The two-point case

When only two gravitons are involved, with𝑙propagators between them (diagram(a) in Section3.3.3), the associated modular graph functions read:

𝐷_𝑙(𝜏) ∶=

∫_𝜏𝐺₁(𝜉, 𝜏)^𝑙𝑑𝜉 𝜏₂.

5See [33] and [94].

6Conjecture 1 is equivalent to Conjecture 2 in this case because Brown proved [21] that periods of mixed Tate motives overℤare (up to inverting𝜋) MZVs.

The behaviour of this integral as 𝜏₂ tends to infinity was already studied in [48].

Using the same ideas we generalize that result giving the following expansion of the functions𝐷_𝑙’s, which is of course a special case of Theorem1.4.1:

Theorem 4.2.1. For every𝑙≥2

𝐷_𝑙(𝜏) = ∑

𝜇,𝜈≥0

𝑑^(𝜇,𝜈)

𝑙 (𝜋𝜏₂)𝑞^𝜇𝑞^𝜈, (4.7)

where for every𝜇, 𝜈 ≥0

𝑑_𝑙^(𝜇,𝜈)(𝜋𝜏₂) =

2𝑙−1∑

𝑗=0

𝑎^(𝜇,𝜈)_𝑗 (𝜋𝜏₂)^𝑙−𝑗

is a Laurent polynomial with coefficients𝑎^(𝜇,𝜈)_𝑗 belonging to the algebra of conical sums.

Proof. Using (3.20) and doing the change of variables 𝑥 = 𝜉₂∕𝜏₂, we want to compute

𝐷_𝑙(𝜏) = 1 2^𝑙

∑

𝑙=𝑟+𝑚

𝑙!

𝑟!𝑚!

(𝜋𝜏₂)^𝑟

2^𝑚 ∫_𝜏B₂(𝑥)^𝑟𝑃(𝜉, 𝜏)^𝑚𝑑𝜉₁𝑑𝑥,

where we recall that𝜏denotes the complex torus associated with the lattice𝜏ℤ+ℤ.

Since, by the definition (3.21), 𝑃(𝑧, 𝜏)^𝑚 = ∑

𝑛∈ℤ^𝑚 𝑘∈ℤ^𝑚₀

1

|𝑘|𝑒^{2𝜋𝑖∑𝑖𝑘𝑖(𝑛𝑖𝜏1+𝜉1)}𝑒^{−2𝜋𝜏2∑𝑖|𝑘𝑖||𝑛𝑖−𝑥|}, (4.8)

whereℤ₀∶=ℤ⧵{0}and|𝑘|∶=|𝑘₁|⋯|𝑘_𝑚|, we have

∫_ℝ∕ℤ𝑃(𝜉, 𝜏)^𝑚𝑑𝜉₁= ∑

𝑛∈ℤ^𝑚 𝑘∈ℤ^𝑚₀

𝛿₀(𝑘)

|𝑘| 𝑒^{2𝜋𝑖(𝑘⋅𝑛)𝜏1}𝑒^{−2𝜋𝜏2(∑|𝑘𝑖||𝑛𝑖−𝑥|)},

where, for any𝑣∈ℤ^𝑚 and for any𝑎∈ℤ,𝛿_𝑎(𝑣) = 1if the sum of the coordinates is𝑎, and is zero otherwise. Moreover,𝑥⋅𝑦denotes the standard inner product ofℝ^𝑚. In the interval[0,1],

B₂(𝑥)^𝑟= ∑

𝑎+𝑏+𝑐=𝑟

𝑟!

𝑎!𝑏!𝑐!

(−1)^𝑏 6^𝑐 𝑥^2𝑎+𝑏. Therefore we have

𝐷_𝑙(𝜏) = 1 4^𝑙

∑

𝑎+𝑏+𝑐+𝑚=𝑙

𝑙!(2𝜋𝜏₂)^𝑙−𝑚 𝑎!𝑏!𝑐!𝑚!

(−1)^𝑏 6^𝑐

∑

𝑛∈ℤ^𝑚 𝑘∈ℤ^𝑚₀

𝛿₀(𝑘)

|𝑘| 𝑒^{2𝜋𝑖(𝑘⋅𝑛)𝜏1}

∫

1 0

𝑥^2𝑎+𝑏𝑒^{−2𝜋𝜏2(∑|𝑘𝑖||𝑛𝑖−𝑥|)}𝑑𝑥. (4.9)

To compute the last integral, let us fix𝑛₁,…, 𝑛_𝑚. Since𝑥∈ [0,1]we have

|𝑛_𝑖−𝑥|=

{ 𝑛_𝑖−𝑥 if𝑛_𝑖>0 𝑥−𝑛_𝑖 if𝑛_𝑖≤0, so∑|𝑘_𝑖||𝑛_𝑖−𝑥|=∑|𝑘_𝑖||𝑛_𝑖|+𝑥∑

sgn(−𝑛_𝑖)|𝑘_𝑖|.

By repeated integration by parts one easily finds that for any𝑐∈ℝ_>0,𝛽 ∈ℝ⧵{0}

and𝑀 ∈ℕ

∫

𝑐

0

𝑥^𝑀𝑒^−𝛽𝑥𝑑𝑥= 𝑀! 𝛽^𝑀+1 −

∑𝑀 𝑗=0

(𝑀)_𝑗

𝛽^𝑗+1𝑐^𝑀−𝑗𝑒^−𝛽𝑐, (4.10) where(𝑀)_𝑗 = 𝑀(𝑀 − 1)⋯(𝑀 −𝑗 + 1)is the descending Pochhammer symbol. In our case𝛽 = 2𝜋𝜏₂∑

sgn(−𝑛_𝑖)|𝑘_𝑖|,𝑐= 1and𝑀 = 2𝑎+𝑏. Note that∑

sgn(−𝑛_𝑖)|𝑘_𝑖|can be equal to zero. If it is not zero, since|𝑛_𝑖|+sgn(−𝑛_𝑖) =|𝑛_𝑖− 1|, by (4.10) we get, as a result of the integral,

(2𝑎+𝑏)!𝑒^{−2𝜋𝜏2∑|𝑘𝑖||𝑛𝑖|} (2𝜋𝜏₂∑

sgn(−𝑛𝑖)|𝑘_𝑖|)^2𝑎+𝑏+1 −

2𝑎+𝑏∑

𝑗=0

(2𝑎+𝑏)!𝑒^{−2𝜋𝜏2∑|𝑘𝑖||𝑛𝑖−1|} (2𝑎+𝑏−𝑗)!(2𝜋𝜏₂∑

sgn(−𝑛𝑖)|𝑘_𝑖|)^𝑗+1, while if∑

sgn(−𝑛_𝑖)|𝑘_𝑖|= 0then we just get 𝑒^{−2𝜋𝜏2∑|𝑘𝑖||𝑛𝑖|}

2𝑎+𝑏+ 1 .

In both cases, once we fix𝑛₁,…, 𝑛_𝑚 and𝑘₁,…, 𝑘_𝑚, putting everything together we are left with an expression of the kind

𝑐^(𝑝,𝑞)(𝜋𝜏₂)𝑒^{2𝜋𝑖𝑝𝜏1}𝑒^{−2𝜋𝑞𝜏2}, where 𝑝 = ∑

𝑘_𝑖𝑛_𝑖 ∈ ℤ, 𝑞 is a non-negative integer⁷ equal to either ∑|𝑘_𝑖||𝑛_𝑖− 1| or ∑

|𝑘_𝑖||𝑛_𝑖|, and𝑐^(𝑝,𝑞)(𝜋𝜏₂) is a Laurent polynomial with rational (explicitly deter-mined) coefficients whose maximum power is𝑙and minimum power is1 −𝑙.⁸

Note that𝑒^{2𝜋𝑖𝑝𝜏1}𝑒^{−2𝜋𝑞𝜏2} =𝑞^𝜇𝑞^𝜈, with𝜇 = (𝑞+𝑝)∕2and𝜈 = (𝑞−𝑝)∕2. Therefore we would like to show that𝑞 ≥ |𝑝|and that𝑝 ≡2 𝑞 (this notation is a shorthand for the standard𝑝≡𝑞mod 2), in order to have that𝜇and𝜈are non-negative integers. If𝑞 =

∑|𝑘_𝑖||𝑛_𝑖|the claim is trivial, so we have to take care only of the case𝑞 =∑|𝑘_𝑖||𝑛_𝑖−1|. Note that𝑘_𝑖𝑛_𝑖≡2|𝑘_𝑖||𝑛_𝑖|≡2|𝑘_𝑖|(1 +|𝑛_𝑖− 1|)and that∑

|𝑘_𝑖|≡2

∑𝑘_𝑖= 0, so 𝑝=∑

𝑘_𝑖𝑛_𝑖≡2

∑|𝑘_𝑖|+∑

|𝑘_𝑖||𝑛_𝑖− 1|≡2𝑞.

Moreover|𝑝|=|∑

𝑘_𝑖𝑛_𝑖|+|∑

𝑘_𝑖|≤|∑

𝑘_𝑖(𝑛_𝑖− 1)|≤∑|𝑘_𝑖||𝑛_𝑖− 1|=𝑞.

To conclude our proof we have to analyse more carefully the rational coefficients of𝑐̃^(𝜇,𝜈)(𝜋𝜏₂), which are obtained by the𝑐^(𝑝,𝑞)(𝜋𝜏₂)’s.

Any fixed𝑞 = ∑|𝑘_𝑖||𝑛_𝑖−𝜀|(𝜀= 0,1) can be obtained with just finitely many 𝑚-tuples(𝑛₁,…, 𝑛_𝑚), because|𝑛_𝑖−𝜀|≤𝑞 for any𝑖: otherwise, since for every𝑖|𝑘_𝑖|≥1, we would have∑

|𝑘_𝑖||𝑛_𝑖−𝜀|≥ ∑

|𝑛_𝑖−𝜀| ≥𝑞. This means that for any(𝜇, 𝜈), which is uniquely determined by a couple(𝑝, 𝑞), one has to consider a finite rational linear combination of sums of the kind

𝑇(𝑚, 𝑛, 𝛼) ∶=∑

𝑘

𝛿₀(𝑘)

|𝑘|(∑

sgn(−𝑛_𝑖)|𝑘_𝑖|)^𝛼, where𝑘∈ℤ^𝑚₀ are such that∑

sgn(−𝑛_𝑖)|𝑘_𝑖|≠ 0and𝛼is a positive integer. Note that the function𝑇 only depends on𝑛_𝑖∕|𝑛_𝑖|.

7This notation may generate some confusion, because𝑞also denotes e(𝜏). What we mean will always be clear from the context.

8When𝑚 = 0and𝑚 = 1one can see that there is no contribution to the negative powers, and therefore one can assume that𝑚≥2to get this lower bound.

Since we can split the sum defining𝑇 as a sum over cones such that the factors in the denominator are either bigger or smaller than zero, it follows that our coefficients are linear combinations of conical sums.

□

Specializing carefully this computation to the (𝑝, 𝑞) = (0,0) case, one gets the proof, first obtained in [48], of Proposition3.3.2:

Proof of prop.3.3.2.Note that, since𝑞 =∑|𝑘_𝑖||𝑛_𝑖−𝜀|with𝜀= 0or= 1, the only possible𝑚-tuples(𝑛₁,…, 𝑛_𝑚)which can give𝑞= 0are(0,…,0)and(1,… 1), and with them also𝑝= 0, because∑

𝑘_𝑖= 0. Using this and looking carefully⁹at the proof of the previous theorem one is then lead to the formula given in the statement, except for the hypergeometric coefficient of the leading term, which is more elegant than the term obtained with the integration process described in the proof, and can be deduced by making use of the identity

1

𝑥(𝑥+ 1)⋯(𝑥+𝑛) =

∑𝑛 𝑗=0

(−1)^𝑗 𝑗!(𝑛−𝑗)!

1 𝑥+𝑗, easily obtained by partial fraction decomposition.

□

As remarked in the previous chapter, the function𝑆(𝑚, 𝛼)was proven by Zagier in [83] to be equal to an explicit linear combination of MZVs, allowing to algorith-mically compute in terms of MZVs the non-exponentially small part of𝐷_𝑙:

Proposition 4.2.1(Zagier).

𝑆(𝑚, 𝛼) =𝑚!∑

𝑟≥0

∑

𝑎₁,…,𝑎_𝑟∈{1,2}

𝑎₁+⋯+𝑎_𝑟=𝑚+2

2^{2(𝑟+1)−𝑚−𝛼}𝜁(𝑎₁,…, 𝑎_𝑟, 𝛼+ 2). (4.11)

We will not repeat the proof here, because it is essentially contained in the more complicated proof of Theorem4.3.2, that can be found in AppendixA.

Unfortunately, knowing the Laurent polynomial𝑑_𝑙 =𝑑_𝑙^(0,0)is not enough to per-form the integration over the moduli space of complex tori, so one would like to un-derstand better the behaviour of the functions𝐷_𝑙. This can be achieved by looking at the more general Theorem4.2.1, because it allows us to predict other coefficients of the expansion (4.7).

To make an example, let us recall that the non-holomorphic Eisenstein series are defined, for𝑠∈ℂwithℜ(𝑠)>1and𝜏 ∈ℍ, by

𝐸(𝑠, 𝜏) = (𝜏₂

𝜋

)𝑠 ∑

(𝑚,𝑛)∈ℤ²⧵{(0,0)}

1

|𝑚𝜏+𝑛|^2𝑠. (4.12) Its expansion at the cusp is given, setting𝑦=𝜋𝜏₂, by

𝐸(𝑛, 𝜏) = (−1)^𝑛−1 B_2𝑛

(2𝑛)!(4𝑦)^𝑛+ 4(2𝑛− 3)!

(𝑛− 2)!(𝑛− 1)!𝜁(2𝑛− 1)(4𝑦)^1−𝑛

+ 2

(𝑛− 1)!

∑

𝑁≥1

𝑁^𝑛−1𝜎_1−2𝑛(𝑁)(𝑞^𝑁+𝑞^𝑁)

∑𝑛−1 𝑚=0

(𝑛+𝑚− 1)!

𝑚!(𝑛−𝑚− 1)!(4𝑁 𝑦)^−𝑚, (4.13)

9As explained in [48], one exploits the fact that𝐵₂(1 −𝑥) = 𝐵₂(𝑥) in order to get the nice look-ing formula in the corollary instead of the more complicated one that we would naively get just by performing the same steps as in the theorem’s proof.

whereB_𝑛 is the 𝑛-th Bernoulli number, and𝜎_𝑘(𝑁) = ∑

𝑑|𝑁𝑑^𝑘 is a finite power sum running over the positive divisors of𝑁.

As we have mentioned in the last chapter, using the series representations (3.27) for𝐷_𝑙(𝜏) and (4.12) for the Eisenstein series one can immediately see that 𝐷₂(𝜏) = 𝐸(2, 𝜏)∕16.

Let us briefly see how can we get the same result by comparing the expansion given by Theorem4.2.1and the expansion (4.13), which becomes in this case

𝐸(2, 𝜏) = 𝑦²

45+ 𝜁(3)

𝑦 + 2∑

𝑁≥1

(2𝑁+𝑦⁻¹)𝜎₋₃(𝑁)(𝑞^𝑁 +𝑞^𝑁).

Using the intermediate step (4.9) in the proof of the theorem, for 𝑚 = 0 we get (1∕16)(𝑦²∕45), which is the leading term of the non-exponential part, for 𝑚 = 1we get zero, and for𝑚= 2(so𝑎=𝑏=𝑐= 0) we get

1 16

∑

𝑘∈ℤ⧵{0}

𝑛∈ℤ

𝑒^{2𝜋𝑖𝑝𝜏1}𝑒^{−2𝜋𝜏2(|𝑘|(|𝑛1|+|𝑛2|))}

|𝑘|² ∫

1 0

𝑒^{−2𝜋𝜏2𝑥|𝑘|(sgn(−𝑛1)+sgn(−𝑛2))}𝑑𝑥,

where we denote𝑝 = 𝑘(𝑛₁ −𝑛₂). When 𝑝 = 0, i.e. when 𝑞 and𝑞 have the same power in the expansion given by the theorem, then𝑛₁ = 𝑛₂, which implies that the argument of the exponential in the integral is never zero. Therefore, splitting the sum into the𝑛 ≥ 1 part and the𝑛 ≤ 0 part, one gets two telescoping sums, both giving as a result

∑

𝑘∈ℤ⧵{0}

1 64|𝑘|³𝑦.

We conclude that the variable𝜏₁does not appear only in the non-exponentially small part of𝐷₂(𝜏), which is the Laurent polynomial𝑑₂(𝑦) = (𝑦²∕45 +𝜁(3)∕𝑦)∕16. This fits with the expansion of𝐸(2, 𝜏).

Moreover, if𝑝 > 0one gets again telescoping sums in𝑛, but now we sum only over finitely many𝑘, which are the divisors of𝑝. We leave as an exercise to the reader to verify that one gets exactly the same expansion as we get for the non-holomorphic Eisenstein series.

In general it is too messy to repeat the same game as above and explicitly get the full expansion for other𝐷_𝑙’s, except maybe for𝐷₃(𝜏), which is already known, as we have mentioned in eq. (3.31), to be equal to(𝐸(3, 𝜏) +𝜁(3))∕64. However, it is in principle possible to algorithmically get the coefficient of𝑞^𝜇𝑞^𝜈for any fixed(𝜇, 𝜈), which would allow to check conjectures on the full expansion or to numerically ap-proximate the functions very precisely (the sum (4.7) converges much faster than the sum (3.27)).

Im Dokument Elliptic multiple zeta values, modular graph functions and genus 1 superstring scattering amplitudes (Seite 59-63)