n−1
2
))
by C3. With this coefficient we have terms of the following type:
(
2a1
e1
) n∏
i=2
(
2ai
oi
)
· e1 · C3.
Applying Lemma A.1 to all these terms, we obtain
∑
o1+···+on
=2g−4+n
n∏
i=1
(
2ai
oi
)
· (o1 + · · ·+ on)C3 =
∑
o1+···+on
=2g−4+n
n∏
i=1
(
2ai
oi
)
· (2g − 4 + n)C3.
Applying Lemma A.1 again, we obtain
∑
o1+···+on
=2g−2+n
n∏
i=1
(
2ai
oi
)
· (2g − 4 + n)C3.
A.3.4 Fourth group of terms
Denote −2
(
n−2
1
)
by C4. With this coefficient we have terms of the following type:
(
2a1
e1
)(
2a2
e2
)(
2a3
e3
) n∏
i=4
(
2ai
oi
)
· 3e1e2e3C4,
24 E. Garcia-Failde, R. Kramer, D. Lewański and S. Shadrin
(
2a1
e1
) n∏
i=2
(
2ai
oi
)
· e1o2(o2 − 1)C4,
−
(
2a1
e1
) n∏
i=2
(
2ai
oi
)
· e1(e1 − 1)o2C4,
−
(
2a1
e1
) n∏
i=2
(
2ai
oi
)
· e1o2o3C4.
Applying Lemma A.1 to the first two lines, we obtain
∑
o1+···+on
=2g−2+n
n∏
i=1
(
2ai
oi
)
·
(∑
i<j
oioj
)
· (o1 + · · ·+ on − 2)C4
=
∑
o1+···+on
=2g−2+n
n∏
i=1
(
2ai
oi
)
·
(∑
i<j
oioj
)
· (2g − 4 + n)C4.
Applying Lemma A.1 to the last two lines, we obtain
−
∑
i<j
∑
ei+ej+
∑
`∈{1,...,n}\{i,j} o`
=2g−2+n
(
2ai
ei
)(
2aj
ej
) n∏
`=1
` 6=i,j
(
2ai
oi
)
· eiej ·
ei + ej +
∑
`∈{1,...,n}\{i,j}
o`
C4
= −
∑
i<j
∑
ei+ej+
∑
`∈{1,...,n}\{i,j} o`
=2g−2+n
(
2ai
ei
)(
2aj
ej
) n∏
`=1
`6=i,j
(
2ai
oi
)
· eiej · (2g − 4 + n)C4.
It follows from Lemma A.1 that
∑
o1+···+on
=2g−2+n
n∏
i=1
(
2ai
oi
)
·
∑
i<j
oioj
−
∑
i<j
∑
ei+ej
+
∑
`∈{1,...,n}\{i,j} o`
=2g−2+n
(
2ai
ei
)(
2aj
ej
) n∏
`=1
` 6=i,j
(
2ai
oi
)
· eiej = 0.
Hence, the total sum of all terms with the coefficient C4 is equal to 0.
A.3.5 Final computation
In order to complete the proof of the case t = 3 it is sufficient to observe that
(2g − 2 + n)(2g − 3 + n)(2g − 4 + n)C1 + (2g − 4 + n)(2g − 5 + n)C2 + (2g − 4 + n)C3
= −(2g − 4 + n)(2g − 5 + n)
[((
n− 1
3
)
−
(
n− 1
2
))
(2g + n) + 4
(
n− 1
2
)]
.
A.4 Case n = t = 4
In this case a1 + · · ·+ a4 = 2g + 1. We have the following formula for P4,4:
P4,4
(2g − 1)!
=
4∑
k=1
(−1)k(2g − 3 + k)!
k!(2g − 1)!
∑
I1t···tIk
={1,...,4}
∑
e1,...,ek∈(2Z)≥0
e1+···+ek=2g+4
k∏
j=1
(
2a[Ij ]
ej
)
(ej − 1)!!
(ej + 1− 2|Ij |)!!
.
Half-Spin Tautological Relations and Faber’s Proportionalities of Kappa Classes 25
Note that if k = 1, then (2g − 3 + k)! = (2g − 2)!. But then this term looks like
(
2a1 + 2a2 + 2a3 + 2a4
2g + 4
)
(2g + 3)(2g + 1)(2g − 1),
and the last factor here still allows us to extract the common coefficient of (2g− 1)!. With that
remark we see that every term in the expression for
P4,4
(2g−1)! above is multiplied by a quadratic
polynomial in e1, . . . , ek.
Applying the Chu–Vandermonde identity in the same way as in the previous cases, we obtain
terms of the following type:
4∏
i=1
(
2ai
ei
)
· 0, −
(
2a1
e1
)(
2a2
e2
)(
2a3
o3
)(
2a4
o3
)
· 2e1e2, −
4∏
i=1
(
2ai
oi
)
· o1(o1 − 1).
Applying Lemma A.1 to all these terms, we obtain
−
∑
o1+o2+o3
+o4=2g
4∏
i=1
(
2ai
oi
)
· (o1 + o2 + o3 + o4)(o1 + o2 + o3 + o4 − 1) = −(2g − 1) · A4
(2g − 1)!
,
which confirms this case of the proposition.
A.5 Case n = t = 5
In this case a1 + · · ·+ a5 = 2g + 2. We have the following formula for P5,5:
P5,5
(2g − 1)!
=
5∑
k=1
(−1)k(2g − 3 + k)!
k!(2g − 1)!
∑
I1t···tIk
={1,...,5}
∑
e1,...,ek∈(2Z)≥0
e1+···+ek=2g+6
k∏
j=1
(
2a[Ij ]
ej
)
(ej − 1)!!
(ej + 1− 2|Ij |)!!
.
Note that if k = 1, then (2g − 3 + k)! = (2g − 2)!. But then this term looks like
(
2a1 + 2a2 + 2a3 + 2a4 + 2a5
2g + 6
)
(2g + 5)(2g + 3)(2g + 1)(2g − 1),
and the last factor here still allows us to extract the common coefficient of (2g − 1)!. With
that remark we see that every term in the expression for
P5,5
(2g−1)! above is multiplied by a cubic
polynomial in e1, . . . , ek.
Applying the Chu–Vandermonde identity in the same way as in the previous cases, we obtain
terms of the following type:
−
(
2a1
e1
)(
2a2
e2
)(
2a3
e3
)(
2a4
o3
)(
2a5
o5
)
· 6e1e2e3,
−
(
2a1
e1
)(
2a2
o2
) 5∏
i=3
(
2ai
oi
)
· 3e1o2(o2 − 1),
−
(
2a1
e1
) 5∏
i=2
(
2ai
oi
)
· e1(e1 − 1)(e1 − 2).
Applying Lemma A.1 to all these terms, we obtain
−
∑
o1+o2+o3
+o4+o5=2g+1
5∏
i=1
(
2ai
oi
)
·
(
5∑
i=1
oi
)(
5∑
i=1
oi − 1
)(
5∑
i=1
oi − 2
)
= (2g − 1) · A5
(2g − 1)!
,
which confirms this case of the proposition.
26 E. Garcia-Failde, R. Kramer, D. Lewański and S. Shadrin
A.6 Case n = 5, t = 4
In this case a1 + · · ·+ a5 = 2g + 2. We have the following formula for P5,4:
P5,4
(2g − 1)!
=
5∑
k=1
(−1)k(2g − 3 + k)!
k!(2g − 1)!
×
∑
I1t···tIk
={1,...,5}
∑
e1,...,ek∈(2Z)≥0
e1+···+ek=2g+6
k∑
`=1
k∏
j=1
(
2a[Ij ]
ej − δ`j
)
(ej − 1)!!
(ej + 1− 2|Ij |)!!
.
Here we can divide by (2g − 1)! for the same reason as in the case n = t = 5, and after that we
can consider the coefficient of every term in this expression to be a cubic polynomial in ej − δ`j .
We apply the Chu–Vandermonde identity in the same way as in the previous cases, and we
obtain two groups of terms (the sum of the bottom arguments in the binomial coefficients in
these terms is equal to 2g + 5).
The first group of terms consists of
20 terms of the type
(
2a1
e1
)(
2a2
e2
) 5∏
i=3
(
2ai
oi
)
· (−6)e21e2,
10 terms of the type
(
2a1
e1
)(
2a2
e2
) 5∏
i=3
(
2ai
oi
)
· 42e1e2,
30 terms of the type
(
2a1
e1
)(
2a2
e2
)(
2a3
o3
) 5∏
i=4
(
2ai
oi
)
· (−6)e1e2o1.
Taking into account that e1 + e2 + o3 + o4 + o5 = 2g + 5, we see that the sum of all this terms
is equal to
10 terms of the type
(
2a1
e1
)(
2a2
e2
) 5∏
i=3
(
2ai
oi
)
· (−6)e1e2(2g − 2). (A.1)
The second group of terms consists of
5 terms of the type
(
2a1
o1
) 5∏
i=2
(
2ai
oi
)
· (−3)o1(o1 − 1)(o1 − 7),
20 terms of the type
(
2a1
o1
)(
2a2
o2
) 5∏
i=3
(
2ai
oi
)
· (−3)o1(o1 − 1)o2.
Taking into account that o1 + o2 + o3 + o4 + o5 = 2g + 5, we see that the sum of all this terms
is equal to
5 terms of the type
(
2a1
o1
) 5∏
i=2
(
2ai
oi
)
· (−3)o1(o1 − 1)(2g − 2). (A.2)
We apply Lemma A.1 to (A.1) and (A.2), and this gives us
∑
o1+o2+o3
+o4+o5=2g+1
5∏
i=1
(
2ai
oi
)
· (−3)
(
5∑
i=1
oi
)(
5∑
i=1
oi − 1
)
= −3(2g − 2) · A5
(2g − 1)!
,
which confirms the proposition in this case. This concludes the proof of the proposition. �
Half-Spin Tautological Relations and Faber’s Proportionalities of Kappa Classes 27
Acknowledgments
We thank A. Buryak, J. Schmitt and D. Zagier for useful comments on the first version of
the paper. R.K., D.L., and S.S. were supported by the Netherlands Organization for Scientific
Research. D.L. was also supported by the Max Planck Gesellschaft. E.G.-F. was supported by
the Max Planck Gesellschaft and by the Labex Mathematics Hadamard. She is also grateful
for the research stay at the University of Amsterdam, which made the beginning of this work
possible. We thank the anonymous referees for many useful remarks.
References
[1] Buryak A., Shadrin S., A new proof of Faber’s intersection number conjecture, Adv. Math. 228 (2011),
22–42, arXiv:0912.5115.
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https://doi.org/10.4171/103-1/12
1 Introduction
1.1 Half-spin relations
1.2 Faber's intersection numbers conjecture
1.3 Organization of the paper
2 Definition of half-spin relations
3 A combinatorial identity from half-spin relations
4 Psi-classes of negative degree
4.1 Formal negative degrees of psi-classes
4.2 Q-polynomials and a refined string equation
4.3 Applying formal negative degrees of -classes to the combinatorial identity
5 The main combinatorial identity and its structure
5.1 Polynomials vanishing in the integer points of some simplices
5.2 Combinatorial reduction of the identity for ai=1
5.3 A conjectural refinement of the identity
5.4 An equivalent formulation of the conjecture
A Proof of the main combinatorial identity for several cases
A.1 Case t=1
A.2 Case t=2
A.3 Case t=3
A.3.1 First group of terms
A.3.2 Second group of terms
A.3.3 Third group of terms
A.3.4 Fourth group of terms
A.3.5 Final computation
A.4 Case n=t=4
A.5 Case n=t=5
A.6 Case n=5, t=4
References
|
| id | nasplib_isofts_kiev_ua-123456789-210308 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T21:25:06Z |
| publishDate | 2019 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Garcia-Failde, E. Kramer, R. Lewański, D. Shadrin, S. 2025-12-05T09:31:15Z 2019 Half-Spin Tautological Relations and Faber's Proportionalities of Kappa Classes / E. Garcia-Failde, R. Kramer, D. Lewański, S. Shadrin // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 22 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 14H10; 05A10 arXiv: 1902.02742 https://nasplib.isofts.kiev.ua/handle/123456789/210308 https://doi.org/10.3842/SIGMA.2019.080 We employ the 1/2-spin tautological relations to provide a particular combinatorial identity. We show that this identity is a statement equivalent to Faber's formula for proportionalities of kappa-classes on Mg, g≥2. We then prove several cases of the combinatorial identity, providing a new proof of Faber's formula for those cases. We thank A. Buryak, J. Schmitt, and D. Zagier for useful comments on the first version of the paper. R.K., D.L., and S.S. were supported by the Netherlands Organization for Scientific Research. D.L. was also supported by the Max Planck Gesellschaft. E.G.-F. was supported by the Max Planck Gesellschaft and by the Labex Mathematics Hadamard. She is also grateful for the research stay at the University of Amsterdam, which made the beginning of this work possible. We thank the anonymous referees for many useful remarks. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Half-Spin Tautological Relations and Faber's Proportionalities of Kappa Classes Article published earlier |
| spellingShingle | Half-Spin Tautological Relations and Faber's Proportionalities of Kappa Classes Garcia-Failde, E. Kramer, R. Lewański, D. Shadrin, S. |
| title | Half-Spin Tautological Relations and Faber's Proportionalities of Kappa Classes |
| title_full | Half-Spin Tautological Relations and Faber's Proportionalities of Kappa Classes |
| title_fullStr | Half-Spin Tautological Relations and Faber's Proportionalities of Kappa Classes |
| title_full_unstemmed | Half-Spin Tautological Relations and Faber's Proportionalities of Kappa Classes |
| title_short | Half-Spin Tautological Relations and Faber's Proportionalities of Kappa Classes |
| title_sort | half-spin tautological relations and faber's proportionalities of kappa classes |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210308 |
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