Double Box Motive
The motive associated with the second Symanzik polynomial of the double-box two-loop Feynman graph with generic masses and momenta is shown to be an elliptic curve.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 17 (2021), 048, 12 pages
Double Box Motive
Spencer BLOCH
Department of Mathematics, The University of Chicago,
Eckhart Hall, 5734 S University Ave, Chicago IL, 60637, USA
E-mail: spencer bloch@yahoo.com
URL: http://www.math.uchicago.edu/~bloch/
Received March 20, 2021, in final form May 04, 2021; Published online May 13, 2021
https://doi.org/10.3842/SIGMA.2021.048
Abstract. The motive associated to the second Symanzik polynomial of the double-box
two-loop Feynman graph with generic masses and momenta is shown to be an elliptic curve.
Key words: Feynman amplitude; elliptic curve; double-box graph; cubic hypersurface
2020 Mathematics Subject Classification: 14C30; 14D07; 32G20
1 Introduction
Cubic hypersurfaces in algebraic geometry are sort of analogous to the baby bear’s porridge
in the tale of Goldilocks and the three bears. Hypersurfaces of degrees one and two are too cold
and uninteresting, while degrees four and higher are too hot to handle. Degree three is just right.
The author thanks Dirk Kreimer and Pierre Vanhove for explaining the importance for physics
of two-loop Feynman graphs whose amplitudes are (mixed, or relative) periods of certain singular
cubic hypersurfaces. In particular, the paper [6] suggests the central role the singularities of the
second Symanzik hypersurface play. The basic setup of iterated subdivision of the two-loop
sunset grew out of collaboration with Vanhove [1]. The kite graph (written (2, 1, 2) in the
notation explained in Appendix A) also leads to an elliptic curve. The situation for the kite is
more elementary since one can fall back on the classical theory of cubic threefolds with isolated
double points in P4. Vanhove has been able to calculate the j-invariant for the resulting elliptic
curve. A similar calculation for the double box looks difficult. The author was also influenced
by unpublished work of Matt Kerr classifying Feynman motives associated to a number of other
two-loop diagrams.
This paper focuses on what is perhaps the most striking example, the massive second Syman-
zik motive associated to the double box graph, see Figure 1. The massive second Symanzik in
our case is a (singular) cubic hypersurface X3,1,3 in P6. For a smooth cubic hypersurface X ⊂ P6,
the Hodge structure is the Hodge structure associated with the cohomology in middle dimension,
in this case H5(X,Q). It is known [5, p. 16], that H5(X,Q) ∼= H1(A,Q(−2)) where A is a certain
abelian variety of dimension 21. For the double box hypersurface with generic momenta and
masses, H5(X3,1,3,Q) is a mixed Hodge structure with weights ≤ 5. We will construct a specific
resolution of singularities π : Z → X3,1,3. We will show F 3H5(Z) = C and F 4H5(Z) = (0).
From this it will follow that as a Hodge structure, H5(Z,Q) ∼= H1(E,Q(−2)) for a suitable
elliptic curve E. (We do not specify the elliptic curve.) The dimension of the abelian variety
drops from 21 to 1!
This paper is a contribution to the Special Issue on Algebraic Structures in Perturbative Quan-
tum Field Theory in honor of Dirk Kreimer for his 60th birthday. The full collection is available at
https://www.emis.de/journals/SIGMA/Kreimer.html
mailto:spencer_bloch@yahoo.com
http://www.math.uchicago.edu/~bloch/
https://doi.org/10.3842/SIGMA.2021.048
https://www.emis.de/journals/SIGMA/Kreimer.html
2 S. Bloch
1
3
7
4
5
62
1
3
7
4
5
62 = (3, 1, 3)↔ subdivided
sunset graph:
Figure 1. Double box graph.
From the structure of the map π we will deduce our main result
Theorem 1.1. With the above notation,
H1(E,Q(−2)) ∼= grW5 H
5(X3,1,3,Q).
The proof is given in three steps.
Step 1. We construct the resolution of singularities Z → X3,1,3 and we show F 3H5(Z,C) ∼= C.
This is Theorem 5.1 below.
Step 2. We show F 4H5(Z,C) = (0), so the Hodge type of H5(Z) is (0, 0, 1, 1, 0, 0), from which
it follows that H5(Z) ∼= H1(E,Q(−2)) for a suitable elliptic curve.
Step 3. We show H5(Z,Q) ∼= grW5 H
5(X3,1,3,Q).
The study of motives associated to two-loop graphs involves a lot of detailed algebraic geom-
etry. The author can only hope he has gotten the details right! Another approach, which might
yield more information in this case about the elliptic curve, could be based on the study of linear
spaces contained in the second Symanzik variety. For example, projecting from a double point
on the variety realizes a singular cubic hypersurface of dimension n as (birational to) the blowup
of an intersection of a quadric and a cubic hypersurface on Pn [2]. When n = 5, this intersection
will be a (singular) Fano threefold, and one may expect a rough dictionary
intermediate jacobian of Fano↔ lines on Fano↔ planes on cubic fivefold
↔ intermediate jacobian of cubic fivefold.
The weight 5 piece of H5 for the double box X (with generic parameters) is seen to have Hodge
type (2, 3), (3, 2) which suggests the role of P2’s. Namely, we might expect a correspondence
of the form
H1(C)
α∗−→ H1(P )
β∗−→
(
W5H
5(X)
)
(2).
Here P
α−→ C is a family of P2’s parametrized by a curve C. A deeper understanding of the
elliptic curve arising from H5 of the double box (2.1) should come from a study of the Fano
Double Box Motive 3
variety of planes on (2.1). (Presumably this Fano is a surface, see [2, Section 1].) This is
a wonderful problem! In the case of the kite graph (2, 1, 2) one is able to write the elliptic curve
as an intersection of two quadrics in P3. I do not know if similar methods will work for the
double box. Another approach, suggested by the referee, would be to study the Picard–Fuchs
equation associated to H5 of the double box (2.1), relating it if possible to the Picard–Fuchs
equation for a family of elliptic curves.
More generally it would be interesting to generalize motivic considerations to cover all 2-loop
graphs with generic parameters. A physicist in the audience at the Newton institute suggested
that the non-planar case would be more subtle. I asked him to email me so we could correspond,
but he never did, so I cannot give him credit, but he is certainly correct. A calculation for the
graph (3, 2, 2) shows that F 3H5 (resolution of (3, 2, 2)) has dimension ≥ 5. On the other hand,
the graphs (n, 1, n) seem better behaved.
2 Singularities of X3,1,3
We focus now on singularities for the double box (for details, see Proposition A.1)
X3,1,3 : 0 = Ψ3,1,3 = Q(x5, x6, x7)(x1 + x2 + x3 + x4)
+Q′(x1, x2.x3)(x4 + x5 + x6 + x7) + x4A. (2.1)
Here A has degree 2 and is linear separately in each variable x1, . . . , x7.
Proposition 2.1. The singular locus X3,1,3,sing = C q C ′ q S, where
C : Q(x5, . . . , x7) = 0 = x1 = x2 = · · · = x4,
C ′ : Q′(x1, . . . , x3) = 0 = x4 = x5 = · · · = x7.
Here C (resp. C ′) is a smooth quadric in P2. Finally S is a finite set of ordinary double points
on X3,1,3 defined by
Q(x5, x6, x7) = Q′(x1, x2, x3) = x4 = x1 + x2 + x3 = x5 + x6 + x7 = A = 0.
The double points S are disjoint from C and C ′.
Proof. The reader can easily check that points of C, C ′, and S are singular on X3,1,3. We can
understand points of S by first imposing the linear relations x4 = x1 + x2 + x3 = x5 + x6
+x7 = 0, so
S : Q(x5, x6,−x5 − x6) = Q′(x1, x2,−x1 − x2)
= A(x1, x2,−x1 − x2, 0, x5, x6,−x5 − x6) = 0
in P3
x1,x2,x5,x6 . �
Remark 2.2. It will be important in the sequel that points in S all lie on the hyperplane x4 = 0.
3 Blowings up
We blow up C q C ′ and S on P6 and on X3,1,3, obtaining the diagram
Z −−−−→ Wy yσ=blow S
Y −−−−→ Vy yπ=blow C q C′
X3,1,3 −−−−→ P6
4 S. Bloch
The exceptional divisor in V is written E q E′ with
E = P
(
N∨C⊂P6
)
, E′ = P
(
N∨C′⊂P6
)
.
Both conormal bundles have the form N∨ = O(−1)4 ⊕O(−2). Since S is disjoint from E qE′,
the map π ◦ σ is itself a blowup, with exceptional fibre
q6P5 q E q E′.
Lemma 3.1. Z is a smooth divisor on W .
Proof. The points of S are ordinary double points on X3,1,3. Since the fibres of Z over these
points are smooth quadrics, it suffices to show Y ∩(EqE′) is smooth. The situation is symmetric
so we can focus on Y ∩ E. We have C : x1 = x2 = x3 = x4 = Q(x5, x6, x7) = 0. Let I ⊂ OP6
be the ideal of C. Since Ψ3,1,3 vanishes to order 2 along C, Ψ defines a map OP6(−3) → I2.
The image of this map defines a homogeneous sheaf of ideals of degree 2 for the graded sheaf
of algebras
⊕
Ik/Ik+1 with corresponding variety E ∩ Y .
To avoid having to keep track of the grading on I/I2, write yi = xi/x5. Write q =
Q(x5, x6, x7)/x
2
5 for the coordinate in the fibre. In contrast q′(y1, y2, y3) = Q′(x1, x2, x3)/x
2
5.
The defining equation locally looks like
q(y1 + y2 + y3 + y4) + q′(y1, y2, y3)(1 + y6 + y7) + y4L(y1, y2, y3, y4). (3.1)
Here L is defined (compare the definition of A in (A.1)).
L(y1, y2, y3, y4) = (y1, y2, y3, y4)
a15 a16 a17
a25 a26 a27
a35 a36 a37
a45 a46 a47
1
y6
y7
.
Note that (3.1) does not vanish to order ≥ 2 at any point of E ∩ Y . (It may clarify to remark
that E ∩ Y is not smooth over C. Indeed, at the point of C defined by 1 + y6 + y7 = 0, the
equation in the fibre becomes q(y1 + y2 + y3 + y4) + y4L(y1, y2, y3, y4) = 0 and this can vanish
to order 2.) �
Smoothness of Z permits us to calculate the Hodge filtration on H6(W − Z) using the pole
order filtration [3, Chapter II, Proposition 3.13]. W is obtained by blowing up points and
smooth rational curves on P6, so all cohomology classes on W are Hodge. In particular the
Gysin sequence yields
F 3H5(Z) ∼= F 4H6(W − Z) ∼= H2
(
W,Ω4
W (Z)
d−→ Ω5
W (2Z)
d−→ Ω6
W (3Z)
)
. (3.2)
4 Leray
To evaluate F 3H5(Z) in (3.2), we will use the Leray spectral sequence for π : W → P6.
Proposition 4.1. We have Riπ∗Ω
j
W (kZ) = (0), i, k ≥ 1, j ≥ 0.
Proof. For i ≥ 1 it is clear that the support of Riπ∗Ω
j
W (kZ) is contained in C q C ′ q S.
Lemma 4.2. For i ≥ 1, Riπ∗Ω
j
W (kZ) is killed by the ideal J ⊂ OP6 of functions vanishing
on C q C ′ q S.
Double Box Motive 5
Proof of lemma. As a consequence of Zariski’s formal function theorem,
Riπ∗Ω
j
W (kZ)∧ ∼= lim←−
n
Riπ∗
(
Ωj
W (kZ)⊗OW /Jn
)
.
Since completion is faithfully flat, it will suffice to show the individual sheaves Riπ∗
(
Ωj
W (kZ)⊗
OW /Jn
)
are killed by J . For this it will suffice to show
Riπ∗
(
Ωj
W (kZ)⊗ Jn−1/Jn
)
= (0), i ≥ 1, n ≥ 2, j, k ≥ 0.
Since X3,1,3 vanishes to order 2 along CqC ′qS, it follows that locally, the divisor Z ·exceptional
divisor corresponds to the line bundle OP(N∨)(2) where N∨ is the conormal bundle of CqC ′qS.
Thus we need to show
Riπ∗
(
Ωj
V ⊗OP(N∨)(2k + n− 1)
)
= (0).
For j = 0 this is standard because we are computing cohomology in degree > 0 of a positive
multiple of the tautological bundle on projective space. For j > 0, we have exact sequences
(writing E for the exceptional divisor)
0→ OE(−E)→ Ω1
W |E → Ω1
E → 0,
0→ Ωj−1
P(N∨)(1)→ Ωj
W |E → Ωj
P(N∨) → 0. (4.1)
It therefore suffices to show Riπ∗
(
Ω`
P(N∨)(m)
)
= (0) for i,m ≥ 1, ` > 0. We have a filtration
on Ω`
P(N∨) locally over the base with graded pieces Ωq
P(N∨)/CqC′qS , q = ` − 1, `. The assertion
is local on the base, so finally we have to check that H i
(
P3,Ωq
P3(m)
)
= (0) for i,m ≥ 1.
This is standard, because Ωj
Ps admits a resolution by direct sums of line bundles O(−r) with
r ≤ s+ 1. �
We return now to the proof of Proposition 4.1. Since for i ≥ 1, Riπ∗
(
Ωj
W (kZ)
)
is supported
on C q C ′ q S, it follows from the lemma that
Riπ∗
(
Ωj
W (kZ)
) ∼= Ri
(
π|π−1(CqC′qS)
)
∗
(
Ωj
W (kZ)
)
|(π−1(CqC′qS).
Finally, it follows from (4.1) that Riπ∗(Ω
j
W (kZ)) = (0) for k ≥ 1, proving Proposition 4.1. �
In order to evaluate (3.2), we need finally to calculate π∗Ω
3+m
W (mZ) for m = 1, 2, 3. Note that
sections of π∗Ω
3+m
W (mZ) coincide with sections ω of Ω3+m
P6 (mX) such that π∗(ω) has no pole
along the exceptional divisor. First consider what happens over the ordinary double points S.
Here Ψ3,1,3 vanishes to order 2 so a pullback of a section of Ω3+m
P6 (mX) gets a pole of order 2m
along the exceptional divisor coming from the pullback of mX. On the other hand if the singular
point is defined by z1 = · · · = z6 = 0, then writing dzj = d(zi(zj/zi)) it follows that any form
dzi1 ∧ · · · ∧ dzi3+m downstairs pulls back to a form vanishing to at least order 2 + m along the
exceptional divisor upstairs. For m = 1, 2 the pole order 2m− (2 +m) upstairs is ≤ 0 so there
is no pole. On the other hand, for m = 3 we get a pole of order 1 along the exceptional divisor.
Let I(S) ⊂ OP6 be the ideal of functions vanishing on S. Then
I(S)Ω6
P6(3X)
have no poles on the exceptional divisor lying over S.
Over the curves C and C ′ local defining equations for the singular set have the form z1 =
· · · = z5 = 0. A complete set of parameters locally on P6 becomes z1, . . . , z5, t for some t.
6 S. Bloch
Locally, a 4-form downstairs pulls back to have a zero of order at least 2 on the exceptional
divisor upstairs, from which it follows that
π∗Ω
4
W (Z) = Ω4
P6(X).
Sections of Ω4
P6 pull back to vanish to order 4 on the exceptional divisor, so writing I ⊂ OP6 for
the ideal of functions vanishing on C q C ′, we have
π∗Ω
6
W (3Z) = (I(S)) ∩ I2Ω6
P6(3X).
Finally, over C q C ′, the situation for π∗Ω
5
W (2X) is more complicated. Let I ⊂ OP6 be the
ideal of C q C ′. Write
Ω1
P6,I := IΩ1
P6 + dI = ker
(
Ω1
P6 → Ω1
CqC′
)
We have
π∗Ω
5
W (2Z) = Ω5
P6,I(2X) := Image
( 5∧
Ω1
P6,I → Ω5
P6
)
(2X).
Since cohomology in degree > 0 of Ωi
Pn(j) vanishes for j > 0, we conclude in (3.2) that
F 3H5(Z) ∼= H2
(
W,Ω4
P6(X)→ Ω5
P6,I(2X)→
(
I(S) ∩ I2
)
Ω6
P6(3X)
)
∼= H1
(
P6,Ω5
P6,I(2X)→
(
I(S) ∩ I2
)
Ω6
P6(3X)
)
.
5 The computation of F 3H5(Z)
Theorem 5.1. With notation as above, we have F 3H5(Z) ∼= C.
Proof.
Lemma 5.2. H0
(
Ω5
P6,I(2X)
)
= (0).
Proof of lemma. We have Ω5
P6(2X) ∼= TP6(−1). Also IΩ5
P6 ⊂ Ω5
P6,I(2X), so
TP6 |CqC′(−1) � Ω5
P6(2X)/Ω5
P6,I(2X).
Locally Ω5
P6,I is generated by IΩ5
P6 and dz1 ∧ · · · ∧ dz5. It follows that
Ω5
P6(2X)/Ω5
P6,I(2X) ∼= NCqC′/P6(−1) = TP6(−1)|CqC′/TCqC′(−1).
We build a diagram (ignore for the moment the δi and the bottom line which will only be used
in the next lemma):
0 −−→ H0
(
TP6(−1)
) a−−→ H0
(
C q C ′, NCqC′/P6(−1)
) b−−→ H1
(
Ω5
P6,I(6)
)
−−→ 0yδ1 yδ2 yδ3
H0
(
OP6(2)
) a′−−→ H0
(
OP6/I2(2)
) b′−−→ H1
(
I2OP6(2)
)
−−→ 0
(5.1)
Recall that I is the ideal of C q C ′, where C is defined in homogeneous coordinates by x1 =
x2 = x3 = x4 = Q(x5, x6, x7) = 0 and C ′ is defined by x4 = x5 = x6 = x7 = Q′(x1, x2, x3) = 0.
The kernel of a′ above consists of homogeneous polynomials of degree 2 on P6 which vanish to
Double Box Motive 7
degree 2 both on C and on C ′. I.e., ker(a′) = Cx24. Also b′ is surjective because H1
(
OP6(2)
)
= (0)
and b is surjective because H1(P6, TP6(−1)) = (0).
To see that a is injective, note the Euler sequence
0→ OP6(−1)→
6⊕
0
C
∂
∂xi
→ TP6(−1)→ 0
yields
H0
(
P6, TP6(−1)
)
=
6⊕
0
C
∂
∂xi
.
Recall C : x1 = x2 = x3 = x4 = Q(x5, x6, x7) = 0. The projection of a onto NC/P6(−1) is given
by a(∂/∂xi) = ∂/∂xi, i = 1, 2, 3, 4 and a(∂/∂xj) = (∂Q/∂xj)∂/∂Q, j = 5, 6, 7. This map is
injective. Since H0
(
Ω5
P6,I(2X)
)
= (0) ⊂ ker(a), the lemma follows. �
Lemma 5.3. The arrow
H1
(
P6,Ω5
P6,I(2X)
) d−→ H1
(
P6, (I(S) ∩ I2)Ω6
P6(3X)
)
is injective.
Proof of lemma. The remaining arrows in (5.1) are defined as follows. We identify
H1
(
TP6(−1)
)
=
6⊕
0
∂
∂xi
as above, and define
δ1
(
∂
∂xi
)
=
∂(Ψ3,1,3)
∂xi
∈ H0
(
OP6(2)
)
.
Note that in fact δ1(∂/∂xi) ∈ H0
(
IOP6(2)
)
so we get a well-defined map
TP6(−1)|C → OP6/I2(2).
Tangent vectors along C stabilize powers of I so in fact the map factors through NC/P6(−1)→
OP6/I2(2), defining δ2 in (5.1). Finally δ3 exists because the top row in (5.1) is exact.
It is straightforward to check that the diagram
H1
(
P6,Ω5
P6,I(2X)
) d−−−−→ H1
(
P6,
(
I(S) ∩ I2
)
Ω6
P6(3X)
)yδ3 y
H1
(
I2OP6(2)
)
H1
(
P6, I2Ω6
P6(3X)
)
is commutative, so the lemma will follow if we show δ3 is injective. It is convenient to isolate
this statement as a separate sublemma.
Lemma 5.4 (sublemma). With reference to diagram (5.1), we have
Image(δ2) ∩ Image(a′) = Image(a′ ◦ δ1).
As a consequence, δ3 is injective.
8 S. Bloch
Proof. Fix C : x1 = x2 = x3 = x4 = Q(x5, x6, x7) = 0 to be one component of C. Let I ⊃ I be
the ideal of C ⊂ C. The normal bundle with a −1 twist is NC/P6(−1) ∼= O4
C ⊕OC(1). Here we
have to be careful because OC(1) is a line bundle of degree 2 on the conic C so h0(OC(1)) = 3
and the composition
H0
(
TP6(−1)
) a−→ H0
(
C, NC/P6(−1)
) proj−−→ H0
(
C,NC/P6(−1)
)
is an isomorphism of vector spaces of dimension 7.
Suppose now we have in the sublemma that δ2(u) = a′(v). By the above, we can mod-
ify u (resp. v) by some a(w) (resp. δ1(w)) so that u is trivial on C, that is a′(v) ∈ I2(2) =
(x1, x2, x3, x4)
2. By assumption, a′(v) ∈ δ2(HC′/P6(−1)) which means we can write (here
Ψ := Ψ3,1,3)
a′(v) = c4
∂Ψ
∂x4
+ · · ·+ c7
∂Ψ
∂x7
+ c8
∂Q′
∂x1
∂Ψ
∂Q′
+ c9
∂Q′
∂x2
∂Ψ
∂Q′
+ c10
∂Q′
∂x3
∂Ψ
∂Q′
= c4
∂Ψ
∂x4
+ · · ·+ c7
∂Ψ
∂x7
+
(
c8
∂Q′
∂x1
+ c9
∂Q′
∂x2
+ c10
∂Q′
∂x3
)
(x4 + x5 + x6 + x7). (5.2)
We want to show this expression cannot be a non-trivial quadric in x1, . . . , x4. The monomials
which appear fall into 4 classes:
(123)2, (123)(456), (4)(456), (567)2.
(Here, e.g., (123)2 refers to monomials xixj with 1 ≤ i, j ≤ 3.) The only terms in (567)2 appear
in (5.2) with coefficient c4 so we must have c4 = 0. We have for i = 5, 6, 7
∂Ψ
∂xi
=
∂Q
∂xi
(x1 + x2 + x3 + x4) +Q′ + x4
∂A
∂xi
.
The terms Q′ and x4∂A/∂xi lie in (1234)2. For the terms in (5.2) not in (1234)2 to cancel we
must have for some constant K
c5
∂Q
∂x5
+
c6∂Q
∂x6
+
c7∂Q
∂x7
= K(x5 + x6 + x7),
c8
∂Q′
∂x8
+ c9
∂Q′
∂x9
+ c10
∂Q′
∂x10
= −K(x1 + x2 + x3). (5.3)
It follows from (5.2) and (5.3) that a′(v) in (5.2) will involve a term K(x5 +x6 +x7) which does
not cancel. This proves Lemma 5.4. �
The assertion (needed to prove Lemma 5.3) that δ3 is injective is now just a diagram chase. �
Lemma 5.5. H0
(
P6, I(S) ∩ I2Ω6
P6(3X3,1,3)
) ∼= C.
Proof of lemma. We will show that H0
(
P6, I2Ω6
P6(3X3,1,3)
) ∼= C and that the generating
section also vanishes on points of S. In fact, I2Ω6
P6(3X3,1,3) ∼= I2P6(2). Note I2 = (IC ∩ IC′)2,
so we are looking for homogenous forms of degree 2 which vanish to order 2 on C and on C ′.
The only such forms lie in C · x24. Finally, note that x4 vanishes on S. �
Theorem 5.1 follows by combining Lemmas 5.2, 5.3, and 5.5. �
Double Box Motive 9
6 The Hodge structure in more detail
In this section, we justify steps 2 and 3 from the introduction. A general reference is [7].
Proposition 6.1. With notation as above, we have F 4H5(Z,C) = (0).
Proof. Again from [3, Chapter II, Proposition 3.13], we have
H1
(
W,Ω5
W (Z)
d−→ Ω6
W (2Z)
) ∼= F 5H6(W − Z,C) ∼= F 4H5(Z,C).
It will suffice to show
H0
(
W,Ω6
W (2Z)
)
= 0 = H1
(
W,Ω5
W (Z)
)
.
Z is birational with the cubic X3,1,3 ⊂ P6, and a section of Ω6
W with only a pole of order 2
along Z would give rise to a section of Ω6
P6 with only a pole of order 2 along X3,1,3. Since
Ω6
P6
∼= OP6(−7) there are no such sections.
It remains to show H1
(
W,Ω5
W (Z)
)
= (0). By duality, it suffices to show H5
(
W,Ω1
W (−Z)
)
= (0). Recall W is obtained from P6 by blowing up C q C ′ q S. The exceptional divisors
are E, E′,
∐
i P5 ⊂W . Define Ξ = Ω1
W /π
∗Ω1
P6 . We claim
Ξ ∼= Ω1
E/C ⊕ Ω1
E′/C′ ⊕
⊕
i
Ω1
P5 .
Indeed there is a natural surjective map just by restricting 1-forms from W to the exceptional
divisors. Consider the local structure along E. Locally C : z1 = · · · = z5 = 0 in P6. Locally
upstairs, we have E : z1 = 0 and Ω1
W = π∗Ω1
P6 +
∑
OWd(zi/z1). Note
z1d(zi/z1) = dzi − (zi/z1)dz1 ∈ π∗Ω1
P6 .
This implies that π∗Ω1
P6 ⊃ OW (−E)Ω1
W , so Ω1
W |E � Ξ. Also the conormal bundle N∨(E/W ) =
OW (−E)/OW (−2E) of E ⊂W lies in Ω1
W |E . Similarly, the conormal bundle N∨C/P6 lies in Ω1
P6 |C ,
and the pullback π∗N∨C/P6 → N∨(E/W ) is surjective. It follows that
Ω1
E =
(
Ω1
W |E
)
/N∨(E/W ) � Ξ.
The proof of (6.1) is now straightforward.
We have the exact sequence
0→ π∗Ω1
P6(−Z)→ Ω1
W (−Z)→ Ξ(−Z)→ 0. (6.1)
In the Picard group of W we have −Z = −π∗(X3,1,3) + 2E + 2E′+ 2
∑
s∈S P5
s. We compute the
direct images
Rπ∗
(
π∗Ω1
P6(−Z)
)
= Ω1
P6(−3)⊗Rπ∗
(
OW
(
2E + 2E′ + 2
∑
s∈S
P5
s
))
. (6.2)
The calculus of intersection theory on blowups [4, Section II.7], yields E · E = [OE(−1)], the
dual of the tautological relatively ample bundle on the projective bundle E over C (and similarly
for E′ and the P5
s). Applying Rπ∗ to the exact sequences
0→ OW
(
E + E′ +
∑
P5
s
)
→ OW
(
2E + 2E′ + 2
∑
P5
s
)
→ OE(−2)⊕OE′(−2)⊕
⊕
S
OP5(−2)→ 0
10 S. Bloch
and the similar sequence
0→ OW → OW
(
E + E′ +
∑
P5
s
)
→ OE(−1)⊕OE′(−1) +
∑
OP5
s
(−1)→ 0
and recalling that for a projective space bundle P π−→ S with fibre Pm we have Rπ∗OP(−n) = (0)
for 1 ≤ n ≤ m, we can see that
Rπ∗
(
OW
(
2E + 2E′ + 2
∑
s∈S
P5
s
))
∼= Rπ∗(OW ). (6.3)
We want to use these results to prove H5
(
W,Ω1
W (−Z)
)
= (0). From (6.1), it will suffice to show
H5
(
W,π∗Ω1
P6(−Z)
)
= (0) = H5(W,Ξ(−Z)).
From (6.2) and (6.3) we get
H5
(
W,π∗Ω1
P6(−Z)
)
= H5
(
P6,Ω1
P6(−3)⊗Rπ∗OW
)
. (6.4)
We claim Rπ∗(OW ) ∼= OP6 , i.e., Riπ∗OW = (0) for i ≥ 1. This can be checked locally on P6,
so for notational simplicity we assume W is P6 blown up along C. (Of course, R0π∗OW = OP6 .
Also Riπ∗(OE) = (0) for i ≥ 1.) The ideal sheaf OW (−E) is relatively ample (a basic fact about
blowing up [4, Section II.7]) so Riπ∗(OW (−nE)) = (0) for n � 0 and i ≥ 1. We have
(
note
E · E = [OE(−1)]
)
Riπ∗
(
OW (−nE)
)
→ Riπ∗
(
OW (−(n− 1)E)
)
→ Riπ∗
(
OE(n− 1)
)
.
By descending induction we conclude Riπ∗(OW (−nE)) = (0) for n ≥ 0 and i ≥ 1.
Finally, from (6.4) we get
H5
(
W,π∗Ω1
P6(−Z)
)
= H5
(
P6,Ω1
P6(−3)
)
.
Dualizing and twisting the Euler sequence yields
0→ Ω1
P6(−3)→
⊕
7
OP6(−4)→ OP6(−3)→ 0,
from which we conclude the desired vanishing for H5
(
W,π∗Ω1
P6(−Z)
)
.
We need finally to check vanishing for H5(W,Ξ(−Z)), with Ξ as in (6.1). It suffices to show
for example Rπ∗
(
Ω1
E/C(−2)
)
= (0). We have 0 → Ω1
E/C →
⊕
5OE(−1) → OE → 0, and the
sheaves OE(−2) and OE(−3) are fibrewise OP4(−k), k = 2, 3 and have vanishing cohomology
along the fibres in all degrees. �
Proposition 6.2 (verification of Step 3 at the end of Section 1). We have
H5(Z,Q) ∼= grW5 H
5(X3,1,3,Q).
Proof. We consider the Leray spectral sequence (writing πZ = π|Z : Z → X3,1,3).
Ep,q2 = Hp
(
X3,1,3, R
qπZ,∗QZ
)
⇒ Hp+q(Z,Q).
The fibres of πZ are connected, so π∗QZ
∼= QX3,1,3 and E5,0
2 = H5(X3,1,3,Q). We can stratify
X3,1,3 so the fibres consist either of one point or a (possibly singular) 3-dimensional complex
quadric (over CqC ′) or a 4-dimensional complex quadric (over the isolated singular set S). The
only odd q ≤ 5 representing a possible non-zero cohomology of a fibre is H3 which can occur
in isolated singular fibres for the 3-dimensional quadric over C q C ′. But then H2
(
R3
)
= (0).
Thus, the only Ep,q2 which contributes to H5(F ) is E5,0
2 = H5(X3,1,3,Q)→ H5(F,Q). This map
is necessarily surjective. It factors through grW5 H
5(X3,1,3,Q), proving Step 3. �
Double Box Motive 11
A The symanzik polynomials
Let (m, 1, n) denote the 2-loop graph obtained by subdividing the 3 edges of the sunset graph
(1, 1, 1) into m (resp. 1, resp. n) edges. We associate edge variables x1, . . . , xm (resp. xm+1,
resp. xm+2, . . . , xm+n+1) to the subdivided edges in the evident way. The first Symanzik poly-
nomial is the quadratic polynomial
φm,1,n(x1, . . . , xm, xm+1, xm+2, . . . , xm+n+1) =
∑
xixj
with the sum being taken over all pairs xi 6= xj such that cutting edges i and j renders the
graph a tree. One checks
φm,1,n = xm+1(x1 + · · ·+ xm + xm+2 + · · ·+ xm+n+1) +
( m∑
i=1
xi
)(m+n+1∑
j=m+2
xj
)
.
The (massless) second Symanzik polynomial ψm,1,n has degree 3 in the edge variables. The
coefficients are quadratic in the momentum flows wµ ∈ CD where µ runs over the vertices
of the graph. A monomial xixjxk appears in ψm,1,n if cutting the edges i, j, k in (m, 1, n)
reduces the graph to a 2-tree. Here 2-tree means a subgraph with no loops and exactly 2 con-
nected components. (Connected components may be isolated vertices.) The coefficient of xixjxk
in ψm,1,n is computed as follows. Let C be one of the two connected components of the cut graph
(m, 1, n)− {i, j, k}. The coefficient of xixjxk in ψm,1,n is
ci,j,k =
( ∑
µ∈vert(C)
wµ
)2
.
Here all vertex momenta are oriented to point inward, and the square refers to the quadratic
form on CD. Because the sum of graph momenta vanishes, switching the choice of connected
component C does not change ci,j,k.
A reminder: the geometric results in this paper are predicated on the coefficients ci,j,k being
sufficiently generic. In particular, for the (3, 1, 3) double box case, when D = 1 the ci,j,k are
never sufficiently generic. They are sufficiently generic if D = 4 and the momentum vectors wµ
have generic entries.
Finally, the massive second Symanzik is given by
Ψm,1,n :=
(∑
m2
ixi
)
φm,1,n + ψm,1,n.
Proposition A.1. For generic values of masses mi and momentum vectors, the massive second
Symanzik can be written
Ψm,1,n(x1, . . . , xm+n+1) = Q(x1, . . . , xm)(xm+1 + · · ·+ xm+n+1)
+Q′(xm+2, . . . , xm+n+1)(x1 + · · ·+ xm+1) + xm+1A.
Here Q and Q′ are homogeneous of degree 2, and
A = (x1, . . . , xm+1)
a1,m+1 a1,m+2 · · · a1,m+n+1
a2,m+1 · · · · · · a2,m+n+1
am,m+1
...
...
...
0 am+1,2 · · · am+1,m+n+1
xm+1
xm+2
...
xm+n+1
. (A.1)
The proof of the proposition is straightforward using a computer. The author will leave
a program on his web page in the Publications folder.
Note that the terms in A have generic coefficients. They are linear in x1, . . . , xm+1 and also
in xm+1, . . . , xm+n+1 and in xm+1. The quadrics Q and Q′ have generic coefficients.
12 S. Bloch
References
[1] Bloch S., Vanhove P., The elliptic dilogarithm for the sunset graph, J. Number Theory 148 (2015), 328–364,
arXiv:1309.5865.
[2] Collino A., The Abel–Jacobi isomorphism for the cubic fivefold, Pacific J. Math. 122 (1986), 43–55.
[3] Deligne P., Équations différentielles à points singuliers réguliers, Lecture Notes in Math., Vol. 163, Springer-
Verlag, Berlin – New York, 1970.
[4] Hartshorne R., Algebraic geometry, Graduate Texts in Mathematics, Vol. 52, Springer-Verlag, New York –
Heidelberg, 1977.
[5] Huybrechts D., The geometry of cubic hypersurfaces, Preprint, 2020.
[6] Kreimer D., The master two-loop two-point function. The general case, Phys. Lett. B 273 (1991), 277–281.
[7] Peters C.A.M., Steenbrink J.H.M., Mixed Hodge structures, Ergebnisse der Mathematik und ihrer Grenzge-
biete. 3. Folge. A Series of Modern Surveys in Mathematics, Vol. 52, Springer-Verlag, Berlin, 2008.
https://doi.org/10.1016/j.jnt.2014.09.032
https://arxiv.org/abs/1309.5865
https://doi.org/10.2140/pjm.1986.122.43
https://doi.org/10.1007/BFb0061194
https://doi.org/10.1007/BFb0061194
https://doi.org/10.1007/978-1-4757-3849-0
https://doi.org/10.1016/0370-2693(91)91684-N
https://doi.org/10.1007/978-3-540-77017-6
1 Introduction
2 Singularities of X{3,1,3}
3 Blowings up
4 Leray
5 The computation of F3H5(Z)
6 The Hodge structure in more detail
A The symanzik polynomials
References
|
| id | nasplib_isofts_kiev_ua-123456789-211301 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-16T04:01:32Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Bloch, Spencer 2025-12-29T11:06:01Z 2021 Double Box Motive. Spencer Bloch. SIGMA 17 (2021), 048, 12 pages 1815-0659 2020 Mathematics Subject Classification: 14C30; 14D07; 32G20 arXiv:2105.06132 https://nasplib.isofts.kiev.ua/handle/123456789/211301 https://doi.org/10.3842/SIGMA.2021.048 The motive associated with the second Symanzik polynomial of the double-box two-loop Feynman graph with generic masses and momenta is shown to be an elliptic curve. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Double Box Motive Article published earlier |
| spellingShingle | Double Box Motive Bloch, Spencer |
| title | Double Box Motive |
| title_full | Double Box Motive |
| title_fullStr | Double Box Motive |
| title_full_unstemmed | Double Box Motive |
| title_short | Double Box Motive |
| title_sort | double box motive |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211301 |
| work_keys_str_mv | AT blochspencer doubleboxmotive |