TAUTOLOGICAL EQUATIONS IN GENUS 2 VIA INVARIANCE THEOREM
D. ARCARA AND Y.-P. LEE
Abstract. We verify the Invariance Conjectures of tautological equations [7] in genus two.
In particular, a uniform derivation of all known genus two equations is given.
0. Introduction
The purpose of this paper is to verify the genus two case of Invariance Conjectures of tautological equations proposed in [7]. In particular, applying Theorem 5 in [8] (i.e. Conjecture 1 in
[7]) and E. Getzler’s Hodge numbers calculations [4], we are able to give a uniform derivation of
all known genus two tautological equations: Mumford–Getzler’s equation, Getzler’s equation [4]
and Belorousski–Pandharipande’s equation [2]. This, combined with [5] in genus one and [1] in
genus three, shows that this method generates and proves all known tautological equations.
0.1. Tautological rings. One reference for tautological rings, which is close to the spirit of the
present paper, is R. Vakil’s survey article [10].
Let Mg,n be the moduli stacks of n-pointed smooth genus g curves. They have modular compactifications Mg,n , the moduli stacks of stable curves, introduced by P. Deligne, D. Mumford
and F. Knudsen. Mg,n are proper, irreducible, smooth Deligne–Mumford stacks. The Chow
rings A∗ (Mg,n ) over Q are isomorphic to the Chow rings of their coarse moduli spaces. The
tautological rings R∗ (Mg,n ) are subrings of A∗ (Mg,n ), or subrings of H 2∗ (Mg,n ) via cycle maps,
generated by some “geometric classes” which will be described below.
The first type of geometric classes are the boundary strata. Mg,n have natural stratification
by topological types. The second type of geometric classes are the Chern classes of tautological
vector bundles. These includes cotangent classes ψi , Hodge classes λk and κ-classes κl .
To give a precise definition of the tautological rings, some natural morphisms between moduli
stacks of curves will be used. The forgetful morphisms
(1)
fti : Mg,n+1 → Mg,n
forget one of the n + 1 marked points. The gluing morphisms
(2)
Mg1 ,n1 +1 × Mg2 ,n2 +1 → Mg1 +g2 ,n1 +n2 ,
Mg−1,n+2 → Mg,n ,
glue two marked points to form a curve with a new node. Note that the boundary strata are
the images (of the repeated applications) of the gluing morphisms, up to factors in Q due to
automorphisms.
Definition 1. The system of tautological rings {R∗ (Mg,n )}g,n is the smallest system of Q-unital
subalgebra (containing classes of type one and two, and is) closed under push-forwards via the
forgetful and gluing morphisms.
The study of the tautological rings is one of the central problems in moduli of curves. The
readers are referred to [10] and references therein for many examples and motivation. Note that
the tautological rings are defined by generators and relations. Since the generators are explicitly
given, the study of tautological rings is the study of relations of tautological classes.
The second author is partially supported by NSF and AMS Centennial Fellowship.
1
2
D. ARCARA AND Y.-P. LEE
0.2. Invariance Conjectures. Here some ingredients in [7] and [8] will be briefly reviewed.
The strata can be conveniently presented by their (dual) graphs, which can be described as
follows. To each stable curve C with marked points, one can associate a dual graph Γ. Vertices
of Γ correspond to irreducible components. They are labeled by their geometric genus. Assign
an edge joining two vertices each time the two components intersect. To each marked point, one
draws an half-edge incident to the vertex, with the same label as the point. Now, the stratum
corresponding to Γ is the closure of the subset of all stable curves in Mg,n which have the
same topological type as C. For each dual graph Γ, one can decorate the graph by assigning a
monomial, or more generally a polynomial, of ψ to each half-edge and κ classes to each vertex.
The tautological classes in Rk (Mg,n ) can be represented by Q-linear combinations of decorated
graphs. Since there is no κ, λ-classes involved in this paper, they will be left out of discussions
below.
For typesetting reasons, it is more convenient to denote a decorated graph by another notation,
inspired by Gromov–Witten theory, called gwi. Given a decorated graph Γ.
• For the vertices of Γ of genus g1 , g2 , . . ., assign a product of “brackets” hig1 hig2 . . .. To
simplify the notations, hi := hi0 .
• Assign each half-edge a symbol ∂ ∗ . The external half-edges use super-indices ∂ x , ∂ y , . . .,
corresponding to their labeling. For each pair of half-edges coming from one and the same
edge, the same super-index will by used, denoted by Greek letters (µ, ν, . . ..) Otherwise,
all half-edges should use different super-indices.
• For each decoration to a half-edge a by ψ-classes ψ k , assign a subindex to the corresponding half-edge ∂ka .
• For each a given vertex hig with m half-edges, n external half-edges, an insertion is
placed in the vertex h∂kx1 ∂ky2 . . . ∂kµ′ ∂kν′′ . . .ig .
Example. Let Γ be the following graph.
z ψw2
•g = 2
xy
g = 0•
•
g=1
The corresponding gwi is:
h∂ x ∂ y ∂ µ ih∂ z ∂2w ∂ ν i2 h∂ µ ∂ ν i1 .
The key tool employed in this paper is the existence of linear operators
(3)
•
rl : Rk (Mg,n ) → Rk−l+1 (Mg−1,n+2 ),
l = 1, 2, . . . ,
where the symbol • denotes the moduli of possibly disconnected curves. The existence is proved
in [8] Theorem 5, originally Invariance Conjecture 1 in [7]. rl is defined as an operation on the
decorated graphs. The output graphs have two more markings, which are denoted by i, j. In
TAUTOLOGICAL EQUATIONS IN GENUS 2 VIA INVARIANCE THEOREM
3
terms of gwis,
rl (h∂kµ′ . . .ig′ . . . h∂kµ′′ . . .ig′′ )
1 i
=
h∂k′ +l . . .ig′ . . . h∂kj ′′ . . .ig′′ + h∂kj ′ . . .ig′ h∂ki ′′ +l . . .ig′′
2
1
+ (−1)l−1 h∂kj ′ +l . . .ig′ . . . h∂ki ′′ . . .ig′′ + h∂ki ′ . . .ig′ h∂kj ′′ +l . . .ig′′ + . . .
2
l−1
1 X
i
j µ
(−1)m+1 h∂l−1−m
∂m
∂k′ . . .ig′ −1 . . . h∂kµ′′ . . .ig′′ + . . .
+
2 m=0
(4)
l−1
1 X
i
j µ
(−1)m+1 h∂kµ′ . . .ig′ . . . h∂l−1−m
∂m
∂k′′ . . .ig′′ −1
2 m=0


g′
l−1
X
1 X
i
j
(−1)m+1
∂kµ′ . . . h∂l−1−m
ig h∂m
ig′ −g  h∂kµ′′ . . .ig′′ + . . .
+
2 m=0
g=0


g′′
l−1
X
X
1 µ
j
i
+ h∂k′ . . .ig′ 
(−1)m+1
ig′′ −g  ,
∂kµ′′ . . . h∂l−1−m
ig′′ h∂m
2
m=0
g=0
+
j
i
ig2 ) means that the half-edge insertions ∂kµ . . . acts on
where the notation ∂kµ . . . (h∂l−1−m
ig1 h∂m
j
i
the product of vertices h∂l−1−m ig1 h∂m ig2 by Leibniz rule. Note that h. . .i−1 := 0.
Remark. In terms of graphical operations, the first two lines stand for “cutting edges”; the
middle two for “genus reduction”; the last two for “splitting vertices”. These are explained in
[7].
There are three Invariance Conjectures proposed in [7]. Invariance Conjecture 1 has been
stated above. The remaining two conjectures are
Invariance Conjecture 2. If rl (E) = 0 for all l, then E = 0 is a tautological equation.
Invariance Conjecture 3. Invariance Conjecture 2 will produce all tautological equations inductively.
0.3. The algorithm of finding tautological equations. A general algorithm of finding the
tautological equations, based on Conjectures 2 and 3, is explained in [7] Section 2. Since these
remain conjectural, one possible alternative to the general scheme is to
• Calculate the rank of tautological ring Rk (Mg,n ) to see if there is any new equation. If
so, write this new equation as
X
E=
cm Γm = 0.
• Apply invariance equation (3)
r1 (E) = 0,
to obtain the coefficients cm .
•
Note that r1 (Γ) lies in Rk (Mg−1,n+1 ), whose structure is known by induction. In the case of
g = 2, one has the comfort of knowing genus one tautological rings are completely determined
by E. Getzler [3] and in preparation.
0.4. Main results.
Theorem 1. Invariance Conjectures hold for (g, n, k) = (2, 1, 2), (2, 2, 2), (2, 3, 2). In particular, a uniform derivation of all known genus two tautological equations is given by invariance
condition (3).
4
D. ARCARA AND Y.-P. LEE
Remark. As explained in [7] and [8], our calculation in terms of gwis can be translated verbatim
into one for any (axiomatic) Gromov–Witten theories. Therefore, it completes (the write-up of)
a proof of Virasoro conjecture in the semisimple case and of Witten’s conjecture (on spin curves
and Gelfand–Dickey hierarchies), both up to genus two.
Acknowledgement. We wish to thank A. Bertram, E. Getzler, A. Givental, R. Pandharipande,
and R. Vakil for many useful discussions. The final stage of this work was done during the
second author’s visit to NCTS, whose hospitality is greatly appreciated.
1. Mumford–Getzler’s equation in M2,1
In all calculations below, we will employ the “gwi” notations for decorated graphs. It is
explained in [7] that gwis are equivalent to the decorated graphs, or a tautological class. The
notations are obviously inspired by Gromov–Witten invariants.
1.1. Tautological classes of R2 (M2,1 ). There are 8 boundary strata of codimension ≤ 2 in
M2,1 : 1 stratum in codimension 0, 2 strata in codimension 1, and 5 strata in codimension 2.
If we insert ψ classes, the 2 boundary strata in codimension 1 produce 5 different tautological
elements in codimension 2. Note that the κ-classes can be expressed in terms of boundary and
ψ-classes in genus two. So the only decoration one would need is the ψ-classes.
Here is a list of all the 11 strata with ψ classes in codimension 2:
h∂2x i2 , h∂ x ∂1µ ∂ µ i1 , h∂1x ∂ µ ∂ µ i1 , h∂ x ∂1µ i1 h∂ µ i1 , h∂ x ∂ µ i1 h∂1µ i1 , h∂1x ∂ µ i1 h∂ µ i1 ,
h∂ µ i1 h∂ x ∂ µ ∂ ν ∂ ν i, h∂ x ∂ µ i1 h∂ µ ∂ ν ∂ ν i, h∂ µ ∂ ν i1 h∂ x ∂ µ ∂ ν i, h∂ µ i1 h∂ ν i1 h∂ x ∂ µ ∂ ν i,
h∂ x ∂ µ ∂ µ ∂ ν ∂ ν i.
The 5 strata with ψ-classes can be written in terms of the 5 strata without ψ-classes using
TRR’s, and therefore we only have 6 terms which could be independent. A general element can
be written as
E =c1 h∂2x i2 + c2 h∂ µ i1 h∂ x ∂ µ ∂ ν ∂ ν i + c3 h∂ x ∂ µ i1 h∂ µ ∂ ν ∂ ν i
+c4 h∂ µ ∂ ν i1 h∂ x ∂ µ ∂ ν i + c5 h∂ µ i1 h∂ ν i1 h∂ x ∂ µ ∂ ν i + c6 h∂ x ∂ µ ∂ µ ∂ ν ∂ ν i.
1.2. Calculating r1 (E). Throughout this paper, the labelings i, j are assumed to be symmetrized
for l odd, and anti-symmetrized for l even.
h∂2x i2
1
1
7→ − h∂ µ i1 h∂ x ∂ µ ∂ ν ih∂ i ∂ j ∂ ν i − h∂ x ∂ µ ∂ µ ∂ ν ih∂ i ∂ j ∂ ν i
2
48
1 x µ ν i j µ ν
1 j
− h∂ ∂ ∂ ih∂ ∂ ∂ ∂ i − h∂ i1 h∂ x ∂ µ ∂ ν ih∂ i ∂ µ ∂ ν i
24
24
h∂ µ i1 h∂ x ∂ µ ∂ ν ∂ ν i 7→
1 i µ µ x j ν ν
h∂ ∂ ∂ ih∂ ∂ ∂ ∂ i + h∂ j i1 h∂ i ∂ µ ∂ ν ih∂ x ∂ µ ∂ ν i
24
1
− h∂ i ∂ j ∂ µ ih∂ x ∂ µ ∂ ν ∂ ν i − h∂ µ i1 h∂ x ∂ i ∂ µ ih∂ j ∂ ν ∂ ν i
2
h∂ x ∂ µ i1 h∂ µ ∂ ν ∂ ν i 7→ h∂ µ i1 h∂ i ∂ x ∂ µ ih∂ j ∂ ν ∂ ν i +
1 i x µ µ j ν ν
h∂ ∂ ∂ ∂ ih∂ ∂ ∂ i
24
1
− h∂ i ∂ j ∂ x ∂ µ ih∂ µ ∂ ν ∂ ν i − h∂ i i1 h∂ j ∂ x ∂ µ ih∂ µ ∂ ν ∂ ν i
2
h∂ µ ∂ ν i1 h∂ x ∂ µ ∂ ν i 7→ 2h∂ µ i1 h∂ i ∂ µ ∂ ν ih∂ x ∂ j ∂ ν i +
1 i µ µ ν x j ν
h∂ ∂ ∂ ∂ ih∂ ∂ ∂ i
12
1
− h∂ i ∂ j ∂ µ ∂ ν ih∂ x ∂ µ ∂ ν i − h∂ i i1 h∂ j ∂ µ ∂ ν ih∂ x ∂ µ ∂ ν i
2
h∂ µ i1 h∂ ν i1 h∂ x ∂ µ ∂ ν i 7→
1 ν
h∂ i1 h∂ i ∂ µ ∂ µ ih∂ x ∂ j ∂ ν i − h∂ ν i1 h∂ i ∂ j ∂ µ ih∂ x ∂ µ ∂ ν i
12
TAUTOLOGICAL EQUATIONS IN GENUS 2 VIA INVARIANCE THEOREM
5
h∂ x ∂ µ ∂ µ ∂ ν ∂ ν i 7→ 4h∂ i ∂ µ ∂ ν ih∂ x ∂ j ∂ µ ∂ ν i − 2h∂ i ∂ x ∂ µ ∂ µ ih∂ j ∂ ν ∂ ν i
1.3. Setting r1 (E) = 0. Now we will pick a basis, and set its coordinates to zero.
(5)
h∂ µ i1 h∂ x ∂ µ ∂ ν ih∂ i ∂ j ∂ ν i :
1
− c1 + 2c4 − c5 = 0.
2
(6)
h∂ µ i1 h∂ x ∂ i ∂ µ ih∂ j ∂ ν ∂ ν i :
−c2 + c3 +
(7)
h∂ i i1 h∂ j ∂ x ∂ µ ih∂ µ ∂ ν ∂ ν i :
−
1
c5 = 0.
12
1
c1 + c2 − c3 − c4 = 0.
24
1
1
c2 + c3 − 2c6 = 0.
24
24
The remaining terms are related to each via WDVV as follows:
(8)
h∂ x ∂ i ∂ µ ∂ µ ih∂ j ∂ ν ∂ ν i :
h∂ x ∂ i ∂ j ∂ µ ih∂ µ ∂ ν ∂ ν i
=h∂ x ∂ µ ∂ µ ∂ ν ih∂ i ∂ j ∂ ν i + 2h∂ x ∂ µ ∂ ν ih∂ i ∂ j ∂ µ ∂ ν i − 2h∂ x ∂ i ∂ µ ih∂ j ∂ µ ∂ ν ∂ ν i,
h∂ x ∂ i ∂ µ ∂ ν ih∂ j ∂ µ ∂ ν i
=h∂ x ∂ µ ∂ µ ∂ ν ih∂ i ∂ j ∂ ν i + h∂ x ∂ µ ∂ ν ih∂ i ∂ j ∂ µ ∂ ν i − h∂ x ∂ i ∂ µ ih∂ j ∂ µ ∂ ν ∂ ν i.
Therefore, among the 5 vectors, only 3 of them are linearly independent.
(9)
(10)
h∂ x ∂ µ ∂ µ ∂ ν ih∂ i ∂ j ∂ ν i :
h∂ x ∂ µ ∂ ν ih∂ i ∂ j ∂ µ ∂ ν i :
−
1
1
1
c1 − c2 − c3 + 4c6 = 0.
48
2
2
−
1
1
c1 − c4 − c3 + 4c6 = 0.
24
2
1
c4 − 4c6 = 0.
12
The system of equations (5), (6), (7), (8), (9), (10), and (11) has a unique solution (up to
scaling)
1
1
7
1
13
c 1 , c3 =
c 1 , c4 = − c 1 , c5 = − c 1 , c6 = −
c1
c2 = −
240
240
10
10
960
We therefore obtain that, if we let c1 = −1,
13 µ
1
−h∂2x i2 +
h∂ i1 h∂ x ∂ µ ∂ ν ∂ ν i −
h∂ x ∂ µ i1 h∂ µ ∂ ν ∂ ν i
240
240
1
7
1
+ h∂ µ ∂ ν i1 h∂ x ∂ µ ∂ ν i + h∂ µ i1 h∂ ν i1 h∂ x ∂ µ ∂ ν i +
h∂ x ∂ µ ∂ µ ∂ ν ∂ ν i = 0,
10
10
960
which is Mumford–Getzler’s equation.
(11)
h∂ x ∂ i ∂ µ ih∂ j ∂ µ ∂ ν ∂ ν i :
c3 +
1.4. Checking r2 (E) = 0. Let us now calculate r2 (E).
h∂2x i2 7→ −
1
h∂ i ∂ µ ∂ µ ih∂ x ∂ α ∂ j ih∂ α ∂ ν ∂ ν i
576
1 j µ µ i α ν x α ν
1
h∂ ∂ ∂ ih∂ ∂ ∂ ih∂ ∂ ∂ i − h∂ j ∂ x ∂ µ ih∂ i ∂ ν ∂ ν ih∂ α ∂ α ∂ µ i
24
24
1
h∂ µ ∂ ν i1 h∂ x ∂ µ ∂ ν i 7→ − h∂ j ∂ ν ∂ µ ih∂ i ∂ α ∂ α ih∂ x ∂ ν ∂ µ i
24
h∂ x ∂ µ ∂ µ ∂ ν ∂ ν i 7→ −h∂ j ∂ µ ∂ µ ih∂ i ∂ α ∂ ν ih∂ x ∂ α ∂ ν i − h∂ j ∂ µ ∂ µ ih∂ i ∂ α ∂ ν ih∂ x ∂ α ∂ ν i.
h∂ µ ∂ µ ∂ ν ih∂ x ∂ ν i1 7→
The other graphs all have r2 (Γ) = 0.
6
D. ARCARA AND Y.-P. LEE
Therefore, r2 (E) = 0 as
1
1
1
1
c1 + c3 + c3 + c4 − c6 − c6 = 0.
576
24
24
24
2. Getzler’s equation in M2,2
2.1. Tautological classes in M2,2 of codimension 2. A general linear combination of codimension 2 tautological classes in M2,2 is, after removing the linearly dependent classes from the
induced equations (TRR’s and Mumford–Getzler’s),
E =c1 h∂1x ∂1y i2 + c2 h∂1µ i2 h∂ x ∂ y ∂ µ i + c3 h∂ x ∂ y ∂ µ ∂ µ ∂ ν ∂ ν i + c4 h∂ µ ∂ µ ∂ ν i1 h∂ x ∂ y ∂ ν i
+c5 h∂ x ∂ y ∂ µ ∂ µ ∂ ν ih∂ ν i1 + c6 h∂ x ∂ µ ∂ µ ∂ ν ih∂ y ∂ ν i1 + c7 h∂ y ∂ µ ∂ µ ∂ ν ih∂ x ∂ ν i1
+c8 h∂ µ ∂ µ ∂ ν ih∂ x ∂ y ∂ ν i1 + c9 h∂ x ∂ µ ∂ ν i1 h∂ y ∂ µ ∂ ν i + c10 h∂ y ∂ µ ∂ ν i1 h∂ x ∂ µ ∂ ν i
+c11 h∂ µ ∂ ν i1 h∂ x ∂ y ∂ µ ∂ ν i + c12 h∂ µ ∂ ν i1 h∂ x ∂ y ∂ µ ih∂ ν i1 + c13 h∂ x ∂ y ∂ µ ∂ ν ih∂ µ i1 h∂ ν i1
+c14 h∂ x ∂ µ ∂ ν ih∂ µ i1 h∂ y ∂ ν i1 + c15 h∂ y ∂ µ ∂ ν ih∂ µ i1 h∂ x ∂ ν i1 .
2.2. Setting r1 (E) = 0. The routine calculation of r1 (E) is omitted. Again, a basis will be
chosen and the components set to zero.
(12)
h∂ x ∂ i i1 h∂ y ∂ µ ∂ ν ih∂ ν ∂ j ∂ µ i :
c7 − c8 − c9 = 0.
(13)
h∂ y ∂ i i1 h∂ x ∂ µ ∂ ν ih∂ ν ∂ j ∂ µ i :
c6 − c8 − c10 = 0.
(14)
h∂ x ∂ µ i1 h∂ y ∂ i ∂ µ ih∂ j ∂ ν ∂ ν i :
−c7 + c8 +
1
c15 = 0.
24
(15)
h∂ y ∂ µ i1 h∂ x ∂ i ∂ µ ih∂ j ∂ ν ∂ ν i :
−c6 + c8 +
1
c14 = 0.
24
1
c2 − c8 = 0.
240
(16)
h∂ i ∂ µ i1 h∂ x ∂ y ∂ j ih∂ µ ∂ ν ∂ ν i :
−
(17)
h∂ i ∂ µ i1 h∂ x ∂ y ∂ µ ih∂ j ∂ ν ∂ ν i :
−c4 +
(18)
h∂ i i1 h∂ µ i1 h∂ x ∂ y ∂ ν ih∂ ν ∂ j ∂ µ i :
−2c1 − c2 + 2c13 − c14 − c15 = 0.
(19)
h∂ µ ∂ ν i1 h∂ x ∂ y ∂ i ih∂ j ∂ µ ∂ ν i :
(20)
h∂ µ i1 h∂ ν i1 h∂ x ∂ i ∂ µ ih∂ y ∂ j ∂ ν i :
(21)
1
c12 = 0.
24
1
c2 + c4 + c4 − c11 = 0.
10
−c1 − 2c13 + c14 + c15 = 0.
h∂ µ i1 h∂ ν i1 h∂ x ∂ y ∂ i ih∂ j ∂ µ ∂ ν i :
7
c2 + c12 − c13 = 0.
10
(22)
h∂ i ∂ µ ∂ µ ∂ ν ih∂ x ∂ y ∂ j ih∂ ν i1 :
13
1
c2 + c4 − c5 + c12 = 0.
240
24
(23)
h∂ i ∂ x ∂ µ ∂ µ ih∂ j ∂ y ∂ ν ih∂ ν i1 :
−
1
1
c1 − c5 + c6 + c15 = 0.
24
24
(24)
h∂ i ∂ y ∂ µ ∂ µ ih∂ j ∂ x ∂ ν ih∂ ν i1 :
−
1
1
c1 − c5 + c7 + c14 = 0.
24
24
(25)
h∂ i ∂ µ ∂ µ ih∂ x ∂ y ∂ j ∂ ν ih∂ ν i1 :
(26)
h∂ x ∂ y ∂ i ih∂ j ∂ µ ∂ µ ∂ ν ∂ ν i :
−c5 + c8 +
1
c13 = 0.
12
1
1
c2 − c3 + c4 = 0.
960
24
TAUTOLOGICAL EQUATIONS IN GENUS 2 VIA INVARIANCE THEOREM
(27)
(28)
h∂ x ∂ i ∂ µ ∂ µ ih∂ y ∂ j ∂ ν ∂ ν i :
−
7
1
1
1
c1 − 2c3 + c6 + c7 = 0.
576
24
24
h∂ x ∂ y ∂ i ∂ µ ∂ µ ih∂ j ∂ ν ∂ ν i :
−2c3 +
1
1
c5 + c8 = 0.
24
24
The 7 vectors
h∂ x ∂ i ∂ µ ih∂ y ∂ µ ∂ ν ∂ ν ih∂ j i1 , h∂ x ∂ i ∂ µ ∂ ν ih∂ y ∂ µ ∂ ν ih∂ j i1 , h∂ x ∂ y ∂ µ ih∂ i ∂ µ ∂ ν ∂ ν ih∂ j i1 ,
h∂ x ∂ y ∂ µ ∂ ν ih∂ i ∂ µ ∂ ν ih∂ j i1 , h∂ x ∂ y ∂ i ∂ µ ih∂ µ ∂ ν ∂ ν ih∂ j i1 , h∂ x ∂ µ ∂ ν ih∂ y ∂ i ∂ µ ∂ ν ih∂ j i1 ,
h∂ x ∂ µ ∂ µ ∂ ν ih∂ y ∂ i ∂ ν ih∂ j i1
are related by WDVV equations. There are 4 independent vectors.
1
(29)
h∂ x ∂ i ∂ µ ih∂ y ∂ µ ∂ ν ∂ ν ih∂ j i1 : − c1 − c7 + 2c5 − c11 − c6 = 0.
12
(30)
(31)
(32)
h∂ x ∂ i ∂ µ ∂ ν ih∂ y ∂ µ ∂ ν ih∂ j i1 :
h∂ x ∂ y ∂ µ ih∂ i ∂ µ ∂ ν ∂ ν ih∂ j i1 :
−
−
1
c1 − c9 + 2c5 − c11 + c10 − 2c6 = 0.
12
1
1
c2 − c4 − c5 + c12 + c11 − c10 + c6 = 0.
24
24
h∂ x ∂ y ∂ i ∂ µ ih∂ µ ∂ ν ∂ ν ih∂ j i1 :
−c8 − c10 + c6 = 0.
All of the other remaining terms are related to each other via WDVV, TRR’s and Getzler’s
genus one equation. After applying the above equations, one can write them in terms of a basis.
(33)
(34)
h∂ x ∂ i ∂ µ ih∂ y ∂ j ∂ ν ih∂ µ ∂ ν i1 :
1
1
c1 + c2 + 12c4 − 12c5 + 12c7 + 12c10 − c12 = 0.
2
2
h∂ x ∂ y ∂ µ ih∂ i ∂ j ∂ ν ih∂ µ ∂ ν i1 :
(35) h∂ y ∂ i ∂ µ ih∂ j ∂ µ ∂ ν ih∂ x ∂ ν i1 :
(36)
c1 + c2 + 20c4 − 24c5 + 24c7 + 2c9 + 26c10 − 2c11 = 0.
1
1
−c1 − c2 − 10c4 + 12c5 − 12c7 + 2c9 − 12c10 − c15 = 0.
2
2
1
1
−c1 − c2 − 10c4 + 12c5 − 12c7 − 10c10 − c14 = 0.
2
2
h∂ x ∂ i ∂ µ ih∂ j ∂ µ ∂ ν ih∂ y ∂ ν i1 :
(37) h∂ x ∂ y ∂ µ ih∂ i ∂ µ ∂ ν ih∂ j ∂ ν i1 :
−4c1 − 2c2 − 20c4 + 24c5 − 24c7 − 2c9 − 26c10 + 2c11 = 0.
(38) h∂ x ∂ i ∂ µ ih∂ y ∂ j ∂ µ ∂ ν ih∂ ν i1 :
1
−3c1 − c2 − 8c4 + 12c5 − 12c7 + 2c9 − 10c10 − c12 − c13 = 0.
2
(39) h∂ y ∂ i ∂ j ∂ µ ih∂ x ∂ µ ∂ ν ih∂ ν i1 :
3
1
1
c1 + c2 + 10c4 − 12c5 + 12c7 − 2c9 + 12c10 + c13 − c14 = 0.
2
2
2
(40)
(41) h∂ x ∂ y ∂ i ∂ µ ih∂ j ∂ µ ∂ ν ih∂ ν i1 :
(42)
(43)
(44)
(45)
3
1
1
1
− c1 − c2 + 2c4 + 2c9 − c12 − c15 = 0.
2
2
2
2
h∂ y ∂ i ∂ µ ih∂ x ∂ j ∂ µ ∂ ν ih∂ ν i1 :
1
c1 + c2 + 8c4 − 12c5 + 12c7 + 12c10 + 2c11 + c12 − c13 = 0.
2
h∂ x ∂ y ∂ ν ∂ µ ih∂ µ ∂ i ∂ j ∂ ν i :
h∂ x ∂ i ∂ ν ∂ µ ih∂ µ ∂ y ∂ j ∂ ν i :
h∂ x ∂ ν ∂ ν ∂ µ ih∂ µ ∂ y ∂ i ∂ j i :
5
5
1
c2 + c4 − c11 = 0.
24
6
12
1
1
5
1
− c1 − c2 + 4c3 − c4 − c7 − c10 + c11 = 0.
8
24
6
12
−
1
5
1
1
1
1
c1 − c2 − c4 + c5 − c6 − c7 − c11 = 0.
24
24
6
2
2
12
h∂ x ∂ i ∂ µ ih∂ µ ∂ y ∂ j ∂ ν ∂ ν i :
−
1
1
5
5
c1 − c2 − c4 − c10 = 0.
16
48
12
12
8
(46)
D. ARCARA AND Y.-P. LEE
h∂ y ∂ i ∂ µ ih∂ µ ∂ x ∂ j ∂ ν ∂ ν i :
−
5
1
5
1
1
1
c1 − c2 − c4 + c5 − c7 + c9 − c10 − c11 = 0.
48
16
4
12
2
12
(47)
h∂ y ∂ ν ∂ µ ih∂ µ ∂ x ∂ i ∂ j ∂ ν i :
1
5
1
1
1
1
c1 + c2 + c4 − c5 + c7 − c9 + c10 + c11 = 0.
24
24
6
2
2
12
(48)
h∂ i ∂ ν ∂ µ ih∂ µ ∂ x ∂ y ∂ j ∂ ν i :
1
5
1
1
c1 + c2 + 4c3 + c4 − 2c5 + c7 + c10 + c11 = 0.
12
12
3
12
1
1
5
1
1
1
1
c1 − c2 − c4 + c5 − c7 − c8 − c10 − c11 = 0.
24
24
6
2
2
2
12
Solving equations (12)-(49), we can write all of the coefficients in terms of c1 :
1
23
1
1
1
c 1 , c4 =
c 1 , c5 = −
c 1 , c6 = − c 1 , c7 = − c 1 ,
c2 = −3c1 , c3 = −
576
30
240
48
48
1
1
1
7
4
c8 =
c1 , c9 = − c1 , c10 = − c1 , c11 = − c1 , c12 = c1 ,
80
30
30
30
5
13
4
4
c13 = − c1 , c14 = − c1 , c15 = − c1
10
5
5
and these are the coefficients of Getzler’s equation in M2,2 .
(49)
h∂ ν ∂ ν ∂ µ ih∂ µ ∂ x ∂ y ∂ i ∂ j i :
−
2.3. Checking r2 (E) = 0. Again, one has to pick a basis and check all components vanish.
1
1
c2 + c4 − c5 − c8 + c12 = 0.
20
24
1
1
1
h∂ j ∂ x ∂ y ih∂ i ∂ µ ∂ ν ∂ ν ih∂ µ ∂ α ∂ α i :
c2 − c3 + c4 − c8 = 0.
1152
24
24
1
1
h∂ j ∂ x ∂ y ih∂ i ∂ µ ∂ ν ih∂ µ ∂ ν ∂ α ∂ α i :
c2 − 2c3 + c4 = 0.
480
12
1
1
1
h∂ j i1 h∂ x ∂ y ∂ µ ih∂ i ∂ µ ∂ ν ih∂ ν ∂ α ∂ α i : − c1 − c2 − c4 + c5 − c8 − c9 − c10 + c12 = 0.
12
24
24
1
1
1
1
1
h∂ µ i1 h∂ j ∂ ν ∂ ν ih∂ i ∂ µ ∂ α ih∂ x ∂ y ∂ α i :
c1 + c2 − c4 − c5 + c8 + c12 + c14 + c15 = 0.
12
24
24
24
24
1
1
µ
j y µ
i ν α
x ν α
h∂ i1 h∂ ∂ ∂ ih∂ ∂ ∂ ih∂ ∂ ∂ i : − c1 − c5 + c8 + c10 + c15 = 0.
24
24
1
1
µ
j x µ
i ν α
y ν α
h∂ i1 h∂ ∂ ∂ ih∂ ∂ ∂ ih∂ ∂ ∂ i : − c1 − c5 + c8 + c9 + c14 = 0.
24
24
1
1
1
1
j x µ µ
i ν α
y ν α
c1 − 2c3 + c6 + c8 + c9 = 0.
h∂ ∂ ∂ ∂ ih∂ ∂ ∂ ih∂ ∂ ∂ i : −
576
24
24
24
1
1
1
1
h∂ j ∂ y ∂ µ ∂ µ ih∂ i ∂ ν ∂ α ih∂ x ∂ ν ∂ α i : −
c1 − 2c3 + c7 + c8 + c10 = 0.
576
24
24
24
The 7 vectors
h∂ i ∂ µ ∂ µ ∂ ν ih∂ x ∂ y ∂ ν ih∂ j ∂ α ∂ α i, h∂ x ∂ µ ∂ ν ∂ ν ih∂ i ∂ y ∂ µ ih∂ j ∂ α ∂ α i,
h∂ µ i1 h∂ j ∂ x ∂ y ih∂ i ∂ µ ∂ ν ih∂ ν ∂ α ∂ α i :
h∂ y ∂ µ ∂ ν ∂ ν ih∂ i ∂ x ∂ µ ih∂ j ∂ α ∂ α i, h∂ x ∂ y ∂ µ ∂ ν ih∂ i ∂ µ ∂ ν ih∂ j ∂ α ∂ α i,
h∂ i ∂ x ∂ y ∂ µ ih∂ µ ∂ ν ∂ ν ih∂ j ∂ α ∂ α i, h∂ i ∂ x ∂ µ ∂ ν ih∂ y ∂ µ ∂ ν ih∂ j ∂ α ∂ α i,
h∂ i ∂ y ∂ µ ∂ ν ih∂ x ∂ µ ∂ ν ih∂ j ∂ α ∂ α i
are related by WDVV equations. 4 of them are linear independent.
1
1
1
1
1
h∂ i ∂ µ ∂ µ ∂ ν ih∂ x ∂ y ∂ ν ih∂ j ∂ α ∂ α i :
c1 +
c2 − 2c3 + c8 + c9 + c10 = 0.
288
576
12
24
24
1
1
1
c6 − c8 − c9 = 0.
h∂ x ∂ µ ∂ ν ∂ ν ih∂ i ∂ y ∂ µ ih∂ j ∂ α ∂ α i :
24
24
24
1
1
1
h∂ y ∂ µ ∂ ν ∂ ν ih∂ i ∂ x ∂ µ ih∂ j ∂ α ∂ α i :
c7 − c8 − c10 = 0.
24
24
24
1
1
1
1
1
c1 − 4c3 + c8 + c9 + c10 + c11 = 0.
h∂ x ∂ y ∂ µ ∂ ν ih∂ i ∂ µ ∂ ν ih∂ j ∂ α ∂ α i :
288
6
24
24
24
TAUTOLOGICAL EQUATIONS IN GENUS 2 VIA INVARIANCE THEOREM
9
All of the remaining strata are related by WDVV equations. Only 3 of them are linearly
independent.
h∂ x ∂ j ∂ µ ih∂ µ ∂ y ∂ i ∂ ν ih∂ ν ∂ α ∂ α i :
5
1
1
1
−4c3 − c4 + c8 + c9 + c10 − c11 = 0.
6
6
6
12
h∂ x ∂ j ∂ α ∂ µ ih∂ µ ∂ i ∂ ν ih∂ ν ∂ y ∂ α i :
5
5
− c4 − c9 = 0.
6
6
h∂ y ∂ j ∂ α ∂ µ ih∂ µ ∂ i ∂ ν ih∂ ν ∂ x ∂ α i :
5
5
− c4 − c10 = 0.
6
6
Therefore, r2 (E) = 0.
2.4. Calculating r3 (E). Since l = 3 case is new, the calculation is presented.
h∂1x ∂1y i2 7→
1
1
− h∂ y ∂ ν ∂ j ih∂ x ∂ α ∂ µ ih∂ α ∂ ν ∂ i ih∂ µ ∂ β ∂ β i − h∂ y ∂ ν ∂ µ ih∂ x ∂ α ∂ j ih∂ α ∂ ν ∂ i ih∂ µ ∂ β ∂ β i
24
24
1
1
− h∂ x ∂ α ∂ j ih∂ α ∂ i ∂ µ ih∂ y ∂ β ∂ ν ih∂ β ∂ µ ∂ ν i − h∂ y ∂ α ∂ j ih∂ α ∂ i ∂ µ ih∂ x ∂ β ∂ ν ih∂ β ∂ µ ∂ ν i
24
24
1
1
− h∂ i ∂ µ ∂ j ih∂ y ∂ β ∂ ν ih∂ x ∂ α ∂ ν ih∂ α ∂ β ∂ µ i − h∂ i ∂ µ ∂ j ih∂ y ∂ β ∂ ν ih∂ x ∂ α ∂ ν ih∂ α ∂ β ∂ µ i
24
24
1
1
+ h∂ α ∂ y ∂ β ih∂ β ∂ x ∂ µ ih∂ µ ∂ j ∂ ν ih∂ ν ∂ i ∂ α i + h∂ x ∂ µ ∂ α ih∂ ν ∂ y ∂ β ih∂ β ∂ µ ∂ j ih∂ ν ∂ i ∂ α i
48
48
1
1
+ h∂ x ∂ µ ∂ α ih∂ µ ∂ j ∂ ν ih∂ α ∂ y ∂ β ih∂ β ∂ ν ∂ i i + h∂ α ∂ y ∂ β ih∂ β ∂ j ∂ ν ih∂ x ∂ µ ∂ α ih∂ µ ∂ ν ∂ i i
48
48
1
1
+ h∂ j ∂ ν ∂ α ih∂ α ∂ y ∂ β ih∂ β ∂ x ∂ µ ih∂ µ ∂ ν ∂ i i + h∂ j ∂ ν ∂ α ih∂ x ∂ µ ∂ α ih∂ i ∂ y ∂ β ih∂ β ∂ µ ∂ ν i
48
48
1
1
+
h∂ y ∂ µ ∂ α ih∂ µ ∂ x ∂ ν ih∂ ν ∂ i ∂ α ih∂ j ∂ β ∂ β i +
h∂ x ∂ ν ∂ α ih∂ y ∂ µ ∂ α ih∂ µ ∂ ν ∂ i ih∂ j ∂ β ∂ β i
576
576
1
+
h∂ x ∂ µ ∂ ν ih∂ µ ∂ i ∂ ν ih∂ y ∂ α ∂ β ih∂ α ∂ j ∂ β i
576
h∂1µ i2 h∂ x ∂ y ∂ µ i 7→
1
1
h∂ i ∂ α ∂ µ ih∂ α ∂ β ∂ β ih∂ µ ∂ ν ∂ ν ih∂ x ∂ y ∂ j i +
h∂ i ∂ α ∂ ν ih∂ α ∂ β ∂ ν ih∂ β ∂ µ ∂ µ ih∂ x ∂ y ∂ j i
−
5760
960
1
1
− h∂ µ ∂ j ∂ β ih∂ β ∂ i ∂ ν ih∂ ν ∂ α ∂ α ih∂ x ∂ y ∂ µ i − h∂ i ∂ ν ∂ j ih∂ α ∂ µ ∂ β ih∂ β ∂ ν ∂ α ih∂ x ∂ y ∂ µ i
24
24
1
1
+ h∂ µ ∂ β ∂ α ih∂ β ∂ j ∂ ν ih∂ ν ∂ i ∂ α ih∂ x ∂ y ∂ µ i + h∂ j ∂ ν ∂ α ih∂ µ ∂ β ∂ α ih∂ β ∂ ν ∂ i ih∂ x ∂ y ∂ µ i
48
48
1
+
h∂ µ ∂ ν ∂ α ih∂ ν ∂ i ∂ α ih∂ j ∂ β ∂ β ih∂ x ∂ y ∂ µ i
576
h∂ x ∂ y ∂ µ ∂ µ ∂ ν ∂ ν i 7→
+4h∂ i ∂ ν ∂ β ih∂ β ∂ µ ∂ ν ih∂ µ ∂ α ∂ j ih∂ α ∂ x ∂ y i − 2h∂ i ∂ α ∂ µ ih∂ α ∂ β ∂ µ ih∂ β ∂ x ∂ y ih∂ j ∂ ν ∂ ν i
−4h∂ i ∂ α ∂ ν ih∂ α ∂ β ∂ µ ih∂ β ∂ x ∂ y ih∂ j ∂ µ ∂ ν i − 4h∂ i ∂ α ∂ ν ih∂ α ∂ β ∂ µ ih∂ β ∂ x ∂ µ ih∂ j ∂ y ∂ ν i
−4h∂ i ∂ α ∂ ν ih∂ α ∂ β ∂ µ ih∂ β ∂ y ∂ µ ih∂ j ∂ x ∂ ν i − h∂ i ∂ α ∂ ν ih∂ α ∂ β ∂ ν ih∂ β ∂ µ ∂ µ ih∂ j ∂ x ∂ y i
+4h∂ i ∂ µ ∂ α ih∂ α ∂ x ∂ y ih∂ j ∂ ν ∂ β ih∂ β ∂ µ ∂ ν i + 2h∂ i ∂ µ ∂ α ih∂ α ∂ x ∂ µ ih∂ j ∂ ν ∂ β ih∂ β ∂ y ∂ ν i
+4h∂ i ∂ ν ∂ α ih∂ α ∂ x ∂ µ ih∂ j ∂ ν ∂ β ih∂ β ∂ y ∂ µ i
10
D. ARCARA AND Y.-P. LEE
h∂ µ ∂ µ ∂ ν i1 h∂ x ∂ y ∂ ν i 7→
1
1
+ h∂ i ∂ ν ∂ β ih∂ β ∂ µ ∂ j ih∂ µ ∂ α ∂ α ih∂ x ∂ y ∂ ν i + h∂ i ∂ µ ∂ β ih∂ β ∂ ν ∂ µ ih∂ ν ∂ α ∂ α ih∂ x ∂ y ∂ j i
12
24
1 i β µ j α ν α β µ x y ν
i α ν
α β µ
β j µ
x y ν
−h∂ ∂ ∂ ih∂ ∂ ∂ ih∂ ∂ ∂ ih∂ ∂ ∂ i + h∂ ∂ ∂ ih∂ ∂ ∂ ih∂ ∂ ∂ ih∂ ∂ ∂ i
2
1 i β ν j α µ α β µ x y ν
1
+ h∂ ∂ ∂ ih∂ ∂ ∂ ih∂ ∂ ∂ ih∂ ∂ ∂ i − h∂ i ∂ α ∂ µ ih∂ α ∂ β ∂ β ih∂ j ∂ µ ∂ ν ih∂ x ∂ y ∂ ν i
2
12
1 i α ν α β β j µ µ x y ν
1
− h∂ ∂ ∂ ih∂ ∂ ∂ ih∂ ∂ ∂ ih∂ ∂ ∂ i + h∂ i ∂ α ∂ α ih∂ j ∂ β ∂ ν ih∂ β ∂ µ ∂ µ ih∂ x ∂ y ∂ ν i
24
24
h∂ µ ∂ µ ∂ ν ih∂ x ∂ y ∂ ν i1 7→
1
+ h∂ µ ∂ µ ∂ j ih∂ i ∂ y ∂ β ih∂ β ∂ ν ∂ x ih∂ ν ∂ α ∂ α i − h∂ µ ∂ µ ∂ ν ih∂ i ∂ α ∂ ν ih∂ α ∂ β ∂ y ih∂ β ∂ j ∂ x i
24
1
1
+ h∂ µ ∂ µ ∂ ν ih∂ i ∂ β ∂ y ih∂ j ∂ α ∂ ν ih∂ α ∂ β ∂ x i + h∂ µ ∂ µ ∂ ν ih∂ i ∂ β ∂ ν ih∂ j ∂ α ∂ y ih∂ α ∂ β ∂ x i
2
2
1 µ µ ν i α x α β β j y ν
1
− h∂ ∂ ∂ ih∂ ∂ ∂ ih∂ ∂ ∂ ih∂ ∂ ∂ i − h∂ µ ∂ µ ∂ ν ih∂ i ∂ α ∂ y ih∂ α ∂ β ∂ β ih∂ j ∂ x ∂ ν i
24
24
1
1
− h∂ µ ∂ µ ∂ ν ih∂ i ∂ α ∂ ν ih∂ α ∂ β ∂ β ih∂ j ∂ x ∂ y i + h∂ µ ∂ µ ∂ ν ih∂ i ∂ α ∂ α ih∂ j ∂ β ∂ ν ih∂ β ∂ x ∂ y i
24
24
h∂ x ∂ µ ∂ ν i1 h∂ y ∂ µ ∂ ν i 7→
1
+ h∂ i ∂ ν ∂ β ih∂ β ∂ µ ∂ x ih∂ µ ∂ α ∂ α ih∂ y ∂ j ∂ ν i − h∂ i ∂ α ∂ ν ih∂ α ∂ β ∂ µ ih∂ β ∂ j ∂ x ih∂ y ∂ µ ∂ ν i
12
1
1
+ h∂ i ∂ β ∂ µ ih∂ j ∂ α ∂ ν ih∂ α ∂ β ∂ x ih∂ y ∂ µ ∂ ν i + h∂ i ∂ β ∂ ν ih∂ j ∂ α ∂ µ ih∂ α ∂ β ∂ x ih∂ y ∂ µ ∂ ν i
2
2
1 i α x α β β j µ ν y µ ν
1
− h∂ ∂ ∂ ih∂ ∂ ∂ ih∂ ∂ ∂ ih∂ ∂ ∂ i − h∂ i ∂ α ∂ µ ih∂ α ∂ β ∂ β ih∂ j ∂ x ∂ ν ih∂ y ∂ µ ∂ ν i
24
12
1
+ h∂ i ∂ α ∂ α ih∂ j ∂ ν ∂ β ih∂ β ∂ x ∂ µ ih∂ y ∂ µ ∂ ν i
24
h∂ y ∂ µ ∂ ν i1 h∂ x ∂ µ ∂ ν i 7→
1
+ h∂ i ∂ ν ∂ β ih∂ β ∂ µ ∂ y ih∂ µ ∂ α ∂ α ih∂ x ∂ j ∂ ν i − h∂ i ∂ α ∂ ν ih∂ α ∂ β ∂ µ ih∂ β ∂ j ∂ y ih∂ x ∂ µ ∂ ν i
12
1
1
+ h∂ i ∂ β ∂ µ ih∂ j ∂ α ∂ ν ih∂ α ∂ β ∂ y ih∂ x ∂ µ ∂ ν i + h∂ i ∂ β ∂ ν ih∂ j ∂ α ∂ µ ih∂ α ∂ β ∂ y ih∂ x ∂ µ ∂ ν i
2
2
1 i α y α β β j µ ν x µ ν
1
− h∂ ∂ ∂ ih∂ ∂ ∂ ih∂ ∂ ∂ ih∂ ∂ ∂ i − h∂ i ∂ α ∂ µ ih∂ α ∂ β ∂ β ih∂ j ∂ y ∂ ν ih∂ x ∂ µ ∂ ν i
24
12
1
+ h∂ i ∂ α ∂ α ih∂ j ∂ ν ∂ β ih∂ β ∂ y ∂ µ ih∂ x ∂ µ ∂ ν i
24
The other graphs all have r3 (Γ) = 0.
2.5. Checking r3 (E) = 0. Now the r3 (E) has only a few independent strata.
1
1
1
1
1
c1 +
c2 − 2c3 + c8 + c9 + c10 = 0.
288
576
12
24
24
1
1
1
h∂ i ∂ x ∂ y i :
c2 − c3 + c4 − c8 = 0.
1152
24
24
1
1
1
1
c1 + 2c3 − c8 − c9 − c10 = 0.
h∂ i ∂ x ∂ µ ih∂ µ ∂ ν ∂ ν ih∂ j ∂ y ∂ α ih∂ α ∂ β ∂ β i :
576
12
24
24
h∂ i ∂ µ ∂ µ i :
TAUTOLOGICAL EQUATIONS IN GENUS 2 VIA INVARIANCE THEOREM
11
The remaining strata, which are all equivalent as codimension 4 classes in M1,4 , have coefficient
1
1
− c1 − c2 = 0.
8
24
3. The Belorousski-Pandharipande equation in M2,3
3.1. Strata in M2,3 of codimension 2. Throughout this section, we will always assume that
the three external labelings x, y, z are symmetrized.
Using Mumford–Getzler’s and Getzler’s equations for genus 2, all genus 2 terms with more
than one descendent can be rewritten in terms of the others. Also, using TRR’s, all genus 0 or
1 terms with a descendent can be rewritten in terms of the others. We are therefore left with 21
strata which could be independent. 1
A general linear combination is
X
c1 h∂ x ∂ µ ∂ ν ih∂ µ ∂ y ∂ z ih∂ ν i2 + c2 h∂ x ∂ y ∂ z ∂ µ ih∂1µ i2 + c3 h∂1x ∂ µ i2 h∂ µ ∂ y ∂ z i
E=
S3 (x,y,z)
+c4 h∂ x ∂1µ i2 h∂ µ ∂ y ∂ z i + c5 h∂ x ∂ y ∂ z ∂ µ ∂ ν ih∂ µ i1 h∂ ν i1
+c6 h∂ x ∂ y ∂ µ ∂ ν ih∂ µ i1 h∂ ν ∂ z i1 + c7 h∂ x ∂ µ ∂ ν ih∂ µ i1 h∂ ν ∂ y ∂ z i1
+c8 h∂ x ∂ µ i1 h∂ µ ∂ y ∂ ν ih∂ ν ∂ z i1 + c9 h∂ x ∂ y ∂ z ∂ µ ih∂ µ ∂ ν i1 h∂ ν i1
+c10 h∂ x ∂ µ i1 h∂ µ ∂ ν i1 h∂ ν ∂ y ∂ z i + c11 h∂ µ i1 h∂ µ ∂ x ∂ ν i1 h∂ ν ∂ y ∂ z i
+c12 h∂ x ∂ y ∂ z ∂ µ ∂ µ ∂ ν ih∂ ν i1 + c13 h∂ x ∂ y ∂ µ ∂ µ ∂ ν ih∂ ν ∂ z i1
+c14 h∂ x ∂ µ ∂ µ ∂ ν ih∂ ν ∂ y ∂ z i1 + c15 h∂ x ∂ y ∂ z ∂ µ i1 h∂ µ ∂ ν ∂ ν i
+c16 h∂ x ∂ y ∂ z ∂ µ ∂ ν ih∂ µ ∂ ν i1 + c17 h∂ x ∂ y ∂ µ ∂ ν ih∂ µ ∂ ν ∂ z i1
+c18 h∂ x ∂ µ ∂ ν ih∂ µ ∂ ν ∂ y ∂ z i1 + c19 h∂ x ∂ y ∂ z ∂ µ ih∂ µ ∂ ν ∂ ν i1
+c20 h∂ x ∂ y ∂ µ ih∂ µ ∂ ν ∂ ν ∂ z i1 + c21 h∂ x ∂ y ∂ z ∂ µ ∂ µ ∂ ν ∂ ν i = 0,
3.2. Setting r1 (E) = 0. Again a basis is chosen for the output graphs of r1 (E), and the coefficients are set to zero.
(50)
h∂ x ∂ µ ∂ j ih∂ µ ∂ y ∂ z ih∂1i i2 :
(51)
h∂ x ∂ y ∂ i ih∂ z ∂ µ ∂ j ih∂1µ i2 :
(52)
h∂ x ∂ y ∂ i i1 h∂ z ∂ j ∂ µ ih∂ µ ∂ ν ∂ ν i :
(53)
(54)
(55)
(56)
(57)
(58)
(59)
c1 + c2 = 0.
−3c2 + 3c3 + c4 = 0.
c14 − 3c15 − c18 = 0.
1
c7 − c14 + c15 + c15 + c15 = 0.
24
h∂ x ∂ y ∂ µ i1 h∂ z ∂ i ∂ µ ih∂ j ∂ ν ∂ ν i :
h∂ x ∂ i ∂ µ i1 h∂ µ ∂ ν ∂ ν ih∂ y ∂ z ∂ j i :
−
1
1
c3 −
c4 − 3c15 = 0.
80
240
h∂ x ∂ i ∂ µ i1 h∂ y ∂ z ∂ µ ih∂ j ∂ ν ∂ ν i :
h∂ x ∂ µ ∂ ν i1 h∂ i ∂ µ ∂ ν ih∂ y ∂ z ∂ j i :
1
c11 = 0.
24
1
1
c3 + c4 − c17 + c20 + c20 = 0.
30
10
h∂ i ∂ µ ∂ ν i1 h∂ x ∂ µ ∂ ν ih∂ y ∂ z ∂ j i :
h∂ i ∂ µ ∂ µ i1 h∂ x ∂ y ∂ ν ih∂ ν ∂ z ∂ j i :
h∂ µ ∂ ν ∂ ν i1 h∂ x ∂ i ∂ µ ih∂ y ∂ z ∂ j i :
−c20 +
−
1
c3 − c18 = 0.
30
c19 − c20 = 0.
1
c3 − 2c19 − c19 + c20 = 0.
30
1We rewrote the strata here in the same order as in the Belorousski-Pandharipande equation, with the extra
strata h∂ x ∂ y ∂ z ∂ µ ∂ µ ∂ ν ∂ ν i, which does not appear in the equation, at the end.
12
D. ARCARA AND Y.-P. LEE
(60)
h∂ x ∂ i i1 h∂ µ i1 h∂ y ∂ z ∂ ν ih∂ ν ∂ j ∂ µ i :
−c4 + c6 − 2c7 + c10 − c11 = 0.
(61)
h∂ x ∂ µ i1 h∂ ν i1 h∂ i ∂ µ ∂ ν ih∂ y ∂ z ∂ j i :
7
4
c3 + c4 − c6 + c10 + c11 = 0.
5
5
(62)
(63)
(64)
(65)
h∂ x ∂ µ i1 h∂ µ ∂ y ∂ i ih∂ ν i1 h∂ ν ∂ z ∂ j i :
h∂ i ∂ µ i1 h∂ µ ∂ x ∂ y ih∂ ν i1 h∂ ν ∂ z ∂ j i :
(67)
h∂ µ ∂ ν i1 h∂ i ∂ µ ∂ ν ih∂ x ∂ y ∂ z ∂ j i :
(70)
(71)
(72)
−c3 + c10 − c11 = 0.
h∂ i ∂ µ i1 h∂ µ i1 h∂ x ∂ y ∂ ν ih∂ ν ∂ z ∂ j i :
h∂ µ ∂ ν i1 h∂ µ i1 h∂ x ∂ i ∂ ν ih∂ y ∂ z ∂ j i :
(69)
4
c3 − c7 = 0.
5
h∂ i ∂ µ i1 h∂ ν i1 h∂ x ∂ µ ∂ ν ih∂ y ∂ z ∂ j i :
(66)
(68)
−2c6 + c7 + c7 + 2c8 = 0.
4
− c3 − 2c9 − c9 + c11 = 0.
5
,
1
c2 − c16 + c19 + c19 = 0.
10
h∂ i ∂ µ i1 h∂ µ ∂ ν ∂ ν ih∂ x ∂ y ∂ z ∂ j i : , −
h∂ x ∂ µ i1 h∂ i ∂ µ ∂ ν ∂ ν ih∂ y ∂ z ∂ j i :
1
c2 − c15 = 0.
240
1
13
1
c3 +
c4 + c10 − c13 + c20 = 0.
48
240
24
h∂ µ i1 h∂ ν i1 h∂ i ∂ µ ∂ ν ih∂ x ∂ y ∂ z ∂ j i :
h∂ µ i1 h∂ i ∂ µ ∂ ν ∂ ν ih∂ x ∂ y ∂ z ∂ j i :
h∂ x ∂ y ∂ i ih∂ j ∂ z ∂ µ ∂ µ ∂ ν ∂ ν i :
c9 − c11 = 0.
7
c2 − c5 + c9 = 0.
10
1
13
c2 + c9 − c12 + c19 = 0.
240
24
1
1
1
c3 +
c4 + c20 + 3c21 = 0.
576
960
24
(73)
h∂ µ ∂ µ ∂ i ih∂ j ∂ x ∂ y ∂ z ∂ ν ∂ ν i :
1
1
c12 + c15 + 2c21 = 0.
24
24
(74)
h∂ x ∂ y ∂ z ∂ i ih∂ j ∂ µ ∂ µ ∂ ν ∂ ν i :
1
1
c2 + c19 + c21 = 0.
960
24
1
1
c13 + c14 + 6c21 = 0.
24
24
The coefficients of the Belorousski-Pandharipande equation are the only solution of the equations (50)-(75).
(75)
h∂ x ∂ µ ∂ µ ∂ i ih∂ j ∂ y ∂ z ∂ ν ∂ ν i :
3.3. Checking r2 (E) = 0. Since the whole output graphs are far too numerous, we shall present
the the coefficients of the following four (disconnected) graphs are zero:
h∂ i ∂ x i1 h∂ j ∂ y ∂ µ ih∂ µ ∂ z ∂ ν ih∂ ν ∂ α ∂ α i,
h∂ i ∂ µ i1 h∂ µ ∂ x ∂ y ih∂ j ∂ z ∂ ν ih∂ ν ∂ α ∂ α i
h∂ µ ∂ ν i1 h∂ i ∂ µ ∂ ν ih∂ j ∂ x ∂ α ih∂ α ∂ y ∂ z i,
h∂ x ∂ µ i1 h∂ µ ∂ i ∂ y ih∂ j ∂ z ∂ ν ih∂ ν ∂ α ∂ α i
In the following, the graphs Γ appearing in BP equation such that r2 (Γ) contain any of the
above four graphs will be listed.
h∂ x ∂ µ ∂ ν ih∂ µ ∂ y ∂ z ih∂ ν i2 7→
1 µ ν
h∂ ∂ i1 h∂ i ∂ µ ∂ ν ih∂ j ∂ x ∂ α ih∂ α ∂ y ∂ z i + . . . .
10
1 i µ
h∂ ∂ i1 h∂ µ ∂ x ∂ y ih∂ j ∂ z ∂ ν ih∂ ν ∂ α ∂ α i + . . .
24
1 i x
h∂ ∂ i1 h∂ j ∂ y ∂ µ ih∂ µ ∂ z ∂ ν ih∂ ν ∂ α ∂ α i + . . .
h∂ x ∂1µ i2 h∂ µ ∂ y ∂ z i 7→
24
h∂1x ∂ µ i2 h∂ µ ∂ y ∂ z i 7→
TAUTOLOGICAL EQUATIONS IN GENUS 2 VIA INVARIANCE THEOREM
h∂ x ∂ µ i1 h∂ µ ∂ y ∂ ν ih∂ ν ∂ z i1 7→ −
13
1 x µ
h∂ ∂ i1 h∂ µ ∂ i ∂ y ih∂ j ∂ z ∂ ν ih∂ ν ∂ α ∂ α i + . . .
12
h∂ x ∂ µ i1 h∂ µ ∂ ν i1 h∂ ν ∂ y ∂ z i 7→
1
1
− h∂ i ∂ x i1 h∂ j ∂ y ∂ µ ih∂ µ ∂ z ∂ ν ih∂ ν ∂ α ∂ α i − h∂ i ∂ µ i1 h∂ µ ∂ x ∂ y ih∂ j ∂ z ∂ ν ih∂ ν ∂ α ∂ α i + . . .
24
24
h∂ x ∂ y ∂ µ ∂ µ ∂ ν ih∂ ν ∂ z i1 7→
−h∂ i ∂ x i1 h∂ j ∂ y ∂ µ ih∂ µ ∂ z ∂ ν ih∂ ν ∂ α ∂ α i + 2h∂ x ∂ µ i1 h∂ µ ∂ i ∂ y ih∂ j ∂ z ∂ ν ih∂ ν ∂ α ∂ α i + . . .
h∂ x ∂ y ∂ z ∂ µ i1 h∂ µ ∂ ν ∂ ν i 7→
−6h∂ x ∂ µ i1 h∂ µ ∂ i ∂ y ih∂ j ∂ z ∂ ν ih∂ ν ∂ α ∂ α i + 3h∂ i ∂ x i1 h∂ j ∂ y ∂ µ ih∂ µ ∂ z ∂ ν ih∂ ν ∂ α ∂ α i + . . .
h∂ x ∂ y ∂ z ∂ µ ∂ ν ih∂ µ ∂ ν i1 7→ h∂ µ ∂ ν i1 h∂ i ∂ µ ∂ ν ih∂ j ∂ x ∂ α ih∂ α ∂ y ∂ z i + . . .
h∂ x ∂ µ ∂ ν ih∂ µ ∂ ν ∂ y ∂ z i1 7→
2h∂ i ∂ x i1 h∂ j ∂ y ∂ µ ih∂ µ ∂ z ∂ ν ih∂ ν ∂ α ∂ α i − 2h∂ x ∂ µ i1 h∂ µ ∂ i ∂ y ih∂ j ∂ z ∂ ν ih∂ ν ∂ α ∂ α i + . . .
h∂ x ∂ y ∂ µ ih∂ µ ∂ ν ∂ ν ∂ z i1 7→
−2h∂ µ ∂ ν i1 h∂ i ∂ µ ∂ ν ih∂ j ∂ x ∂ α ih∂ α ∂ y ∂ z i + h∂ i ∂ x i1 h∂ j ∂ y ∂ µ ih∂ µ ∂ z ∂ ν ih∂ ν ∂ α ∂ α i
+h∂ i ∂ µ i1 h∂ µ ∂ x ∂ y ih∂ j ∂ z ∂ ν ih∂ ν ∂ α ∂ α i + . . .
Checking the coefficients:
1
1
c4 − c10 − c13 + 3c15 + 2c18 + c20 = 0.
24
24
1
1
c3 − c10 + c20 = 0.
h∂ i ∂ µ i1 h∂ µ ∂ x ∂ y ih∂ j ∂ z ∂ ν ih∂ ν ∂ α ∂ α i :
24
24
1
h∂ µ ∂ ν i1 h∂ i ∂ µ ∂ ν ih∂ j ∂ x ∂ α ih∂ α ∂ y ∂ z i :
c10 + c16 − 2c20 = 0.
10
1
h∂ x ∂ µ i1 h∂ µ ∂ i ∂ y ih∂ j ∂ z ∂ ν ih∂ ν ∂ α ∂ α i : − c8 + 2c13 − 6c15 − 2c18 = 0.
12
h∂ i ∂ x i1 h∂ j ∂ y ∂ µ ih∂ µ ∂ z ∂ ν ih∂ ν ∂ α ∂ α i :
3.4. Checking r3 (E) = 0. In the same spirit as the case l = 2, only the following four disconnected output graphs will be presented here.
h∂ x ∂ y ∂ z ∂ i ih∂ j ∂ α ∂ µ ih∂ µ ∂ α ∂ ν ih∂ ν ∂ β ∂ β i,
h∂ i i1 h∂ x ∂ y ∂ µ ih∂ µ ∂ z ∂ ν ih∂ ν ∂ i ∂ α ih∂ α ∂ β ∂ β i,
h∂ i ∂ x ∂ µ ∂ µ ih∂ y ∂ z ∂ ν ih∂ ν ∂ j ∂ α ih∂ α ∂ β ∂ β i,
h∂ µ i1 h∂ µ ∂ i ∂ x ih∂ y ∂ z ∂ ν ih∂ ν ∂ j ∂ α ih∂ α ∂ β ∂ β i.
1
h∂ x ∂ µ ∂ ν ih∂ µ ∂ y ∂ z ih∂ ν i2 7→ − h∂ j i1 h∂ x ∂ µ ∂ ν ih∂ µ ∂ y ∂ z ih∂ i ∂ α ∂ ν ih∂ α ∂ β ∂ β i + . . .
24
h∂ x ∂ y ∂ z ∂ µ ih∂1µ i2 7→
1
1
−
h∂ x ∂ y ∂ z ∂ j ih∂ i ∂ α ∂ µ ih∂ α ∂ β ∂ β ih∂ µ ∂ ν ∂ ν i +
h∂ x ∂ y ∂ z ∂ j ih∂ i ∂ α ∂ ν ih∂ α ∂ β ∂ ν ih∂ β ∂ µ ∂ µ i + . . .
5760
960
h∂1x ∂ µ i2 h∂ µ ∂ y ∂ z i 7→
1
− h∂ j i1 h∂ x ∂ µ ∂ β ih∂ β ∂ i ∂ ν ih∂ ν ∂ α ∂ α ih∂ µ ∂ y ∂ z i −
24
1
− h∂ x ∂ ν ∂ j ih∂ ν i1 h∂ i ∂ α ∂ µ ih∂ α ∂ β ∂ β ih∂ µ ∂ y ∂ z i −
24
1 j
h∂ i1 h∂ i ∂ ν ∂ µ ih∂ α ∂ x ∂ β ih∂ β ∂ ν ∂ α ih∂ µ ∂ y ∂ z i
24
1
h∂ x ∂ ν ∂ ν ∂ j ih∂ i ∂ α ∂ µ ih∂ α ∂ β ∂ β ih∂ µ ∂ y ∂ z i + . . .
576
h∂ x ∂1µ i2 h∂ µ ∂ y ∂ z i 7→
1
− h∂ j i1 h∂ µ ∂ x ∂ β ih∂ β ∂ i ∂ ν ih∂ ν ∂ α ∂ α ih∂ µ ∂ y ∂ z i −
24
1
+ h∂ µ i1 h∂ µ ∂ i ∂ x ih∂ ν ∂ β ∂ α ih∂ β ∂ j ∂ α ih∂ ν ∂ y ∂ z i +
24
1 j
h∂ i1 h∂ i ∂ ν ∂ x ih∂ α ∂ µ ∂ β ih∂ β ∂ ν ∂ α ih∂ µ ∂ y ∂ z i
24
1
h∂ i ∂ µ ∂ µ ∂ x ih∂ ν ∂ β ∂ α ih∂ β ∂ j ∂ α ih∂ ν ∂ y ∂ z i + . . .
576
14
D. ARCARA AND Y.-P. LEE
h∂ x ∂ µ ∂ ν ih∂ µ i1 h∂ ν ∂ y ∂ z i1 7→
1 µ
h∂ i1 h∂ µ ∂ x ∂ j ih∂ i ∂ z ∂ α ih∂ α ∂ ν ∂ y ih∂ ν ∂ β ∂ β i + . . .
24
h∂ µ i1 h∂ µ ∂ x ∂ ν i1 h∂ ν ∂ y ∂ z i 7→
1
1 j
h∂ i1 h∂ i ∂ ν ∂ µ ih∂ µ ∂ α ∂ x ih∂ α ∂ β ∂ β ih∂ ν ∂ y ∂ z i − h∂ µ i1 h∂ j ∂ x ∂ µ ih∂ i ∂ α ∂ ν ih∂ α ∂ β ∂ β ih∂ ν ∂ y ∂ z i + . . .
24
24
h∂ x ∂ y ∂ z ∂ µ ∂ µ ∂ ν ih∂ ν i1 7→
h∂ j i1 h∂ i ∂ µ ∂ ν ih∂ ν ∂ α ∂ µ ih∂ α ∂ β ∂ z ih∂ β ∂ x ∂ y i − 3h∂ µ i1 h∂ j ∂ x ∂ µ ih∂ i ∂ α ∂ ν ih∂ α ∂ β ∂ ν ih∂ β ∂ y ∂ z i + . . .
h∂ x ∂ µ ∂ µ ∂ ν ih∂ ν ∂ y ∂ z i1 7→
1 j x µ µ i z ν ν α y α β β
h∂ ∂ ∂ ∂ ih∂ ∂ ∂ ih∂ ∂ ∂ ih∂ ∂ ∂ i + . . .
24
h∂ x ∂ y ∂ z ∂ µ i1 h∂ µ ∂ ν ∂ ν i 7→
1
− h∂ x ∂ y ∂ z ∂ j ih∂ i ∂ α ∂ µ ih∂ α ∂ β ∂ β ih∂ µ ∂ ν ∂ ν i − h∂ j i1 h∂ i ∂ α ∂ µ ih∂ α ∂ β ∂ z ih∂ β ∂ x ∂ y ih∂ µ ∂ ν ∂ ν i
24
1
+3h∂ µ i1 h∂ µ ∂ i ∂ x ih∂ j ∂ β ∂ α ih∂ β ∂ y ∂ z ih∂ α ∂ ν ∂ ν i + h∂ i ∂ x ∂ µ ∂ µ ih∂ j ∂ β ∂ α ih∂ β ∂ y ∂ z ih∂ α ∂ ν ∂ ν i + . . .
8
h∂ x ∂ µ ∂ ν ih∂ µ ∂ ν ∂ y ∂ z i1 7→
−h∂ j i1 h∂ x ∂ µ ∂ ν ih∂ i ∂ α ∂ z ih∂ α ∂ β ∂ y ih∂ β ∂ µ ∂ ν i + 2h∂ µ i1 h∂ µ ∂ i ∂ y ih∂ x ∂ α ∂ ν ih∂ j ∂ β ∂ z ih∂ β ∂ α ∂ ν i
1
+ h∂ i ∂ y ∂ µ ∂ µ ih∂ x ∂ α ∂ ν ih∂ j ∂ β ∂ z ih∂ β ∂ α ∂ ν i + . . .
12
h∂ x ∂ y ∂ z ∂ µ ih∂ µ ∂ ν ∂ ν i1 7→
1 x y z j i ν µ µ α ν α β β
h∂ ∂ ∂ ∂ ih∂ ∂ ∂ ih∂ ∂ ∂ ih∂ ∂ ∂ i + . . .
24
h∂ x ∂ y ∂ µ ih∂ µ ∂ ν ∂ ν ∂ z i1 7→
1
− h∂ x ∂ y ∂ µ ih∂ i ∂ α ∂ µ ih∂ α ∂ β ∂ β ih∂ j ∂ ν ∂ ν ∂ z i − h∂ x ∂ y ∂ µ ih∂ j i1 h∂ i ∂ α ∂ z ih∂ α ∂ β ∂ ν ih∂ β ∂ µ ∂ ν i
24
1
+h∂ µ i1 h∂ µ ∂ i ∂ z ih∂ j ∂ β ∂ ν ih∂ β ∂ ν ∂ α ih∂ x ∂ y ∂ α i + h∂ i ∂ z ∂ µ ∂ µ ih∂ j ∂ β ∂ ν ih∂ β ∂ ν ∂ α ih∂ x ∂ y ∂ α i + . . .
24
Checking the coefficients:
h∂ x ∂ y ∂ z ∂ i ih∂ j ∂ α ∂ µ ih∂ µ ∂ α ∂ ν ih∂ ν ∂ β ∂ β i :
1
1
1
c2 − c15 + c19 = 0.
1152
24
24
1
1
1
1
c1 − c3 − c4 + c11 +c12 −c15 −c18 −c20 = 0.
24
12
12
24
h∂ i i1 h∂ x ∂ y ∂ µ ih∂ µ ∂ z ∂ ν ih∂ ν ∂ i ∂ α ih∂ α ∂ β ∂ β i :
−
h∂ i ∂ x ∂ µ ∂ µ ih∂ y ∂ z ∂ ν ih∂ ν ∂ j ∂ α ih∂ α ∂ β ∂ β i :
−
1
1
1
1
1
c3 +
c4 + c14 + c15 + c18 = 0.
576
576
24
8
12
h∂ µ i1 h∂ µ ∂ i ∂ x ih∂ y ∂ z ∂ ν ih∂ ν ∂ j ∂ α ih∂ α ∂ β ∂ β i :
−
1
1
1
1
c3 + c4 + c7 − c11 −3c12 +3c15 +2c18 +c20 = 0.
24
24
24
24
3.5. Checking r4 (E) = 0. Since l = 4 case is new, the calculation is presented.
h∂ x ∂ µ ∂ ν ih∂ µ ∂ y ∂ z ih∂ ν i2 7→
1
h∂ x ∂ µ ∂ j ih∂ µ ∂ y ∂ z ih∂ i ∂ β ∂ ν ih∂ β ∂ γ ∂ γ ih∂ ν ∂ α ∂ α i
−
5760
1
+
h∂ x ∂ µ ∂ j ih∂ µ ∂ y ∂ z ih∂ i ∂ β ∂ α ih∂ β ∂ γ ∂ α ih∂ γ ∂ ν ∂ ν i
960
1
+
h∂ x ∂ µ ∂ ν ih∂ µ ∂ y ∂ z ih∂ i ∂ β ∂ ν ih∂ β ∂ γ ∂ γ ih∂ j ∂ α ∂ α i
576
TAUTOLOGICAL EQUATIONS IN GENUS 2 VIA INVARIANCE THEOREM
h∂1x ∂ µ i2 h∂ µ ∂ y ∂ z i 7→
1
h∂ i ∂ ν ∂ α ih∂ α ∂ β ∂ x ih∂ β ∂ γ ∂ γ ih∂ µ ∂ µ ∂ ν ih∂ j ∂ y ∂ z i
−
1920
1
h∂ x ∂ µ ∂ ν ih∂ i ∂ ν ∂ α ih∂ α ∂ β ∂ µ ih∂ β ∂ γ ∂ γ ih∂ j ∂ y ∂ z i
+
720
1
+
h∂ i ∂ ν ∂ α ih∂ α ∂ β ∂ ν ih∂ β ∂ γ ∂ µ ih∂ γ ∂ x ∂ µ ih∂ j ∂ y ∂ z i
576
1
+
h∂ x ∂ µ ∂ α ih∂ α ∂ i ∂ β ih∂ β ∂ γ ∂ γ ih∂ j ∂ ν ∂ ν ih∂ µ ∂ y ∂ z i
576
1
+
h∂ i ∂ β ∂ µ ih∂ γ ∂ x ∂ α ih∂ α ∂ β ∂ γ ih∂ j ∂ ν ∂ ν ih∂ µ ∂ y ∂ z i
576
1
+
h∂ i ∂ β ∂ µ ih∂ β ∂ γ ∂ γ ih∂ x ∂ α ∂ ν ih∂ α ∂ j ∂ ν ih∂ µ ∂ y ∂ z i
576
h∂ x ∂1µ i2 h∂ µ ∂ y ∂ z i 7→
1
−
h∂ i ∂ x ∂ α ih∂ α ∂ β ∂ µ ih∂ β ∂ γ ∂ γ ih∂ µ ∂ ν ∂ ν ih∂ j ∂ y ∂ z i
5760
1
+
h∂ i ∂ x ∂ α ih∂ α ∂ β ∂ ν ih∂ β ∂ γ ∂ ν ih∂ γ ∂ µ ∂ µ ih∂ j ∂ y ∂ z i
960
1
+
h∂ µ ∂ x ∂ α ih∂ α ∂ i ∂ β ih∂ β ∂ γ ∂ γ ih∂ j ∂ ν ∂ ν ih∂ µ ∂ y ∂ z i
576
1
+
h∂ i ∂ β ∂ x ih∂ γ ∂ µ ∂ α ih∂ α ∂ β ∂ γ ih∂ j ∂ ν ∂ ν ih∂ µ ∂ y ∂ z i
576
1
+
h∂ i ∂ β ∂ x ih∂ β ∂ γ ∂ γ ih∂ µ ∂ α ∂ ν ih∂ α ∂ j ∂ ν ih∂ µ ∂ y ∂ z i
576
h∂ x ∂ y ∂ z ∂ µ i1 h∂ µ ∂ ν ∂ ν i 7→
1
+ h∂ i ∂ α ∂ ν ih∂ ν ∂ β ∂ α ih∂ β ∂ γ ∂ z ih∂ γ ∂ x ∂ y ih∂ j ∂ µ ∂ µ i
24
1 i y α α β x β γ γ j z µ µ ν ν
− h∂ ∂ ∂ ih∂ ∂ ∂ ih∂ ∂ ∂ ih∂ ∂ ∂ ih∂ ∂ ∂ i
8
1
− h∂ i ∂ µ ∂ α ih∂ α ∂ β ∂ x ih∂ β ∂ γ ∂ γ ih∂ j ∂ y ∂ z ih∂ µ ∂ ν ∂ ν i
8
1
+ h∂ i ∂ β ∂ x ih∂ β ∂ γ ∂ γ ih∂ j ∂ α ∂ µ ih∂ α ∂ y ∂ z ih∂ µ ∂ ν ∂ ν i
8
1
+ h∂ i ∂ β ∂ µ ih∂ β ∂ γ ∂ γ ih∂ j ∂ α ∂ z ih∂ α ∂ x ∂ y ih∂ µ ∂ ν ∂ ν i
24
1
+ h∂ i ∂ β ∂ µ ih∂ β ∂ γ ∂ z ih∂ γ ∂ x ∂ y ih∂ j ∂ α ∂ α ih∂ µ ∂ ν ∂ ν i
24
h∂ x ∂ µ ∂ ν ih∂ µ ∂ ν ∂ y ∂ z i1 7→
1
− h∂ x ∂ µ ∂ ν ih∂ i ∂ ν ∂ α ih∂ α ∂ β ∂ µ ih∂ β ∂ γ ∂ γ ih∂ j ∂ y ∂ z i
24
1
− h∂ x ∂ µ ∂ ν ih∂ i ∂ x ∂ α ih∂ α ∂ β ∂ y ih∂ β ∂ γ ∂ γ ih∂ j ∂ µ ∂ ν i
24
1
+ h∂ x ∂ µ ∂ ν ih∂ i ∂ β ∂ y ih∂ β ∂ γ ∂ γ ih∂ j ∂ α ∂ z ih∂ α ∂ µ ∂ ν i
12
1
+ h∂ x ∂ µ ∂ ν ih∂ i ∂ β ∂ z ih∂ β ∂ γ ∂ y ih∂ γ ∂ µ ∂ ν ih∂ j ∂ α ∂ α i
24
15
16
D. ARCARA AND Y.-P. LEE
h∂ x ∂ y ∂ µ ih∂ µ ∂ ν ∂ ν ∂ z i1 7→
1
+ h∂ x ∂ y ∂ j ih∂ i ∂ α ∂ ν ih∂ ν ∂ β ∂ α ih∂ β ∂ γ ∂ z ih∂ γ ∂ µ ∂ µ i
24
1
− h∂ x ∂ y ∂ µ ih∂ i ∂ z ∂ α ih∂ α ∂ β ∂ µ ih∂ β ∂ γ ∂ γ ih∂ j ∂ ν ∂ ν i
24
1
− h∂ x ∂ y ∂ µ ih∂ i ∂ ν ∂ α ih∂ α ∂ β ∂ ν ih∂ β ∂ γ ∂ γ ih∂ j ∂ µ ∂ z i
24
1
+ h∂ x ∂ y ∂ µ ih∂ i ∂ β ∂ µ ih∂ β ∂ γ ∂ γ ih∂ j ∂ α ∂ z ih∂ α ∂ ν ∂ ν i
24
1
+ h∂ x ∂ y ∂ µ ih∂ i ∂ β ∂ z ih∂ β ∂ γ ∂ γ ih∂ j ∂ α ∂ ν ih∂ α ∂ µ ∂ ν i
24
1
+ h∂ x ∂ y ∂ µ ih∂ i ∂ β ∂ z ih∂ β ∂ γ ∂ ν ih∂ γ ∂ µ ∂ ν ih∂ j ∂ α ∂ α i
24
The other graphs all have r4 (Γ) = 0.
Now let’s check all the coefficients:
1
1
1
1
1
x µ ν
c1 +
c3 +
c4 + c15 + c18 = 0.
h∂ ∂ ∂ ih∂ µ ∂ y ∂ z ih∂ i ∂ β ∂ ν ih∂ β ∂ γ ∂ γ ih∂ j ∂ α ∂ α i :
576
288
288
12
24
1
1
1
1
1
c3 +
c4 − c15 − c18 + c20 = 0.
h∂ i ∂ ν ∂ α ih∂ α ∂ β ∂ x ih∂ β ∂ γ ∂ γ ih∂ µ ∂ µ ∂ ν ih∂ j ∂ y ∂ z i :
384
1152
8
24
24
1
1
1
h∂ x ∂ µ ∂ j ih∂ µ ∂ y ∂ z ih∂ i ∂ β ∂ ν ih∂ β ∂ γ ∂ γ ih∂ ν ∂ α ∂ α i :
c1 + c15 − c20 = 0.
1152
24
24
1
1
1
1
c3 −
c4 − c15 − c18 = 0.
h∂ j ∂ x ∂ µ ih∂ µ ∂ ν ∂ ν ih∂ i ∂ y ∂ α ih∂ α ∂ β ∂ z ih∂ β ∂ γ ∂ γ i :
576
576
4
8
3.6. Conclusion. By Lemma 1 in [7], rl (E) = 0 for l ≥ 3 (respectively 4, 5) for (g, n, k) =
(2, 1, 2), (respectively (2, 2, 2), (2, 3, 2)). Therefore, Invariance Conjectures 1 and 2 are proved.
By a Betti number calculation of E. Getzler [4], they are the only tautological equations for the
corresponding (g, n, k). Therefore, Invariance Conjecture 3 also holds.
References
[1] D. Arcara, Y.-P. Lee, Tautological equation in M3,1 via invariance constraint, math.AG/0503184.
[2] P. Belorousski, R. Pandharipande, A descendent relation in genus 2, Ann. Scuola Norm. Sup. Pisa Cl. Sci.
(4) 29 (2000), no. 1, 171–191.
[3] E. Getzler, Intersection theory on M1,4 and elliptic Gromov-Witten invariants, J. Amer. Math. Soc. 10
(1997), no. 4, 973–998.
[4] E. Getzler, Topological recursion relations in genus 2, Integrable systems and algebraic geometry
(Kobe/Kyoto, 1997), 73–106, World Sci. Publishing, River Edge, NJ, 1998.
[5] A. Givental, Y.-P. Lee, preliminary draft.
[6] Y.-P. Lee, Witten’s conjecture and Virasoro conjecture up to genus two, math.AG/0310442. To appear
in the proceedings of the conference ”Gromov-Witten Theory of Spin Curves and Orbifolds”, Contemp.
Math., AMS.
[7] Y.-P. Lee, Invariance of tautological equations I: conjectures and applications, math.AG/0604318.
[8] Y.-P. Lee, Invariance of tautological equations II: Gromov–Witten theory, math.AG/0605708.
[9] Y.-P. Lee, Witten’s conjecture, Virasoro conjecture, and invariance of tautological relations,
math.AG/0311100.
[10] R. Vakil, The moduli space of curves and Gromov-Witten theory, math.AG/0602347.
Department of Mathematics, University of Utah, Salt Lake City, UT 84112-0090, USA
E-mail address: arcara@math.utah.edu
E-mail address: yplee@math.utah.edu