Let $G$ be a primitive log-divergent Feynman graph, $E_{G}$ the collection of
its edges, and $n_{G}=|E_{G}|$, as in Section 2.4. Recall that $x_{e}$ denotes
the Schwinger parameter associated to $e\in E_{G}$, and $\Psi_{G}$, $I_{G}$,
and $X_{G}$ denote the first graph polynomial, the Feynman integral, and the
graph hypersurface, as given in (2.12), (2.20), and (2.21), respectively.
Denote by $\omega_{G}$ and $\sigma$ the integrand and the domain of
integration of $I_{G}$. Since $\omega_{G}$ is a top-degree algebraic
differential form on $\mathbb{P}^{n_{G}-1}\backslash X_{G}$, and
$\partial\sigma$ is contained in the union $D$ of the coordinate hyperplanes
$\\{x_{e}=0$, $e\in E_{G}\\}$, we may intuitively try to build the motive
$I_{G}^{\rm m}$ on the relative $H$-system

$\displaystyle H^{n_{G}-1}\big{(}\mathbb{P}^{n_{G}-1}\backslash
X_{G},D\backslash(D\cap X_{G})\big{)}.$ (5.35)

However, this naïve attempt fails whenever the hypersurface $X_{G}$ intersects
the integration cycle $\sigma$ non-trivially, implying the presence of non-
negligible singularities. Whenever singularities are present, $\sigma$ does
not define an element in the corresponding naïve relative Betti homology
group. To successfully build the motive $I_{G}^{\rm m}$ in the presence of
singularities, the blow up technique is applied.

A linear subvariety $L\subset\mathbb{P}^{n_{G}-1}$ defined by the vanishing of
a subset of the set of Schwinger parameters is called a coordinate linear
space, while its subspace of real points with non-negative coordinates is
denoted by

$\displaystyle L(\mathbb{R}_{\geq 0})=\\{[x_{e}]_{e\in E_{G}}\in
L\,|\,x_{e}\in\mathbb{R}_{\geq 0}\\}.$ (5.36)

Since the coefficients of $\Psi_{G}$ are positive, the locus of problematic
singularities is

$\displaystyle\sigma\cap X_{G}(\mathbb{C})=\bigcup_{L\subset
X_{G}}L(\mathbb{R}_{\geq 0}),$ (5.37)

where the union is taken over all coordinate linear spaces $L\subset X_{G}$.

######  Remark 5.2.

The coordinate linear spaces $L\subset X_{G}$ are in one-to-one correspondence
with the subgraphs $\gamma\subset G$ such that $l_{\gamma}>0$. It follows that

$\displaystyle\sigma\cap X_{G}(\mathbb{C})=\bigcup_{\gamma\subset
G}L_{\gamma}(\mathbb{R}_{\geq 0}),$ (5.38)

where the union is taken over all subgraphs $\gamma\subset G$ with
$l_{\gamma}>0$, and $L_{\gamma}$ is the linear subvariety of
$\mathbb{P}^{n_{G}-1}$ defined by the equations $\\{x_{e}=0,e\in
E_{\gamma}\\}$.

The following theorem is proven, and an explicit algorithmic construction of
the blow ups is given, by Bloch, Esnault and Kreimer [5].

######  Theorem 5.3.

Let $G$ be a primitive log-divergent Feynman graph. There exists a tower

$\displaystyle\pi\colon\ P=P_{r}\rightarrow P_{r-1}\rightarrow\dots\rightarrow
P_{1}\rightarrow P_{0}=\mathbb{P}^{n_{G}-1}$ (5.39)

such that, for each $i=1,\dots,r$, $P_{i}$ is obtained by blowing up $P_{i-1}$
along the strict transform of a coordinate linear space $L_{i}\subset X_{G}$,
and the following conditions hold:

  * $(1)$

The pulled-back differential $\hat{\omega}_{G}=\pi^{*}\omega_{G}$ has no poles
along the exceptional divisors associated to the blow ups.

  * $(2)$

Let $B$ be the total transform of $D$ in $P$, that is

$\displaystyle B=\pi^{-1}(D)=\pi^{-1}\bigg{(}\bigcup_{e\in
E_{G}}\\{x_{e}=0\\}\bigg{)}.$ (5.40)

Then, $B\subset P$ is a normal crossing divisor such that none of the non-
empty intersections of its irreducible components is contained in the strict
transform $Y_{G}$ of $X_{G}$ in $P$.

  * $(3)$

The strict transform of $\sigma$ in $P$ does not meet $Y_{G}$, that is

$\displaystyle\hat{\sigma}\cap Y_{G}(\mathbb{C})=\varnothing.$ (5.41)

As a consequence of Theorem 5.3, the motive $I_{G}^{\rm m}$ associated to any
primitive log-divergent Feynman graph $G$ can be written explicitly. Being
$\partial\hat{\sigma}\subset B\backslash(B\cap Y_{G})$, the domain of
integration defines the class

$\displaystyle[\hat{\sigma}]\in H_{n_{G}-1}^{{\rm B}}(P\backslash
Y_{G},B\backslash(B\cap Y_{G}))$ (5.42)

called Betti framing, while the integrand defines the class

$\displaystyle[\hat{\omega}_{G}]\in H^{n_{G}-1}_{\rm dR}(P\backslash
Y_{G},B\backslash(B\cap Y_{G}))$ (5.43)

called de Rham framing. Brown and Doryn [16] present a method for explicit
computation of the framings on the cohomology of Feynman graph hypersurfaces.
Then, the de Rham and Betti system of realisations
$H_{G}=H^{n_{G}-1}(P\backslash Y_{G},B\backslash(B\cap Y_{G}))$ is called the
graph $H$-system,505050The graph $H$-system is also explicitly known in the
general case of renormalised amplitudes of single-scale graphs due to the work
of Brown and Kreimer [18], who paved the way for the rigorous investigation of
divergent Feynman graphs and their renormalised amplitudes from an algebro-
geometric perspective. and the motivic Feynman integral $I_{G}^{\rm m}$ is
given by

$\displaystyle I_{G}^{\rm
m}=\big{[}H_{G},[\hat{\omega}_{G}],[\hat{\sigma}]\big{]}^{\rm m}.$ (5.44)

Indeed, the pairing of the classes $[\hat{\omega}_{G}]$ and $[\hat{\sigma}]$
yields the period

$\displaystyle\int_{\hat{\sigma}}\hat{\omega}_{G}=\int_{\hat{\sigma}}\pi^{*}(\omega_{G})=\int_{\pi_{*}(\hat{\sigma})}\omega_{G}=\int_{\sigma}\omega_{G}=I_{G}$
(5.45)

by the equivalence relation under change of variables in $\mathcal{P}^{\rm
m}$.

######  Example 5.4.

Adopting the notation

$\displaystyle\mathcal{P}_{\rm log}=\mathbb{Q}\langle I_{G}\,|\,G\text{ is a
primitive log-divergent Feynman graph}\rangle,$ (5.46)
$\displaystyle\mathcal{P}_{\phi^{4}}=\mathbb{Q}\big{\langle}I_{G}\,|\,G\text{
is a primitive log-divergent Feynman graph in $\phi^{4}$
theory}\big{\rangle},$ (5.47)

we observe that the sequence of inclusions
$\mathcal{P}_{\phi^{4}}\subset\mathcal{P}_{\rm log}\subset\mathcal{P}$ is
preserved after promoting periods to periods of motives, that is
$\mathcal{P}_{\phi^{4}}^{\rm m}\subset\mathcal{P}_{\rm log}^{\rm
m}\subset\mathcal{P}^{\rm m}$.

Many concrete results on the structure of $\mathcal{P}_{\rm log}$ follow from
the study of the corresponding motivic space $\mathcal{P}_{\rm log}^{\rm m}$.
For example, the following proposition on graph $H$-systems is proven by Brown
[14] within the formalism of motivic Feynman integrals.

######  Proposition 5.5.

Let $G$ be a primitive log-divergent Feynman graph. If $G$ has a single
vertex, that is $v_{G}=1$, or if $G$ has a single loop, that is $l_{G}=1$,
then its $H$-system $H_{G}$ is isomorphic to the pure Hodge–Tate system
$\mathbb{Q}(0)$.

We observe that Proposition 5.5 makes no restriction on the physicality of the
graph $G$, which can have arbitrary vertex-degrees, so that $I_{G}$ belongs to
$\mathcal{P}_{\rm log}$, but not necessarily to $\mathcal{P}_{\phi^{4}}$.

###  5.4 Tannakian formalism

We briefly introduce the fundamentals of the theory of Tannakian categories,
following the more detailed and comprehensive exposition by Deligne et al
[29]. The concept of a Tannakian category was first introduced by Saavedra
Rivano [54] to encode the properties of the category $\mathop{\rm
Rep}\nolimits_{\mathbb{K}}(G)$ of the finite-dimensional $\mathbb{K}$-linear
representations of an affine group scheme $G$ over a field $\mathbb{K}$. Let
us recall some preliminary notions in category theory. Let $\mathbb{K}$ be a
subfield of $\mathbb{C}$.

######  Definition 5.6.

A $\mathbb{K}$-linear category $\mathcal{C}$ is an additive category such
that, for each pair of objects $X,Y\in\text{Ob}(\mathcal{C})$, the group
$\mathop{\rm Hom}_{\mathcal{C}}(X,Y)$ is a $\mathbb{K}$-vector space and the
composition maps are $\mathbb{K}$-bilinear.

######  Definition 5.7.

Let $\mathcal{C}$ be a $\mathbb{K}$-linear category endowed with a
$\mathbb{K}$-bilinear functor
$\otimes\colon\mathcal{C}\times\mathcal{C}\rightarrow\mathcal{C}$.

  * $(a)$

An associativity constraint for $(\mathcal{C},\otimes)$ is a natural
transformation

$\displaystyle\phi=\phi_{\cdot,\cdot,\cdot}\colon\
\cdot\otimes(\cdot\otimes\cdot)\longrightarrow(\cdot\otimes\cdot)\otimes\cdot$
(5.48)

such that the following two conditions hold:

    * $(a.1)$

For all $X,Y,Z\in\text{Ob}(\mathcal{C})$, the map $\phi_{X,Y,Z}$ is an
isomorphism.

    * $(a.2)$

For all $X,Y,Z,T\in\text{Ob}(\mathcal{C})$, the following diagram commutes:

${X\otimes(Y\otimes(Z\otimes T))}$${X\otimes((Y\otimes Z)\otimes
T)}$${(X\otimes Y)\otimes(Z\otimes T)}$${(X\otimes(Y\otimes Z))\otimes
T}$${((X\otimes Y)\otimes Z)\otimes
T.}$$\scriptstyle{\text{Id}\otimes\phi_{Y,Z,T}}$$\scriptstyle{\phi_{X,Y,Z\otimes
T}}$$\scriptstyle{\phi_{X,Y\otimes Z,T}}$$\scriptstyle{\phi_{X\otimes
Y,Z,T}}$$\scriptstyle{\phi_{X,Y,Z}\otimes\text{Id}}$ (5.52)

  * $(b)$

A commutativity constraint for $(\mathcal{C},\otimes)$ is a natural
transformation

$\displaystyle\psi=\psi_{\cdot,*}\colon\cdot\otimes*\longrightarrow*\otimes\cdot$
(5.53)

such that the following two conditions hold:

    * $(b.1)$

For all $X,Y\in\text{Ob}(\mathcal{C})$, the map $\psi_{X,Y}$ is an
isomorphism.

    * $(b.2)$

For all $X,Y\in\text{Ob}(\mathcal{C})$, the following composition is the
identity:

$\displaystyle\psi_{Y,X}\circ\psi_{X,Y}\colon\ X\otimes Y\longrightarrow
X\otimes Y.$ (5.54)

  * $(c)$

An associativity constraint and a commutativity constraint are compatible if,
for all $X,Y,\allowbreak Z\in\text{Ob}(\mathcal{C})$, the following diagram
commutes:

${X\otimes(Y\otimes Z)}$${(X\otimes Y)\otimes Z}$${X\otimes(Z\otimes
Y)}$${Z\otimes(X\otimes Y)}$${(X\otimes Z)\otimes Y}$${(Z\otimes X)\otimes
Y.}$$\scriptstyle{\phi_{X,Y,Z}}$$\scriptstyle{\text{Id}\otimes\psi_{Y,Z}}$$\scriptstyle{\psi_{X\otimes
Y,Z}}$$\scriptstyle{\phi_{X,Z,Y}}$$\scriptstyle{\phi_{X,Z,Y}}$$\scriptstyle{\psi_{X,Z}\otimes\text{Id}}$
(5.58)

  * ($d)$

A pair $(U,u)$ consisting of an object $U\in\text{Ob}(\mathcal{C})$ and an
isomorphism $u\colon U\rightarrow U\otimes U$ is an identity object if the
functor $X\mapsto U\otimes X$ is an equivalence of categories.

######  Definition 5.8.

A $\mathbb{K}$-linear tensor category is a tuple
$(\mathcal{C},\otimes,\phi,\psi)$ consisting of a $\mathbb{K}$-linear category
$\mathcal{C}$, a $\mathbb{K}$-bilinear functor
$\otimes\colon\mathcal{C}\times\mathcal{C}\rightarrow\mathcal{C}$, and
compatible associativity and commutativity constraints $\phi$, $\psi$ such
that $\mathcal{C}$ contains an identity object.

######  Definition 5.9.

An object $L\in\text{Ob}(\mathcal{C})$ is invertible if the functor $X\mapsto
L\otimes X$ is an equivalence of categories. Equivalently, $L$ is invertible
if and only if there exists an object $L^{\prime}\in\text{Ob}(\mathcal{C})$
such that $L\otimes L^{\prime}\simeq\mathbf{1}$. Then, $L^{\prime}$ is also
invertible.

######  Definition 5.10.

Let $(\mathcal{C},\otimes)$ be a $\mathbb{K}$-linear tensor category, where we
omit the constraints $\phi$, $\psi$ for simplicity, and let
$X,Y\in\text{Ob}(\mathcal{C})$. Assume that there exists an object
$Z\in\text{Ob}(\mathcal{C})$ such that, for all $T\in\text{Ob}(\mathcal{C})$,
the functors $T\mapsto\mathop{\rm Hom}(T,Z)$ and $T\mapsto\mathop{\rm
Hom}(T\otimes X,Y)$ admit a functorial isomorphism

$\displaystyle\mathop{\rm Hom}(T,Z)\xlongrightarrow{\sim}\mathop{\rm
Hom}(T\otimes X,Y).$ (5.59)

In this case, the functor $T\mapsto\mathop{\rm Hom}(T\otimes X,Y)$ is said to
be representable and the object $Z$ is called the internal Hom between the
objects $X$ and $Y$. It is alternatively written as $\underline{\mathop{\rm
Hom}}(X,Y)$ and it is unique up to isomorphism.

######  Definition 5.11.

The dual of an object $X\in\text{Ob}(\mathcal{C})$ is defined as
$X^{\vee}=\underline{\mathop{\rm Hom}}(X,\mathbf{1})$. If $X^{\vee}$ and
${(X^{\vee})}^{\vee}$ exist, then there is a natural morphism
$X\mapsto{\big{(}X^{\vee}\big{)}}^{\vee}$ and the object $X$ is reflexive if
such a morphism is an isomorphism.

######  Definition 5.12.

A $\mathbb{K}$-linear tensor category $(\mathcal{C},\otimes)$ is rigid if the
following conditions hold:

  * (1)

For all $X,Y\in\text{Ob}(\mathcal{C})$, $\underline{\mathop{\rm Hom}}(X,Y)$
exists.

  * (2)

For all $X_{1},X_{2},Y_{1},Y_{2}\in\text{Ob}(\mathcal{C})$, the natural
morphism

$\displaystyle\underline{\mathop{\rm
Hom}}(X_{1},Y_{1})\otimes\underline{\mathop{\rm
Hom}}(X_{2},Y_{2})\longrightarrow\underline{\mathop{\rm Hom}}(X_{1}\otimes
X_{2},Y_{1}\otimes Y_{2})$ (5.60)

is an isomorphism.

  * (3)

All objects are reflexive.

######  Definition 5.13.

A Tannakian category over the field $\mathbb{K}$ is a rigid abelian
$\mathbb{K}$-linear tensor category $\mathcal{T}$ such that
$\text{End}(\mathbf{1})=\mathbb{K}$, and there exists an exact faithful
$\mathbb{K}$-linear tensor functor
$\omega\colon\mathcal{T}\rightarrow\text{Vec}_{\mathbb{K}}$, where
$\text{Vec}_{\mathbb{K}}$ is the category of finite-dimensional vector spaces
over $\mathbb{K}$. Any such functor is called a fibre functor.

######  Example 5.14.

The category $\text{Vec}_{\mathbb{K}}$ of finite-dimensional
$\mathbb{K}$-vector spaces, together with the identity functor, is a Tannakian
category over $\mathbb{K}$.

######  Example 5.15.

The category $\text{GrVec}_{\mathbb{K}}$ of finite-dimensional graded
$\mathbb{K}$-vector spaces, together with the forgetful functor
$\omega\colon\text{GrVec}_{\mathbb{K}}\rightarrow\text{Vec}_{\mathbb{K}}$,
sending $(V,(V_{n})_{n\in\mathbb{Z}})$ to $V$, is a Tannakian category over
$\mathbb{K}$.

######  Example 5.16.

The category $\mathop{\rm Rep}\nolimits_{\mathbb{K}}(G)$ of finite-dimensional
$\mathbb{K}$-linear representations of an abstract group $G$, together with
the functor $\omega\colon\mathop{\rm
Rep}\nolimits_{\mathbb{K}}(G)\rightarrow\text{Vec}_{\mathbb{K}}$ that forgets
the action of $G$, is a Tannakian category over $\mathbb{K}$.

Let us fix a Tannakian category $\mathcal{T}$ over $\mathbb{K}$ and a fibre
functor $\omega$ of $\mathcal{T}$. Let $R$ be a $\mathbb{K}$-algebra. We
denote by $\underline{\mathop{\rm Aut}}^{\otimes}(\omega)(R)$ the collection
of families $(\lambda_{X})_{X\in\text{Ob}(\mathcal{T})}$ of $R$-linear
automorphisms

$\displaystyle\lambda_{X}\colon\
\omega(X)\otimes_{\mathbb{K}}R\longrightarrow\omega(X)\otimes_{\mathbb{K}}R$
(5.61)

which are compatible with the tensor structure and functorial. Here,
compatibility with the tensor structure and functoriality mean515151In the
given diagrams, all unlabelled tensor products are over $\mathbb{K}$ and all
unlabelled arrows are the natural isomorphisms. that:

  * (1)

For all $X_{1},X_{2}\in\text{Ob}(\mathcal{T})$, the following diagram
commutes:

${\omega(X_{1}\otimes X_{2})\otimes R}$${\omega(X_{1}\otimes X_{2})\otimes
R}$${\omega(X_{1})\otimes\omega(X_{2})\otimes
R}$${\omega(X_{1})\otimes\omega(X_{2})\otimes R}$${(\omega(X_{1})\otimes
R)\otimes_{R}(\omega(X_{2})\otimes R)}$${(\omega(X_{1})\otimes
R)\otimes_{R}(\omega(X_{2})\otimes R).}$$\scriptstyle{\lambda_{X_{1}\otimes
X_{2}}}$$\scriptstyle{\lambda_{X_{1}}\otimes_{R}\lambda_{X_{2}}}$ (5.65)

  * (2)

The following diagram commutes:

${\omega(\mathbf{1})\otimes R}$${\omega(\mathbf{1})\otimes
R}$${R}$${R.}$$\scriptstyle{\lambda_{\mathbf{1}}}$Id (5.68)

  * (3)

For all $X,Y\in\text{Ob}(\mathcal{T})$ and for every morphism
$\alpha\in\mathop{\rm Hom}(X,Y)$, the following diagram commutes:

${\omega(X)\otimes R}$${\omega(X)\otimes R}$${\omega(Y)\otimes
R}$${\omega(Y)\otimes
R.}$$\scriptstyle{\lambda_{X}}$$\scriptstyle{\omega(\alpha)\otimes\text{Id}}$$\scriptstyle{\omega(\alpha)\otimes\text{Id}}$$\scriptstyle{\lambda_{Y}}$
(5.71)

Deligne et al [29] proved that all Tannakian categories are categories of
finite-dimensional linear representations of a pro-algebraic group.

######  Theorem 5.17.

Let $\mathcal{T}$ be a Tannakian category over $\mathbb{K}$ with a fibre
functor $\omega$.

  * $(1)$

The functor $R\mapsto\underline{\mathop{\rm Aut}}^{\otimes}(\omega)(R)$ is
representable by an affine group scheme over $\mathbb{K}$, which is denoted as
$\underline{\mathop{\rm Aut}}^{\otimes}(\omega)$ or $G^{\omega}$, and is
called the Tannaka group of the pair $(\mathcal{T},\omega)$.

  * $(2)$

For every $X\in\text{Ob}(\mathcal{T})$, the group $\underline{\mathop{\rm
Aut}}^{\otimes}(\omega)$ acts naturally on $\omega(X)$ and the functor

${\mathcal{T}}$${\mathop{\rm
Rep}\nolimits_{\mathbb{K}}(G^{\omega}),}$${X}$${\omega(X)}$$\scriptstyle{G^{\omega}}$
(5.74)

sending $X$ to the vector space $\omega(X)$ with this action of
$\underline{\mathop{\rm Aut}}^{\otimes}(\omega)$, is an equivalence of
categories.

Given a second fibre functor $\omega^{\prime}$, we analogously define
$\underline{\mathop{\rm Isom}}^{\otimes}(\omega,\omega^{\prime})(R)$ to be the
collection of families $(\tau_{X})_{X\in\text{Ob}(\mathcal{T})}$ of $R$-linear
isomorphisms

$\displaystyle\tau_{X}\colon\
\omega(X)\otimes_{\mathbb{K}}R\longrightarrow\omega^{\prime}(X)\otimes_{\mathbb{K}}R$
(5.75)

which are compatible with the tensor structure and functorial. Deligne et al
[29] proved the following result.

######  Theorem 5.18.

Let $\mathcal{T}$ be a Tannakian category over $\mathbb{K}$ with two fibre
functors $\omega$ and $\omega^{\prime}$. The functor
$R\mapsto\underline{\mathop{\rm Isom}}^{\otimes}(\omega,\omega^{\prime})(R)$
is representable by an affine scheme over $\mathbb{K}$, which is denoted as
$\underline{\mathop{\rm Isom}}^{\otimes}(\omega,\omega^{\prime})$, and is a
right torsor under $\underline{\mathop{\rm Aut}}^{\otimes}(\omega)$ and a left
torsor under $\underline{\mathop{\rm Aut}}^{\otimes}(\omega^{\prime})$.

######  Remark 5.19.

In what follows, we denote $\mathop{\rm
Aut}^{\otimes}(\omega)=\underline{\mathop{\rm
Aut}}^{\otimes}(\omega)(\mathbb{C})$ the group of $\mathbb{C}$-linear
automorphisms of the fibre functor $\omega$, and analogously $\mathop{\rm
Isom}\nolimits^{\otimes}(\omega,\omega^{\prime})=\underline{\mathop{\rm
Isom}}^{\otimes}(\omega,\omega^{\prime})(\mathbb{C})$ the group of
$\mathbb{C}$-linear isomorphisms between the fibre functors $\omega$ and
$\omega^{\prime}$.

###  5.5 Motivic Galois theory

Recall the notions of pure and mixed $H$-systems associated with algebraic
varieties over $\mathbb{Q}$ given in Sections 3.4 and 3.5. On the one hand,
the algebraic de Rham and Betti cohomologies of a smooth projective
$\mathbb{Q}$-variety are fundamentally described by a pure $H$-system. On the
other hand, applying the resolution of singularities by Hironaka [40, 41], the
algebraic de Rham and Betti cohomologies of an arbitrary quasi-projective
$\mathbb{Q}$-variety can be expressed in terms of the cohomologies of smooth
projective varieties, and since pure $H$-systems of different weights get
mixed in this expression, they are fundamentally described by a mixed system
of realisations. Because we specifically look at the application of the theory
of motives to the theory of periods, it is sufficient to our purpose to work
with the partial realisation of Grothendieck’s notion of motives provided by
$H$-systems525252In the literature on motivic periods, mixed de Rham and Betti
systems of realisations are sometimes called motives themselves.. However, it
turns out to be necessary and fruitful to enhance the naïve construction given
in Section 4.2 to its rigorous category-theoretic formulation.

Recall that $\mathbf{MHSy}(\mathbb{Q})$ is the category of mixed $H$-systems
over $\mathbb{Q}$, and $\omega_{{\rm B}}$, $\omega_{\rm dR}$ are its two
forgetful functors arising from the Betti and de Rham realisations,
respectively. All the defining properties of a Tannakian category, encoded in
Definition 5.13, apply to $\mathbf{MHSy}(\mathbb{Q})$, thus justifying the use
of the Tannakian machinery in the study of motivic periods.

######  Proposition 5.20.

$\mathbf{MHSy}(\mathbb{Q})$ is a Tannakian category over $\mathbb{Q}$ and the
functors $\omega_{{\rm B}}$, $\omega_{\rm dR}$ are fibre functors.

In what follows, we write $\mathcal{H}=\mathbf{MHSy}(\mathbb{Q})$. The pro-
algebraic group $\mathop{\rm Aut}^{\otimes}(\omega_{\rm dR})$ is denoted by
$G^{\rm dR}$ and called the motivic Galois group. Observe that $G^{\rm dR}(H)$
is a group in ${\rm GL}(\omega_{\rm dR}(H))$ for every object
$H\in\text{Ob}(\mathcal{H})$. Following Theorem 5.17, the category of mixed
$H$-systems is equivalent to the category of finite-dimensional
$\mathbb{Q}$-linear representations of the motivic Galois group, that is

$\displaystyle\mathcal{H}\simeq\mathop{\rm
Rep}\nolimits_{\mathbb{Q}}\big{(}G^{\rm dR}\big{)}.$ (5.76)

######  Remark 5.21.

We observe that the motivic Galois group can alternatively be realised via
Betti cohomology as $G^{{\rm B}}=\mathop{\rm Aut}^{\otimes}(\omega_{{\rm B}})$
and the corresponding category of finite-dimensional $\mathbb{Q}$-linear
representations is still the same category of mixed $H$-systems $\mathcal{H}$.

In Tannakian formalism, the space of motivic periods $\mathcal{P}^{\rm m}$ is
expressed as

$\displaystyle\mathcal{P}^{\rm
m}=\mathbb{Q}\big{\langle}[H,\omega,\sigma]^{\rm
m}\,|\,H\in\text{Ob}(\mathcal{H}),\;\omega\in\omega_{\rm
dR}(H),\,\sigma\in\omega_{{\rm B}}(H)^{\vee}\big{\rangle}$ (5.77)

with implicit factorisation modulo bilinearity and functoriality. Recall that

$\displaystyle H=(H_{\rm dR},\,H_{{\rm B}},\,\mathop{\rm
comp}\nolimits_{H}\colon H_{\rm
dR}\otimes_{\mathbb{Q}}\mathbb{C}\xrightarrow{\sim}H_{{\rm
B}}\otimes_{\mathbb{Q}}\mathbb{C}),$ (5.78)

where $H_{\rm dR}=\omega_{\rm dR}(H)$, $H_{{\rm B}}=\omega_{{\rm B}}(H)$ are
finite-dimensional $\mathbb{Q}$-vector spaces and $\mathop{\rm
comp}\in\mathop{\rm Isom}\nolimits^{\otimes}(\omega_{\rm dR},\omega_{{\rm
B}})$ is a $\mathbb{C}$-linear isomorphism. In this framework, an alternative
but equivalent description of motivic periods follows.

######  Proposition 5.22.

$\mathcal{P}^{\rm m}$ is isomorphic to the space of regular functions on the
affine $\mathbb{Q}$-scheme $\mathop{\rm Isom}\nolimits^{\otimes}(\omega_{\rm
dR},\omega_{{\rm B}})$.

We observe that such an isomorphism is explicitly written as

$\displaystyle\mathcal{P}^{\rm m}$
$\displaystyle\xlongrightarrow{\sim}\mathcal{O}\big{(}\mathop{\rm
Isom}\nolimits^{\otimes}(\omega_{\rm dR},\omega_{{\rm B}})\big{)},$ (5.79)
$\displaystyle{}[H,\omega,\sigma]^{\rm m}$
$\displaystyle\longmapsto\big{[}(\lambda_{X})_{X\in\text{Ob}(\mathcal{H})}\mapsto(\sigma\otimes_{\mathbb{Q}}\text{Id}_{\mathbb{C}})\circ\lambda_{H}\circ(\omega\otimes
2\pi{\rm i})\big{]},$ (5.80)

where we have

$\displaystyle\begin{array}[]{rcccl}\omega_{\rm
dR}(H)\otimes_{\mathbb{Q}}\mathbb{C}&\xlongrightarrow{\lambda_{H}}&\omega_{{\rm
B}}(H)\otimes_{\mathbb{Q}}\mathbb{C}&\xlongrightarrow{\sigma\otimes_{\mathbb{Q}}\text{Id}_{\mathbb{C}}}&\mathbb{C},\\\\[4.30554pt]
\omega&\longmapsto&\lambda_{H}(\omega)&\longmapsto&\sigma(\lambda_{H}(\omega)).\end{array}$
(5.83)

Following Theorem 5.17, the motivic Galois group $G^{\rm dR}$ has a natural
action on $\mathop{\rm Isom}\nolimits^{\otimes}(\omega_{\rm dR},\omega_{{\rm
B}})$ which is written as

$\displaystyle G^{\rm dR}\otimes\mathop{\rm
Isom}\nolimits^{\otimes}(\omega_{\rm dR},\omega_{{\rm
B}})\longrightarrow\mathop{\rm Isom}\nolimits^{\otimes}(\omega_{\rm
dR},\omega_{{\rm B}})$ (5.84)

and which induces a dual coaction on the corresponding space of regular
functions $\mathcal{P}^{\rm m}$, that is

$\displaystyle\Delta\colon\ \mathcal{P}^{\rm m}$
$\displaystyle\longrightarrow\mathcal{O}\big{(}G^{\rm
dR}\big{)}\otimes\mathcal{P}^{\rm m},$ (5.85)
$\displaystyle{}[H,\omega,\sigma]^{\rm m}$
$\displaystyle\longmapsto\sum_{i=1}^{n}\big{[}H,\omega,e_{i}^{\vee}\big{]}^{\rm
dR}\otimes[H,e_{i},\sigma]^{\rm m},$ (5.86)

where $\\{e_{i}\\}_{i=1,\dots,n}$ is a basis of $\omega_{\rm dR}(H)$, and
$\big{\\{}e_{i}^{\vee}\big{\\}}_{i=1,\dots,n}$ denotes the associated vector
dual basis of $\omega_{{\rm B}}(H)^{\vee}$, as introduced in Section 4.3.
Here, $[H,e_{i},\sigma]^{\rm m}\in\mathcal{P}^{\rm m}$ is called a Galois
conjugate of the motivic period $[H,\omega,\sigma]^{\rm m}$, while
$\big{[}H,\omega,e_{i}^{\vee}\big{]}^{\rm dR}\in\mathcal{O}\big{(}G^{\rm
dR}\big{)}$ is called a de Rham period. We denote by $\mathcal{P}^{\rm
dR}=\mathcal{O}\big{(}G^{\rm dR}\big{)}$ the space of regular functions on the
motivic Galois group, and we call it the space of de Rham periods. The
coaction $\Delta$ is known as the Galois coaction.

######  Remark 5.23.

Note that the space of de Rham periods is naturally a Hopf algebra, while the
space of motivic periods is not, thus making the coaction intrinsically
asymmetric. Indeed, the Galois coaction turns the finite-dimensional
$\mathbb{Q}$-vector space $\mathcal{P}^{\rm m}$ into a comodule over the Hopf
algebra $\mathcal{P}^{\rm dR}$. Moreover, there is a canonical single-valued
map that associates a number to each de Rham period, thus representing the de
Rham analogue of the period map. For a detailed discussion we refer to Brown
[14, 15].

######  Example 5.24.

Consider the motivic logarithm $\log(z)^{\rm m}$ for
$z\in\mathbb{Q}\backslash\\{0,1\\}$. Following Section 4.4.2, we have

$\displaystyle\log(z)^{\rm
m}=\bigg{[}H^{1}(\mathbb{G}_{m},\\{1,z\\}),\bigg{[}\frac{{\rm
d}x}{x}\bigg{]},[\gamma_{1}]\bigg{]}^{\rm m}.$ (5.87)

Let us denote $H=H^{1}(\mathbb{G}_{m},\\{1,z\\})$ for simplicity. Adopting the
canonical choice of bases, as in Example 4.8, the period matrix of $H$ is

$\displaystyle\begin{pmatrix}1&\log(z)\\\ 0&2\pi{\rm i}\end{pmatrix}\\!.$
(5.88)

Direct application of the prescription in (5.85) gives the explicit
decomposition

$\displaystyle\Delta\bigg{[}H,\bigg{[}\frac{{\rm
d}x}{x}\bigg{]},[\gamma_{1}]\bigg{]}^{\rm m}=\bigg{[}H,\bigg{[}\frac{{\rm
d}x}{x}\bigg{]},\bigg{[}\frac{{\rm d}x}{z-1}\bigg{]}^{\vee}\bigg{]}^{\rm
dR}\otimes\bigg{[}H,\bigg{[}\frac{{\rm
d}x}{z-1}\bigg{]},[\gamma_{1}]\bigg{]}^{\rm m}$ (5.89)
$\displaystyle\hphantom{\Delta\bigg{[}H,\bigg{[}\frac{{\rm
d}x}{x}\bigg{]},[\gamma_{1}]\bigg{]}^{\rm m}=}{}+\bigg{[}H,\bigg{[}\frac{{\rm
d}x}{x}\bigg{]},\bigg{[}\frac{{\rm d}x}{x}\bigg{]}^{\vee}\bigg{]}^{\rm
dR}\otimes\bigg{[}H,\bigg{[}\frac{{\rm
d}x}{x}\bigg{]},[\gamma_{1}]\bigg{]}^{\rm m}.$ (5.90)

Observing that $\big{[}\frac{{\rm d}x}{z-1}\big{]}^{\vee}=[\gamma_{1}]$ and
$\big{[}\frac{{\rm d}x}{x}\big{]}^{\vee}=[\gamma_{2}]$, and identifying the de
Rham periods

$\displaystyle\bigg{[}H,\bigg{[}\frac{{\rm d}x}{x}\bigg{]},\bigg{[}\frac{{\rm
d}x}{z-1}\bigg{]}^{\vee}\bigg{]}^{\rm dR}=\log(z)^{\rm
dR},\qquad\bigg{[}H,\bigg{[}\frac{{\rm d}x}{x}\bigg{]},\bigg{[}\frac{{\rm
d}x}{x}\bigg{]}^{\vee}\bigg{]}^{\rm dR}=(2\pi{\rm i})^{\rm dR},$ (5.91)

we find that the expression (5.90) is equivalent to

$\displaystyle\Delta\log(z)^{\rm m}=\log(z)^{\rm dR}\otimes 1^{\rm
m}+(2\pi{\rm i})^{\rm dR}\otimes\log(z)^{\rm m},$ (5.92)

where $1^{\rm m}$ and $\log(z)^{\rm m}$ are the Galois conjugates of
$\log(z)^{\rm m}$.

######  Example 5.25.

As for $\log(z)^{\rm m}$, the Galois coaction of the motivic multiple zeta
value $\zeta(\mathbf{s})^{\rm m}$ can be computed explicitly. For example,
from the period matrix (5.34), we have that

$\displaystyle\Delta\zeta(2)^{\rm m}=\zeta(2)^{\rm dR}\otimes 1^{\rm
m}+\big{(}(2\pi{\rm i})^{\rm dR}\big{)}^{2}\otimes\zeta(2)^{\rm m}.$ (5.93)

As powers of $(2\pi{\rm i})^{\rm dR}$ naturally appear among de Rham
conjugates, the Galois coaction is often intended with an implicit
factorisation535353The operation of factorisation of a $H$-system modulo
$2\pi{\rm i}$ is called a Tate twist, and it can indeed be formally defined in
terms of the Hodge–Tate systems introduced in Example 3.27. of
$\mathcal{P}^{\rm dR}$ modulo the ideal generated by $(2\pi{\rm i})^{\rm dR}$.
Let us assume so. For $n\geq 1$, we have that

$\displaystyle\Delta\zeta(2)^{\rm m}=\zeta(2)^{\rm dR}\otimes 1^{\rm m}+1^{\rm
dR}\otimes\zeta(2)^{\rm m},$ (5.94) $\displaystyle\Delta\zeta(2n+1)^{\rm
m}=\zeta(2n+1)^{\rm dR}\otimes 1^{\rm m}+1^{\rm dR}\otimes\zeta(2n+1)^{\rm
m},$ (5.95) $\displaystyle\Delta(\zeta(2)^{\rm m}\zeta(2n+1)^{\rm
m})=\zeta(2n+1)^{\rm dR}\otimes\zeta(2)^{\rm m}+1^{\rm dR}\otimes\zeta(2)^{\rm
m}\zeta(2n+1)^{\rm m}.$ (5.96)

###  5.6 Coaction conjecture

Let us look at the example of scalar massless $\phi^{4}$ quantum field theory
and consider the Galois coaction restricted to $\mathcal{P}_{\phi^{4}}^{\rm
m}$. This is a priori valued in the whole space $\mathcal{P}^{\rm
dR}\otimes\mathcal{P}^{\rm m}$. However, after computing every known
$\phi^{4}$-period with loop orders at most $7$ and several $\phi^{4}$-periods
with higher loop orders, and explicitly verifying that in each case the Galois
coaction preserves the space $\mathcal{P}_{\phi^{4}}^{\rm m}$, Panzer and
Schnetz545454Panzer and Schnetz [52] explicitly computed the first examples of
$\phi^{4}$-amplitudes which are not MZVs. Such numbers are polylogarithms at
2nd and 6th roots of unity. The coaction conjecture is verified for them as
well. [52] proposed the following conjecture, known as the coaction
conjecture.

######  Conjecture 5.26.

The Galois coaction closes on $\phi^{4}$-periods. In other words, the Galois
conjugates of a $\phi^{4}$-period are also $\phi^{4}$-periods, that is

$\displaystyle\Delta\big{(}\mathcal{P}_{\phi^{4}}^{\rm
m}\big{)}\subseteq\mathcal{P}^{\rm dR}\otimes\mathcal{P}_{\phi^{4}}^{\rm m}.$
(5.97)

Such a conjecture implies the existence of a fundamental hidden symmetry
underlying the class of $\phi^{4}$-periods that we do not yet properly
understand. Indeed, the unexpected observations by Panzer and Schnetz, and the
resulting conjecture, have greatly stimulated research, motivating the search
for a mathematical mechanism able to distinguish $\phi^{4}$-periods from
periods of all graphs, and thus explain this surprising evidence.

Some advancements in this direction have already been made. Conjecture 5.26 is
the strongest among several reformulations of its statement obtained by
suitably enlarging the space of amplitudes under consideration. A weaker
version of the coaction conjecture has been proven by Brown [14]. To a scalar
Feynman graph $G$, we associate the finite-dimensional $\mathbb{Q}$-vector
space $\mathcal{P}_{a}^{\rm m}(G)$ consisting of the motivic realisations of
all affine integrals of globally-defined algebraic differential forms on the
usual integration domain $\sigma$, called the affine periods of $G$. They
include convergent affine integrals of the form

$\displaystyle\int_{0}^{\infty}\cdots\int_{0}^{\infty}\frac{q}{\Psi_{G}^{k}}\bigg{|}_{x_{n_{G}}=1}{\rm
d}^{n_{G}-1}x,$ (5.98)

where $k\geq 1$ is an integer, and $q$ is a polynomial in
$\mathbb{Q}[x_{1},\dots,x_{n_{G}-1}]$. However, the denominator of the
integrand of an affine period of $G$ can also possibly involve linear factors
of the form $\sum_{e\in E_{\gamma}}x_{e}$, where $\gamma$ is a subgraph of
$G$.

######  Theorem 5.27.

The Galois group acts on $\mathcal{P}_{a}^{\rm m}(G)$, that is
$\Delta(\mathcal{P}_{a}^{\rm m}(G))\subseteq\mathcal{P}^{\rm
dR}\otimes\mathcal{P}_{a}^{\rm m}(G)$.

We write $\mathcal{P}_{a}^{\rm m}$ to denote the space spanned by all
$\mathcal{P}_{a}^{\rm m}(G)$ for any555555We observe that restricting to
$\phi^{4}$-graphs does not change the resulting space $\mathcal{P}_{a}^{\rm
m}$, which is sometimes also denoted by $\mathcal{P}_{\tilde{\phi}^{4}}^{\rm
m}$. scalar Feynman graph $G$. Since direct computations show that
$\mathcal{P}_{a}^{\rm m}\simeq\mathcal{P}_{\phi^{4}}^{\rm m}$ at low loop
orders, Theorem 5.27 directly supports Conjecture 5.26.

######  Remark 5.28.

The affine motive $H_{a}$ of a scalar Feynman graph $G$ is defined analogously
to the projective $H$-system $H=H^{n_{G}-1}(P\backslash
Y_{G},B\backslash(B\cap Y_{G}))$, described in Section 5.3, after replacing
the projective space $P$ with an affine open subspace $A\subset P$ obtained by
removing its hyperplanes with strictly positive coefficients. The inclusion
$A\backslash Y_{G}\hookrightarrow P\backslash Y_{G}$ induces a morphism of
objects $H\rightarrow H_{a}$ in $\mathcal{H}$, which gives an equivalence at
the level of motivic periods. Every motivic period of $H$ is also a motivic
period of $H_{a}$, but the contrary is not true. Many periods of $H_{a}$ are
not periods of Feynman graphs in the ordinary sense.

Let $G$ be a scalar Feynman graph. We define the generalised Feynman integrals
associated to $G$ as the projective integrals of parametric form565656Note
that the ordinary Feynman integral (2.17) of $G$ can be written in this form.

$\displaystyle\int_{\sigma}\frac{p\,\Omega}{\Psi_{G}^{k}\,\Xi_{G}^{h}(\\{p_{j},m_{e}\\})},$
(5.99)

where $k,h\geq 1$ are integers, and $p$ is a homogeneous polynomial in
$\mathbb{Q}[\\{x_{e}\\}]$ of degree $kl_{G}+h(l_{G}+1)-n_{G}$, so that the
overall integrand is homogeneous of degree zero. Although possibly divergent,
generalised Feynman integrals are periods, and they can be promoted to their
motivic realisations after being suitably regularised. We define the finite-
dimensional $\mathbb{Q}$-vector space $\mathcal{P}_{g}^{\rm m}$ to be the
space of motivic realisations of regularised versions of generalised Feynman
integrals. An analogous statement to Conjecture 5.26 is proposed by Brown
[14].

######  Conjecture 5.29.

$\mathcal{P}_{g}^{\rm m}$ is stable under the Galois coaction, that is
$\Delta(\mathcal{P}_{g}^{\rm m})\subseteq\mathcal{P}^{\rm
dR}\otimes\mathcal{P}_{g}^{\rm m}$.

###  5.7 Weights and the small graph principle

Recall the notions of Hodge and weight filtrations introduced in Sections 3.4
and 3.5. For $H\in\text{Ob}(\mathcal{H})$, the $\mathbb{Q}$-vector space
$\omega_{\rm dR}(H)$ is equipped with a Hodge filtration $F^{\bullet}$ and a
weight filtration $W^{\rm dR}_{\bullet}$, while the $\mathbb{Q}$-vector space
$\omega_{{\rm B}}(H)$ is provided with a weight filtration $W^{{\rm
B}}_{\bullet}$ only. Mixed $H$-systems, contrary to pure ones, do not have a
well-defined weight. However, their graded quotients with respect to the
weight filtration do possess a pure structure of definite weight. These
properties are used to define a notion of weight for motivic periods.

######  Definition 5.30.

The weight filtration on $\omega_{\rm dR}(H)$ induces a weight filtration on
the space of motivic periods by

$\displaystyle W_{\bullet}^{\rm dR}\mathcal{P}^{\rm
m}=\mathbb{Q}\big{\langle}[H,\omega,\sigma]^{\rm m}\,|\,\omega\in
W_{\bullet}^{\rm dR}\big{\rangle}.$ (5.100)

We denote $W=W^{\rm dR}$ for simplicity. A given motivic period
$[H,\omega,\sigma]^{\rm m}$ is said to have weight at most $n$ if it belongs
to $W_{n}\mathcal{P}^{\rm m}$, and to have weight $n$ if it is non-zero in the
graded quotient ${\rm Gr}^{W}_{n}\mathcal{P}^{\rm m}=W_{n}\mathcal{P}^{\rm
m}/W_{n-1}\mathcal{P}^{\rm m}$.

######  Remark 5.31.

We observe that the weight of motivic periods can alternatively, but
equivalently, be defined from the Betti realisation via the weight filtration
induced on $\mathcal{P}^{\rm m}$ by $W^{{\rm B}}$.

######  Example 5.32.

Consider $H=H^{1}(\mathbb{G}_{m},\\{1,z\\})$ for
$z\in\mathbb{Q}\backslash\\{0,1\\}$. Its weight filtration in the de Rham
realisation is

$\displaystyle W_{-1}=0\subseteq W_{0}=W_{1}=\mathbb{Q}(0)\subseteq
W_{2}=H^{1}(\mathbb{G}_{m},\\{1,z\\}).$ (5.101)

Observing that $0,1\in W_{0}$ and $2\pi{\rm i},\log(z)\in W_{2}$, the weight
of each entry of the period matrix of $H$ is determined. Indeed, $0$, $1$ are
periods of weight zero, while $2\pi{\rm i}$, $\log(z)$ have weight $2$.

######  Example 5.33.

The weight filtration can be used to systematically study
$\mathcal{P}_{\phi^{4}}^{\rm m}$ weight by weight. For example, direct
computation in low weight shows that $W_{0}\mathcal{P}_{\phi^{4}}^{\rm
m}=W_{1}\mathcal{P}_{\phi^{4}}^{\rm m}=W_{2}\mathcal{P}_{\phi^{4}}^{\rm
m}=\mathbb{Q}(0)$.

More generally, the following proposition is due to Brown [14].

######  Proposition 5.34.

Let $G$ be a primitive log-divergent Feynman graph. Every Galois conjugate of
its motivic Feynman integral $I_{G}^{\rm m}$ which has weight up to $2$ is a
period of $\mathbb{Q}(0)$, that is a rational number.

The Galois conjugates of Feynman periods are expected to satisfy the following
conjecture by Brown [14], known as small graph principle.575757The small graph
principle is also conjectured to hold for regularised versions of generalised
Feynman integrals associated to arbitrary scalar Feynman graphs.

######  Conjecture 5.35.

Let $G$ be a primitive log-divergent Feynman graph. Denote by
$[H_{G},\omega_{G},\sigma]^{\rm m}$ the motivic realisation of its Feynman
integral $I_{G}^{\rm m}$. The elements on the right-hand side of the coaction
formula for $\Delta[H_{G},\omega_{G},\sigma]^{\rm m}$ can be expressed in the
form

$\displaystyle\prod_{i}[H_{\gamma_{i}},\omega_{\gamma_{i}},\sigma]^{\rm m},$
(5.102)

where the product runs over a subset $\\{\gamma_{i}\\}$ of the set of
subgraphs and quotient graphs of $G$.

The small graph principle implies that the Galois conjugates of weight at most
$k$ of the motivic amplitude of a primitive log-divergent Feynman graph are
associated to its sub-quotient graphs585858Sub-quotient graphs are formally
obtained by contracting and deleting edges of the original graph. with at most
$k+1$ edges. In other words, when interested in Feynman periods of weight at
most $k$, it suggests to look at graphs with up to $k+1$ edges. As a
consequence, the topology of a given graph constrains the Galois theory of its
amplitudes.

######  Example 5.36.

Consider the system of realisations $H=H^{1}(\mathbb{G}_{m},\\{1,p\\})$ for
$p\in\mathbb{Q}\backslash\\{0,1\\}$ prime. Following Example 5.32,
$\log(p)^{\rm m}$ is a period of $H$ with weight 2. Then, the small graph
principle suggests that any $\log(p)^{\rm m}$ appearing in the right-hand side
of the coaction formula for a given $\phi^{4}$-periods comes from graphs with
at most three edges. Proposition 5.5 implies that all two-edge graphs are
trivial, meaning that the associated graph motive $H_{G}$ is the Hodge–Tate
system $\mathbb{Q}(0)$, which does not have $\log(p)^{\rm m}$ in its period
matrix. Writing down all possible graphs with three edges, we get the graphs
shown in Fig. 16 along with their first graph polynomials in the Schwinger
parameters.

(a)

(b)

(c)

(d)

Figure 16: Feynman graphs with 3 edges and their first graph polynomials.

The two outer graphs (a) and (d) are also trivial by Proposition 5.5, while
the two middle graphs (b) and (c) satisfy
$H_{G}=\mathbb{Q}(0)\oplus\mathbb{Q}(-1)$, which does not have $\log(p)^{\rm
m}$ as a period. Indeed, $\log(p)$ cannot be obtained as an integral with a
denominator equal to either of the first graph polynomials (b) or (c). We
conclude that $\log(p)^{\rm m}$ cannot be a Galois conjugate of a
$\phi^{4}$-period. By equation (5.92), we derive that $\log(p)^{\rm
m}\notin\mathcal{P}_{\phi^{4}}^{\rm m}$. Note that this is consistent with
Proposition 5.34.

######  Example 5.37.

Direct computation by Panzer and Schnetz [52] shows that all
$\phi^{4}$-periods of loop order up to 6 are $\mathbb{Q}$-linear combinations
of multiple zeta values. Following the small graph principle, we graphically
order the set of MZVs by weight as

${1}$${\zeta(2)}$${\zeta(3)}$${\zeta(2)^{2}}$${\zeta(5)}$${\zeta(3)^{2}}$${\zeta(7)}$${\zeta(3,5)}$${\cdots}$${\zeta(2)\zeta(3)}$${\zeta(2)^{3}}$${\zeta(2)\zeta(5)}$${\zeta(2)\zeta(3)^{2}}$${\zeta(2)^{2}\zeta(3)}$${\vdots}$
(5.106)

As a consequence of the coaction conjecture and explicit expressions of the
coaction formula, as the ones in equation (5.94), $\zeta(2)^{\rm
m}\notin\mathcal{P}_{\phi^{4}}^{\rm m}$ implies that all MZVs which are linear
in $\zeta(2)$ cannot be $\phi^{4}$-periods. Analogously,
$\big{(}\zeta(2)^{2}\big{)}^{\rm m}\notin\mathcal{P}_{\phi^{4}}^{\rm m}$
implies that all MZVs which are quadratic in $\zeta(2)$ are not
$\phi^{4}$-periods. The set of MZVs that can appear as $\phi^{4}$-periods
would then be reduced to

${1}$${\zeta(3)}$${\zeta(5)}$${\zeta(3)^{2}}$${\zeta(7)}$${\zeta(3,5)}$${\cdots}$${\zeta(2)^{3}}$${\vdots}$
(5.109)

However, the statements $\zeta(2),\zeta(2)^{2}\notin\mathcal{P}_{\phi^{4}}$
are only conjectural. Precisely, it is conjectured that
$\zeta(2)^{k}\notin\mathcal{P}_{\phi^{4}}$ for $k\leq 5$, while
$\zeta(2)^{k}\in\mathcal{P}_{\phi^{4}}$ for $k\geq 6$. We observe that such
statements rely on the control over weight drops, as conjectured by Panzer and
Schnetz [52], which excludes low weight MZVs coming from high loop order
graphs.

From similar considerations, other highly non-trivial constraints at all loop
orders in perturbation theory can be derived using the Galois coaction and
weight filtrations. For example, by Conjecture 5.26, whenever it is shown that
a given period is not a $\phi^{4}$-period, we conjecturally deduce that all
periods that have the given one among their Galois conjugates cannot appear in
$\mathcal{P}_{\phi^{4}}$ either.

######  Remark 5.38.

Structures even more fundamental than those captured by the coaction
conjecture and the small graph principle underly the space of motivic periods
of Feynman graphs. Although not being sufficiently explored in the literature,
the notion of operad in the category of motives imposes strong constraints on
the admissible periods and it should be the object of further investigation.
The operad structure underlying the space of motivic Feynman integrals is
interestingly the same structure governing the renormalisation group equation.
Kaufmann and Ward [45] provide details on related notions in category theory.

##  6 Conclusions

Originally providing a framework for re-organising and re-interpreting much of
the previous knowledge on Feynman integrals, the theory of motivic periods has
revealed unexpected features, placing restrictions on the set of numbers which
can occur as amplitudes and paving the way for a more comprehensive
understanding of their general structure. Indeed, the coaction conjecture
gives new constraints at each loop order, which in turn propagate to all
higher loop orders because of the recursive structure inherent in perturbative
quantum field theories. At the same time, the small graph principle makes
finite computations at low-loop into all-order results.

Assume to deal with a Feynman integral of the form $\int_{\sigma}\omega$ in
$\mathcal{P}$. The general prescription for its investigation via the theory
of motivic periods can be summarised as follows:

  * (1)

Associate a motivic representation $[H,\omega,\sigma]^{\rm m}$ to the integral
$\int_{\sigma}\omega$, deriving explicitly the corresponding algebraic de Rham
and Betti system of realisations, and cohomology and homology classes.

  * (2)

Use all the known information about the mixed system of realisations $H$ to
derive explicit filtrations.

  * (3)

Write down the period matrix of $H$.

  * (4)

Apply the Galois coaction and derive the Galois conjugates.

  * (5)

Apply the theory of weights of mixed Hodge structures to reduce the
calculation of the Galois conjugates to the study of motivic periods of small
graphs.

  * (6)

Analyse explicitly the few admissible small graphs and eliminate the excluded
periods, sometimes called holes.

  * (7)

Possibly use other known symmetries of the specific example at hand to draw
conclusions.

This picture is, however, extensively conjectural. The very first step of
replacing periods with their motivic version requests the validity of the
period conjecture. Moreover, even disregarding the conjectural status of
current statements, the present state of understanding of motivic amplitudes
is still far from building a theory. Although the given general prescription
for the investigation of motivic Feynman integrals has been particularly
fruitful for massless scalar $\phi^{4}$ quantum field theory, further
advancements are needed to enlarge the reach of current results.

Speculating in full generality, consider the whole class of Feynman integrals
in perturbative quantum field theory. We expect them to have a natural motivic
representation, and thus to generate a space $\mathcal{M}$ of motivic periods,
a space $\mathcal{A}$ of de Rham periods, and a corresponding coaction
$\Delta\colon\mathcal{M}\longrightarrow\mathcal{P}^{\rm
dR}\otimes\mathcal{P}^{\rm m}$. A potential coaction principle would then
state that $\Delta(\mathcal{M})\subseteq\mathcal{A}\otimes\mathcal{M}$. Being
$\mathcal{A}$ a Hopf algebra, we could canonically introduce the group $C$ of
homomorphisms from $\mathcal{A}$ to any commutative ring. It would follow that
the coaction principle can be recast in terms of the group action
$C\times\mathcal{M}\longrightarrow\mathcal{M}$, that is, the space of
amplitudes is stable under the action of the group $C$, often referred to as
cosmic Galois group. This speculative construction, that broadly reproduces
the general prescription summarised above, motivates a programme of research
leading towards a systematic study of scattering amplitudes via the
representation theory of groups.

Although practically harder than the $\phi^{4}$-case, like-minded attempts are
already on the way to gather information about the numbers that come from
evaluating other classes of Feynman integrals.

  * $(a)$

Towards a general motivic description of scalar quantum field theories, Abreu
et al. [1, 2, 3] give evidence suggesting that scalar Feynman integrals of
small graphs with non-trivial masses and momenta satisfy similar properties to
$\phi^{4}$-periods. A diagrammatic coaction for specific families of integrals
appearing in the evaluation of scalar Feynman diagrams, such as multiple
polylogarithms and generalised hypergeometric functions, is proposed and a
connection between this diagrammatic coaction and graphical operations on
Feynman diagrams is conjectured. At one-loop order, a fully explicit and very
compact representation of the coaction in terms of one-loop integrals and
their cuts is found. Moreover, Brown and Dupont [17] investigate a rigorous
theory of motives associated to certain hypergeometric integrals.

  * $(b)$

A subsequent generalisation arises transitioning from scalar quantum field
theories to gauge theories. The problem of dealing with much more involved
parametric integrands which are not explicitly expressed in terms of the
Symanzik polynomials of the associated Feynman graphs has only recently been
tackled. A combinatoric and graph-theoretic approach to Schwinger parametric
Feynman integrals in quantum electrodynamics by Golz [36] has revealed that
the parametric integrands can be explicitly written in terms of new types of
graph polynomials related to specific subgraphs. The tensor structure of
quantum electrodynamics is given a diagrammatic interpretation. The resulting
significant simplification of the integrands paves the way for a systematic
motivic description of gauge theories.

  * $(c)$

In the same research direction, a high-precision computation of the 4-loop
contribution to the electron anomalous magnetic moment $g-2$ by Laporta [49]
shows the presence of polylogarithmic parts with fourth and sixth roots of
unity. This result is conjecturally recast into the motivic $f$-alphabet by
Schnetz [59], giving a more compact expression which explicitly reveals a
Galois structure. In this work, the $\mathbb{Q}$-vector spaces of Galois
conjugates of the $g-2$ are conjectured up to weight four.

As a final remark, we mention that scattering amplitudes do not appear
exclusively in perturbative quantum field theory. Among other settings, there
are string perturbation theory and $\mathcal{N}=4$ super Yang–Mills theory. In
each of these theories, after suitably defining the space of integrals or
amplitudes595959In various modern approaches to $\mathcal{N}=4~{}\text{SYM}$,
including the bootstrap method, on-shell techniques, and the amplituhedron,
the amplitude is constructed independently of the Feynman graphs. In these
settings, the coaction principle operates on the entire amplitude, contrary to
the case of perturbative quantum field theory, where it operates graph by
graph. under consideration, a version of the coaction principle is expected to
hold and some promising preliminary results have already been found. We refer
to the work of Schlotterer, Stieberger and Taylor [57, 66] and subsequent
developments for superstring perturbation theory, and to the work of Caron-
Huot et al. [21, 22] for the planar limit of $\mathcal{N}=4$ super Yang–Mills
theory.

### Acknowledgements

I thank Francis Brown and Lionel Mason for useful discussions. I thank the
three anonymous referees for their detailed reports that have provided a
valuable guide in improving the paper. Finally, I thank Evgeny Mukhin and the
rest of the organisers of the Conference on Representation Theory and
Integrable Systems (ETH Zürich, 2019) for the opportunity to speak and to
contribute to the special issue. This work is partially supported by the
Italian Department of Education, Research and University (Torno Subito
13474/19.09.2018 POR-Lazio-FSE/2014-2020) and the Swiss National Centre of
Competence in Research SwissMAP (NCCR 51NF40-141869 The Mathematics of
Physics).

## References

  * [1] Abreu S., Britto R., Duhr C., Gardi E., Diagrammatic Hopf algebra of cut Feynman integrals: the one-loop case, J. High Energy Phys. 2017 (2017), no. 12, 090, 73 pages, arXiv:1704.07931.
  * [2] Abreu S., Britto R., Duhr C., Gardi E., Matthew J., Coaction for Feynman integrals and diagrams, PoS Proc. Sci. (2018), PoS(LL2018), 047, 14 pages, arXiv:1808.00069.
  * [3] Abreu S., Britto R., Duhr C., Gardi E., Matthew J., From positive geometries to a coaction on hypergeometric functions, J. High Energy Phys. 2020 (2020), no. 2, 122, 44 pages, arXiv:1910.08358.
  * [4] André Y., Une introduction aux motifs (motifs purs, motifs mixtes, périodes), Panoramas et Synthèses, Vol. 17, Société Mathématique de France, Paris, 2004.
  * [5] Bloch S., Esnault H., Kreimer D., On motives associated to graph polynomials, Comm. Math. Phys. 267 (2006), 181–225, arXiv:math.AG/0510011.
  * [6] Blümlein J., Broadhurst D.J., Vermaseren J.A.M., The multiple zeta value data mine, Comput. Phys. Comm. 181 (2010), 582–625, arXiv:0907.2557.
  * [7] Bogner C., Weinzierl S., Periods and Feynman integrals, J. Math. Phys. 50 (2009), 042302, 16 pages, arXiv:0711.4863.
  * [8] Bogner C., Weinzierl S., Feynman graph polynomials, Internat. J. Modern Phys. A 25 (2010), 2585–2618, arXiv:1002.3458.
  * [9] Bott R., Tu L.W., Differential forms in algebraic topology, Graduate Texts in Mathematics, Vol. 82, Springer-Verlag, New York – Berlin, 1982.
  * [10] Broadhurst D., Multiple zeta values and modular forms in quantum field theory, in Computer Algebra in Quantum Field Theory, Texts Monogr. Symbol. Comput., Springer, Vienna, 2013, 33–73.
  * [11] Broadhurst D.J., Kreimer D., Knots and numbers in $\phi^{4}$ theory to $7$ loops and beyond, Internat. J. Modern Phys. C 6 (1995), 519–524, arXiv:hep-ph/9504352.
  * [12] Broadhurst D.J., Kreimer D., Association of multiple zeta values with positive knots via Feynman diagrams up to $9$ loops, Phys. Lett. B 393 (1997), 403–412, arXiv:hep-th/9609128.
  * [13] Brown F., On the decomposition of motivic multiple zeta values, in Galois–Teichmüller theory and arithmetic geometry, Adv. Stud. Pure Math., Vol. 63, Math. Soc. Japan, Tokyo, 2012, 31–58, arXiv:1102.1310.
  * [14] Brown F., Feynman amplitudes, coaction principle, and cosmic Galois group, Commun. Number Theory Phys. 11 (2017), 453–556, arXiv:1512.06409.
  * [15] Brown F., Notes on motivic periods, Commun. Number Theory Phys. 11 (2017), 557–655, arXiv:1512.06410.
  * [16] Brown F., Doryn D., Framings for graph hypersurfaces, arXiv:1301.3056.
  * [17] Brown F., Dupont C., Lauricella hypergeometric functions, unipotent fundamental groups of the punctured Riemann sphere, and their motivic coactions, arXiv:1907.06603.
  * [18] Brown F., Kreimer D., Angles, scales and parametric renormalization, Lett. Math. Phys. 103 (2013), 933–1007, arXiv:1112.1180.
  * [19] Brown F., Schnetz O., A K3 in $\phi^{4}$, Duke Math. J. 161 (2012), 1817–1862, arXiv:1006.4064.
  * [20] Brown F., Schnetz O., Single-valued multiple polylogarithms and a proof of the zig-zag conjecture, J. Number Theory 148 (2015), 478–506, arXiv:1208.1890.
  * [21] Caron-Huot S., Dixon L.J., Drummond J.M., Dulat F., Foster J., Gürdoğan O., von Hippel M., McLeod A.J., Papathanasiou G., The Steinmann cluster bootstrap for ${\mathcal{N}}=4$ super Yang–Mills amplitudes, PoS Proc. Sci. (2020), PoS(CORFU2019), 003, 37 pages, arXiv:2005.06735.
  * [22] Caron-Huot S., Dixon L.J., Dulat F., von Hippel M., McLeod A.J., Papathanasiou G., The cosmic Galois group and extended Steinmann relations for planar ${\mathcal{N}}=4$ SYM amplitudes, J. High Energy Phys. 2019 (2019), no. 9, 061, 65 pages, arXiv:1906.07116.
  * [23] Colmez P., Serre J.P. (Editors), Correspondance Grothendieck–Serre, Documents Mathématiques (Paris), Vol. 2, Société Mathématique de France, Paris, 2001.
  * [24] De Rham G., Sur l’analysis situs des variétés à $n$ dimensions, 1931, available ar http://www.numdam.org/item?id=THESE_1931__129__1_0.
  * [25] Deligne P., Théorie de Hodge. I, in Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1, 1971, 425–430.
  * [26] Deligne P., Théorie de Hodge. II, Inst. Hautes Études Sci. Publ. Math. 40 (1971), 5–57.
  * [27] Deligne P., Théorie de Hodge. III, Inst. Hautes Études Sci. Publ. Math. 44 (1974), 5–77.
  * [28] Deligne P., Goncharov A.B., Groupes fondamentaux motiviques de Tate mixte, Ann. Sci. École Norm. Sup. (4) 38 (2005), 1–56, arXiv:math.NT/0302267.
  * [29] Deligne P., Milne J.S., Ogus A., Shih K., Hodge cycles, motives, and Shimura varieties, Lecture Notes in Math., Vol. 900, Springer-Verlag, Berlin – New York, 1982.
  * [30] Demazure M., Motifs des variétés algébriques, in Séminaire Bourbaki, Vol. 1991/92, Exp. No. 364–381, Lecture Notes in Math., Vol. 180, Springer-Verlag, Berlin – Heidelberg, 1971, 19–38.
  * [31] Dyson F.J., The radiation theories of Tomonaga, Schwinger, and Feynman, Phys. Rev. 75 (1949), 486–502.
  * [32] Dyson F.J., Divergence of perturbation theory in quantum electrodynamics, Phys. Rev. 85 (1952), 631–632.
  * [33] Elvang H., Huang Y., Scattering amplitudes, arXiv:1308.1697.
  * [34] Feynman R.P., Space-time approach to quantum electrodynamics, Phys. Rev. 76 (1949), 769–789.
  * [35] Gil J.I.B., Fresán J., Multiple zeta values: from numbers to motives, available at https://javier.fresan.perso.math.cnrs.fr/mzv.pdf.
  * [36] Golz M., Parametric quantum electrodynamics, Ph.D. Thesis, Humboldt University, Berlin, 2018.
  * [37] Grothendieck A., On the de Rham cohomology of algebraic varieties, Inst. Hautes Études Sci. Publ. Math. 29 (1966), 95–103.
  * [38] Hartshorne R., Algebraic geometry, Graduate Texts in Mathematics, Vol. 52, Springer-Verlag, New York – Heidelberg, 1977.
  * [39] Henn J.M., Lectures on differential equations for Feynman integrals, J. Phys. A: Math. Theor. 48 (2015), 153001, 35 pages, arXiv:1412.2296.
  * [40] Hironaka H., Resolution of singularities of an algebraic variety over a field of characteristic zero. I, Ann. of Math. 79 (1964), 109–203.
  * [41] Hironaka H., Resolution of singularities of an algebraic variety over a field of characteristic zero. II, Ann. of Math. 79 (1964), 205–326.
  * [42] Hodge W.V.D., The theory and applications of harmonic integrals, Cambridge University Press, Cambridge, Macmillan Company, New York, 1941.
  * [43] Huber A., Müller-Stach S., Periods and Nori motives, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, Vol. 65, Springer, Cham, 2017.
  * [44] Kashiwara M., Schapira P., Categories and sheaves, Grundlehren der Mathematischen Wissenschaften, Vol. 332, Springer-Verlag, Berlin, 2006.
  * [45] Kaufmann R.M., Ward B.C., Feynman categories, Astérisque 387 (2017), vii+161 pages, arXiv:1312.1269.
  * [46] Kleiman S.L., Motives, in Algebraic Geometry, Oslo 1970 (Proc. Fifth Nordic Summer-School in Math., Oslo, 1970), 1972, 53–82.
  * [47] Kontsevich M., Zagier D., Periods, in Mathematics Unlimited – 2001 and Beyond, Springer, Berlin, 2001, 771–808.
  * [48] Kreimer D., Renormalization and knot theory, J. Knot Theory Ramifications 6 (1997), 479–581, arXiv:hep-th/9412045.
  * [49] Laporta S., High-precision calculation of the 4-loop contribution to the electron $g-2$ in QED, Phys. Lett. B 772 (2017), 232–238, arXiv:1704.06996.
  * [50] Laporta S., Remiddi E., The Analytical value of the electron $g-2$ at order $\alpha^{3}$ in QED, Phys. Lett. B 379 (1996), 283–291, arXiv:hep-ph/9602417.
  * [51] Murre J.P., Nagel J., Peters C.A.M., Lectures on the theory of pure motives, University Lecture Series, Vol. 61, Amer. Math. Soc., Providence, RI, 2013.
  * [52] Panzer E., Schnetz O., The Galois coaction on $\phi^{4}$ periods, Commun. Number Theory Phys. 11 (2017), 657–705, arXiv:1603.04289.
  * [53] Petermann A., Fourth order magnetic moment of the electron, Helv. Phys. Acta 30 (1957), 407–408.
  * [54] Saavedra Rivano N., Catégories Tannakiennes, Lecture Notes in Math., Vol. 265, Springer-Verlag, Berlin – New York, 1972.
  * [55] Salam A., Divergent integrals in renormalizable field theories, Phys. Rev. 84 (1951), 426–431.
  * [56] Salam A., Overlapping divergences and the $S$-matrix, Phys. Rev. 82 (1951), 217–227.
  * [57] Schlotterer O., Stieberger S., Motivic multiple zeta values and superstring amplitudes, J. Phys. A: Math. Theor. 46 (2013), 475401, 37 pages, arXiv:1205.1516.
  * [58] Schnetz O., Quantum periods: a census of $\phi^{4}$-transcendentals, Commun. Number Theory Phys. 4 (2010), 1–47, arXiv:0801.2856.
  * [59] Schnetz O., The Galois coaction on the electron anomalous magnetic moment, Commun. Number Theory Phys. 12 (2018), 335–354, arXiv:1711.05118.
  * [60] Schnetz O., Numbers and functions in quantum field theory, Phys. Rev. D 97 (2018), 085018, 20 pages, arXiv:1606.08598.
  * [61] Serone M., Spada G., Villadoro G., The power of perturbation theory, J. High Energy Phys. 2017 (2017), no. 5, 056, 41 pages, arXiv:1702.04148.
  * [62] Serre J.P., Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier (Grenoble) 6 (1956), 1–42.
  * [63] Shanks D., Dihedral quartic approximations and series for $\pi$, J. Number Theory 14 (1982), 397–423.
  * [64] Smirnov V.A., Feynman integral calculus, Springer-Verlag, Berlin, 2006.
  * [65] Srednicki M., Quantum field theory, Cambridge University Press, Cambridge, 2010\.
  * [66] Stieberger S., Taylor T.R., Closed string amplitudes as single-valued open string amplitudes, Nuclear Phys. B 881 (2014), 269–287, arXiv:1401.1218.
  * [67] ’t Hooft G., Veltman M., Regularization and renormalization of gauge fields, Nuclear Phys. B 44 (1972), 189–213.
  * [68] Voevodsky V., Triangulated categories of motives over a field, in Cycles, Transfers, and Motivic Homology Theories, Ann. of Math. Stud., Vol. 143, Princeton University Press, Princeton, NJ, 2000, 188–238.
  * [69] Voisin C., Hodge theory and complex algebraic geometry. I, Cambridge Studies in Advanced Mathematics, Vol. 76, Cambridge University Press, Cambridge, 2002.
  * [70] Waldschmidt M., Transcendence of periods: the state of the art, Pure Appl. Math. Q. 2 (2006), 435–463.
  * [71] Weibel C.A., An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, Vol. 38, Cambridge University Press, Cambridge, 1994\.
  * [72] Weinberg S., High-energy behavior in quantum field-theory, Phys. Rev. 118 (1960), 838–849.
  * [73] Zee A., Quantum field theory in a nutshell, Princeton University Press, Princeton, NJ, 2003.