conical sum, we can rewrite the matrix as
⎛⎜
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎝ 1
⋮ ⋱
1 ⋯ 1
1 ⋯ 1 0
⋮ ⋮ 𝐵 ⋮
1 ⋯ 1 0
1 ⋯ 1 1 ⋯ 1 1
1 ⋯ 1 0 ⋯ 0 1
0 ⋯ 0 0
⋮ ⋮ 𝐶 ⋮
0 ⋯ 0 0
⎞⎟
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎠
where𝐵 and𝐶 are matrices with, from left to right, a string of ones followed by a string of zeros in every row, such that the length of the string of ones increases in 𝐵with the increase of the row’s index and decreases in 𝐶. At this point we almost have a matrix belonging to, the only problem being the row in the middle of the form𝑟= 1,…,1,0,…,0,1.
Note now that a partial fraction operation on the sum of the kind 1
𝑙𝑖(𝑥)𝑙𝑗(𝑥) = 1
𝑙𝑖(𝑥) (𝑙𝑖+𝑙𝑗)(𝑥) + 1 𝑙𝑗(𝑥) (𝑙𝑖+𝑙𝑗)(𝑥)
is reflected on the matrix just by substituting the𝑖-th or the𝑗-th row by the sum of the 2. Hence if we do this sum operation on𝑟together with the row immediately below we get the sum of 2 matrices, one belonging to (when𝑟is deleted) and one such that the sub-matrix below𝑟is strictly smaller (after interchanging 𝑟with the new row obtained as a sum). Iterating this process one finally gets that𝑟is the last row in the matrix, and in this case the matrix belongs toand we are done.
□
where the notation employed for the exponential or the factorial of a vector was introduced in the proof of Theorem4.3.1. The integration over the real part of the 𝜉𝑖’s gives (recall the definition of the incidence matrix(Γ𝑖,𝛽)from Section3.3.4)
∑
𝑘1,…,𝑘𝑙∈ℤ⧵{0}
𝑛1,…,𝑛𝑙∈ℤ
∏𝑁 𝑖=1
𝛿0 (∑𝑙
𝛽=1
Γ𝑖,𝛽𝑘𝛽 )∏𝑙
𝛼=1
exp(2𝜋𝑖𝜏1𝑘𝛼𝑛𝛼)
|𝑘𝛼| ×
×∫(ℝ∕ℤ)𝑁−1
∏
1≤𝑖<𝑗≤𝑁
B2(𝑥𝑖−𝑥𝑗)𝑟𝑖,𝑗exp (
− 2𝜋𝜏2(𝑡𝑖,𝑗∑+𝑚𝑖,𝑗
𝛼=𝑡𝑖,𝑗
|𝑘𝛼||𝑛𝛼−𝑥𝑖+𝑥𝑗|))
𝑑𝑥1⋯𝑑𝑥𝑁−1, where𝑡𝑖,𝑗 denote certain integers in {1,…, 𝑙}, explicitly determined by an explicit choice of labelling variables when we sum all𝑙𝑖,𝑗’s propagators to obtain𝑙 propaga-tors. Let us call𝐼the integral appearing in the last step. We have
𝐼= ∑
𝑎+𝑏+𝑐=𝑟
𝑟!
𝑎!𝑏!𝑐!
(−1)𝑏 6𝑐
∏
1≤𝑖<𝑗≤𝑁
exp (
− 2𝜋𝜏2(𝑡𝑖,𝑗∑+𝑚𝑖,𝑗
𝛼=𝑡𝑖,𝑗
|𝑘𝛼||𝑛𝛼−𝑥𝑖+𝑥𝑗|))
×
×( ∑
1≤𝑖<𝑗≤𝑁∫𝑃𝑖,𝑗
) ∏
1≤𝑖<𝑗≤𝑁
|𝑥𝑖−𝑥𝑗|𝑀𝑖,𝑗𝑒−𝛾1𝑥1⋯𝑒−𝛾𝑁−1𝑥𝑁−1𝑑𝑥1⋯𝑑𝑥𝑁−1, (4.16) where𝑃𝑖,𝑗 is the path0 ≤ 𝜎𝑖,𝑗(𝑥1) ≤ ⋯ ≤𝜎𝑖,𝑗(𝑥𝑁−1) ≤1, for𝜎𝑖,𝑗 a permutation of the 𝑁− 1variables𝑥1,…, 𝑥𝑁−1,𝑀𝑖,𝑗 ∶= 2𝑎𝑖,𝑗+𝑏𝑖,𝑗 and the𝛾𝑖’s will depend on the path chosen.
Let us consider the path associated with the identity of the symmetric group 𝑆𝑁−1, i.e0≤𝑥1≤⋯≤𝑥𝑁−1. In this case, for every𝑖, we have
𝛾𝑖= 2𝜋𝜏2𝛿 (∑𝑙
𝛽=1
Γ𝑖,𝛽sgn(−𝑛𝛽)|𝑘𝛽|) .
The integral on this path reduces to a linear combination of iterated integrals of the kind
∫[0,1]
𝑥𝑄𝑁𝑁−1−1𝑒−𝛾𝑁−1𝑥𝑁−1𝑑𝑥𝑁−1⋯𝑥𝑄11𝑒−𝛾1𝑥1𝑑𝑥1,
where the𝑄𝑖’s are non negative integers. One can solve the integral by repeatedly using integration by parts, and all the possible exponentials involved in the result are 𝑒−𝛾𝑁−1, 𝑒−(𝛾𝑁−2+𝛾𝑁−1),…, 𝑒−(𝛾1+⋯+𝛾𝑁−1). Multiplying them by the exponential in front of the integral in formula (4.16) tells us what are all the possible integers𝑞in the terms of the kind𝑒−2𝜋𝑞𝜏2.
It is not difficult to see that the argument used in the two-point case works for all of these𝑞’s, and that nothing new happens to the Laurent polynomials involved and to their coefficients, which are therefore expressible as conical sums.
□
We do not write down here an explicit formula for the Laurent polynomial part of the functions𝐷Γ(𝜏), because it gets really complicated already in the four-point case and does not really allow one to work with it. Indeed, the same method explained in the previous section to explicitly write down the conical sums as integrals produces, already in the four-point case, matrices with coefficients strictly bigger than 1, whose computation in terms of special values of polylogarithms goes beyond the current limits ofHyperInt.
Chapter 5
Elliptic multiple zeta values
Let us fix the notation for this chapter, which is the same as that of Chapter2. First of all, we will come back to denoting tuples byx= (𝑥1,…, 𝑥𝑛). We will consider𝜏 ∈ℍ, 𝑞=e(𝜏)1and𝜏 =ℂ∕(𝜏ℤ+ℤ). Moreover, for𝜉∈𝜏, we will write𝑟𝜏(𝜉) ∶=ℑ(𝜉)∕ℑ(𝜏).
Recall also from Chapter2the Kronecker function 𝐹(𝜉, 𝛼, 𝜏) ∶= 𝜃′(0, 𝜏)𝜃(𝜉+𝛼, 𝜏)
𝜃(𝜉, 𝜏)𝜃(𝛼, 𝜏) =∑
𝑛≥0
𝑓𝑛(𝜉, 𝜏)(2𝜋𝑖𝛼)𝑛−1 (5.1) where𝛼is a formal variable, and the modified real analytic
Ω(𝜉, 𝛼, 𝜏) ∶=e(𝑟𝜏(𝜉)𝛼)𝐹(𝜉, 𝛼, 𝜏) =∑
𝑛≥0
𝜔𝑛(𝜉, 𝜏)(2𝜋𝑖𝛼)𝑛. (5.2) We have already seen in Chapter2that
𝑓𝑛(𝜉, 𝜏) =
⎧⎪
⎨⎪
⎩
2𝜋𝑖 ∶ 𝑛= 0
𝜋cot(𝜋𝜉) − 2𝜋𝑖∑
𝑚≥1
(e(𝑚𝜉) −e(−𝑚𝜉)) ∑
𝑝≥1𝑞𝑚𝑝 ∶ 𝑛= 1
2𝜋𝑖 (𝑛−1)!
(B𝑛
𝑛 −∑
𝑚≥1
(e(𝑚𝜉) + (−1)𝑛e(−𝑚𝜉)) ∑
𝑝≥1𝑝𝑛−1𝑞𝑚𝑝)
∶ 𝑛≥2,
(5.3)
and by definition𝜔𝑛(𝜉, 𝜏) =∑𝑛 𝑘=0
𝑟𝜏(𝜉)𝑘
𝑘! 𝑓𝑛−𝑘(𝜉, 𝜏). Finally, recall the KZB form (2.47):
𝜔𝐾𝑍𝐵(𝑥0, 𝑥1;𝜉, 𝜏) = −𝜈(𝜉)𝑥0+(
ad𝑥0Ω(𝜉,ad𝑥0, 𝜏)𝑑𝜉)
(𝑥1) (5.4)
= −𝜈(𝜉)𝑥0+∑
𝑛≥0
(𝜔𝑛(𝜉)𝑑𝜉)
(2𝜋𝑖)𝑛ad𝑛𝑥
0(𝑥1), (5.5) where𝜈(𝜉) ∶=𝑑𝑟𝜏(𝜉), ad𝑥0(⋅) = [𝑥0,⋅]and
ad𝑛𝑥
0(𝑥1) = [𝑥0,[𝑥0,[⋯[𝑥0
⏟⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏟
𝑛
, 𝑥1]]⋯].
Elliptic multiple zeta values were defined about five years ago by Enriquez in the context of his work on elliptic associators (see [30], [43], [44]). Immediately after-wards, some physicists realized that these functions naturally appear as coefficients of genus 1 open superstring amplitudes [14].
We have mentioned in Chapter2that multiple zeta values can be seen as the coef-ficients of the Drinfel’d associator, a power series in two non-commutative variables which describes the regularized monodromies of the KZ differential equation (2.14).
Enriquez’s elliptic analogue of the Drinfel’d associator is given by a pair of power series(𝐴(𝑥0, 𝑥1;𝜏), 𝐵(𝑥0, 𝑥1;𝜏)) in two non-commutative variables 𝑥0, 𝑥1 describing
1Recall thate(𝑥) ∶= exp(2𝜋𝑖𝑥).
the regularized monodromies of an elliptic analogue of the KZ equation related to the KZB form (5.4), called theKZB differential equation2, that we will not discuss here.
The upshot is that (a slightly modified version3 of) the elliptic associator can be de-fined as
𝐴(𝑥0, 𝑥1, 𝜏) = lim
𝜖→0(−2𝜋𝑖𝜖)ad𝑥0(𝑥1)exp [
∫
1−𝜖 𝜖
𝜔𝐾𝑍𝐵(𝑥0, 𝑥1;𝜉, 𝜏) ]
(−2𝜋𝑖𝜖)−ad𝑥0(𝑥1)
, (5.6)
𝐵(𝑥0, 𝑥1, 𝜏) = lim
𝜖→0(−2𝜋𝑖𝜖𝜏)ad𝑥0(𝑥1)
exp [
∫
(1−𝜖)𝜏 𝜖𝜏
𝜔𝐾𝑍𝐵(𝑥0, 𝑥1;𝜉, 𝜏) ]
(−2𝜋𝑖𝜖𝜏)−ad𝑥0(𝑥1)
. (5.7) One can show (see [65], [52]) that these two limits exist and are finite. In order to define elliptic MZVs, Enriquez4considered the following modified formal series:
𝐴𝐸𝑛𝑟(𝑥0, 𝑥1, 𝜏) = lim
𝜖→0(−2𝜋𝑖𝜖)ad𝑥0(𝑥1)
exp [
∫
1−𝜖 𝜖
ad𝑥0Ω(𝜉,ad𝑥0, 𝜏)(𝑥1)𝑑𝜉 ]
(−2𝜋𝑖𝜖)−ad𝑥0(𝑥1)
, (5.8) 𝐵𝐸𝑛𝑟(𝑥0, 𝑥1, 𝜏) =
= lim
𝜖→0(−2𝜋𝑖𝜖)ad𝑥0(𝑥1)
exp [
∫
(1−𝜖)𝜏 𝜖𝜏
ad𝑥0Ω(𝜉,ad𝑥0, 𝜏)(𝑥1)𝑑𝜉 ]
(−2𝜋𝑖𝜖)−ad𝑥0(𝑥1)
. (5.9) Once again, these two limits exist and are finite. In analogy with the genus zero case,𝐴𝐸𝑛𝑟(𝑥0, 𝑥1, 𝜏)and𝐵𝐸𝑛𝑟(𝑥0, 𝑥1, 𝜏)can be considered as the generating series of two families of functions on the complex upper-half plane: Enriquez called them elliptic analogues of multiple zeta values.
It is important to remark, as we already mentioned in the introduction, that En-riquez defined them as the coefficients (with respect to ad𝑛𝑥1
0(𝑥1)⋯ad𝑛𝑥𝑟
0(𝑥1)) of the modified pair(𝐴𝐸𝑛𝑟(𝑥0, 𝑥1;𝜏), 𝐵𝐸𝑛𝑟(𝑥0, 𝑥1;𝜏)), while for instance Matthes considered the coefficients (with respect to monomials in the non-commutative variables 𝑥0 and𝑥1) of the elliptic associator (𝐴(𝑥0, 𝑥1;𝜏), 𝐵(𝑥0, 𝑥1;𝜏)) [65]. This second choice is in some sense more natural, because by Theorem 2.3.1 (extended to tangential base points) it giveshomotopy invariantiterated integrals on the paths[0,1]and[0, 𝜏].
This happens to be the case also for the coefficients of (5.8), because 𝜈(𝜉) ≡ 0 on the straight path[0,1], but it is not the case for the coefficients of (5.9). However, since on the straight path[0, 𝜏]we have𝜈(𝜉) =𝑑𝜉∕𝜏, Matthes’s elliptic MZVs can be expressed as certain (homotopy invariant)ℚ[𝜏±1]-linear combinations of Enriquez’s elliptic MZVs.
In this chapter we will not use most of what we have just said, and try to give a self-contained analytic theory of elliptic MZVs, without referring to the elliptic asso-ciator. In particular, we will deduce new explicit results on the asymptotic expansion and the modular behaviour of B-elliptic MZVs. At the end of the chapter, we will compare elliptic MZVs with special values of multi-valued and single-valued elliptic polylogarithms.
2KZB is the acronym of Knizhnik-Zamolodchikov-Bernard.
3See [65].
4Enriquez presented these generating series in a different way [44]. Here we prefer to follow [65], Definition 3.4.1, but we warn the reader of a typo in formula (3.48) therein, which is corrected in our formula (5.9).