2.1 Periods of cohomology with coefficients

Let $k\subset \mathbb {C}$ be a field, and let $\Sigma =\{\sigma _0,\ldots , \sigma _n\}$ be distinct points in k with $\sigma _0=0$ . Write

$$ \begin{align*}X_{\Sigma}= \mathbb{A}_k^1 \backslash \Sigma .\end{align*} $$

We consider a tuple $\underline {s}=(s_0,\ldots ,s_n)$ of complex numbers that we shall often assume to be generic, meaning that we have

(13) $$ \begin{align} \{s_0, s_1, \ldots, s_n, s_0+s_1+\cdots+s_n\} \cap \mathbb{Z} = \varnothing . \end{align} $$

An alternative point of view, that we do not develop here, would be to treat the $s_i$ as formal variables (see §3.5).

2.1.1 Algebraic de Rham cohomology

For any subfield $k\subset \mathbb {C}$ , denote by

$$ \begin{align*}k^{\mathrm{dR}}_{\underline{s}} = k(s_0,\ldots, s_n).\end{align*} $$

The algebraic de Rham cohomology groups that we will consider are $k^{\mathrm {dR}}_{\underline {s}}$ -vector spaces with a natural $\mathbb Q^{\mathrm {dR}}_{\underline {s}}$ -structure. Define the following logarithmic 1-forms on $\mathbb P^1_k$ :

(14) $$ \begin{align} \omega_i= \frac{dx}{x- \sigma_i} \qquad \mbox{ for } \quad i = 0,\ldots, n, \end{align} $$

which have residue $0$ or $1$ at points of $\Sigma $ , and $-1$ at $\infty $ . They form a basis of the space of global logarithmic forms $\Gamma (\mathbb P^1_k, \Omega ^1_{\mathbb P^1_k} ( \log \Sigma \cup \{\infty \}))$ , which maps isomorphically to $H_{\mathrm {dR}}^1(X_{\Sigma }/k) $ .

Definition 2.1. Let $\mathcal {O}_{X_\Sigma }$ denote the trivial rank-one bundle on $X_{\Sigma } \times _{k} k_{\underline {s}}^{\mathrm {dR}}$ , and consider the following logarithmic connection upon it:

$$ \begin{align*}\nabla_{\underline{s}} : \mathcal{O}_{X_\Sigma} \longrightarrow \Omega^1_{X_{\Sigma}} \qquad \mbox{ given by } \qquad \nabla_{\underline{s}} = d + \sum_{i=0}^n s_i\, \omega_i .\end{align*} $$

It is automatically integrable since $X_{\Sigma }$ has dimension one, and is in fact closely related to the abelianization of the canonical connection on the de Rham unipotent fundamental group of $X_\Sigma $ . Consider the algebraic de Rham cohomology groups

$$ \begin{align*}H^r_{\mathrm{dR}}(X_{\Sigma}, \nabla_{\underline{s}}) = H^r_{\mathrm{dR}}(X_{\Sigma}, (\mathcal{O}_{X_{\Sigma}}, \nabla_{\underline{s}})),\end{align*} $$

which are finite-dimensional $k_{\underline {s}}^{\mathrm {dR}}$ -vector spaces. The fact that the $s_i$ are generic implies, by [Reference DeligneD1, Prop. II.3.13], that one has a logarithmic comparison theorem for $(\mathcal {O}_{X_\Sigma },\nabla _{\underline {s}})$ . Since the cohomology of $X_\Sigma $ is spanned by global logarithmic forms, this implies (see [Reference Esnault, Schechtman and ViehwegESV]) that the cohomology groups $H^r_{\mathrm {dR}}(X_\Sigma ,\nabla _{\underline {s}})$ are computed by the complex of global logarithmic forms

$$ \begin{align*}0\longrightarrow k_{\underline{s}}^{\mathrm{dR}}\stackrel{\nabla_{\underline{s}}}{\longrightarrow} k_{\underline{s}}^{\mathrm{dR}}\,\omega_0\oplus \cdots \oplus k_{\underline{s}}^{\mathrm{dR}}\,\omega_n \longrightarrow 0,\end{align*} $$

where $\nabla _{\underline {s}}(1)=\sum _{i=0}^ns_i\omega _i$ . Again, by genericity of the $s_i$ , $H^r_{\mathrm {dR}}(X_{\Sigma }, \nabla _{\underline {s}})$ vanishes for $r\neq 1$ , and $H^1_{\mathrm {dR}}(X_{\Sigma }, \nabla _{\underline {s}})$ has dimension n. It is generated by the $\omega _i$ subject to the single relation

(15) $$ \begin{align}\sum_{i=0}^n s_i \, \omega_i = 0. \end{align} $$

Since the forms $\omega _i$ have rational residues, they in fact define a natural $\mathbb Q^{\mathrm {dR}}_{\underline {s}}$ -structure on $H^1_{\mathrm {dR}}(X_{\Sigma }, \nabla _{\underline {s}})$ which we shall denote by $H^1_{\varpi }(X_{\Sigma }, \nabla _{\underline {s}})$ . We therefore have

(16) $$ \begin{align} H^1_{\varpi}(X_{\Sigma}, \nabla_{\underline{s}}) \cong \left(\bigoplus_{i=0}^n \mathbb Q^{\mathrm{dR}}_{\underline{s}} \omega_i\right) / \ \, \mathbb Q^{\mathrm{dR}}_{\underline{s}}\,\sum_{i=0}^n s_i\omega_i. \end{align} $$

We shall use the following basis for (16):

(17) $$ \begin{align} \{ - s_i \omega_i, \mbox{ for } i= 1,\ldots, n\}. \end{align} $$

2.1.2 Betti (co)homology

We introduce the subfield of $\mathbb {C}$ defined by

$$ \begin{align*}\mathbb Q_{\underline{s}}^{\mathrm{B}}=\mathbb Q(e^{2\pi is_0},\ldots,e^{2\pi is_n}).\end{align*} $$

Definition 2.2. Let $\mathcal {L}_{\underline {s}}$ denote the rank-one local system of $\mathbb Q^{\mathrm {B}}_{\underline {s}}$ -vector spaces on the complex points $X_{\Sigma }(\mathbb {C})= \mathbb {C} \backslash \Sigma $ defined as

$$ \begin{align*}\mathcal{L}_{\underline{s}} = \mathbb Q^{\mathrm{B}}_{\underline{s}}\; x^{-s_0} \prod_{k=1}^n (1-x \sigma_k^{-1})^{-s_k} .\end{align*} $$

The local system $\mathcal {L}_{\underline {s}}$ has monodromy $e^{-2\pi is_k}$ around the point $\sigma _k$ . After extending scalars to $\mathbb {C}$ , it is identified with the horizontal sections of the (analytified) connexion $\nabla _{\underline {s}}$ on the trivial vector bundle of rank one on $X_\Sigma (\mathbb {C})$ :

$$ \begin{align*}\mathcal{L}_{\underline{s}}\otimes_{\mathbb Q_{\underline{s}}^{\mathrm{B}}} \mathbb{C} \cong \left( \mathcal{O}^{\mathrm{an}}_{X_\Sigma}\right)^{\nabla_{\underline{s}}}.\end{align*} $$

We will be interested in its cohomology

$$ \begin{align*}H_{\mathrm{B}}^r(X_{\Sigma}, \mathcal{L}_{\underline{s}})= H^r(\mathbb{C}\backslash \Sigma, \mathcal{L}_{\underline{s}}) \ \cong \ H_r(\mathbb{C} \backslash \Sigma, \mathcal{L}_{\underline{s}}^\vee)^{\vee},\end{align*} $$

where $\mathcal {L}_{\underline {s}}^\vee $ is the dual local system

$$ \begin{align*}\mathcal{L}_{\underline{s}}^\vee=\mathbb Q^{\mathrm{B}}_{\underline{s}}\; x^{s_0}\prod_{k=1}^n (1-x\sigma_k^{-1})^{s_k}.\end{align*} $$

The genericity assumption (13) implies that $\mathcal {L}_{\underline {s}}$ and $\mathcal {L}_{\underline {s}}^\vee $ have nontrivial monodromy around every point of $\Sigma $ and around $\infty $ , which implies that the natural map

$$ \begin{align*}H_r( \mathbb{C} \backslash \Sigma, \mathcal{L}_{\underline{s}}^\vee) \longrightarrow H^{\mathrm{lf}}_r( \mathbb{C} \backslash \Sigma, \mathcal{L}_{\underline{s}}^\vee)\end{align*} $$

from ordinary homology to locally finite homology is an isomorphism. Its inverse is sometimes called regularization. An easy computation shows that all homology is concentrated in degree one and $H_1^{\mathrm {lf}}(\mathbb {C} \backslash \Sigma , \mathcal {L}_{\underline {s}}^\vee )$ has rank n. It has a basis consisting of the (classes of the) locally finite chains

(18) $$ \begin{align} \delta_i \otimes x^{s_0} \prod_{k=1}^n (1-x \sigma_k^{-1})^{s_k} ,\end{align} $$

for $i=1,\ldots ,n$ , where $\delta _i: (0,1) \rightarrow \mathbb {C} \backslash \Sigma $ is a smooth path from $0$ to $\sigma _i$ that can be extended to a smooth path $\overline {\delta _i}:[0,1]\to \mathbb {C}$ , and $x^{s_0}\prod _{k=1}^n(1-x\sigma _k^{-1})^{s_k}$ denotes some choice of section of $\mathcal {L}_{\underline {s}}^\vee $ on $\delta _i$ .

Remark 2.3. In all that follows, we assume that the choices of representatives are such that $\delta _i$ does not wind infinitely around $0$ and $\sigma _i$ , that is, that the argument of $\delta _i(t)$ is bounded as t approaches $0$ and that the argument of $\delta _i(t)-\sigma _i$ is bounded as t approaches $1$ . This assumption will ensure the convergence of the integrals considered below.

2.1.3 Comparison isomorphism

There is a canonical isomorphism [Reference DeligneD1, §6]

(19) $$ \begin{align} \mathrm{comp}_{\mathrm{B}, \mathrm{dR}}(\underline{s}): H^1_{\mathrm{dR}}(X_{\Sigma}, \nabla_{\underline{s}}) \otimes_{k^{\mathrm{dR}}_{\underline{s}}} \mathbb{C} \overset{\sim}{\longrightarrow} H^1_{\mathrm{B}}(X_{\Sigma}, \mathcal{L}_{\underline{s}}) \otimes_{\mathbb Q^{\mathrm{B}}_{\underline{s}}} \mathbb{C}, \end{align} $$

whose restriction to the $\mathbb Q_{\underline {s}}^{\mathrm {dR}}$ -structure we shall denote by

(20) $$ \begin{align} \mathrm{comp}_{\mathrm{B}, \varpi}(\underline{s}): H^1_{\varpi}(X_{\Sigma}, \nabla_{\underline{s}}) \otimes_{\mathbb Q^{\mathrm{dR}}_{\underline{s}}} \mathbb{C} \overset{\sim}{\longrightarrow} H^1_{\mathrm{B}}(X_{\Sigma}, \mathcal{L}_{\underline{s}}) \otimes_{\mathbb Q^{\mathrm{B}}_{\underline{s}}} \mathbb{C}. \end{align} $$

Assuming (13), we can identify Betti cohomology $H_{\mathrm {B}}^1(X_{\Sigma }, \mathcal {L}_{\underline {s}})$ with the dual of locally finite homology, which leads to a bilinear pairing

$$ \begin{align*}H_1^{\mathrm{lf}}(\mathbb{C}\backslash {\Sigma}, \mathcal{L}_{\underline{s}}^\vee) \times H^1_{\varpi}(X_{\Sigma}, \nabla_{\underline{s}}) \longrightarrow \mathbb{C}.\end{align*} $$

It is well-known that the comparison isomorphism is computed by integration when it makes sense.

Lemma 2.4. Assuming (13), a matrix representative for the comparison isomorphism in the bases (17) and (18) is the $(n\times n)$ period matrix $L_{\Sigma }$ with entries

$$ \begin{align*}(L_{\Sigma})_{ij} = -s_j \int_{\delta_i} x^{s_0} \prod_{k=1}^n (1-x \sigma_k^{-1})^{s_k} \frac{dx}{x-\sigma_j},\end{align*} $$

provided that $\mathrm {Re}\, s_0>-1$ and $\mathrm {Re}\, s_i> \begin {cases} -1, & \mbox { if }\; i\neq j\ , \\ 0, & \mbox { if }\; i=j. \end {cases} $

Proof. By definition, the pairing that we wish to compute is

$$ \begin{align*}\langle\, \delta_i \otimes x^{s_0} \prod_{k=1}^n (1-x \sigma_k^{-1})^{s_k} , \, \mathrm{comp}_{\mathrm{B},\mathrm{dR}}(\underline{s})(-s_j\omega_j)\,\rangle = -s_j \int_{\delta_i}x^{s_0}\prod_{k=1}^n(1-x\sigma_k^{-1})^{s_k}\widetilde{\omega}_j,\end{align*} $$

where $\widetilde {\omega }_j$ is a smooth form on $\mathbb {C}\setminus \Sigma $ with compact support, representing the cohomology class of $\omega _j=dz/(z-\sigma _j)$ . In other words, we have

$$ \begin{align*}\widetilde{\omega}_j-\omega_j = \nabla_{\underline{s}}\phi = d\phi + \sum_{k=0}^ns_k\, \phi \, d\log(z-\sigma_k),\end{align*} $$

where $\phi $ is a smooth function on $\mathbb {P}^1(\mathbb {C})$ . Since $\widetilde {\omega }_j$ vanishes in the neighborhood of $\Sigma \cup \infty $ , taking residues in the previous equation along points of $\Sigma \cup \infty $ implies that $\phi $ vanishes at every $\sigma _k$ , $k\neq j$ , including at $\sigma _0=0$ . Now, we only need to prove that the integral

$$ \begin{align*}\int_{\delta_i}x^{s_0} \prod_{k=1}^n (1-x \sigma_k^{-1})^{s_k} \,\nabla_{\underline{s}}\phi\end{align*} $$

vanishes. Since $x^{s_0}\prod _{k=1}^n (1-x \sigma _k^{-1})^{s_k} \,\nabla _{\underline {s}}\phi = d(x^{s_0}\prod _{k=1}^n (1-x \sigma _k^{-1})^{s_k} \,\phi )$ , this integral is computed by Stokes’ theorem and we need to prove that $x^{s_0}\prod _{k=1}^n(1-x\sigma _k^{-1})^{s_k}\phi $ vanishes at $0$ and at $\sigma _i$ .

• – At $0$ , this amounts to proving that $\delta (t)^{s_0}\phi (\delta (t))$ goes to $0$ when $t\to 0$ . Since $\phi $ is smooth and vanishes at $0$ , we have $ |\phi (\delta (t)) | < C |\delta (t)| $ for some constant C, and so

  $$ \begin{align*}|\delta(t)^{s_0}\phi(\delta(t))| < C |\delta(t)^{s_0+1}|=C |\delta(t)|^{\mathrm{Re}(s_0)+1}\exp(-\mathrm{Im}(s_0)\mathrm{arg}(\delta(t))).\end{align*} $$
  By assumption, $\mathrm {Re}(s_0)+1>0$ and $\mathrm {arg}(\delta (t))$ is bounded as t approaches zero (see Remark 2.3), which implies that the limit of $\delta (t)^{s_0+1}$ , and also $\delta (t)^{s_0}\phi (\delta (t))$ , is zero when $t\rightarrow 0$ .

• – At $\sigma _i$ , the same argument gives the desired vanishing, by using the fact that $\phi $ vanishes at $\sigma _i$ if $i\neq j$ .

The result follows.

2.2 Intersection pairings

For generic $s_i$ (13), the natural map $H_1(\mathbb {C}\backslash \Sigma , \mathcal {L}_{\underline {s}}^\vee )\rightarrow H_1^{\mathrm {lf}}(\mathbb {C}\backslash \Sigma , \mathcal {L}_{\underline {s}}^\vee )$ is an isomorphism. Poincaré duality gives an isomorphism

$$ \begin{align*}H_1^{\mathrm{lf}}(\mathbb{C}\backslash \Sigma, \mathcal{L}_{\underline{s}}^\vee) \simeq H_1(\mathbb{C}\backslash \Sigma, \mathcal{L}_{\underline{s}})^\vee \simeq H_1(\mathbb{C}\backslash \Sigma, \mathcal{L}_{-\underline{s}}^\vee)^\vee,\end{align*} $$

where we set $-\underline {s}=(-s_0,\ldots , -s_n)$ . By combining these two isomorphisms, we get a perfect pairing, called the Betti intersection pairing [Reference Kita and YoshidaKY1, §2], [Reference Cho and MatsumotoCM], [Reference MizeraMY, §2]:

$$ \begin{align*}\langle \ , \, \rangle_{\mathrm{B}} : H_1(\mathbb{C}\backslash \Sigma, \mathcal{L}_{-\underline{s}}^\vee) \otimes_{\mathbb Q_{\underline{s}}^{\mathrm{B}}} H_1(\mathbb{C}\backslash \Sigma, \mathcal{L}_{\underline{s}}^\vee) \longrightarrow \mathbb Q_{\underline{s}}^{\mathrm{B}},\end{align*} $$

or dually in cohomology:

(21) $$ \begin{align} \langle \ , \, \rangle^{\mathrm{B}} : H^1_{\mathrm{B}}(X_\Sigma, \mathcal{L}_{-\underline{s}}) \otimes_{\mathbb Q_{\underline{s}}^{\mathrm{B}}} H^1_{\mathrm{B}}(X_\Sigma, \mathcal{L}_{\underline{s}}) \longrightarrow \mathbb Q_{\underline{s}}^{\mathrm{B}}. \end{align} $$

The de Rham counterpart is the de Rham intersection pairing [Reference Cho and MatsumotoCM], [Reference MatsumotoMa1]:

(22) $$ \begin{align} \langle \ , \, \rangle^{\mathrm{dR}} : H^1_{\mathrm{dR}}(X_\Sigma,\nabla_{-\underline{s}}) \otimes_{k_{\underline{s}}^{\mathrm{dR}}} H^1_{\mathrm{dR}}(X_\Sigma,\nabla_{\underline{s}}) \longrightarrow k_{\underline{s}}^{\mathrm{dR}}, \end{align} $$

which comes from Poincaré duality and the fact that the natural map

(23) $$ \begin{align} H^1_{\mathrm{dR},\mathrm{c}}(X_\Sigma,\nabla_{\underline{s}}) \longrightarrow H^1_{\mathrm{dR}}(X_\Sigma,\nabla_{\underline{s}}) \end{align} $$

is an isomorphism if the $s_i$ are generic, where the subscript $\mathrm {c}$ denotes compactly supported cohomology. The map (22) respects the natural $\mathbb Q_{\underline {s}}^{\mathrm {dR}}$ -structures and induces

$$ \begin{align*}\langle\ , \, \rangle^{\varpi}:H^1_{\varpi}(X_\Sigma,\nabla_{-\underline{s}}) \otimes_{\mathbb Q_{\underline{s}}^{\mathrm{dR}}} H^1_{\varpi}(X_\Sigma,\nabla_{\underline{s}}) \longrightarrow \mathbb Q_{\underline{s}}^{\mathrm{dR}}.\end{align*} $$

We set

(24) $$ \begin{align} \nu_i = \frac{dx}{x-\sigma_i} - \frac{dx}{x} \qquad \mbox{ for } \quad 1\leq i \leq n \end{align} $$

and use the same notation for the corresponding class in $H^1_{\varpi }(X_\Sigma ,\nabla _{-\underline {s}})$ .

Lemma 2.5. For all $1\leq i, j \leq n$ , and $s_0,\ldots , s_n$ satisfying (13),

(25) $$ \begin{align} \left\langle \nu_i , \omega_j \right\rangle^{\mathrm{dR}} = -\frac{1}{s_i} \, \mathbf{1}_{i=j} . \end{align} $$

Proof. By definition, the de Rham intersection pairing that we wish to compute is

$$ \begin{align*}\langle \nu_i,\omega_j \rangle^{\mathrm{dR}} = \frac{1}{2\pi i}\iint_{\mathbb P^1(\mathbb{C})}\widetilde{\nu}_i\wedge\omega_j,\end{align*} $$

where $\widetilde {\nu }_i$ is a smooth form on $\mathbb {C}\setminus \Sigma $ with compact support, representing the cohomology class of $\nu _i$ . In other words, we have

$$ \begin{align*}\widetilde{\nu}_i-\nu_i =\nabla_{-\underline{s}}\phi = d\phi -\sum_{k=0}^ns_k\,\phi\, d\log(x-\sigma_k),\end{align*} $$

where $\phi $ is a smooth function on $\mathbb P^1(\mathbb {C})$ . Since $\widetilde {\nu }_i$ vanishes in the neighborhood of $\Sigma \cup \infty $ , taking residues along points of $\Sigma \cup \infty $ implies that $\phi $ vanishes at every $\sigma _k$ , $k\notin \{ 0,i\}$ , and at $\infty $ , and

$$ \begin{align*}\phi(0) = -\frac{1}{s_0} \; , \; \phi(\sigma_i)=\frac{1}{s_i} .\end{align*} $$

By noticing that $\nu _i\wedge \omega _j=0$ and $d\log (x-\sigma _k)\wedge \omega _j=0$ , we thus get

$$ \begin{align*}\langle \nu_i,\omega_j\rangle^{\mathrm{dR}} = \frac{1}{2\pi i}\iint_{\mathbb P^1(\mathbb{C})} d\phi\wedge \omega_j = \frac{1}{2\pi i}\iint_{\mathbb P^1(\mathbb{C})} d(\phi\,\omega_j).\end{align*} $$

By Stokes, this last integral can be computed as the limit when $\varepsilon $ goes to zero of

$$ \begin{align*}-\frac{1}{2\pi i}\int_{\partial P_\varepsilon} \phi\,\omega_j,\end{align*} $$

where $P_\varepsilon $ is the complement in $\mathbb P^1(\mathbb {C})$ of $\varepsilon $ -disks around the points of $\Sigma \cup \infty $ , and the sign comes from the orientation of $\partial P_\varepsilon $ . By using the fact that $\phi (\infty )=0$ and $\omega _j$ is regular at every $\sigma _k$ , $k\neq j$ , a local computation (variant of Cauchy’s formula) thus gives

$$ \begin{align*}\langle \nu_i , \omega_j\rangle^{\mathrm{dR}} = -\mathrm{Res}_{\sigma_j}(\phi\,\omega_j) =-\frac{1}{s_i}\mathbf{1}_{i=j}.\end{align*} $$

This lemma implies that the dual basis to (17) is given by the classes

$$ \begin{align*}\nu_i \;\;\in \;\; H^1_{\varpi}(X_\Sigma,\nabla_{-\underline{s}}) \qquad \mbox{ for } i=1,\ldots,n .\end{align*} $$

Remark 2.6. The Betti and de Rham intersection pairings are compatible with the comparison isomorphism, which leads to quadratic relations among periods of cohomology with coefficients, known in the literature as twisted period relations [Reference Cho and MatsumotoCM], [Reference GotoG]. In matrix form, they read:

(26) $$ \begin{align} {}^tL_\Sigma(-\underline{s})\, I^{\mathrm{B}}_\Sigma(\underline{s})\,L_\Sigma(\underline{s}) = 2\pi i\,I^{\mathrm{dR}}_\Sigma(\underline{s}), \end{align} $$

where $I^{\mathrm {B}}_\Sigma (\underline {s})$ and $I^{\mathrm {dR}}_\Sigma (\underline {s})$ are the matrices of the Betti and de Rham intersection pairings (21) and (22), respectively.

2.3 Single-valued periods of cohomology with coefficients

We can define and compute a period pairing on de Rham cohomology classes by transporting complex conjugation.

2.3.1 Definition of the single-valued period map

Let $\overline {\Sigma }=\{\overline {\sigma _0},\overline {\sigma _1},\ldots ,\overline {\sigma _n}\}$ denote the complex conjugates of the points in $\Sigma $ . We have an anti-holomorphic diffeomorphism

$$ \begin{align*}\mathrm{conj}:\mathbb{C}\backslash \overline{\Sigma} \longrightarrow \mathbb{C}\backslash \Sigma\end{align*} $$

given by complex conjugation. We note that the induced map $H_1(\mathbb {C}\backslash \overline {\Sigma })\rightarrow H_1(\mathbb {C}\backslash \Sigma )$ sends the class of a positively oriented loop around $\overline {\sigma _j}$ to the class of a negatively oriented loop around $\sigma _j$ . Since a rank one local system on $\mathbb {C}\backslash \Sigma $ (resp. $\mathbb {C}\backslash \overline {\Sigma }$ ) is equivalent to a representation of the abelian group $H_1(\mathbb {C}\backslash \Sigma )$ (resp. $H_1(\mathbb {C}\backslash \overline {\Sigma })$ ), we see that we have an isomorphism of local systems:

(27) $$ \begin{align} \mathrm{conj}^*\mathcal{L}_{\underline{s}}\simeq \mathcal{L}_{-\underline{s}}. \end{align} $$

We thus get a morphism of local systems on $\mathbb {C}\backslash \Sigma $ :

$$ \begin{align*}\mathcal{L}_{\underline{s}} \longrightarrow \mathrm{conj}_*\mathrm{conj}^*\mathcal{L}_{\underline{s}} \simeq \mathrm{conj}_*\mathcal{L}_{-\underline{s}},\end{align*} $$

which at the level of cohomology induces a morphism of $\mathbb Q_{\underline {s}}^{\mathrm {B}}$ -vector spaces

$$ \begin{align*}F_\infty: H^1_{\mathrm{B}}(\mathbb{C}\backslash \Sigma, \mathcal{L}_{\underline{s}}) \longrightarrow H^1_{\mathrm{B}}(\mathbb{C}\backslash \overline{\Sigma},\mathcal{L}_{-\underline{s}}).\end{align*} $$

We call $F_\infty $ the real Frobenius or Frobenius at the infinite prime. We will use the notation $F_\infty (\underline {s})$ when we want to make the dependence on $\underline {s}$ explicit. One checks that the Frobenius is involutive: $F_{\infty }(-\underline {s}) F_{\infty }(\underline {s}) =\mathrm {id} $ .

Remark 2.7. The isomorphism (27) is induced by the trivialisation of the tensor product $\mathrm {conj}^*\mathcal {L}_{\underline {s}}\otimes \mathcal {L}_{\underline {s}}$ given by the section

$$ \begin{align*}g_{\underline{s}} = |z|^{-2s_0}\prod_{k=1}^n|1-z\sigma_k^{-1}|^{-2s_k}.\end{align*} $$

Thus, the homological real Frobenius

$$ \begin{align*}F_\infty:H_1^{\mathrm{B}}(\mathbb{C}\setminus \overline{\Sigma},\mathcal{L}_{\underline{s}}) \longrightarrow H_1^{\mathrm{B}}(\mathbb{C}\setminus \Sigma,\mathcal{L}_{-\underline{s}})\end{align*} $$

is computed at the level of representatives by the formula

$$ \begin{align*}\delta\otimes z^{-s_0}\prod_{k=1}^n(1-z\overline{\sigma_k}^{-1})^{-s_k} \mapsto \overline{\delta}\otimes \overline{z}^{-s_0}\prod_{k=1}^n(1-\overline{z}\,\overline{\sigma_k}^{-1})^{-s_k} \, g_{\underline{s}}^{-1} = \overline{\delta}\otimes z^{s_0}\prod_{k=1}^n(1-z\sigma_k^{-1})^{s_k}.\end{align*} $$

Remark 2.8. A morphism similar to $F_\infty $ was considered in [Reference Hattori and KimuraHY] and leads to similar formulae but has a different definition and a different interpretation. Our definition only uses the action of complex conjugation on the complex points of the variety $X_\Sigma $ relative to two complex conjugate embeddings of k in $\mathbb {C}$ , whereas the definition in [Reference Hattori and KimuraHY] conjugates the field of coefficients of the local systems, which requires the $s_i$ to be real. Note that our definition does not require the assumption that the $s_i \in \mathbb {R}$ .

In the rest of this article, however, we often assume that the $s_i$ are real. (This is an unnatural assumption and would not be necessary if the $s_i$ were treated as formal variables; see §3.5.) In this way, the complex conjugate of the field $k_{\underline {s}}^{\mathrm {dR}}$ inside $\mathbb {C}$ is the field $\overline {k}(s_1,\ldots ,s_n)$ . We use the notation $(-)\otimes _{k_{\underline {s}}^{\mathrm {dR}}}\overline {\mathbb {C}}$ for the tensor product with $\mathbb {C}$ , viewed as a $k_{\underline {s}}^{\mathrm {dR}}$ -vector space via the complex conjugate embedding. We thus have an additional $\mathbb {C}$ -linear comparison isomorphism:

$$ \begin{align*}\mathrm{comp}_{\overline{\mathrm{B}},\mathrm{dR}}(\underline{s}):H^1_{\mathrm{dR}}(X_\Sigma,\nabla_{\underline{s}})\otimes_{k_{\underline{s}}^{\mathrm{dR}}}\overline{\mathbb{C}} \longrightarrow H^1_{\mathrm{B}}(\mathbb{C}\setminus \overline{\Sigma},\mathcal{L}_{\underline{s}})\otimes_{\mathbb Q_{\underline{s}}^{\mathrm{B}}}\mathbb{C}.\end{align*} $$

Definition 2.9. Assume that the $s_i$ are real. The single-valued period map is the $\mathbb {C}$ -linear isomorphism

$$ \begin{align*}\mathbf{s} : H^1_{\mathrm{dR}}(X_\Sigma,\nabla_{\underline{s}})\otimes_{k^{\mathrm{dR}}_{\underline{s}}}\mathbb{C}\longrightarrow H^1_{\mathrm{dR}}(X_\Sigma,\nabla_{-\underline{s}})\otimes_{k^{\mathrm{dR}}_{\underline{s}}}\overline{\mathbb{C}}\end{align*} $$

defined as the composite

$$ \begin{align*}\mathbf{s} = \mathrm{comp}_{\overline{\mathrm{B}},\mathrm{dR}}^{-1}(-\underline{s}) \circ (F_\infty\otimes \mathrm{id}) \circ \mathrm{comp}_{\mathrm{B},\mathrm{dR}}(\underline{s}) .\end{align*} $$

In other words, it is defined by the commutative diagram

We will use the notation $\mathbf {s}(\underline {s})$ when we want to make the dependence on $\underline {s}$ explicit.

The single-valued period map is a transcendental comparison isomorphism that is naturally interpreted at the level of analytic de Rham cohomology via the isomorphisms

$$ \begin{align*}H^1_{\mathrm{dR}}(X_\Sigma,\nabla_{\underline{s}})\otimes_{k^{\mathrm{dR}}_{\underline{s}}}\mathbb{C} \simeq H^1_{\mathrm{dR, an}}(\mathbb{C}\backslash\Sigma, (\mathcal{O}_{\mathbb{C}\backslash \Sigma},\nabla_{\underline{s}}))\end{align*} $$

and

$$ \begin{align*}H^1_{\mathrm{dR}}(X_\Sigma,\nabla_{-\underline{s}})\otimes_{k^{\mathrm{dR}}_{\underline{s}}}\overline{\mathbb{C}} \simeq H^1_{\mathrm{dR, an}}(\mathbb{C}\backslash\overline{\Sigma}, (\mathcal{O}_{\mathbb{C}\backslash \overline{\Sigma}},\nabla_{-\underline{s}})).\end{align*} $$

To avoid any confusion, we use the coordinate $w=\overline {z}$ on $\mathbb {C}\backslash \overline {\Sigma }$ .

Lemma 2.10. Assume that the $s_i$ are real. In analytic de Rham cohomology, the single-valued period map is induced by the morphism of smooth de Rham complexes

$$ \begin{align*} \mathbf{s}_{\mathrm{an}} : (\mathcal{A}^{\bullet}_{\mathbb{C}\backslash \Sigma},\nabla_{\underline{s}}) &\longrightarrow & \mathrm{conj}_*(\mathcal{A}^{\bullet}_{\mathbb{C}\backslash \overline{\Sigma}}, \nabla_{-\underline{s}}) \end{align*} $$

given on the level of sections by

$$ \begin{align*}\mathcal{A}^{\bullet}_{\mathbb{C}\backslash\Sigma}(U)\,\ni\, \omega \quad \mapsto \quad |w|^{2s_0}\prod_{k=1}^n|1-w\overline{\sigma}_k^{-1}|^{2s_k}\; \mathrm{conj}^*(\omega)\,\in\, \mathcal{A}^{\bullet}_{\mathbb{C}\backslash\overline{\Sigma}}(\overline{U}).\end{align*} $$

Proof. Let $P=|w|^{2s_0}\prod _{k=1}^n|1-w\overline {\sigma }_k^{-1}|^{2s_k}$ . We first check that $\mathbf {s}_{\mathrm {an}}$ is a morphism of complexes:

$$ \begin{align*} \nabla_{-\underline{s}}(\mathbf{s}_{\mathrm{an}}(\omega)) & = \nabla_{-\underline{s}}(P\,\mathrm{conj}^*(\omega)) \\ & = P\left(\left(\sum_{k=0}^ns_k\,d\log(w-\overline{\sigma_k}) + \sum_{k=0}^ns_k\,d\log(\overline{w}-\sigma_k)\right)\wedge\mathrm{conj}^*(\omega) + d(\mathrm{conj}^*(\omega))\right) \\ & \qquad -\sum_{k=0}^ns_k\, d\log(w-\overline{\sigma_k})\wedge (P \,\mathrm{conj}^*(\omega)) \\ & = P\left(\sum_{k=0}^n s_k\,d\log(\overline{w}-\sigma_k)\wedge\mathrm{conj}^*(\omega)+d(\mathrm{conj}^*(\omega))\right) \\ & = P\,\mathrm{conj}^*\left(\sum_{k=0}^n s_k\, d\log(z-\sigma_k)\wedge\omega+ d\omega \right) \\ & = \mathbf{s}_{\mathrm{an}}(\nabla_{\underline{s}}(\omega)). \end{align*} $$

On the level of horizontal sections, we compute

$$ \begin{align*} \mathbf{s}_{\mathrm{an}}\left(z^{-s_0}\prod_{k=1}^n(1-z\sigma_k^{-1})^{-s_k}\right) & = |w|^{2s_0}\prod_{k=1}^n|1-w\overline{\sigma}_k^{-1}|^{2s_k}\, \overline{w}^{-s_0}\prod_{k=1}^n(1-\overline{w}\sigma_k^{-1})^{-s_k} \\ &= w^{s_0}\prod_{k=1}^n(1-w\overline{\sigma_k}^{-1})^{s_k}. \end{align*} $$

Thus, $\mathbf {s}_{\mathrm {an}}$ induces the morphism $\mathcal {L}_{\underline {s}}\rightarrow \mathrm {conj}_*\mathcal {L}_{-\underline {s}}$ and the result follows.

2.3.2 Integral formula for single-valued periods

We derive a formula for the single-valued period map $\mathbf {s}$ using the de Rham intersection pairing (22), that is, for the single-valued period pairing,

(28) $$ \begin{align} H^1_{\mathrm{dR}}(X_\Sigma,\nabla_{\underline{s}}) \times H^1_{\mathrm{dR}}(X_\Sigma,\nabla_{\underline{s}}) \longrightarrow \mathbb{C} \;\;\; ,\;\;\; (\nu ,\omega)\,\mapsto\, \langle \nu, \mathbf{s}\,\omega\rangle^{\mathrm{dR}} . \end{align} $$

Note that this pairing is $k_{\underline {s}}^{\mathrm {dR}}$ -linear in each argument, where in the first slot the complex conjugate embedding of $k_{\underline {s}}^{\mathrm {dR}}$ inside $\mathbb {C}$ is understood: this means that we have, for $a,b\in k_{\underline {s}}^{\mathrm {dR}}$ ,

$$ \begin{align*}\langle a\nu , \mathbf{s}\, b\omega\rangle^{\mathrm{dR}} = \overline{a}b\,\langle \nu,\mathbf{s}\,\omega\rangle^{\mathrm{dR}}.\end{align*} $$

Proposition 2.11. Assume that $s_0,\ldots , s_n$ are real and generic (13). Let $\omega , \nu \in \Gamma (\mathbb P^1_k, \Omega ^1_{\mathbb P^1_k} (\log \Sigma \cup \infty ))$ , and write $\omega $ and $\nu $ for their classes in $H^1_{\mathrm {dR}} (X_{\Sigma }, \nabla _{\underline {s}})$ . Assume that $\nu $ has no pole at $\infty $ . Then the single-valued pairing is

(29) $$ \begin{align} \langle \nu , \, \mathbf{s}\, \omega \rangle^{\mathrm{dR}} = - \frac{1}{2\pi i} \iint_{\mathbb{C}} |z|^{2s_0} \prod_{k=1}^n |1- z \sigma_k^{-1} |^{2s_k} \, \overline{ \nu} \wedge \omega \end{align} $$

whenever $s_i>0$ for all $0\leq i \leq n$ , and $s_0+\cdots +s_n < 1/2$ .

Proof. Let us first note that the integral in (29) converges under the assumptions on $\omega $ , $\nu $ , and the $s_i$ . To check this, pass to local polar coordinates $z=\rho e^{i\theta }$ in the neighbourhood of every point in $\Sigma $ , and verify that $|z|^{2s}\frac {dz\,d\overline {z}}{z\overline {z}}$ is proportional to $\rho ^{2s-1}d\rho \,d\theta $ and is integrable when $\mathrm {Re}(s)>0$ . At $\infty $ , use the fact that $\nu $ has no pole to obtain a local estimate of the form $\rho ^{-2s_0-\cdots -2s_n}d\rho \,d\theta $ , which is integrable for $s_0+\cdots +s_n < 1/2$ .

We use the notation $\omega =\omega _\sigma $ for the smooth form on $\mathbb {C}\setminus \Sigma $ induced by $\omega $ , and $\omega _{\overline {\sigma }}$ for the smooth form on $\mathbb {C}\setminus \overline {\Sigma }$ induced by $\omega $ . (It is obtained by replacing each occurrence of $\sigma _j$ in $\omega $ by a $\overline {\sigma _j}$ .) Then, by definition, we have $\langle \nu ,\mathbf {s}\,\omega \rangle = \langle \nu _{\overline {\sigma }},\mathbf {s}_{\mathrm {an}}\omega _\sigma \rangle $ . By Lemma 2.10 and the definition of the de Rham intersection pairing, this equals the integral

(30) $$ \begin{align} \frac{1}{2\pi i}\iint_{\mathbb P^1(\mathbb{C})}|w|^{2s_0}\prod_{k=1}^n|1-w\overline{\sigma_k}^{-1}|^{2s_k}\, \widetilde{\nu_{\overline{\sigma}}}\wedge \mathrm{conj}^*(\omega_{\sigma}), \end{align} $$

where $\widetilde {\nu _{\overline {\sigma }}}$ is a smooth form on $\mathbb {C}\setminus \overline {\Sigma }$ with compact support, representing the cohomology class of $\nu _{\overline {\sigma }}$ . In other words, we have

$$ \begin{align*}\widetilde{\nu_{\overline{\sigma}}} - \nu_{\overline{\sigma}} = \nabla_{\underline{s}}\phi = d\phi + \sum_{k=0}^ns_k\,\phi\, d\log(w-\overline{\sigma_k}),\end{align*} $$

where $\phi $ is a smooth function on $\mathbb P^1(\mathbb {C})$ . The assumption that $\nu $ has no pole at $\infty $ and the fact that $s_0+\cdots +s_n\neq 0$ imply, by taking residues, that $\phi (\infty )=0$ . We first prove that we may remove the tilde in (30), that is, that the integral

(31) $$ \begin{align} \iint_{\mathbb P^1(\mathbb{C})}|w|^{2s_0}\prod_{k=1}^n|1-w\overline{\sigma_k}^{-1}|^{2s_k}\, \nabla_{\underline{s}}\phi\wedge \mathrm{conj}^*(\omega_{\sigma}) \end{align} $$

vanishes. Its integrand equals

$$ \begin{align*}d\left(|w|^{2s_0}\prod_{k=1}^n|1-w\overline{\sigma_k}^{-1}|^{2s_k}\phi\; \mathrm{conj}^*(\omega)\right)\end{align*} $$

because $d\overline {w}\wedge \mathrm {conj}^*(\omega _\sigma )=\mathrm {conj}^*(dz\wedge \omega _\sigma )=0$ . By Stokes, the integral (31) can be computed as the limit when $\varepsilon $ goes to zero of

$$ \begin{align*}-\int_{\partial P_\varepsilon} |w|^{2s_0}\prod_{k=1}^n|1-w\overline{\sigma_k}^{-1}|^{2s_k}\phi\;\mathrm{conj}^*(\omega),\end{align*} $$

where $P_\varepsilon $ is the complement in $\mathbb P^1(\mathbb {C})$ of $\varepsilon $ -disks around the points of $\overline {\Sigma }\cup \infty $ , and the sign comes from the orientation of $\partial P_\varepsilon $ . The contribution of each point of $\overline {\Sigma }$ vanishes, as can be seen from a computation in a local coordinate, because of the assumption that $s_i>0$ for all i. The contribution of the point $\infty $ also vanishes because of the fact that $\phi (\infty )=0$ and the assumption that $s_0+\cdots +s_n < 1/2$ . Thus, we have

$$ \begin{align*} \langle \nu , \, \mathbf{s} \,\omega \rangle^{\mathrm{dR}} & = \frac{1}{2\pi i}\iint_{\mathbb P^1(\mathbb{C})}|w|^{2s_0}\prod_{k=1}^n|1-w\overline{\sigma_k}^{-1}|^{2s_k}\, \nu_{\overline{\sigma}}\wedge \mathrm{conj}^*(\omega_{\sigma}) \\ & = - \frac{1}{2\pi i}\iint_{\mathbb P^1(\mathbb{C})}|\overline{z}|^{2s_0}\prod_{k=1}^n|1-\overline{z}\overline{\sigma_k}^{-1}|^{2s_k}\, \mathrm{conj}^*(\nu_{\overline{\sigma}})\wedge \omega_{\sigma} \\ & = - \frac{1}{2\pi i}\iint_{\mathbb P^1(\mathbb{C})}|z|^{2s_0} \prod_{k=1}^n |1- z \sigma_k^{-1} |^{2s_k} \, \overline{ \nu_\sigma} \wedge \omega_\sigma . \end{align*} $$

The second equality follows from performing a change of variables via $\mathrm {conj}:\mathbb {C}\backslash \Sigma \rightarrow \mathbb {C}\backslash \overline {\Sigma }$ , which reverses the orientation of $\mathbb {C}$ , hence the minus sign. The third equality relies on $\mathrm {conj}^*(\nu _{\overline {\sigma }}) = \overline {\nu _\sigma }$ , which is obvious. The result follows.

Corollary 2.12. Assume that $s_0,\ldots ,s_n$ are real and generic (13). Then, for all $1\leq i,j\leq n$ , the single-valued Lauricella function $(L_\Sigma ^{\mathbf {s}})_{ij}$ (3) is a single-valued period of cohomology with coefficients:

$$ \begin{align*}(L_\Sigma^{\mathbf{s}})_{ij} = \langle \nu_i, \mathbf{s}(-s_j\omega_j) \rangle^{\mathrm{dR}}\end{align*} $$

for $s_k>0$ for all $0\leq k\leq n$ and $s_0+\cdots +s_n<1/2$ .

Proof. This follows from Proposition 2.11.

Corollary 2.13. Under the assumptions of the previous corollary, we have the matrix equality

(32) $$ \begin{align} L_\Sigma^{\mathbf{s}}(s_0,\ldots,s_n) = L_{\overline{\Sigma}}(-s_0,\ldots,-s_n)^{-1} L_\Sigma(s_0,\ldots,s_n)\ , \end{align} $$

where it is understood that the Lauricella functions $(L_\Sigma )_{ij}$ and $(L_{\overline {\Sigma }})_{ij}$ are computed via choices of paths which are complex conjugate to each other.

Proof. This follows from the definition of the single-valued period homomorphism and Corollary 2.12 since the classes $\nu _i$ are the dual basis, with respect to the de Rham pairing, of the basis $(-s_i\omega _i)$ by Lemma 2.5.

Remark 2.14. The inverse of the matrix $L_{\overline {\Sigma }}(-\underline {s})$ appearing in (32) can be computed in terms of $L_{\overline {\Sigma }}(\underline {s})$ from the twisted period relation (26). This leads to quadratic expressions of the single-valued Lauricella function in terms of ordinary Lauricella functions, which we call double copy formulae. They read, in matrix form,

$$ \begin{align*}L^{\mathbf{s}}_\Sigma(\underline{s}) = \frac{1}{2\pi i}\, {}^tL_{\overline{\Sigma}}(\underline{s})\, I^{\mathrm{B}}_{\overline{\Sigma}}(-\underline{s})\, L_\Sigma(\underline{s}).\end{align*} $$

This is because the matrix $I^{\mathrm {dR}}_{\overline {\Sigma }}(-\underline {s})$ is the identity matrix in the bases $(+s_i\omega _i)$ and $(\nu _i)$ by Lemma 2.5. They are related to double copy formulae in the physics literature such as the Kawai–Lewellen–Tye formula [Reference Kawai, Lewellen and TyeKLT], which was interpreted in the framework of cohomology with coefficients in [Reference Mimachi and YoshidaMi2].

3.1 Minimalist version

Let $k_{\mathrm {dR}} \subset \mathbb {C}$ and $\mathbb Q_{\mathrm {B}} \subset \mathbb {C}$ be two subfields of $\mathbb {C}$ . Consider the $\mathbb Q$ -linear abelian category $\mathcal {T}$ whose objects consist of triples $V=(V_{\mathrm {B}}, V_{\mathrm {dR}}, c)$ where $V_{\mathrm {B}}, V_{\mathrm {dR}}$ are finite-dimensional vector spaces over $\mathbb Q_{\mathrm {B}} $ and $k_{\mathrm {dR}}$ , respectively, and $c: V_{\mathrm {dR}} \otimes _{k_{\mathrm {dR}}} \mathbb {C} \overset {\sim }{\rightarrow } V_{\mathrm {B}} \otimes _{\mathbb Q_{\mathrm {B}}} \mathbb {C}$ is a $\mathbb {C}$ -linear isomorphism. The morphisms $\phi $ in $\mathcal {T}$ are pairs of linear maps $\phi _{\mathrm {B}}$ , $\phi _{\mathrm {dR}}$ compatible with the isomorphisms c. The category $\mathcal {T}$ is Tannakian with two fiber functors

$$ \begin{align*}\omega_{\mathrm{dR}} : \mathcal{T} \longrightarrow \mathrm{Vec}_{k_{\mathrm{dR}}} \qquad \mbox{ and } \qquad \omega_{\mathrm{B}} : \mathcal{T} \longrightarrow \mathrm{Vec}_{\mathbb Q_{\mathrm{B}}}.\end{align*} $$

The ring $\mathcal {P}_{\mathcal {T}}^{\mathfrak {m} } = \mathcal {O}(\mathrm {Isom}^{\otimes }_{\mathcal {T}} (\omega _{\mathrm {dR}},\omega _{\mathrm {B}}))$ is the $(\mathbb Q_{\mathrm {B}}, k_{\mathrm {dR}})$ -bimodule spanned by equivalence classes of matrix coefficients $[V, \sigma , \omega ]^{\mathfrak {m} }$ where $\sigma \in V_{\mathrm {B}}^{\vee }$ and $\omega \in V_{\mathrm {dR}}$ . The $k_{\mathrm {dR}}$ -algebra $\mathcal {P}_{\mathcal {T}}^{\mathfrak {dr}} = \mathcal {O}(\mathrm {Aut}^{\otimes }_{\mathcal {T}} (\omega _{\mathrm {dR}}))$ is spanned by equivalence classes of matrix coefficients $[V, f, \omega ]^{\mathfrak {dr}}$ where $f \in V_{\mathrm {dR}}^{\vee }$ and $\omega \in V_{\mathrm {dR}}$ . The multiplicative structure is given by tensor products. There is a natural coaction

$$ \begin{align*}\Delta: \mathcal{P}^{\mathfrak{m} }_{\mathcal{T}} \longrightarrow \mathcal{P}^{\mathfrak{m} }_{\mathcal{T}} \otimes_{k_{\mathrm{dR}}} \mathcal{P}^{\mathfrak{dr}}_{\mathcal{T}},\end{align*} $$

which expresses $\mathcal {P}^{\mathfrak {m} }_{\mathcal {T}}$ as an algebra comodule over the Hopf algebra $\mathcal {P}^{\mathfrak {dr}}_{\mathcal {T}}$ . It is given by the formula

(33) $$ \begin{align} \Delta [ V, \sigma, \omega]^{\mathfrak{m} } = \sum_{i} \, [V, \sigma, e_i]^{\mathfrak{m} } \otimes [V,e_i^{\vee}, \omega]^{\mathfrak{dr}}, \end{align} $$

where the sum ranges over a $k_{\mathrm {dR}}$ -basis $\{e_i\}$ of $V_{\mathrm {dR}}$ , and $e_i^{\vee }$ denotes the dual basis of $V_{\mathrm {dR}}^{\vee }$ .

Definition 3.1. For generic complex numbers $s_i$ (13), let $k_{\mathrm {dR}}=\mathbb Q^{\mathrm {dR}}_{\underline {s}}$ and $\mathbb Q_{\mathrm {B}} =\mathbb Q^{\mathrm {B}}_{\underline {s}}$ , and define

$$ \begin{align*}M_{\Sigma} = \left( \ H^1_{\mathrm{B}}(X_{\Sigma}, \mathcal{L}_{\underline{s}}) \ , \ H^1_{\varpi}(X_{\Sigma}, \nabla_{\underline{s}}) \ , \ \mathrm{comp}_{\mathrm{B},\varpi}(\underline{s}) \ \right) \quad \in\quad \mathrm{Ob}(\mathcal{T}) .\end{align*} $$

Define a matrix $L_{\Sigma }^{\mathfrak {m} } \in M_{n\times n}( \mathcal {P}^{\mathfrak {m} }_{\mathcal {T}})$ by

$$ \begin{align*}(L_{\Sigma}^{\mathfrak{m} })_{ij} = \Big[ M_{\Sigma} \ , \ \delta_i \otimes x^{s_0} \prod_{k=1}^n (1-x \sigma_k^{-1})^{s_k} \ , \ - s_j \omega_j \Big]^{\mathfrak{m}} ,\end{align*} $$

where the basis elements are given by (17) and (18). We will use the notation $M_\Sigma (\underline {s})$ and $L^{\mathfrak {m}} _{\Sigma }(\underline {s})$ when we want to make the dependence on $\underline {s}$ explicit.

The image of $L_{\Sigma }^{\mathfrak {m} }$ under the period homomorphism

$$ \begin{align*} \mathrm{per} : \mathcal{P}^{\mathfrak{m} }_{\mathcal{T}} &\longrightarrow \mathbb{C} \nonumber \\ \left[(V_{\mathrm{B}}, V_{\mathrm{dR}}, c), \sigma ,\omega\right]^{\mathfrak{m} } & \mapsto \langle \sigma \, , \, c(\omega)\rangle \end{align*} $$

is precisely the matrix of Lauricella functions (when the integral converges; see Lemma 2.4):

$$ \begin{align*}\mathrm{per} \left(L_{\Sigma}^{\mathfrak{m} }\right)_{ij} = - s_j \int_{\delta_i} x^{s_0} \prod_{k=1}^n (1-x \sigma_k^{-1})^{s_k} \frac{dx}{x-\sigma_j}\ \cdot\end{align*} $$

For this reason, we think of $L_{\Sigma }^{\mathfrak {m} }$ as a (global) motivic lift of the matrix of Lauricella functions (1). Define a (global) de Rham version

$$ \begin{align*}(L_{\Sigma}^{\mathfrak{dr}})_{ij} = \left[ M_{\Sigma} \ , \ \nu_i \ , \ - s_j \omega_j \right]^{\mathfrak{dr}} ,\end{align*} $$

where the forms $\nu _i$ , defining classes in $H^1_\varpi (X_\Sigma ,\nabla _{-\underline {s}})$ , were defined in (24). Recall that the de Rham pairing (23) induces an isomorphism $H^1_\varpi (X_\Sigma ,\nabla _{-\underline {s}})\simeq H^1_\varpi (X_\Sigma ,\nabla _{\underline {s}})^\vee $ and that Lemma 2.5 implies that the basis $\{\nu _i\}$ is dual to the basis $\{-s_i\omega _i\}$ .

Example 3.3. For all $n\in \mathbb {Z}$ , one has ‘Tate’ objects

$$ \begin{align*}\mathbb Q_{\underline{s}}(-n) = ( \mathbb Q^{\mathrm{B}}_{\underline{s}}, \mathbb Q^{\mathrm{dR}}_{\underline{s}}, 1\mapsto (2\pi i)^n).\end{align*} $$

The Betti and de Rham pairings (21) and (22), along with their compatibility with the comparison map (19) (a.k.a. the twisted period relations), can be succinctly encoded as a perfect pairing

$$ \begin{align*}M_{\Sigma}(-\underline{s}) \otimes M_{\Sigma}(\underline{s}) \longrightarrow \mathbb Q_{\underline{s}}(-1)\end{align*} $$

in the category $\mathcal {T}$ . Equivalently, $M_{\Sigma }( \underline {s})^{\vee } \simeq M_{\Sigma }(-\underline {s})(1)$ , where $(n)$ is the standard notation for Tate twist, that is, tensor product with $ \mathbb Q_{\underline {s}}(n)$ .

Proposition 3.4. The (global) coaction satisfies

$$ \begin{align*}\Delta L_{\Sigma}^{\mathfrak{m} }= L_{\Sigma}^{\mathfrak{m} }\otimes L_{\Sigma}^{\mathfrak{dr}}.\end{align*} $$

3.2 Version with real Frobenius involutions

In order to incorporate the real Frobenius isomorphism, and hence single-valued periods, into a Tannakian framework, we are obliged to consider a more complicated version of the previous categorical construction. This is somewhat artificial, but is forced upon us by the fact that complex conjugation does not define an involution on $H^{1}_{\mathrm {B}}(X, \mathcal {L}_{\underline {s}})$ , but rather relates the cohomology of $\mathcal {L}_{\underline {s}}$ with that of $\mathcal {L}_{-\underline {s}}$ . For this reason, we must consider two de Rham and two Betti realizations corresponding to coefficients $\nabla _{\underline {s}}$ , $\nabla _{-\underline {s}}$ and $\mathcal {L}_{\underline {s}}$ , $\mathcal {L}_{-\underline {s}}$ , which we denote by superscripts $+/-$ .

Let $s_i \in \mathbb R$ be real numbers satisfying the genericity conditions (13). Let $k_{\mathrm {dR}} \subset \mathbb {C}$ and $\mathbb Q_{\mathrm {B}} \subset \mathbb {C}$ be two subfields of $\mathbb {C}$ . We let $(-)\otimes _{k_{\mathrm {dR}}}\overline {\mathbb {C}}$ denote the tensor product taken with respect to the complex conjugate embedding. One solution is to consider a category $\mathcal {T}_{\infty }$ defined in a similar manner as $\mathcal {T}$ , except that the objects are given by:

1. 1. a pair of finite-dimensional $k_{\mathrm {dR}}$ -vector spaces $V^+_{\mathrm {dR}}$ , $V_{\mathrm {dR}}^-$ ;

2. 2. two pairs of finite-dimensional $\mathbb Q_{\mathrm {B}}$ -vector spaces $V^+_{\mathrm {B}}$ , $V_{\mathrm {B}}^-$ and $V^+_{\overline {\mathrm {B}}}$ , $V_{\overline {\mathrm {B}}}^-$ ;

3. 3. two $\mathbb {C}$ -linear comparison isomorphisms

   $$ \begin{align*} c_{\mathrm{B},\mathrm{dR}}^{+}: V_{\mathrm{dR}}^{+} \otimes_{k_{\mathrm{dR}}} \mathbb{C} & \overset{\sim}{\longrightarrow} V_{\mathrm{B}}^{+} \otimes_{\mathbb Q_{\mathrm{B}}} \mathbb{C} \nonumber \ ,\\ c_{\mathrm{\overline{B}}, \mathrm{dR}}^{+}: V_{\mathrm{dR}}^{+} \otimes_{k_{\mathrm{dR}}} \overline{\mathbb{C}} &\overset{\sim}{\longrightarrow} V_{\mathrm{\overline{B}}}^{+} \otimes_{\mathbb Q_{\mathrm{B}}} \mathbb{C} \ , \nonumber \end{align*} $$
   and another two defined in the same way with all $+$ ’s replaced by $-$ ,

4. 4. two $\mathbb Q_B$ -linear real Frobenius maps

   $$ \begin{align*}V_{B}^{+} \overset{\sim}{\longrightarrow} V_{\overline{B}}^{-} \qquad \mbox{ and } \qquad V_{\overline{B}}^{+} \overset{\sim}{\longrightarrow} V_{B}^{-} ,\end{align*} $$
   and another two with the $+$ ’s and $-$ ’s interchanged. These maps will simply be denoted by $F_{\infty }$ , since the source and, hence, the target will be clear from context. The composition of any two such maps, when defined, is the identity: $F_{\infty } F_{\infty } = \mathrm {id} .$

The morphisms $\phi $ between objects are given by the data of two $k_{\mathrm {\mathrm {dR}}}$ -linear maps $\phi _{\mathrm {dR}}^{+}, \phi _{\mathrm {dR}}^{-}$ , and four $\mathbb Q_{\mathrm {B}}$ -linear maps $\phi _{\mathrm {B}}^{+}$ , $\phi _{\mathrm {B}}^{-}$ , $\phi _{\overline {\mathrm {B}}}^{+}$ , $\phi _{\overline {\mathrm {B}}}^{-}$ which are compatible with $(3)$ and $(4)$ . One checks that this category is Tannakian, equipped with six fiber functors $\omega _{\mathrm {dR}}^{\pm }, \omega _{\mathrm {B}}^{\pm }, \omega _{\overline {\mathrm {B}}}^{\pm }$ (over $k_{\mathrm {dR}}, \mathbb Q_{\mathrm {B}}$ , respectively) which are obtained by forgetting all data except one of the vector spaces in $(1)$ or $(2)$ .

The category $\mathcal {T}_\infty $ admits various rings of periods defined in a similar manner as before. Consider the eight rings of periods:

$$ \begin{align*}\mathcal{P}_{\mathcal{T}_{\infty}}^{\mathrm{B}^{\pm} , {\mathrm{dR}}^{\pm}} = \mathcal{O}( \mathrm{Isom}^{\otimes}_{\mathcal{T}_{\infty}} ( \omega^{\pm}_{\mathrm{dR}}, \omega^{\pm}_{\mathrm{B}}) ) \qquad \mbox{ and } \qquad \mathcal{P}_{\mathcal{T}_{\infty}}^{\mathrm{\overline{B}^{\pm}} , \mathrm{dR}^{\pm}} = \mathcal{O}(\mathrm{Isom}^{\otimes}_{\mathcal{T}_{\infty}} ( \omega^{\pm}_{\mathrm{dR}}, \omega^{\pm}_{\mathrm{\overline{B}}})) ,\end{align*} $$

corresponding to all possible choices of signs. The four comparison isomorphisms define four period homomorphisms:

$$ \begin{align*}\mathrm{per}: \mathcal{P}_{\mathcal{T}_{\infty}}^{ \mathrm{B}^+, \mathrm{dR}^+} \longrightarrow \mathbb{C} \qquad \mbox{ and } \qquad \mathrm{per}: \mathcal{P}_{\mathcal{T}_{\infty}}^{ \mathrm{\overline{B}}^+, \mathrm{dR}^+} \longrightarrow \mathbb{C} \ \end{align*} $$

and similarly with $+$ replaced with $-$ . The four Frobenius maps define isomorphisms:

$$ \begin{align*}F_{\infty} \ : \ \mathcal{P}_{\mathcal{T}_{\infty}}^{\mathrm{B}^{+}, \bullet} \ \cong \ \mathcal{P}_{\mathcal{T}_{\infty}}^{\mathrm{\overline{B}}^{-}, \bullet} \quad \mbox{ and } \quad F_{\infty} \ : \ \mathcal{P}_{\mathcal{T}_{\infty}}^{\mathrm{B}^{-}, \bullet} \ \cong \ \mathcal{P}_{\mathcal{T}_{\infty}}^{\mathrm{\overline{B}}^{+}, \bullet} ,\end{align*} $$

where $\bullet \in \{\mathrm {dR}^+, \mathrm {dR}^-\}.$ By composing with the period homomorphism $\mathrm {per}$ , one obtains a full set of eight period homomorphisms for each of our period rings, for example, $\mathrm {per}\, F_{\infty }: \mathcal {P}_{\mathcal {T}_{\infty }}^{\mathrm {B}^{+}, \mathrm {dR}^-} \rightarrow \mathbb {C}.$

There are also four possible rings of de Rham periods. We shall mainly consider two of them:

$$ \begin{align*}\mathcal{P}_{\mathcal{T}_{\infty}}^{\mathrm{dR}^-, \mathrm{dR}^+} = \mathcal{O}(\mathrm{Isom}^{\otimes}_{\mathcal{T}_{\infty}} ( \omega^{+}_{\mathrm{dR}}, \omega^{-}_{\mathrm{dR}})) \qquad \mbox{ and } \qquad \mathcal{P}_{\mathcal{T}_{\infty}}^{\mathrm{dR}^+, \mathrm{dR}^-} = \mathcal{O}(\mathrm{Isom}^{\otimes}_{\mathcal{T}_{\infty}} ( \omega^{-}_{\mathrm{dR}}, \omega^{+}_{\mathrm{dR}})) .\end{align*} $$

Duality induces a canonical isomorphism $\mathcal {P}_{\mathcal {T}_{\infty }}^{\mathrm {dR}^-, \mathrm {dR}^+} \cong \mathcal {P}_{\mathcal {T}_{\infty }}^{\mathrm {dR}^+, \mathrm {dR}^-}$ which we shall not make use of here. Both of these rings admit a single-valued period map, which can be used to detect the nonvanishing of de Rham periods.

Definition 3.7. There is a homomorphism single-valued period

$$ \begin{align*}\mathbf{s}^{-,+} : \mathcal{P}^{\mathrm{dR}^-, \mathrm{dR}^+}_{\mathcal{T}_{\infty}} \longrightarrow \mathbb{C} \otimes_{k^{\mathrm{dR}}} \overline{\mathbb{C}}\end{align*} $$

defined by the composite

Since $\mathbb {C}$ and $\overline {\mathbb {C}}$ have different $k_{\mathrm {dR}}$ -structures, this map defines a point on the torsor of isomorphisms from $\omega _{\mathrm {dR}}^+$ to $\omega _{\mathrm {dR}}^-$ only after extending scalars to $\mathbb {C}\otimes _{k_{\mathrm {dR}}} \overline {\mathbb {C}}$ . The single-valued map therefore sends $[V, f, \omega ]^{\mathfrak {dr}}$ to $ \langle f , ( c^-_{\mathrm {\overline {B}}, \mathrm {dR}})^{-1} F_{\infty } c^+_{\mathrm {B}, \mathrm {dR}} \, \omega \rangle $ .

However, in the case when $k^{\mathrm {dR}} \subset \mathbb R$ , which is the case that we shall mostly consider, we can compose with the multiplication homomorphism $\mathbb {C} \otimes _{k^{\mathrm {dR}}} \overline {\mathbb {C}} \rightarrow \mathbb {C}$ to obtain a homomorphism

$$ \begin{align*}\mathbf{s}^{-,+} : \mathcal{P}^{\mathrm{dR}^-, \mathrm{dR}^+}_{\mathcal{T}_{\infty}} \longrightarrow \mathbb{C} .\end{align*} $$

There is a variant $\mathbf {s}^{+,-}$ obtained by interchanging all $+$ ’s and $-$ ’s in the above. When it is clear from the context, we drop the superscripts and simply write $\mathbf {s}$ .

The composition of torsors between fiber functors defines several coaction morphisms, including

(34) $$ \begin{align} \Delta : \mathcal{P}^{\mathrm{B}^+,\mathrm{dR}^+}_ {\mathcal{T}_{\infty}} \longrightarrow \mathcal{P}^{\mathrm{B}^+,\mathrm{dR}^-}_ {\mathcal{T}_{\infty}} \otimes_{k_{\mathrm{dR}}} \mathcal{P}^{\mathrm{dR}^-,\mathrm{dR}^+}_ {\mathcal{T}_{\infty}}, \end{align} $$

and likewise with $+, -$ interchanged. The period homomorphisms defined earlier are compatible with composition of torsors between fiber functions. In particular, one has

(35) $$ \begin{align} \mathrm{per} \left(\xi\right)= \left( \mathrm{per}\, F_{\infty} \otimes \mathbf{s} \right)\left( \Delta \xi \right) \qquad \mbox{ for all } \quad \xi \ \in \ \mathcal{P}^{\mathrm{B}^+,\mathrm{dR}^+}_ {\mathcal{T}_{\infty}}. \end{align} $$

Definition 3.9. For $s_i$ real and generic (13), let $k_{\mathrm {dR}}= \mathbb Q^{\mathrm {dR}}_{\underline {s}}$ and $\mathbb Q_{\mathrm {B}}= \mathbb Q^{\mathrm {B}}_{\underline {s}}$ , and define an object of rank n in $\mathcal {T}_{\infty }$ , denoted by $\widetilde {M}_\Sigma $ , whose underlying vector spaces are given by

$$ \begin{align*}V_{\mathrm{dR}}^{+} = H^1_{\varpi}(X_{\Sigma}, \nabla_{+ \underline{s}}) \quad , \quad V_{\mathrm{B}}^{+} = H^1(\mathbb{C}\backslash \Sigma, \mathcal{L}_{+ \underline{s}}) \quad , \quad V_{\mathrm{\overline{B}}}^{+} = H^1(\mathbb{C}\backslash \overline{\Sigma}, \mathcal{L}_{+ \underline{s}})\end{align*} $$

and similarly with all $+$ ’s replaced with $-$ ’s. The Frobenius maps $F_{\infty }$ are induced on Betti cohomology by complex conjugation $\mathrm {conj}^*$ and its inverse. The comparison isomorphisms $ c^{+}_{\mathrm {B}, \mathrm {dR}} , c^{+}_{\mathrm {\overline {B}}, \mathrm {dR}}$ are defined by

$$ \begin{align*}\mathrm{comp}_{\mathrm{B}, \varpi}(+\underline{s}) : H^1_{\varpi}(X_{\Sigma}, \nabla_{+ \underline{s}})\otimes_{k_{\mathrm{dR}}} \mathbb{C} \longrightarrow H^1(\mathbb{C}\backslash \Sigma, \mathcal{L}_{+ \underline{s}}) \otimes_{\mathbb Q_{\mathrm{B}}} \mathbb{C},\end{align*} $$

$$ \begin{align*}\mathrm{comp}_{\mathrm{\overline{B}}, \varpi}(+\underline{s}) : H^1_{\varpi}(X_{\Sigma}, \nabla_{+ \underline{s}})\otimes_{k^{\mathrm{dR}}} \overline{\mathbb{C}} \longrightarrow H^1(\mathbb{C}\backslash \overline{\Sigma}, \mathcal{L}_{+ \underline{s}}) \otimes_{\mathbb Q_{\mathrm{B}}} \mathbb{C},\end{align*} $$

and $c^{-}_{\mathrm {B}, \mathrm {dR}} , c^{-}_{\mathrm {\overline {B}}, \mathrm {dR}}$ are similarly defined by replacing all $+$ ’s with $-$ ’s. Since $k_{\mathrm {dR}}=\mathbb Q^{\mathrm {dR}}_{\underline {s}} \subset \mathbb R$ , the complex conjugate $\overline {\mathbb {C}}$ in the left-hand side of the previous equation can, in fact, be replaced with $\mathbb {C}. $

The image of $\widetilde {M}_{\Sigma }$ under the functor $\mathcal {T}_{\infty } \rightarrow \mathcal {T}$ which forgets all data except for $(V^+_{\mathrm {B}}, V_{\mathrm {dR}}^+, c_{\mathrm {B}, \mathrm {dR}}) $ is the object $M_{\Sigma }$ , which was defined in Definition 3.1.

Let us define an $(n \times n)$ matrix

$$ \begin{align*}\widetilde{L}^{\mathfrak{m} }_{{\Sigma}} \quad \in \quad M_{n\times n} ( \mathcal{P}^{\mathrm{B}^+,\mathrm{dR}^+}_{\mathcal{T}_{\infty}})\end{align*} $$

whose entries are

$$ \begin{align*}\left(\widetilde{L}_{\Sigma}^{\mathfrak{m} }\right)_{ij} = \Big[ \widetilde{M}_{\Sigma} \ , \ \delta_i \otimes x^{ s_0} \prod_{k=1}^n (1-x \sigma_k^{-1})^{ s_k} \ , \ - s_j \omega_j \Big]^{\mathfrak{m} },\end{align*} $$

where the Betti homology class lies in $ H^1_{\mathrm {B}}(X_{\Sigma }, \mathcal {L}_{\underline {s}} ) ^\vee = H_1(\mathbb {C}\setminus \Sigma ,\mathcal {L}_{\underline {s}}^\vee )$ and the de Rham class lies in $H^1_{\varpi }(X_{\Sigma }, \nabla _{\underline {s}})$ . We will use the notation $\widetilde {M}_{\Sigma }(\underline {s})$ and $\widetilde {L}^{\mathfrak {m}} _\Sigma (\underline {s})$ when we want to make the dependence on $\underline {s}$ explicit. The image of $\widetilde {L}^{\mathfrak {m}} _\Sigma $ under the natural map $\mathcal {P}^{\mathrm {B}^+,\mathrm {dR}^+}_{\mathcal {T}_{\infty }} \rightarrow \mathcal {P}^{\mathfrak {m} }_{\mathcal {T}} $ is $L^{\mathfrak {m} }_{\Sigma }$ , and its period is the Lauricella matrix

(36) $$ \begin{align} \mathrm{per} \, \widetilde{L}^{\mathfrak{m} }_{{\Sigma}} = L_{\Sigma} . \end{align} $$

3.3 de Rham motivic version and single-valued periods

Now, let us define a de Rham motivic Lauricella function

$$ \begin{align*}\left(\widetilde{L}_{\Sigma}^{\mathfrak{dr}}\right)_{ij} = \left[ \widetilde{M}_{\Sigma} \ , \ \nu_i \ , \ - s_j \omega_j \right]^{\mathfrak{dr}} \quad \in \quad \mathcal{P}_{\mathcal{T}_{\infty}}^{\mathrm{dR}^-, \mathrm{dR}^+} ,\end{align*} $$

where the de Rham class $ \nu _i $ is to be viewed in $H^1_\varpi (X_\Sigma ,\nabla _{\underline {s}})=H^1_{\varpi }(X_{\Sigma }, \nabla _{-\underline {s}})^{\vee }$ and the second is to be viewed in $H^1_{\varpi }(X_{\Sigma }, \nabla _{\underline {s}})$ . Note that this differs from the earlier definition of the de Rham Lauricella matrix in the ring of de Rham periods of the category $\mathcal {T}$ because the de Rham classes reside in different cohomology groups. This is required so that we may speak of its single-valued period. By Corollary 2.12, the matrix of single-valued periods consists of the single-valued Lauricella hypergeometric functions (3):

$$ \begin{align*}\mathbf{s} ( \widetilde{L}^{\mathfrak{dr}}_{\Sigma}) =L^{\mathbf{s}}_{\Sigma},\end{align*} $$

whenever all the $s_i$ are real and generic and satisfy $ s_i>0$ for all $0\leq i \leq n$ , and $s_0+\cdots + s_n < 1/2$ .

3.4 Coaction

The coaction (34), applied to the motivic Lauricella function, will give rise to a third but closely related quantity, given by the matrix

$$ \begin{align*}\left( \widetilde{L}^{\mathfrak{m} , -}_{\Sigma}(\underline{s})\right)_{ij}= \Big[ \widetilde{M}_{\Sigma}(\underline{s}) \ , \ \delta_i \otimes x^{ s_0} \prod_{k=1}^n (1-x \sigma_k^{-1})^{ s_k} \ , \ s_j \omega_j \Big]^{\mathfrak{m} } \quad \in \quad \mathcal{P}_{\mathcal{T}_{\infty}}^{\mathrm{B}^+, \mathrm{dR}^-},\end{align*} $$

where the de Rham class $ s_j \omega _j$ is in $H^1_{\varpi }(X_{\Sigma }, \nabla _{-\underline {s}})$ . It is the image under $F_{\infty }$ of

$$ \begin{align*}\left(\widetilde{L}^{\mathfrak{m} }_{\overline{\Sigma}}(-\underline{s})\right)_{ij} = \Big[ \widetilde{M}_{\Sigma}(\underline{s}) \ , \ \overline{\delta}_i \otimes x^{-s_0} \prod_{k=1}^n (1-x \sigma_k^{-1})^{-s_k} \ , \ s_j \omega_j \Big]^{\mathfrak{m} } \quad \in \quad \mathcal{P}_{\mathcal{T}_{\infty}}^{\mathrm{\overline{B}}^-, \mathrm{dR}^-} ,\end{align*} $$

where the Betti class is viewed in $H_1(\mathbb {C} \backslash \overline {\Sigma }, \mathcal {L}_{-\underline {s}}^\vee )$ (see Remark 2.7) and the de Rham class $ s_j \omega _j$ is viewed in $H^1_{\varpi }(X_{\Sigma }, \nabla _{-\underline {s}})$ as before. More precisely, we have

$$ \begin{align*}\widetilde{L}^{\mathfrak{m} , -}_{\Sigma}(\underline{s}) = F_{\infty} \widetilde{L}^{\mathfrak{m} }_{\overline{\Sigma}}( - \underline{s}),\end{align*} $$

and hence, by definition of the period homomorphism $\mathrm {per} \, F_{\infty } : \mathcal {P}_{\mathcal {T}_{\infty }}^{\mathrm {B}^+, \mathrm {dR}^-} \rightarrow \mathbb {C} $ , the period is

$$ \begin{align*}\mathrm{per} \, F_{\infty} \left( \widetilde{L}^{\mathfrak{m} , -}_{\Sigma}(\underline{s}) \right)= L_{\overline{\Sigma}} (-\underline{s}) .\end{align*} $$

Corollary 3.10. The coaction (34) satisfies

$$ \begin{align*}\Delta \widetilde{L}^{\mathfrak{m} }_{\Sigma}(\underline{s})= \widetilde{L}^{\mathfrak{m} ,-}_{\Sigma}(\underline{s})\otimes \widetilde{L}^{\mathfrak{dr}}_{\Sigma}(\underline{s}) .\end{align*} $$

Equation (35), applied to the matrix $ \widetilde {L}^{\mathfrak {m} }_{\Sigma }(\underline {s})$ , reduces to the equation

$$ \begin{align*}L_{\Sigma}(\underline{s}) = L_{\overline{\Sigma}}(-\underline{s})\, L^{\mathbf{s}}_{\Sigma}(\underline{s}),\end{align*} $$

which is another way of writing formula (32). Consider the following refinement of the period map:

Indeed, by composing it with the multiplication map $\mathbb {C}\otimes _{\mathbb Q}\mathbb {C}\to \mathbb {C}$ , one recovers the usual period map, by (35). Then, under this map, the elements $(\widetilde {L}^{\mathfrak {m} }_{\Sigma }(\underline {s}))_{ij} $ , for $1\leq i,j\leq n$ , map to

(37) $$ \begin{align} &\sum_{\ell=1}^{n} \left(s_{\ell} \int_{\delta_i} x^{-s_0} \prod_{k=1}^n (1- x \overline{\sigma_k}^{-1})^{-s_k} \frac{dx}{x-\overline{\sigma_{\ell}}} \right) \otimes \nonumber\\ &\qquad\qquad\qquad\frac{s_j }{2\pi i} \left( \iint_{\mathbb{C}} |z|^{ 2s_0} \prod_{k=1}^n |1- z \sigma_k^{-1} |^{ 2s_k} \,\left( \frac{d\overline{z}}{\overline{z}-\overline{\sigma_{\ell}}}- \frac{d \overline{z}}{\overline{z}} \right) \wedge \frac{dz}{z-\sigma_j} \right) . \end{align} $$

The point of this formula is that both sides of the tensor product admit a Taylor expansion in the $s_i$ , which is the subject of the next section.

3.5 Variant

We had to assume that the parameters $s_i$ are real in order to obtain a Tannakian interpretation of the single-valued period homomorphism. This was so that the subfield $\overline {k}(s_0,\ldots ,s_n)\subset \mathbb {C}$ is isomorphic to $k(s_0,\ldots ,s_n)$ , which ensures that there is a comparison map for de Rham cohomology associated with the complex conjugate embedding of k:

$$ \begin{align*}H^1_{\mathrm{dR}}(X_\Sigma,\nabla_{\underline{s}})\otimes_{k_{\underline{s}}^{\mathrm{dR}}}\overline{\mathbb{C}} \longrightarrow H^1_{\mathrm{B}}(\mathbb{C}\backslash\overline{\Sigma},\mathcal{L}_{\underline{s}})\otimes_{\mathbb Q_{\underline{s}}^{\mathrm{dR}}}\mathbb{C}.\end{align*} $$

However, such an assumption is unnatural, for instance, because formulae such as (5) are true for all $s_i\in \mathbb {C}$ for which they make sense. A way to remedy this would be to treat the $s_i$ as formal parameters and work with modules over the polynomial ring $k[s_0,\ldots ,s_n]$ . This would also be needed to build a bridge between the framework of cohomology with coefficients that we discussed in this section and the Taylor expansions that we consider in the rest of this article.

4.1 Renormalization of line integrals

Let $\Sigma =\{\sigma _0,\ldots ,\sigma _n\}$ be distinct complex numbers with $\sigma _0=0$ . We fix an index $i\in \{1,\ldots ,n\}$ and a locally finite path $\delta _i$ in $\mathbb {C}\setminus \Sigma $ from $0$ to $\sigma _i$ . More precisely, $\delta _i:(0,1)\to \mathbb {C}\setminus \Sigma $ is a smooth path that can be extended to a smooth path $\overline {\delta _i}:[0,1]\to \mathbb {C}$ satisfying $\overline {\delta _i}(0)=0$ and $\overline {\delta _i}(1)=\sigma _i$ . As in all this paper, we assume that $\delta _i$ does not wind infinitely around $0$ or $\sigma _i$ (Remark 2.3).

Let $\omega $ be a global meromorphic form on $\mathbb {P}^1(\mathbb {C})$ with logarithmic poles at $\Sigma \cup \{\infty \}$ . We define

$$ \begin{align*}\Omega= x^{s_0}\prod_{k=1}^n(1-x\sigma_k^{-1})^{s_k}\,\omega\end{align*} $$

and view it as a multivalued $1$ -form which depends holomorphically on the parameters $\underline {s}$ , and that we wish to integrate along $\delta _i$ .

Without any assumption on the residues of $\omega $ at $0$ and $\sigma _i$ , the integral of $\Omega $ along $\delta _i$ defines a holomorphic function of the parameters $\underline {s}$ in the domain $\{\operatorname {Re}(s_0),\operatorname {Re}(s_i)>0\}$ .

Definition 4.3. Following [Reference Broedel, Schlotterer, Stieberger and TerasomaBD2], let us define the renormalized version of $\Omega $ with respect to the points $\{0,\sigma _i\}$ to be

$$ \begin{align*}\Omega^{\mathrm{ren}_{i}}= \Omega - \mathrm{Res}_0(\omega)\, x^{s_0}\frac{dx}{x} - \mathrm{Res}_{\sigma_i}(\omega)\Big(\sigma_i^{s_0}\prod_{k\neq i}(1-\sigma_i\sigma_k^{-1})^{s_k}\Big)\, (1-x\sigma_i^{-1})^{s_i} \frac{dx}{x-\sigma_i} \ \cdot\end{align*} $$

Proposition 4.4. We have, for every $\underline {s}$ in the domain $\{\operatorname {Re}(s_0),\operatorname {Re}(s_i)>0\}$ , the equality

(38) $$ \begin{align} \int_{\delta_i}\Omega = \mathrm{Res}_0(\omega)\frac{\sigma_i^{s_0}}{s_0} - \mathrm{Res}_{\sigma_i}(\omega)\frac{\sigma_i^{s_0}\prod_{k\neq i}(1-\sigma_i\sigma_k^{-1})^{s_k}}{s_i} + \int_{\delta_i}\Omega^{\mathrm{ren}_i}.\end{align} $$

The integral of $\Omega ^{\mathrm {ren}_i}$ along $\delta _i$ defines a holomorphic function of the parameters $\underline {s}$ in the domain $\{\operatorname {Re}(s_0),\operatorname {Re}(s_i)>-1\}$ .

Proof. The equality follows from the computations:

$$ \begin{align*}\int_{\delta_i}x^{s_0}\frac{dx}{x} = \frac{\sigma_i^{s_0}}{s_0} \qquad \mbox{and} \qquad \int_{\delta_i}(1-x\sigma_i^{-1})^{s_i}\frac{dx}{x-\sigma_i} = -\frac{1}{s_i}\ \cdot\end{align*} $$

Since $\omega -\mathrm {Res}_0(\omega )\frac {dx}{x}$ does not have a pole at $0$ , the singularities of $\Omega ^{\mathrm {ren}_i}$ at $0$ are at worst of the type $x^{s_0}dx$ and therefore integrable for $\operatorname {Re}(s_0)>-1$ . Likewise, its singularities at $\sigma _i$ are at worst of the type $(1-x\sigma _i^{-1})^{s_i}dx$ and are integrable for $\operatorname {Re}(s_i)>-1$ .

Since $\underline {s}=0$ is in the domain of convergence of the integral of $\Omega ^{\mathrm {ren}_i}$ along $\delta _i$ , we see that (38) yields a Laurent series expansion at $\underline {s}=0$ for the integral of $\Omega $ along $\delta _i$ . We record the following special case, which shows that the Lauricella functions $(L_\Sigma )_{ij}$ admit Taylor expansions around $\underline {s}=0$ . Here, the integrand of $(L_\Sigma )_{ij}$ is understood according to the conventions of Remark 4.1.

Proposition 4.5. We have the following equality:

$$ \begin{align*}(L_\Sigma)_{ij} = \mathbf{1}_{i=j}\Big(\sigma_i^{s_0}\prod_{k\neq i}(1-\sigma_i\sigma_k^{-1})^{s_k}\Big)-s_j\int_{\delta_i}\Omega_j^{\mathrm{ren}_i},\end{align*} $$

where

$$ \begin{align*}\Omega_j^{\mathrm{ren}_i}= \Big(x^{s_0}\prod_{k\neq i}(1-x\sigma_k^{-1})^{s_k} - \mathbf{1}_{i=j}\, \sigma_i^{s_0}\prod_{k\neq i}(1-\sigma_i\sigma_k^{-1})^{s_k}\Big)(1-x\sigma_i^{-1})^{s_i}\frac{dx}{x-\sigma_j},\end{align*} $$

and the integral on the right-hand side defines a holomorphic function of the parameters $\underline {s}$ in the domain $\{\operatorname {Re}(s_0),\operatorname {Re}(s_i)>-1\}$ .

4.2 Renormalization of complex volume integrals

Let $\Sigma =\{\sigma _0,\ldots ,\sigma _n\}$ be distinct complex numbers with $\sigma _0=0$ . Let $\omega $ be a global meromorphic form on $\mathbb {P}^1(\mathbb {C})$ with logarithmic poles at $\Sigma \cup \{\infty \}$ . We define

$$ \begin{align*}\Omega^{\mathbf{s}} \ = \ |z|^{2 s_0} \prod_{k=1}^n |1-z\sigma_k^{-1}|^{2s_k} \, \omega\end{align*} $$

and view it as a smooth $1$ -form on $\mathbb {P}^1(\mathbb {C})\setminus (\Sigma \cup \{\infty \})$ , which depends holomorphically on the parameters $\underline {s}$ . We fix an index $i\in \{1,\ldots ,n\}$ and consider the integral

$$ \begin{align*}-\frac{1}{2\pi i}\iint_{\mathbb{C}}\left(\frac{d\overline{z}}{\overline{z}-\overline{\sigma_i}}-\frac{d\overline{z}}{\overline{z}}\right)\wedge \Omega^{\mathbf{s}}.\end{align*} $$

Without any assumption on the residues of $\omega $ , this integral defines a holomorphic function of the parameters $\underline {s}$ in the domain

$$ \begin{align*}D=\{\mathrm{Re}(s_0),\mathrm{Re}(s_i)>0\; ,\; \mathrm{Re}(s_k)>-1/2 \;\;(k\notin\{0,i\})\; , \; \mathrm{Re}(s_0)+\cdots+\mathrm{Re}(s_n)<1/2\}.\end{align*} $$

One can check this by passing to local coordinates around every point of $\Sigma \cup \{\infty \}$ .

Definition 4.6. Following [Reference Broedel, Schlotterer, Stieberger and TerasomaBD2], let us define the renormalized version of $\Omega ^{\mathbf {s}}$ with respect to the points $\{0,\sigma _i\}$ to be

$$ \begin{align*}\Omega^{\mathbf{s},\mathrm{ren}_{i}}= \Omega^s - \mathrm{Res}_0(\omega)\, |z|^{2s_0}\frac{dz}{z} - \mathrm{Res}_{\sigma_i}(\omega)\Big(|\sigma_i|^{2s_0}\prod_{k\neq i}|1-\sigma_i\sigma_k^{-1}|^{2s_k}\Big)\, |1-z\sigma_i^{-1}|^{2s_i} \frac{dz}{z-\sigma_i}\ \cdot\end{align*} $$

The following lemma will be used several times in the sequel. It is a single-valued analog of the identity $\int _0^{\sigma }z^s\frac {dz}{z}=\frac {\sigma ^s}{s}$ .

Lemma 4.7. For any $\sigma ,s \in \mathbb {C}$ with $\mathrm {Re}(s)>0$ , we have

$$ \begin{align*}-\frac{1}{2\pi i} \iint_{\mathbb{C}} \left(\frac{d\overline{z}}{\overline{z}-\overline{\sigma}} -\frac{d\overline{z}}{\overline{z}} \right)\wedge |z|^{2s}\frac{dz}{z} = \frac{|\sigma|^{2s}}{s}.\end{align*} $$

Proof. One verifies in local polar coordinates that the integral converges. Let

$$ \begin{align*}F = - \frac{1}{s} |z|^{2s} \left( \frac{ d\overline{z}}{\overline{z}-\overline{\sigma}} -\frac{d\overline{z}}{\overline{z}} \right).\end{align*} $$

Its total derivative is the integrand on the left-hand side. By Stokes’s formula applied to the complement of three disks in $\mathbb P^1(\mathbb {C})$ centered at $0$ , $\infty $ , $\sigma $ , it suffices to compute

$$ \begin{align*}-\frac{1}{2\pi i} \int_{D_\varepsilon} F \ = \ \frac{1}{s}\frac{ 1}{2\pi i} \int_{D_\varepsilon} |z|^{2s}\left(\frac{ d\overline{z}}{\overline{z}-\overline{\sigma}} -\frac{d\overline{z}}{\overline{z}} \right),\end{align*} $$

where $D_\varepsilon $ is the union of three circles of radius $\varepsilon $ winding negatively around $0$ , $\infty $ , and $ \sigma $ . When $\varepsilon $ goes to zero, the integral around the first and the second vanish, and the integral around the third gives $2\pi i\,|\sigma |^{2s}$ .

Proposition 4.8. We have, for every $\underline {s}$ in the domain D, the equality

(39) $$ \begin{align} \begin{aligned} -&\frac{1}{2\pi i}\iint_{\mathbb{C}}\left(\frac{d\overline{z}}{\overline{z}-\overline{\sigma_i}}-\frac{d\overline{z}}{\overline{z}}\right)\wedge\Omega^{\mathbf{s}} = \\ & \mathrm{Res}_0(\omega)\frac{|\sigma_i|^{2s_0}}{s_0} - \mathrm{Res}_{\sigma_i}(\omega)\frac{|\sigma_i|^{2s_0}\prod_{k\neq i}|1-\sigma_i\sigma_k^{-1}|^{2s_k}}{s_i} -\frac{1}{2\pi i} \iint_{\mathbb{C}}\left(\frac{d\overline{z}}{\overline{z}-\overline{\sigma_i}}-\frac{d\overline{z}}{\overline{z}}\right)\wedge\Omega^{\mathbf{s},\mathrm{ren}_i}. \end{aligned} \end{align} $$

The integral on the right-hand side defines a holomorphic function of the parameters $\underline {s}$ in the domain

(40) $$ \begin{align} D'=\{\operatorname{Re}(s_k)>-1/2 \;\;(k\in\{0,\ldots,n\})\; ,\; \mathrm{Re}(s_0)+\cdots+\mathrm{Re}(s_n)<1/2\}. \end{align} $$

Proof. The equality follows from the computations

$$ \begin{align*}-\frac{1}{2\pi i}\iint_{\mathbb{C}}\left(\frac{d\overline{z}}{\overline{z}-\overline{\sigma_i}}-\frac{d\overline{z}}{\overline{z}}\right)\wedge |z|^{2s_0}\frac{dz}{z} = \frac{|\sigma_i|^{2s_0}}{s_0} ,\end{align*} $$

and

$$ \begin{align*}-\frac{1}{2\pi i}\iint_{\mathbb{C}}\left(\frac{d\overline{z}}{\overline{z}-\overline{\sigma_i}}-\frac{d\overline{z}}{\overline{z}}\right)\wedge |1-z\sigma_i^{-1}|^{2s_i}\frac{dz}{z-\sigma_i} = - \frac{1}{s_i},\end{align*} $$

which follows from Lemma 4.7 (for the second, apply the change of variables $z\leftrightarrow \sigma -z$ and multiply by $|\sigma |^{-2s}$ ). Since $\omega -\mathrm {Res}_0(\omega )\frac {dx}{x}$ does not have a pole at $0$ , the singularities of $\Omega ^{\mathbf {s},\mathrm {ren}_i}$ at $0$ are at worst of the type $|z|^{2s_0}dz$ and therefore

$$ \begin{align*}\left(\frac{d\overline{z}}{\overline{z}-\overline{\sigma_i}}-\frac{d\overline{z}}{\overline{z}}\right)\wedge\Omega^{\mathbf{s},\mathrm{ren}_i}\end{align*} $$

is integrable around zero for $\operatorname {Re}(s_0)>-1/2$ . Likewise, it is integrable around $\sigma _i$ for $\operatorname {Re}(s_i)>-1/2$ .

Since $\underline {s}=0$ lies in the domain $D'$ , this proposition provides a Laurent series expansion at $\underline {s}=0$ for the integral on the left-hand side. We record the following special case which shows that the single-valued Lauricella functions (3) admit Taylor expansions around $\underline {s}=0$ .

Proposition 4.9. We have the following equality:

$$ \begin{align*}(L^{\mathbf{s}}_\Sigma)_{ij} = \mathbf{1}_{i=j}\Big(|\sigma_i|^{2s_0}\prod_{k\neq i}|1-\sigma_i\sigma_k^{-1}|^{2s_k}\Big)+\frac{s_j}{2\pi i}\iint_{\mathbb{C}}\left(\frac{d\overline{z}}{\overline{z}-\overline{\sigma_i}}-\frac{d\overline{z}}{\overline{z}}\right)\wedge \Omega_j^{\mathbf{s},\mathrm{ren}_i},\end{align*} $$

where

$$ \begin{align*}\Omega_j^{\mathbf{s},\mathrm{ren}_i}= \Big(|z|^{2s_0}\prod_{k\neq i}|1-z\sigma_k^{-1}|^{2s_k} - \mathbf{1}_{i=j}\,|\sigma_i|^{2s_0}\prod_{k\neq i}|1-\sigma_i\sigma_k^{-1}|^{2s_k}\Big)|1-z\sigma_i^{-1}|^{2s_i}\frac{dz}{z-\sigma_j}\ \cdot\end{align*} $$

The integral on the right-hand side defines a holomorphic function of the parameters $\underline {s}$ in the domain $D'$ described in (40).

5.1 Categorical and Tannakian

1. 1. Let $\mathcal {MT}(k)$ denote the category of mixed Tate motives over k. One can replace $\mathcal {MT}(k)$ with a category of Betti and de Rham realizations, with essentially no change to our arguments. The category $\mathcal {MT}(k)$ has a canonical fiber functor $\varpi : \mathcal {MT}(k) \rightarrow \mathrm {Vec}_{\mathbb Q}$ . Let us set

   $$ \begin{align*}G_{\mathcal{MT}(k)}^{\varpi} =\mathrm{Aut}_{\varpi}^{\otimes}\, \mathcal{MT}(k).\end{align*} $$
   It is an affine group scheme over $\mathbb Q$ . Let us denote by $\omega _{\mathrm {B}}: \mathcal {MT}(k) \rightarrow \mathrm {Vec}_{\mathbb Q}$ the Betti realization functor with respect to the given embedding $k \subset \mathbb {C}$ . The de Rham realization functor $\omega _{\mathrm {dR}}: \mathcal {MT}(k)\rightarrow \mathrm {Vec}_k$ is obtained from $\varpi $ by extending scalars: $\omega _{\mathrm {dR}}= \varpi \otimes _{\mathbb Q} k$ .

2. 2. Let $\mathcal {P}^{\mathfrak {m} } = \mathcal {O}(\mathrm {Isom}^{\otimes }_{\mathcal {MT}(k)}(\varpi ,\omega _{\mathrm {B}}))$ denote the $\mathbb Q$ -algebra of motivic periods on $\mathcal {MT}(k)$ . Let

   $$ \begin{align*}\mathcal{P}^{\varpi} = \mathcal{O}(G_{\mathcal{MT}(k)}^{\varpi})\end{align*} $$
   denote the $\mathbb Q$ -algebra of (canonical, i.e., $(\varpi ,\varpi )$ )) de Rham motivic periods. The latter is a graded Hopf algebra, and the former is a graded algebra comodule over it. Denote the corresponding motivic coaction by
   $$ \begin{align*}\Delta: \mathcal{P}^{\mathfrak{m} } \longrightarrow \mathcal{P}^{\mathfrak{m} } \otimes_{\mathbb Q} \mathcal{P}^{\varpi}.\end{align*} $$
   We let $\mathrm {per} : \mathcal {P}^{\mathfrak {m} } \rightarrow \mathbb {C}$ denote the period homomorphism and let $\mathbf {s}:\mathcal {P}^\varpi \to \mathbb {C}$ denote the single-valued period homomorphism (see [Reference Belavin, Polyakov and ZamolodchikovBD1, §2.6]).

3. 3. Let $\mathbb {L} ^{\mathfrak {m} } = [ \mathbb Q(-1), 1_{\mathrm {B}}^{\vee }, 1_{\varpi }]^{\mathfrak {m} }$ , respectively, $\mathbb {L} ^{\varpi } = [ \mathbb Q(-1), 1_{\varpi }^{\vee }, 1_{\varpi }]^{\varpi }$ , denote the Lefschetz motivic period, whose period is $2\pi i$ , respectively, its (canonical) de Rham version.

4. 4. Let $\mathcal {P}^{\mathfrak {m} ,+}\subset \mathcal {P}^{\mathfrak {m}} $ denote the subspace of effective motivic periods. It is spanned by the motivic periods of objects $M\in \mathcal {MT}(k)$ with $W_{-1}M=0$ . It is a nonnegatively graded subalgebra and sub- $\mathcal {P}^\varpi $ -comodule of $\mathcal {P}^{\mathfrak {m}} $ , and contains the Lefschetz motivic period $\mathbb {L} ^{\mathfrak {m}} $ . There is a canonical projection homomorphism

   $$ \begin{align*}\pi^{\mathfrak{m} ,+}_{\varpi} : \mathcal{P}^{\mathfrak{m} ,+} \longrightarrow \mathcal{P}^{\varpi},\end{align*} $$
   which, in particular, sends $\mathbb {L} ^{\mathfrak {m} }$ to zero. It can be defined by composing the coaction $\Delta $ with projection onto the weight-graded zero piece $W_0 \mathcal {P}^{\mathfrak {m} ,+} \cong \mathbb Q$ .

5.2 Geometric

Let $\Sigma = \{\sigma _0, \sigma _1,\ldots , \sigma _n \} \subset \mathbb {A}^1(k)$ be distinct points with $\sigma _0=0$ . Recall that $X_{\Sigma }= \mathbb {A}_k^1 \backslash \Sigma .$

1. 1. Let $t_0 \in T_0\mathbb {A}^1_k$ denote the tangent vector $1$ at $0$ . For each $ i\geq 1$ , let

   $$ \begin{align*}t_{\sigma_i} \quad \in \quad T_{\sigma_i}\mathbb{A}_k^1\end{align*} $$
   be the tangent vector $\sigma _i$ based at the point $\sigma _i$ . Let $\pi _1^{\mathrm {top}}(X_\Sigma (\mathbb {C}),t_i,-t_j)$ denote the fundamental torsor of homotopy classes of paths between the tangential basepoints $t_i$ and $-t_j$ . Concretely, an element of this set is represented by a (piecewise) smooth path $\gamma :(0,1)\to \mathbb {C}\setminus \Sigma $ that can be extended to a path $\overline {\gamma }:[0,1]\to \mathbb {C}$ which is smooth at $0$ and $1$ and satisfies $\overline {\gamma }(0)=\sigma _i$ and $\overline {\gamma }(1)=\sigma _j$ , with prescribed velocities $\overline {\gamma }'(0)=t_i$ and $\overline {\gamma }'(1)=t_j$ .

2. 2. For all $0\leq i,j\leq n$ , denote by

   $$ \begin{align*}{}_i \Pi^{\bullet}_{j} = \pi_1^{\bullet}(X_{\Sigma}, t_i, -t_j) \qquad \mbox{ where } \quad \bullet \in \{\mathrm{B}, \varpi , \mathrm{mot} \}\end{align*} $$
   the Betti, (canonical) de Rham, or motivic fundamental torsor of paths from the tangential basepoint $t_i$ at $\sigma _i$ , to $-t_j$ at $\sigma _j$ . The versions ${}_i\Pi _j^{\mathrm {B}}$ and ${}_i\Pi _j^\varpi $ are affine $\mathbb {Q}$ -schemes; the version ${}_i\Pi _j^{\mathrm {mot}}$ is an affine scheme in the category $\mathcal {MT}(k)$ , which simply means that its affine ring is isomorphic to an algebra ind-object $\mathcal {O}({}_i\Pi _j^{\mathrm {mot}})$ in $\mathcal {MT}(k)$ , whose ( $\mathrm {B}$ , resp. $\varpi $ ) realizations are $\mathcal {O}({}_i\Pi _j^{\mathrm {B}})$ and $\mathcal {O}({}_i\Pi _j^{\varpi })$ . There is a groupoid structure ${}_a \Pi ^{\bullet }_b \times {}_b \Pi ^{\bullet }_c \longrightarrow {}_a \Pi ^{\bullet }_c \ ,$ for all $a,b,c$ in the set of our tangential basepoints (composition of paths). There are maps
   $$ \begin{align*}\gamma \mapsto \gamma^{\mathrm{B}} \ : \ \pi_1^{\mathrm{top}}(X_{\Sigma}(\mathbb{C}), t_i, -t_j) \longrightarrow {}_i \Pi^{\mathrm{B}}_{j}(\mathbb Q),\ \end{align*} $$
   which are Zariski-dense and compatible with the groupoid structure. (They identify ${}_i\Pi _j^{\mathrm {B}}$ as the Malčev, or pro-unipotent, completion of $\pi _1^{\mathrm {top}}(X_\Sigma (\mathbb {C}), t_i,-t_j)$ .)

3. 3. The scheme ${}_i\Pi _j^\varpi $ does not depend on $i,j$ , although the action of the (canonical) motivic Galois group $G_{\mathcal {MT}(k)}^\varpi $ upon it does depend on $i,j$ . On $X_{\Sigma }$ , we considered the logarithmic 1-forms $\omega _i$ for $i=0,\ldots ,n$ (14). Since they have residue $0$ or $\pm 1$ at points of $\Sigma $ , they generate the canonical $\mathbb Q$ -structure (or $\varpi $ -structure) on the de Rham realization of $H^1(X_{\Sigma }) \in \mathrm {Ob} (\mathcal {MT}(k))$ , which we shall denote simply by $H_{\varpi }^1(X_{\Sigma })$ . It is the $\mathbb Q$ -vector space spanned by the classes $\omega _i$ . The affine ring of the de Rham canonical torsor of paths is

   $$ \begin{align*}\mathcal{O}({}_i \Pi_{j}^\varpi) \cong \bigoplus_{n\geq 0} H^1_{\varpi}(X_{\Sigma})^{\otimes n}.\end{align*} $$
   It is isomorphic to the graded tensor coalgebra on $H^1_{\varpi }(X_{\Sigma })$ , equipped with the shuffle product and deconcatenation coproduct. For any commutative unital $\mathbb Q$ -algebra R, the R-points of ${}_i \Pi _{j}^\varpi $ ,
   $$ \begin{align*}{}_i \Pi_{j}^\varpi (R) \ \subset \ R \langle \langle e_0, \ldots, e_n \rangle \rangle,\end{align*} $$
   are the set of group-like formal power series with respect to the continuous coproduct for which the $e_i$ are primitive. They are formal power series
   $$ \begin{align*}S = \sum_{w\in \{e_0,e_1,\ldots, e_n\}^{\times}} S(w) \, w \quad \in \quad R \langle \langle e_0, \ldots, e_n \rangle \rangle,\end{align*} $$
   where the linear extension of the map $w\mapsto S(w)$ is a homomorphism with respect to the shuffle product. The letters $e_i$ , for $1\leq i \leq n$ , are dual to the $\omega _i$ .

4. 4. For all $0 \leq i \leq n$ , let ${}_01_i \in {}_0\Pi _i^\varpi (\mathbb Q)$ denote the canonical $\varpi $ -path. It is defined by the augmentation map $\mathcal {O}({}_0\Pi _i^\varpi ) \rightarrow \mathbb Q$ onto the degree zero component (or by quotienting by the Hodge filtration $F^1$ ). It is the formal power series $1 \in \mathbb Q\langle \langle e_0,\ldots , e_n \rangle \rangle $ consisting only of the empty word.

5. 5. Since the motivic fundamental torsor of paths is the spectrum of an ind-object in the category $\mathcal {MT}(k)$ , there is a canonical universal comparison isomorphism of schemes (see [Reference Brown and DupontB3, §4.1])

   $$ \begin{align*}\mathrm{comp}^{\mathfrak{m} }_{\mathrm{B}, \varpi} \quad : \quad {}_i\Pi^{\mathrm{B}}_j \times_{\mathbb Q} \mathcal{P}^{\mathfrak{m} } \overset{\sim}{\longrightarrow} {}_i\Pi_j^\varpi \times_{\mathbb Q} \mathcal{P}^{\mathfrak{m} },\end{align*} $$
   for all $0\leq i,j \leq n$ , compatible with the groupoid structure. Composing $\mathrm {comp}^{\mathfrak {m} }_{\mathrm {B},\varpi }$ with the period homomorphism gives back the canonical comparison isomorphism whose coefficients are regularized iterated integrals:
   $$ \begin{align*}\mathrm{comp}_{\mathrm{B}, \varpi} \quad : \quad {}_i\Pi^{\mathrm{B}}_j \times_{\mathbb Q} \mathbb{C} \overset{\sim}{\longrightarrow} {}_i\Pi_j^\varpi \times_{\mathbb Q} \mathbb{C}.\end{align*} $$

6.1 Generalized (motivic) associators

Let us fix an index $1\leq i\leq n$ . The generalized associator $\mathcal {Z}^i$ , that we will soon define, is a formal power series that records all the iterated integrals that can be computed on $X_\Sigma $ on a given (homotopy class of) path $\gamma _i$ between the tangential basepoints $t_0$ and $-t_i$ . For this, we will need to fix a class $\gamma _i\in \pi _1^{\mathrm {top}}(X_\Sigma (\mathbb {C}),t_0,-t_i)$ as in §5.2(1). We will need to impose an extra condition on $\gamma _i$ that can be stated as follows. Consider the following maps between fundamental torsors:

(41) $$ \begin{align} \pi_1^{\mathrm{top}}(X_\Sigma(\mathbb{C}),t_0,-t_i) \longrightarrow \pi_1^{\mathrm{top}}(\mathbb{C}\setminus \{0,\sigma_i\},t_0,-t_i) \longrightarrow \pi_1^{\mathrm{top}}(\mathbb{C}\setminus \{\sigma_i\}, 0, -t_i)\simeq \mathbb{Z}. \end{align} $$

The first map is induced by the inclusion $X_\Sigma (\mathbb {C})\hookrightarrow \mathbb {C}\setminus \{0,\sigma _i\}$ . The second map is also induced by the inclusion $\mathbb {C}\setminus \{0,\sigma _i\}\hookrightarrow \mathbb {C}\setminus \{\sigma _i\}$ , where the tangential basepoint $t_0$ becomes the usual basepoint $0$ . There is a canonical element in $\pi _1^{\mathrm {top}}(\mathbb {C}\setminus \{\sigma _i\}, 0, -t_i)$ , namely the homotopy class of the straight path $t\mapsto \sigma _i t$ , which provides the identification of the torsor $\pi _1^{\mathrm {top}}(\mathbb {C}\setminus \{\sigma _i\}, 0, -t_i)$ with the fundamental group $\pi _1^{\mathrm {top}}(\mathbb {C}\setminus \{\sigma _i\}, 0)\simeq \mathbb {Z}$ .

Roughly speaking, it means that $\gamma _i$ ‘does not wind around’ $\sigma _i$ . A representative of an admissible class in $\pi _1^{\mathrm {top}}(X_\Sigma (\mathbb {C}), t_0,-t_i)$ can be constructed as follows: first, travel via a small arc from the tangential basepoint $t_0$ to $\sigma _i\varepsilon $ for a small $\varepsilon>0$ , and then travel via the straight path $t\mapsto \sigma _i t$ toward the tangential basepoint $-t_i$ (avoiding if necessary the points $\sigma _j$ that are on the line segment between $\sigma _i\varepsilon $ and $\sigma _i$ via small arcs). We see in Remark 6.12 that the admissibility of $\gamma _i$ is equivalent to the vanishing of the following regularized iterated integral:

$$ \begin{align*}\int_{\gamma_i}\omega_i=0.\end{align*} $$

We fix an admissible class $\gamma _i$ for every $1\leq i\leq n$ for the remainder of this article.

Definition 6.2. Define a formal power series

$$ \begin{align*}\mathcal{Z}^{i, \mathfrak{m} } \ \in \ {}_0\Pi_i^\varpi (\mathcal{P}^{\mathfrak{m} }) \subset \mathcal{P}^{\mathfrak{m} } \langle \langle e_0,\ldots, e_n\rangle\rangle\end{align*} $$

by $\mathcal {Z}^{i, \mathfrak {m} } = \mathrm {comp}^{\mathfrak {m} }_{\mathrm {B},\varpi } (\gamma ^{\mathrm {B}}_i)$ where $\gamma ^{\mathrm {B}}_i\in {}_0\Pi _i^{\mathrm {B}}(\mathbb Q)$ is the image of $\gamma _i$ as in §5.2(2). It is called a generalized motivic associator.

Since $\mathcal {O}({}_0 \Pi _i^\varpi )$ has weights $\geq 0$ , it follows that $\mathcal {Z}^{i, \mathfrak {m} }$ actually lies in $ {}_0\Pi _i (\mathcal {P}^{\mathfrak {m} ,+})$ . Explicitly,

$$ \begin{align*}\mathcal{Z}^{i, \mathfrak{m} } = \sum_{w\in \{e_0,\ldots, e_n\}^{\times}} \left[ \mathcal{O}(\pi_1^{\mathrm{mot}}(X_{\Sigma}, t_0, -t_i)), \gamma_i^{\mathrm{B}}, w\right]^{\mathfrak{m} } w,\ \end{align*} $$

where the sum is over all words w in $e_i$ , which are in turn dual to words in the $\omega _i$ (14) and hence define an element $w \in \mathcal {O}( {}_0\Pi _i^\varpi )$ . The path $\gamma _i^{\mathrm {B}}$ is viewed as an element in $\mathcal {O}( {}_0\Pi ^{\mathrm {B}}_i)^{\vee }$ .

Explicitly, it is the group-like formal power series

$$ \begin{align*}\mathcal{Z}^i= \sum_{w \in \{e_0,\ldots, e_n\}^{\times}} \left( \int_{\gamma_i} w \right) w \quad \in \quad \mathbb{C} \langle \langle e_0,\ldots, e_n\rangle \rangle,\end{align*} $$

where the sum is over all words w in $\{e_0,\ldots , e_n\}$ , and the integral is the regularized iterated integral (from left to right) of the corresponding word in $\{\omega _0,\ldots , \omega _n\}$ .

When we wish to emphasize the dependence on the variables $e_i$ , we shall write $\mathcal {Z}^{i, \mathfrak {m} }(e_0,\ldots , e_n)$ for $\mathcal {Z}^{i, \mathfrak {m} }$ , and so on.

Example 6.4. Let $\Sigma = \{0,1\}$ and $k= \mathbb Q$ . Then $ \mathcal {Z}^{1,\mathfrak {m} }= \mathcal {Z}^{\mathfrak {m} }$ , where

(42) $$ \begin{align}\mathcal{Z}^{\mathfrak{m} } = \sum_{w \in \{e_0,e_1\}^{\times} } \zeta^{ \mathfrak{m}}(w) w \quad \in \quad \mathcal{P}^{\mathfrak{m} }_{\mathcal{MT}(\mathbb Q)} \langle \langle e_0,e_1\rangle \rangle \end{align} $$

is the motivic Drinfeld associator. Drinfeld’s associator is $\mathcal {Z} = \mathrm {per}\left ( \mathcal {Z}^{\mathfrak {m} } \right ) \in \mathbb R \langle \langle e_0, e_1 \rangle \rangle $ .

6.2 Beta quotients

Let R be any commutative unital $\mathbb Q$ -algebra.

Definition 6.5. Consider the abelianization map

$$ \begin{align*}F\mapsto \overline{F} \quad : \quad R\langle \langle e_0,\ldots, e_n \rangle \rangle \longrightarrow R [[s_0,\ldots, s_n]],\end{align*} $$

which sends $e_i$ to $s_i$ , where the $s_i$ are commuting variables. We shall call $\overline {F}$ the abelianization of F.

Computing the abelianization of a group-like series is easy, as the next lemma shows.

Lemma 6.6. For a group-like series F, we have

$$ \begin{align*}\overline{F} = \exp \left( F(e_0) s_0 + \cdots +F(e_n) s_n \right)= \prod_{k=0}^n \exp (F(e_k) s_k).\end{align*} $$

Proof. Since F is group-like, $F= \exp (\log F))$ , where $\log (F)$ is a Lie series of the form

$$ \begin{align*}\log (F) = \sum_{k=0}^n F(e_k) e_k + \mbox{commutators}.\end{align*} $$

The exponential and the logarithm are taken with respect to the concatenation product, and commute with the abelianization map. Since abelianization sends all commutators to zero, we have $\overline {\log F}=\sum _{k=0}^nF(e_k)s_k$ and the result follows.

Definition 6.7. For any series $F \in R\langle \langle e_0,\ldots , e_n \rangle \rangle $ , let us write

(43) $$ \begin{align} F= F_{\varnothing} + F_0 e_0 + \cdots + F_n e_n, \end{align} $$

where $F_{\varnothing } \in R$ denotes the coefficient of the empty word (constant coefficient) and $F_{j}$ is obtained from F by deleting the last letter from all words ending in $e_j$ . We call the abelianization $\overline {F_j}$ the $j{th}$ beta quotient of F.

For any two series $A, B \in R\langle \langle e_0,\ldots , e_n \rangle \rangle , $ their product satisfies

(44) $$ \begin{align} \left( A B \right)_j = A \left( B_j\right) + A_j B_{\varnothing}. \end{align} $$

We verify that, if F is invertible, then

(45) $$ \begin{align} \overline{(F^{-1})_j} = - \frac{1}{F_{\varnothing}} \frac{ \overline{F_j} }{ \overline{F} }. \end{align} $$

This follows from applying (44) to $F F^{-1} = 1$ , which gives $F (F^{-1})_j + F_j F^{-1}_{\varnothing }=0$ , and then applying the abelianization map. All series F that we shall consider are group-like (for the continuous coproduct on formal power series for which all letters $e_k$ are primitive) and therefore have constant term $F_{\varnothing }= 1$ .

Recall that $F(w)$ denotes (a linear combination of) coefficients of words w in F.

Lemma 6.9. For any series $F\in R\langle \langle e_0,\ldots ,e_n\rangle \rangle $ , we have

where $[ w ] e_j$ denotes the right concatenation of $e_j$ to any linear combination w of words in the letters $e_0,\ldots , e_n$ . The previous expression can also be written

Proof. Notice that

$$ \begin{align*}\overline{F_j} = \sum_w \overline{w} \, F(w e_j) = \sum_{m_0,\ldots, m_n\geq 0 } s_0^{m_0} \cdots s_n^{m_n} \left( \sum_{\overline{w} = s_0^{m_0} \cdots s_n^{m_n} } F(w e_j)\right)\end{align*} $$

and substitute in the expression

6.3 Beta quotients of generalized associators

Definition 6.10. We define the following $n\times n$ matrices of power series:

$$ \begin{align*}(FL_\Sigma^{\mathfrak{m}} )_{ij} = \mathbf{1}_{i=j}\overline{\mathcal{Z}^{i,\mathfrak{m} }} - s_j\overline{\mathcal{Z}^{i,\mathfrak{m} }_j} \;\;\; \in \; \mathcal{P}^{\mathfrak{m}} [[s_0,\ldots,s_n]] .\end{align*} $$

$$ \begin{align*}(FL_\Sigma)_{ij} = \mathbf{1}_{i=j}\overline{\mathcal{Z}^{i}} - s_j\overline{\mathcal{Z}^{i}_j} \;\;\; \in \;\mathbb{C}[[s_0,\ldots,s_n]] .\end{align*} $$

We clearly have $\mathrm {per}(FL_\Sigma ^{\mathfrak {m}} )=FL_\Sigma $ , where $\mathrm {per}:\mathcal {P}^{\mathfrak {m}} \to \mathbb {C}$ denotes the period map of $\mathcal {MT}(k)$ applied coefficientwise. Our next goal (Theorem 6.18) is to prove that $FL_\Sigma $ equals the matrix of Taylor series of the Lauricella functions $L_\Sigma $ (1). (This justifies the notation $FL_\Sigma $ where the letter F stands for formal.) We now compute the abelianization and beta quotients of the generalized associators $\mathcal {Z}^i$ .

We first need a lemma that clarifies the role of the choice of tangential basepoints. For a point $x=\gamma _i(t)$ , for $t\in (0,1)$ , we let $\gamma _i^x$ denote the restriction of $\gamma _i$ to the interval $(0,t)$ . By abuse of notation, we manipulate functions and forms on $\mathbb {C}\setminus \Sigma $ when we actually mean their pullbacks to $(0,1)$ via the path $\gamma _i$ .

Lemma 6.11. For $0\leq k\leq n$ , we have the equalities, for $x=\gamma _i(t)$ with $t\in (0,1)$ :

$$ \begin{align*}\int_{\gamma_i^x}\omega_k = \begin{cases} \log(x), & \mbox{if } k=0\ , \\ \log(1-x\sigma_k^{-1}), & \mbox{if } 1\leq k\leq n. \end{cases}\end{align*} $$

Furthermore, for $x=\sigma _i$ , that is, for $t=1$ , we have

$$ \begin{align*}\int_{\gamma_i}\omega_k = \begin{cases} \log(\sigma_i), & \mbox{if } k=0\ , \\ \log(1-\sigma_i\sigma_k^{-1}), & \mbox{if } 1\leq k\neq i\leq n\ , \\ 0, & \mbox{if } k=i. \end{cases}\end{align*} $$

Proof. For the first claim, the $1\leq k\leq n$ case is clear, since the differential form $\omega _j$ does not have a singularity at $0$ . For $j=0$ , we compute, by definition,

$$ \begin{align*}\int_{\gamma_i^x}\omega_0 = \mathrm{Reg}_{\varepsilon \to 0} \int_\varepsilon^t\gamma_i^*\omega_0=\mathrm{Reg}_{\varepsilon\to 0}(\log(x)-\log(\gamma_i(\varepsilon))) ,\end{align*} $$

where applying $\mathrm {Reg}_{\varepsilon \to 0}$ amounts to formally setting $\log (\varepsilon )=0$ and $\varepsilon =0$ in the logarithmic asymptotic development (see [Reference BrownBGF, Def. 3.237]). Since $\gamma _i$ extends to a smooth function on $[0,1]$ whose derivative at $0$ is $1$ , we have $\gamma _i(\varepsilon )=\varepsilon + O(\varepsilon ^2)$ and thus $\log (\gamma _i(\varepsilon ))=\log (\varepsilon )+O(\varepsilon )$ . We thus have $\mathrm {Reg}_{\varepsilon \to 0}(\log (\gamma _i(\varepsilon )))=0$ and the claim follows.

For the second claim, only the $k=i$ case requires a comment since in the other cases the form $\omega _j$ does not have a singularity at $\sigma _i$ and we can simply pass to the limit $x\to \sigma _i$ in the first statement. We will use the admissibility condition (Definition 6.1). Since $\omega _i$ only has poles at $\sigma _i$ and $\infty $ , its iterated integral along $\gamma _i$ only depends on the image of $\gamma _i$ in $\pi _1^{\mathrm {top}}(\mathbb {C}\setminus \{\sigma _i\}, 0, -t_i)$ . By the admissibility condition, this image is the class of $\widetilde {\gamma _i}(t)=\sigma _it$ and we can compute

$$ \begin{align*}\int_{\gamma_i}\omega_i = \int_{\widetilde{\gamma_i}}\omega_i= \mathrm{Reg}_{\varepsilon \to 0} \int_{\varepsilon}^{1-\varepsilon}\widetilde{\gamma_i}^*\omega_i = \mathrm{Reg}_{\varepsilon\to 0} \int_{\varepsilon}^{1-\varepsilon} \frac{dt}{t-1} = \mathrm{Reg}_{\varepsilon \to 0} \log\left(\frac{\varepsilon}{1-\varepsilon}\right) = 0.\end{align*} $$

The claim follows.

Proposition 6.13. The abelianization of the generalized associator $\mathcal {Z}^i$ is

(46) $$ \begin{align} \overline{\mathcal{Z}^i}= \sigma_i^{s_0} \prod_{k\neq i } (1 - \sigma_i\sigma_k^{-1})^{s_k} . \end{align} $$

For every $0\leq j\leq n$ , the jth beta quotient of $\mathcal {Z}^i$ is

(47) $$ \begin{align} \overline{\mathcal{Z}^i_j} = \int_{\gamma_i} x^{s_0} \prod _{k=1}^n \left(1- x\, \sigma^{-1}_k\right)^{s_k} \frac{dx}{x-\sigma_j} . \end{align} $$

These expressions are formal power series in the $s_i$ obtained by expanding the exponentials as power series and interpreting the various logarithms which appear as coefficients as regularized iterated integrals via Lemma 6.11.

Proof. Since $\mathcal {Z}^i$ is a group-like series, the first claim follows from Lemma 6.6 and the computations of $\mathcal {Z}^i(e_j)$ performed in the second part of Lemma 6.11. The second claim follows from Lemma 6.9 and the equality

which by the first part of Lemma 6.11 equals

$$ \begin{align*}\int_{\gamma_i}\log^{m_0}(x)\prod_{k=1}^n\log^{m_k}(1-x\sigma_k^{-1})\frac{dx}{x-\sigma_j}\ \cdot\end{align*} $$

The claim follows.

Equation (47) in the case $\Sigma =\{0,1\}$ reduces to Drinfeld’s computation of the metabelian quotient of his associator in terms of the usual beta function [Reference DrinfeldD2]. See also [Reference Esnault and LevineE], [Reference LiL] for further developments.

The following lemma will be useful.

Lemma 6.14. For all $m\geq 0$ ,

(48) $$ \begin{align} \int_{\gamma_i} \log^{m}(1-x \sigma_i^{-1}) \frac{dx}{x-\sigma_i} = 0. \end{align} $$

6.4 Comparing $L_\Sigma $ and $FL_\Sigma $

Up until now, we have considered two types of integrals: convergent line integrals of smooth $1$ -forms along a smooth (locally finite) path $\delta _i$ from $0$ to $\sigma _i$ , and regularized iterated integrals of words in the $\omega _j$ ’s along a path $\gamma _i$ from $0$ to $\sigma _i$ with prescribed tangent directions at $0$ and $\sigma _i$ . In order to compare these two different objects, we have to impose some compatibility between $\gamma _i$ and $\delta _i$ . Let us first fix parameters $\underline {s}=(s_0,s_1,\ldots ,s_n)$ . There is a natural map

$$ \begin{align*}h_i:\pi_1^{\mathrm{top}}(X_\Sigma(\mathbb{C}),t_0,-t_i)\longrightarrow H_1^{\mathrm{lf}}(X_\Sigma(\mathbb{C}),\mathcal{L}_{\underline{s}}^\vee)\end{align*} $$

defined in the following way. For a class in $\pi _1^{\mathrm {top}}(X_\Sigma (\mathbb {C}),t_0,-t_i)$ represented by a continuous map $\gamma :(0,1)\to \mathbb {C}\setminus \Sigma $ , its image under $h_i$ has $\gamma $ as its underlying (locally finite) path, together with the section of $\mathcal {L}_{\underline {s}}^\vee $ given by

$$ \begin{align*}x^{s_0}\prod_{k=1}^n(1-x\sigma_k^{-1})^{s_k}=\prod_{k=0}^n\exp\Big(s_k\int_{\gamma^x}\omega_k\Big),\end{align*} $$

which makes sense thanks to Lemma 6.11. One checks that this is well defined.

Let $\delta _i:(0,1)\to \mathbb {C}\setminus \Sigma $ be a smooth map that can be extended to a smooth map $\overline {\delta _i}:[0,1]\to \mathbb {C}$ , and which does not wind infinitely around $0$ or $\sigma _i$ (see Remark 2.3). Let us fix a determination of $\log (\sigma _i)$ for every $1\leq i\leq n$ . In Remark 4.2, we defined out of this data a class $\delta _{i,\underline {s}}\in H_1^{\mathrm {lf}}(X_\Sigma (\mathbb {C}),\mathcal {L}_{\underline {s}}^\vee )$ for every parameter $\underline {s}=(s_0,\ldots ,s_n)\in \mathbb {C}^{n+1}$ . Let $\gamma _i$ be a class in $\pi _1^{\mathrm {top}}(X_\Sigma (\mathbb {C}),t_0,-t_i)$ .

Definition 6.15. We say that $\delta _i$ and $\gamma _i$ are compatible if for every $\underline {s}\in \mathbb {C}^{n+1}$ we have

$$ \begin{align*}h_i(\gamma_i) = \delta_{i,\underline{s}}.\end{align*} $$

Note that this implies that the determinations of $\log (\sigma _i)$ are given by the regularized iterated integrals $\log (\sigma _i)=\int _{\gamma _i}\omega _0$ .

For all $1\leq j\leq n$ , we define

(49) $$ \begin{align} \Omega_j = x^{s_0} \prod_{k=1}^n (1-x \sigma_{k}^{-1})^{s_k}\, \frac{dx}{x-\sigma_j} \ , \end{align} $$

which we interpret as a $1$ -form on $\delta _i$ as in Remark 4.1. Let $\Omega ^{\mathrm {ren}_i}_j$ denote its renormalized versions (Definition 4.3) with respect to $\{0, \sigma _i\}$ , given by

$$ \begin{align*}\Omega_j^{\mathrm{ren},i} = \Big(x^{s_0}\prod_{k\neq i}(1-x\sigma_k^{-1})^{s_k} - \mathbf{1}_{i=j}\, \sigma_i^{s_0}\prod_{k\neq i}(1-\sigma_i\sigma_k^{-1})^{s_k}\Big)(1-x\sigma_i^{-1})^{s_i}\frac{dx}{x-\sigma_j}\ \cdot\end{align*} $$

According to Proposition 4.5, the integral $\int _{\delta _i}\Omega _j^{\mathrm {ren}_i}$ defines a holomorphic function of the parameters $\underline {s}$ around $\underline {s}=0$ and thus has a Taylor expansion. In the next proposition, we identify this Taylor expansion with the jth beta quotient $\overline {\mathcal {Z}^i_j}$ of the generalized associator, defined via regularized iterated integrals along $\gamma _i$ .

Before proving the proposition, we need a technical lemma that allows us to exchange summation and integration.

Lemma 6.17. Let us write $\Omega _j^{\mathrm {ren}_i}$ as a formal power series:

$$ \begin{align*}\Omega^{\mathrm{ren}_i}_j= \sum_{m_0,\ldots, m_n\geq 0} \Omega^{\mathrm{ren}_i}_j(\underline{m})\, \frac{s^{m_0}_0\cdots s^{m_n}_n}{m_0! \cdots m_n!}\ \cdot\end{align*} $$

Then we have the equality of formal power series:

$$ \begin{align*}\int_{\delta_i}\Omega_j^{\mathrm{ren}_i} = \sum_{m_0,\ldots,m_n\geq 0} \int_{\delta_i}\Omega^{\mathrm{ren}_i}_j(\underline{m}) \, \frac{s^{m_0}_0\cdots s^{m_n}_n}{m_0! \cdots m_n!}\ \cdot\end{align*} $$

Proof. Let us write $\delta _i^*\Omega _j^{\mathrm {ren}_i}(\underline {m})=f_{\underline {m}}(t)dt$ . By Fubini, it is enough to prove that the multiple series

(50) $$ \begin{align} \sum_{m_0,\ldots,m_n\geq 0}\int_0^1|f_{\underline{m}}(t)|dt\, \frac{s^{m_0}_0\cdots s^{m_n}_n}{m_0! \cdots m_n!} \end{align} $$

is absolutely convergent for $|\underline {s}|$ small enough, and therefore we need to estimate the integral of $|f_{\underline {m}}(t)|$ . We do so in the case $j=i$ ; the case $j\neq i$ is even simpler. For simplicity, we only treat the case $n=1$ . The general case is similar and is left to the reader. We write $\sigma =\sigma _1$ and $\delta =\delta _1$ . For indices $m_0,m_1\geq 0$ , we have, according to Definition 4.3,

$$ \begin{align*}f_{m_0,m_1}(t) =\frac{\log^{m_0}(\delta(t))-\log^{m_0}(\sigma)}{\delta(t)-\sigma} \log^{m_1}(1-\delta(t)\sigma^{-1}) \,\delta'(t).\end{align*} $$

Note that $\delta '(t)$ is bounded for $t\in (0,1)$ . To prove convergence, we can clearly assume that there exist small closed disks $D_0$ and $D_\sigma $ around $0$ and $\sigma $ , respectively, and constants $0<\alpha <\beta <1$ , such that $\delta (t)\notin D_\sigma $ for all $t\in (0,\beta )$ , and $\delta (t)\notin D_0$ for all $t\in (\alpha ,1)$ . We treat separately the cases $t\in (0,\beta )$ and $t\in (\alpha ,1)$ , and assume that $m_0,m_1\geq 1$ .

• – For $t\in (0,\beta )$ , $\log (1-\delta (t)\sigma ^{-1})$ is bounded and $\delta (t)-\sigma $ is bounded below in absolute value, which gives an estimate

  $$ \begin{align*}|f_{m_0,m_1}(t)| < \left( |\log^{m_0}(\delta(t))| + |\log^{m_0}(\sigma)|\right) A^{m_1}\end{align*} $$
  for some positive constant A. Now, since by assumption (Remark 2.3) the argument of $\delta (t)$ is bounded as t approaches zero, and since $|\log |\delta (t)||$ tends to infinity when t goes to zero, we have $|\log (\delta (t))|<B|\log |\delta (t)||$ for some positive constant B, and therefore since $\delta $ is smooth at $0$ , we have $|\log (\delta (t))|<B'|\log (t)|$ for all $t\in (0,1/2)$ , for some positive constant $B'$ . We deduce the estimate $|\log ^{m_0}(\delta (t))| + |\log ^{m_0}(\sigma )|< (B")^{m_0}|\log (t)|^{m_0}$ for all $t\in (0,\beta )$ , for some positive constant $B"$ . Therefore,
  $$ \begin{align*}|f_{m_0,m_1}(t)| < C^{m_0+m_1}|\log(t)|^{m_0},\end{align*} $$
  for all $t\in (0,\beta )$ , for some positive constant C.

• – For $t\in (\alpha ,1)$ , the quotient $\frac {\log ^{m_0}(\delta (t))-\log ^{m_0}(\sigma )}{\delta (t)-\sigma }$ is bounded by $D^{m_0}$ for some positive constant D, and by the same argument as before (using the assumption of Remark 2.3) we have an estimate $|\log ^{m_1}(1-\delta (t)\sigma ^{-1})|<E^{m_1}|\log (1-t)|^{m_1}$ for all $t\in (\alpha ,1)$ , for some positive constant E. Therefore,

  $$ \begin{align*}|f_{m_0,m_1}(t)|<F^{m_0+m_1}|\log(1-t)|^{m_1},\end{align*} $$
  for all $t\in (\alpha ,1)$ , for some positive constant F.

Putting those two contributions together, we get

$$ \begin{align*}\int_0^1|f_{m_0,m_1}(t)|dt < G^{m_0+m_1}\int_0^1(|\log(t)|^{m_0} + |\log(t)|^{m_1}) dt\end{align*} $$

for some positive constant G. Since $\int _0^1|\log (t)|^mdt=m!$ for all m, we therefore get the estimate

$$ \begin{align*}\int_0^1|f_{m_0,m_1}(t)|dt < G^{m_0+m_1}(m_0!+m_1!)\end{align*} $$

for some positive constant G. Therefore, the series (50) converges for $|s_0|,|s_1|<G^{-1}$ , and the claim follows.