Periods

3.4.3. Single-valued polylogarithms

The monodromies of hyperlogarithmsL_w(z) whenzencircles a singularityσ ∈Σ can be removed through suitable combinations with hyperlogarithms of the complex conjugate z^∗. The Bloch-Wigner dilogarithm D₂ from (2.1.40) is a very important example, and Francis Brown studied the full subalgebra of L({0,1})(z)⊗L({0,1})(¯z)⊗ Z charac-terized by this property of single-valuedness on C\ {0,1} when ¯z = z^∗ are conjugate [47]. They have been called single-valued multiple polylogarithms (SVMP(L)) and also single-valued harmonic polylogarithms (SVHPL).

Such functions occur naturally in certain Feynman integrals and provide a very efficient tool for practical computations [82, 149] (for example, they are a key ingredient to the proof of the zigzag conjecture [61]). Their special values at one are well understood [56].

But Feynman integrals demand more general functions, as was realized for the first time in [65] and later for example also in [81]. In both cases, the differential form d log(z−z) had to be adjoined to form more general (but still single-valued) integrals.¯ Our calculation of graphical functions (together with Oliver Schnetz) revealed even more general (single-valued) iterated integrals, involving also the forms d log(zz¯−z −z),¯ d log(1−zz) and d log(1¯ −z−z). This stimulated a wide extension of the concept of¯ single-valued hyperlogarithms, which is currently actively developed by Oliver Schnetz.

Some examples of our results can be found in [136], and we briefly comment on them also in section 5.4.

Unfortunately, a basis overQis not known in any case of interest; for example it is still not ruled out thatZ =Q[π²]: all multiple zeta values could be polynomials inπ² [173].

On the other hand, the development of motivic periods recently blossomed into several complete characterizations of the algebra of these analogues of our actual periods, for particular choices of Σ. Bases of motivic periods map to generating sets of the associated actual periods, and the main conjectures of the theory of periods imply that indeed they should stay linearly independent and form a basis overQ.

We cannot go into detail on this fascinating and very active subject, but only collect results and references which are particularly important for our applications.

3.5.1. Multiple zeta values and alternating sums

Definition 3.5.1. Letµ_N :=ξ:ξ^N = 1 denote the N’th roots of unity and set Z^(N):= lin_Q{Lin1,...,nr(ξ1, . . . , ξr): everyni∈N,ξi∈µN and (nr, ξr)6= (1,1)}

(3.5.1) to the rational linear combinations of multiple polylogarithms evaluated at such roots.⁷ It is filtered by the weightn₁+· · ·+nr and the depth r.

Equivalently we can characterize this space as the algebra of special values that hy-perlogarithms over the alphabet{0} ∪µ_N take at one: From (3.4.5),

Regz→1L({0} ∪µN)(z) =_(3.3.22)^{Z 1}

0 reg¹₀T({0} ∪µN) =Z^(N) (3.5.2) where the integration path is the straight line from zero to one.

Remark 3.5.2. More generally, therelative periodsofC\Σ are defined as all convergent integrals along smooth pathsγ: (0,1)−→C\Σ withγ(0), γ(1)∈Σ:

P(C\Σ) :=^X

γ

Z

γ

w: w∈T(Σ) neither starts with ω_γ(1) nor ends inω_γ(0)

. (3.5.3) From (3.3.25) and (3.3.22) it follows that for roots of unity Σ ={0} ∪µ_N, the only new period we can get isiπ,P(C\Σ) =Z^(N)[iπ].

The case of multiple zeta valuesZ :=Z⁽¹⁾ was studied and finally understood motivi-cally by Francis Brown [53]. With his results, the conjectures on periods would imply the existence of an isomorphismZ ∼=Q[π²,Lyn({3,5,7, . . .})] of weight-graded algebras.

A concrete result settles a conjecture of Hoffman and provides a small set of generators.

Theorem 3.5.3. Multiple zeta values are generated by the Hoffman elements

Z= lin_Qⁿζ_n1,...,nr: all n₁, . . . , nr∈ {2,3}^o. (3.5.4)

7The condition (nr, ξr)6= (1,1) is equivalent to the convergence of the series (3.4.3).

For our automated computations it is important to make such a statement effective, which means that we need an efficient method to determine explicitly a reduction of any multiple zeta value to this (or any other) conjectural basis. This is achieved by a coproduct-based algorithm [52] which is available as a program [148].

But the generators of Hoffman are not optimal in that they have very high depth for a given weight, and we like to express results with shortest depth possible. A conjecture of Broadhurst and Kreimer [45] on the depth filtration is still open even motivically [54].

For our practical purposes though, we only consider small weights (so far we did not exceed weight 11 in any of our computations) and can therefore harvest the data mine [27], which provides proven reductions to a depth-minimal set of generators.

The data mine also covers alternating sums Z⁽²⁾. For them we know [75]

Theorem 3.5.4. Every alternating sum is a rational linear combination of products of π^2p andLin1,...,nr(1, . . . ,1,−1)for Lyndon words with odd indicesn_i∈O:={1,3,5, . . .}, with the same weight and at most the same depth. In particular we can write

Z⁽²⁾ =Q^hπ²,Lin(1, . . . ,1,−1):n= (n1, . . . , nr)∈Lyn(O)ⁱ. (3.5.5) Conjecturally, there are no further relations and Z⁽²⁾ ∼=Q[π²]⊗T(O) is an isomor-phism of weight-graded and depth-filtered algebras. Note that (3.5.5) and (3.4.5) show that each alternating sum is a hyperlogarithm over {0,1}atz=−1:

Z⁽²⁾= Reg

z→−1L({0,1})(z) =^{Z −1}

0 T({0,1})ω1.

We encounter alternating sums in many Feynman integral computations, but in the final answers for massless integrals these always combined to just multiple zeta valuesZ. We give an example of this phenomenon in section 5.1.2.

3.5.2. Primitive sixth roots of unity

For one period we computed (see section 5.1.3), the spaceZ⁽²⁾ was not enough and we needed sixth roots of unity. Far less data is available on these sums than in the previous cases and no table of reductions to a conjectural basis exists for high weights. A detailed analysis up to weight four was performed by David Broadhurst in [41], who observed that Feynman integrals tend to lie in very special subspaces of Z⁽⁶⁾. One of them is by now well understood due to Deligne [75]. Let ξ₆ :=e^iπ/3 denote a primitive sixth root of unity andξ₆^∗ =ξ₆⁻¹ = 1−ξ₆ its conjugate. We quote

Theorem 3.5.5. DefineZ_D⁽⁶⁾ as the subalgebra ofZ⁽⁶⁾ generated by Lin1,...,nr(z1, . . . , zr) with z₁, . . . , zr ∈ {1, ξ6, ξ₆^∗} such that all products {z_k· · ·zr: 1≤k≤r} are contained either in {1, ξ₆} or in {1, ξ₆^∗}, and(nr, z_r)6= (1,1).

Then each element of Z_D⁽⁶⁾ is a rational linear combination of products of iπ and Lin1,...,nr(1, . . . ,1, ξ6) for Lyndon words (with n₁, . . . , nr > 1), with at most the same total weight and depth:

Z_D⁽⁶⁾=Q[(iπ),Lin(1, . . . ,1, ξ₆):n= (n₁, . . . , n_r)∈Lyn(N\ {1})]. (3.5.6)

Again the main conjectures imply an isomorphismZ_D⁽⁶⁾∼=Q[iπ]⊗T(N\ {1}), respect-ing weight and depth. As for alternatrespect-ing sums, this is an algebra of special values of hyperlogarithms (integrating along the straight line from 0 toξ₆):

Z_D⁽⁶⁾=L({0,1})(ξ6) =Q[iπ]^{Z ξ6}

0 T({0,1})ω1. (3.5.7) Through its definition, Z_D⁽⁶⁾ ⊇ Z contains the multiple zeta values and is closed under complex conjugation. For our application we needed to determine the real and imaginary parts ofZ_D⁽⁶⁾ separately. In depth one we know [120, chapter VII, section 5.3]

Lin

e^2πix+ (−1)ⁿLin

e^−2πix=−(2πi)ⁿ

n! Bn(x) (3.5.8)

in terms of the rational Bernoulli polynomials Bn(x), so any power of iπ has depth one and Re(Li2n(ξ6)) and iIm(Li2n+1(ξ6)) lie in Q[iπ]. On the other hand, the comple-mentary Im(Li2n(ξ6)) and Re(Li2n+1(ξ6)) are expected to be transcendental constants independent ofπ. Indeed, already Lewin noticed that [120, chapter VII, section 3.3]

ReLi2n+1(ξ₆)= 1 2

1−2⁻²ⁿ 1−3⁻²ⁿζ_2n+1. (3.5.9) We generalize this parity result to all multiple polylogarithms in

Proposition 3.5.6. Abbreviate Li~n(ξ6) := Lin1,...,nr(1, . . . ,1, ξ6) and write |~n|:= n₁+

· · ·+n_r for its weight. Then Deligne’s subalgebra coincides with

Z_D⁽⁶⁾ =Q^h(iπ), i^r+|~n|Rei^r+|~n|Li~n(ξ6):~n= (n1, . . . , nr)∈Lyn (N\ {1})ⁱ (3.5.10) and every Li~n(ξ6) has a representation as a polynomial in these generators with less or equal weight and depth.⁸

These generators Re (Li~n(ξ6)) (r+|~n|even) and iIm (Li~n(ξ6)) (r+|~n| odd) have the benefit that their products split into generators (conjecturally bases) of the subspaces Z_D⁽⁶⁾= ReZ_D⁽⁶⁾⊕iImZ_D⁽⁶⁾.

For the proof we need the well-known parity theorem on multiple zeta values [100, 167].

Theorem 3.5.7. Any multiple zeta value ζ_n1,...,nr with r+n odd is a rational linear combination of MZV of smaller depth and products of MZV of smaller weight.

If we write W_N^dZ := lin_Qⁿζ_n1,...,nr: |~n| ≤N andr ≤d^o ⊂ Z for the subspace of MZV with weight≤N and depth ≤d, this theorem saysW_N^dZ ⊆ W_N^d−1Z+ (W_N−1^d Z)²

8So in particular, Re(Li~n(ξ6)) is expressible in terms of words with lower depth than~n and products of lower weight, if~nhas weight and depth of different parity. The analogue holds for the imaginary parts, see also (3.5.13).

for d+n odd. In fact, the statement is more precise and every time we write just (W_N−1^d Z)² we actually mean the more refined combined weight-depth filtration

X

N⁰+N⁰⁰≤N N⁰,N⁰⁰<N

X

d⁰+d⁰⁰≤d

W_N^d00Z·W_N^d0000Z.

The product terms that occur in our proofs manifestly obey this strong form.

Corollary 3.5.8. Let w=ωⁿ₀^r−1ω₁· · ·ω₀ⁿ¹⁻¹ω₁ with weight N =n₁+· · ·+n_r. Then

z→∞Reg Z z

0w∈ W_N^r−1Z+W_N−^r ₁Z[iπ]². (3.5.11) Proof. From lemma 3.3.17 we know ^R₀^z = ^R₁^z?^R₀¹reg¹ where ^R₁^zω₁ = log(1−z) maps intoQ[iπ] under Regz→∞. So up to products, we can replace^R₁^z with ^R₁^zreg₁ and find

z→∞Reg Z _z

0w≡ Reg

z→∞

Z _z

1reg₁? Z ₁

0reg¹(w)≡ Reg

z→∞

Z _z

1reg₁(w)+^{Z 1}

0reg¹(w) mod W_N−1^r Z[iπ]². We apply the inversion f(z) = z⁻¹ to the first summand, which maps w to Φf(w) = (−ω₀)^nr−1(ω1−ω₀)· · ·(−ω₀)ⁿ¹⁻¹(ω1−ω₀) = (−1)^r+Nw+R, where all words inR have depth (number of lettersω₁) less thanr. Therefore

z→∞Reg Z z

0w≡(−1)^{r+NZ 0}

1 reg⁰₁(w) +^{Z 1}

0 reg¹(w) modW :=W_N^r−1Z+W_N−^r ₁Z[iπ]². If r+n is odd, theorem 3.5.7 applies to both summands and we are done. Otherwise, we use Reg_z→0^R₀^z =^R₁⁰reg⁰₁?^R₀¹reg¹≡^R₁⁰reg⁰₁+^R₀¹reg¹ mod W to conclude

z→∞Reg Z z

0 w≡Reg

z→0

Z _z

1reg₁? Z 1

0reg¹(w)≡Reg

z→0

Z z 0 w ≡

3.3.140 mod W.

This says that a regularized limit at infinity Regz→∞L_w(z) of a multiple polyloga-rithm,w∈T({0,1}), is always reducible into products of MZV and MZV of lower depth.

Note that our proof applies also to the alphabet {−1,0}, with the only change that we split the integration at−1 instead of 1. In this case we can dispose of theiπin (3.5.11), as we understand the limitz→ ∞ to the right:⁹

z→∞Reg Z z

0ω₀^nr−1ω₋₁· · ·ωⁿ₀¹⁻¹ω₋₁ ∈ W_N^r−1Z+ W_N^r₋₁Z² whereN =n₁+· · ·+nr. (3.5.12) Proof of proposition 3.5.6. Consider any Li~n(ξ6) = (−1)^rR₀^ξ6wwherew=ω₀^nr−1ω₁· · ·ω₀ⁿ¹⁻¹ω₁ of weight |~n| and let again W := W_|~^r_n|Z_D⁽⁶⁾+W_|~^r_n|−1Z_D⁽⁶⁾² denote the subspace of el-ements with smaller depth and products of elel-ements with lower weight. We apply the inversionf(z) =z⁻¹ to the complex conjugate and then split with lemma 3.3.17 at zero:

(−1)^rLi~n(ξ^∗₆) =^{Z 1/ξ6}

0 w=^{Z ξ6}

∞ Φf(w) =^X

(w)

Z ξ6

0 Φf

w₍₁₎Reg

z→0

Z z

∞Φf

w₍₂₎.

9Note that Reg_z→∞Lω₀ω₋₁(z) =ζ2 is not considered a counterexample here, becauseζ2 =π²/6 is a product (even thoughπ /∈ Z, conjecturally).

Lemma 3.3.17 applies to the second factor,¹⁰ so Li~n(ξ^∗₆)≡(−1)^{rZ ξ6}

0 Φf(w)≡(−1)^r(−1)^{r+|~n|Z ξ6}

0 w≡(−1)^r+|~n|Li~n(ξ6) modW since Φf(w)≡(−1)^r+|~n| up to words of depth < r. Therefore, depending on the parity ofr+|~n|either the real- or imaginary parts are reducible:

W 3Li~n(ξ₆)−(−1)^r+|~n|Li~n(ξ₆^∗) =

(2iIm Li~n(ξ6) when r+|~n|is even and

2 Re Li~n(ξ₆) when r+|~n|is odd. (3.5.13) The claim follows by induction over the weight and depth.

Im Dokument Feynman integrals and hyperlogarithms (Seite 111-116)