An Introduction to Motivic Feynman Integrals

This article gives a short step-by-step introduction to the representation of parametric Feynman integrals in scalar perturbative quantum field theory as periods of motives. The application of motivic Galois theory to the algebro-geometric and categorical structures underlying Feynman graphs is reviewed up to the current state of research. The example of primitive log-divergent Feynman graphs in scalar massless $\phi^4$ quantum field theory is analysed in detail.


Introduction
In Section 1, we describe the graph-theoretic framework for the investigation of the algebraic information contained in the topology of scalar Feynman diagrams. Perturbative quantum field theories possess an inherent algebraic structure, which underlies the combinatorics of recursion governing renormalisation theory, and are thus deeply connected to the theory of graphs.
In Section 2, we broadly review preliminary notions in algebraic geometry and algebraic topology. An algebraic variety over Q is endowed with two distinct rational structures via algebraic de Rham cohomology and Betti cohomology, which are compatible only after complexification. The coexistence of these two cohomologies and their peculiar compatibility are linked to a specific class of complex numbers, known as numeric periods. The cohomology of an algebraic variety is equipped with two filtrations, and the mixed Hodge structure arising from their interaction constitutes the bridge between the theory of numeric periods and the theory of motives.
In Section 3, we introduce the set of periods, lying betweenQ and C, among which are the numbers that come from evaluating parametric Feynman integrals, and we briefly review their remarkable properties. Suitable cohomological structures are exploited to derive non-trivial information about these numbers.
In Section 4, we describe how Feynman integrals are promoted to periods of motives. Technical issues arising from the presence of singularities are tackled by blow up. We adopt the category-theoretic Tannakian formalism where motivic periods, and motivic Feynman integrals in particular, reveal their most intriguing properties. We present an overview of the current progress of research towards the general understanding of the structure of scattering amplitudes via the theory of motivic periods, giving particular attention to recent results in massless scalar φ 4 quantum field theory.

Perturbative Quantum Field Theory
A quantum field theory encodes in its Lagrangian every admissible interaction among particles, but it does it in a way that makes decoding this information a difficult task. Fixed the initial and final states, an interaction process is associated to a probability amplitude, called its Feynman amplitude, which is determined by the set of kinematic and interaction terms in the Lagrangian. However, individual Lagrangian terms correspond to propagators and interaction vertices which can be linked together in infinitely many distinct ways to connect the same pair of initial and final states. Each of these admissible realisations of the same interaction process has to be accounted for in an infinite sum of contributions to the probability amplitude. We associate to each of these possibilities a graphical representation, called its Feynman diagram, whose individual contribution to the probability amplitude is explicitly written in the form of a Feynman integral by applying the formal correspondence between Lagrangian terms and graphical components, which is established by convention through the set of Feynman rules of the theory. It is only the sum of the contributing Feynman integrals to a given process that has a physical meaning and not the individual integrals, which are themselves interrelated by the gauge symmetry of the Lagrangian.
In perturbative quantum field theory, the sum of individual Feynman integrals is a perturbative expansion in some small parameter of the theory, typically a suitable coupling constant. Thus, the Feynman amplitude can be expanded in a formal power series, which has been shown to be divergent 1 by Dyson [29]. The divergence does not, however, undermine the accuracy of predictions that can be made with the theory. Indeed, although a Feynman amplitude receives contributions to any order in perturbation theory, practical calculations are made by truncating the sum at a certain order and directly evaluating only the remaining finitely many terms. Moreover, the explicit calculation of a Feynman amplitude only includes those diagrams which are one-particle irreducible, or 1PI, that is, diagrams which cannot be divided in two by cutting through a single propagator. See Fig. 1. The contribution from a non-1PI diagram at some given order can be expressed as a combination of lower-order 1PI contributions, which have already been accounted for in the formal series.
The leading order terms in the perturbative expansion of a Feynman amplitude are called tree-level contributions. Higher order diagrams are obtained from tree-level diagrams by adding internal loops. Each independent loop in a diagram is associated to an unconstrained momentum and integrals over unconstrained loop momenta are the origin of singularities in Feynman integrals. We distinguish two classes of singularities. The ultraviolet (UV) divergences arise in the limit of infinite loop momentum, a regime that is far beyond the energy scale that we have currently experimental access to and where we expect new physical phenomena to become relevant and corresponding new (a) One-particle irreducible (b) One-particle reducible Figure 1: Examples of 1PI and non-1PI diagrams.
terms to enter the Lagrangian. Sensitivity to the high loop momentum region is treated by means of renormalisation theory. For a renormalizable theory, a suitable adjustment of the Lagrangian parameters allows to systematically re-express the predictions of the theory in terms of renormalized physical couplings, so that they decouple from UV physics. Thus, the theory gives a finite and well-defined relation between physical observables. The infrared (IR) divergences only arise in theories with massless particles as they originate in the limit of infinitesimal loop momentum. They cannot be removed by renormalisation and introduce numerous subtleties in the evaluation of Feynman integrals which we are not dealing with in the present text. For a detailed and comprehensive presentation of perturbative quantum field theory we refer to Zee [64] and Srednicki [58].
Evaluating Feynman integrals over loop momenta has been of fundamental concern since the early days of perturbative quantum field theory. Smirnov [57] summarised more than fifty years of advancements in the field, providing an overview of the most powerful, successful and well-established methods for evaluating Feynman integrals in a systematic way, and at the same time showing how the problem of evaluation has become more and more critical. What could be easily evaluated has, indeed, already been evaluated years ago. Since the first insights into the problem of UV divergences in a quantum field theory presented by Dyson [28], [29], Salam [51], [50] and Weinberg [63], our understanding has vastly improved. Elvang and Huang [30] give a recent overview of the subject, including unitarity methods, BCFW recursion relations, and the methods of leading singularities and maximal cuts. Overlapping divergences can be treated iteratively, thus revealing in the first place the recursive nature of renormalisation theory. However, this combinatorics of subdivergences is only the first hint to a more fundamental algebraic structure inherent in all renormalizable quantum field theories and deeply connected to the theory of graphs 2 .

Feynman Parametrisation
We consider a scalar quantum field theory in an even number D of space-time dimensions with Euclidean metric 3 and allow different propagators to have different mass. A Feynman diagram is a connected directed graph where each edge represents a propagator and is assigned a momentum and a mass and each vertex stands for a tree-level interaction. External half-edges, also known as external legs, represent incoming or outgoing particles, while internal edges are the internal propagators of the diagram. We define the loop number to be the number of independent cycles of the graph. We adopt the convention for which all external legs have arrows pointing inwards, and consequently distinguish incoming and outgoing particles depending on the momentum being positive or negative, respectively. Momentum is positive when it points in the same direction of the arrow of the corresponding directed edge, and it is negative otherwise. We fix momenta on external lines and for each internal loop we choose an arbitrary orientation of the edges which is consistent with momentum conservation at each vertex of the graph and globally. Momentum conservation leaves one unconstrained free momentum variable for each loop. Thus, the loop number is equal to the number of independent loop momentum vectors. We only consider graphs that are one-particle irreducible.
Let G be such a Feynman graph with m external legs, n internal edges and l independent loops. Its Feynman integral, up to numerical prefactors, is where µ is a scale introduced to make the expression dimensionless, k 1 , ..., k l are the independent loop momenta, m 1 , ..., m n are the masses of the internal lines and q 1 , ..., q n are the momenta flowing through them. These can be expressed as where p 1 , ..., p m are the external momenta and λ ji , σ ji ∈ {−1, 0, 1} are constants depending on the particular graph structure. Feynman [31] introduced the well-known manipulation consisting of defining a set of parameters x 1 , ..., x n , called Feynman parameters, one for each internal edge of the graph, and applying the formula with the choice P j = −q 2 j + m 2 j for j = 1, ..., n. Here, Γ is the Euler Gamma function and δ is the Dirac Delta distribution. Indeed, we can write where M is a l×l-matrix with scalar entries, Q is a l-vector with momentum vectors as entries and J is a scalar. M , Q and J can be suitably expressed in terms of the graph parameters {x j , q j , m j } n j=1 . Applying Feynman parametrisation to (1), the l-dimensional integral over the loop momenta becomes an (n − 1)-dimensional integral over the Feynman parameters which is characterised by the polynomials U = det(M ) and F = det(M )(J + QM −1 Q)/µ 2 , called first and second Symanzik polynomials of the Feynman graph, respectively. Notice that the dimension D of space-time, entering the exponents in the integrand of (5), acts as regularisation. We use dimensional regularisation with D = 4 − 2 and small parameter. A detailed description of Feynman parametrisation can be found in Srednicki [58].
Example 1. Consider the generic one-loop diagram with m = n external legs. Its Symanzik polynomials are where the internal momenta are given 4 by q 1 = k, q i = k + p 1 + ... + p i−1 for 1 < i < n and q n = k − p n . Here, k is the unique loop momentum of the graph.

Graph Polynomials
Re-expression of Feynman integrals in parametric form shows that the correspondence between scalar Feynman diagrams and Feynman integrals can be reformulated in different terms. The information contained in a Feynman graph is shared out among its multiple components, which can be identified as the underlying graph structure, the directionality of edges and the various edge labels. If we destructure a Feynman graph in these layers and momentarily neglect the extra information apart from the graph structure, we observe that its integral is insensitive to changes of the graph which leave its topology unaltered. Focusing on the underlying graph topology, the Symanzik polynomials can be suitably re-interpreted and they are commonly called graph polynomials in this context.
Let G be a finite graph without isolated vertices. G is specified by the pair (V G , E G ), where V G is the collection of vertices and E G is the collection of edges. We choose an arbitrary orientation of its edges and define the map where e ∈ E G is an edge and s(e), t(e) ∈ V G are its source and sink endpoints with respect to the edge orientation. Let us extend this map to the following exact sequence where H 0 (G, Z) and H 1 (G, Z) are the zeroth and first homology groups of the graph. As a consequence, the graph loop number l G is related to the number of edges n G , the number of vertices v G and the number of connected components c G by 5 Assume G is a graph of Feynman type, that is, finite, connected and one-particle irreducible. Let the valency of a vertex be the number of edges attached to it. Being interested in the braid pattern of Feynman graphs, we omit both vertices of valency one, corresponding to the source endpoints of external legs, and vertices of valency two, corresponding to mass insertions. To such a graph G we wish to assign an integral I G which corresponds to the one previously defined in (5) when the neglected extra information is re-inserted. We start by associating a variable x e to every internal edge e ∈ E G of the graph. These variables are known as Schwinger parameters and they are the graph-theoretic analogues of Feynman parameters. Let T 1 be the set of spanning trees 6 of G. The first graph polynomial of G is defined as It is a homogeneous polynomial of degree l G in the Schwinger parameters. Note that each monomial of Ψ G has coefficient one, and Ψ G is linear in each Schwinger parameter.
Example 2. The first graph polynomial of the Feynman graph shown in Fig. 2  By construction, the first Symanzik polynomial U of a Feynman graph G does not depend on momenta and masses involved in the diagram, but is only dependent on the graph topology. Indeed, it explicitly identifies with the first graph polynomial Ψ G of the corresponding pure graph structure. The same is not true for the second Symanzik polynomial F, which is a function of external momenta and internal masses. However, we can re-express F in a way that clearly separates the contribution to F coming from the graph topology from its other dependences. To this end, momenta and masses edge labels must re-enter our discussion. Let T 2 be the set of spanning 2-forests of G and P Ti be the set of external momenta of G attached to its tree T i . The second graph polynomial of G is defined as It is a homogeneous polynomial of degree l G + 1 in the Schwinger parameters. Note that, if all internal masses are zero, then Ξ G is linear in each Schwinger parameter. It follows from their definitions that the second Symanzik polynomial and the second graph polynomial of a Feynman graph are, indeed, the same. Moreover, having fixed the momenta of external particles and the masses of internal propagators, we are left with the explicit dependence of F on the graph structure given in terms of spanning 2-forests.
Example 3. To explicitly see how the individual terms in the graph polynomials arise from the knot structure of the diagram, we look closer at the one-loop Feynman graph with m = 4 external legs, also called box diagram 7 , which is shown in Fig. 3. 5 The loop number is equivalently defined as the rank of the first homology group of the graph, while the number of connected components corresponds to the rank of the zeroth homology group of the graph. 6 A graph of zero loop number with k connected components is called a k-forest. When k = 1, the forest is called a tree. Given an arbitrary connected graph G, a spanning k-forest of G is a subgraph T ⊆ G such that V T = V G and T is a k-forest. A spanning k-forest of G is usually denoted by the collection of its trees. A connected graph has always at least one spanning tree. 7 This gives a next-to-leading order contribution to the two-to-two particle scattering process. Srednicki [58] gives a detailed discussion of two particles elastic scattering at one-loop using standard methods in perturbative quantum field theory.  Its Symanzik polynomials are Neglecting mass terms, the remaining monomials correspond to the spanning forests shown in Fig. 4 and Fig. 5. (a) +x 1 x 2 p 2 Thus, the Symanzik or graph polynomials capture the algebraic information contained in the topology of a Feynman diagram and they prove to be the first tool to be used in the tentative investigation of renormalisation theory via the algebraic manipulation of concatenated one-loop integrals. For a more detailed overview of the properties of Feynman graph polynomials we refer to Bogner and Weinzierl [7].

Primitive Log-Divergent φ 4 Graphs
The parametric Feynman integral in (5) can be written in a slightly different notation, which turns out to be particularly useful henceforth. Neglecting prefactors and assuming D = 4, it is equivalent to the projective integral where σ is the real projective simplex given by and Ω is the top-degree differential form on P n G −1 expressed in local coordinates as One can check that the integrand is homogeneous of degree zero, so that the integral in projective space is well-defined and equivalent, under the affine constraint x n G = 1, to the previous parametric integral in affine space. Integral (13) is in general divergent, as singularities may arise if the zero sets of the graph polynomials Ψ G and Ξ G intersect the domain of integration. Graphs satisfying the condition n G = 2l G are called logarithmically divergent and constitute a particularly interesting class of graphs. In fact, their Feynman integral simplifies to where the dependence on the second Symanzik polynomial, and consequently on momenta and masses, has vanished. Being uniquely sensitive to the graph topology, such a Feynman graph describes a so-called single-scale process 8 . For a logarithmically divergent graph G, we define the graph hypersurface as the zero set of its first Symanzik polynomial which describes the singularities of its Feynman integral I G . The following theorem on the convergence of logarithmically divergent graphs is proven by Bloch, Esnault and Kreimer [5].
Theorem 1. Let G be logarithmically divergent. The integral I G converges if and only if every proper subgraph ∅ = γ ⊂ G satisfies the condition n γ > 2l γ .
A logarithmically divergent graph G such that I G is convergent is called primitive log-divergent, or simply primitive. We give particular attention to the class of primitive log-divergent graphs in scalar massless φ 4 quantum field theory. They are called φ 4 -graphs, and have vertices with valency at most four. Feynman amplitudes in φ 4 theory have been computed to much higher loop orders than most other quantum field theories thanks to the work of Broadhurst and Kreimer [10], [11], and Schnetz [53]. Some of the simplest φ 4 graphs are shown in Fig. 6 along with the values of the associated Feynman integrals. Here, ζ is the Riemann zeta function, and P 3,5 = − 216 5 ζ(3, 5) − 81ζ(5)ζ(3) + 522 5 ζ(8).

Multiple Zeta Values
The Riemann zeta function is defined, on the half-plane of complex numbers s ∈ C with Re(s) > 1, by the absolutely convergent series and extended to a meromorphic function on the whole complex plane with a single pole at s = 1. The first tentative attempts to find polynomial relations among zeta values by multiplying terms of the form (18)  The Q-vector space generated by multiple zeta values forms an algebra under the so-called stuffle product. Analytic methods, like partial fraction expansions, provide only a few of the known relations among MZVs. Many more are obtained, although conjecturally, by performing extensive numerical experiments, as described by Blümlein et al [6]. However, enormous progress followed the analytic discovery of a crucial feature of multiple zeta values, that is, beside their representation as infinite series, they admit an alternative representation as iterated integrals over simplices of If s is an admissible multi-index, write r i = s 1 + ... + s i for each i = 1, ..., l and set r 0 = 0. Define the measure ω s on the interior of the simplex ∆ wt(s) by The theorem below is due to Kontsevich.
This different way of writing multiple zeta values yields a new algebra structure associated with the so-called shuffle product. Many other linear relations among MZVs are obtained systematically in this alternative framework. However, it is the comparison of the two coexisting fundamental representations, given by 19 and 22, which contemporarily endow the Q-vector space of MZVs of two distinct algebraic structures, expressed by the stuffle and shuffle products, to be the most productive source of information about these numbers. Relations among MZVs are also and most interestingly derived by such a comparison. For a more detailed discussion of the classical theory of multiple zeta values we refer to the survey article by Fresán and Gil [32].
We observe the remarkable fact that Q-linear combinations of multiple zeta values are ubiquitous in the evaluation of Feynman amplitudes in perturbative quantum field theories. It was conjectured by Broadhurst and Kreimer [10] and then proved by Brown and Schnetz [17] that Feynman integrals of the infinite family of zig-zag graphs (see Fig.  7) in φ 4 theory are certain known rational multiples of the odd values of the Riemann zeta function.
Theorem 3. Let Z l be the zig-zag graph with l loops. Its Feynman integral is Another example is given by the anomalous magnetic moment of the electron in quantum electrodynamics. The tree level Feynman diagram representing a slow-moving electron emitting a photon is depicted in Fig. 8 along with its one-loop correction.  The two-loop correction comes from the contributions of seven distinct two-loop diagrams. The total two-loop Feynman amplitude has been evaluated by Petermann [48], giving 197 144 + 1 2 ζ(2) − 3ζ(2) log(2) + 3 4 ζ(3), which involves the logarithm of 2 and again values of the Riemann zeta function.
Many more examples are given by Broadhurst [9]. Due to a vast amount of evidence, it was believed for a long time that all primitive amplitudes of the form (16) in massless φ 4 theory should be Q-linear combinations of MZVs. Only recently this conjectural statement was proved false. Explicit examples of φ 4 -amplitudes at high loop orders not expressible in terms of multiple zeta values have been found by Panzer and Schnetz [47]. In the same work, explicit computation of all φ 4 -amplitudes with loop order up to 7 suggests that not all MZVs appear among them. For example, no φ 4 -graph is known to evaluate to ζ(2) or ζ(2) 2 . Remarkably, the integral representation of MZVs partially clarify the presence of these numbers in perturbative calculations in quantum field theory. Indeed, both expressions (16) and (22) are suitably interpreted as periods of algebraic varieties.

Singular Homology
We follow the expositions by Weibel [62] and Hartshorne [35]. Let M be a topological space. For each integer n ≥ 0, the standard n-simplex is For each i = 0, ..., n, the face map δ n i : A singular n-chain in M is a continuous 9 map σ : ∆ n st → M . For each n ≥ 0, let be the free abelian group generated by singular n-chains. Elements of C n (M ) are finite Z-linear combinations of the continuous maps σ : ∆ n st → M . For each n ≥ 1, the boundary map ∂ n : C n (M ) → C n−1 (M ) is defined by where the alternating signs in the sum guarantee that boundary maps satisfy the condition ∂ n−1 • ∂ n = 0. The pair (C • (M ), ∂ • ) is called a homological chain complex and is graphically represented as In degree n, chains in the kernel of the boundary map ∂ n are called (closed) cycles and chains in the image of the boundary map ∂ n+1 are called (exact) boundaries.
Example 4. Let M = C * be the punctured complex plane. The singular chains generate the singular homology groups H s 0 (C * , Z) and H s 1 (C * , Z), respectively. These are both free groups of rank one. All the other homology groups vanish.
For each n ≥ 0, the free abelian group of singular n-cochains is defined by Analogously, applying vector duality, the coboundary maps d n : Definition 2. The singular cohomology of the topological space M is the cohomology of the complex (C Definitions 1 and 2 of singular homology and cohomology, given here with respect to Z, extend to other coefficient rings. For our purposes, we almost exclusively work with rational coefficients. This allows us to identify singular cohomology groups with the vector duals of the corresponding singular homology groups, that is 10 Thus, classes in a cohomology group can be interpreted as classes of linear functionals on the corresponding homology group. The singular cohomology of a topological space given by the complex points of an algebraic variety defined over a subfield of C has a name of its own. Definition 3. Let K be a subfield of C and let X be an algebraic variety over K. The Betti cohomology of X is the singular cohomology of the underlying topological space of complex points X(C) equipped with the analytic topology, that is Example 5. Let G m = SpecQ[x, 1/x] be the multiplicative group. G m is an algebraic variety over Q and its underlying topological space of complex points is G m (C) = C * . For each n ≥ 0, the n-th Betti cohomology group of G m is H n B (G m ) = H n s (C * , Q).

Some Properties of Homology
We briefly recall some properties of singular homology and cohomology, assuming the ring of coefficients to be Q.
1) Homotopy invariance. If M 1 and M 2 are homotopically equivalent topological spaces, then H s n (M 1 , Q) H s n (M 2 , Q) for each n ≥ 0. An analogous statement holds for singular cohomology.
For any two open subspaces U, V ⊆ M such that M = U ∪ V , there is a long exact sequence of the following form An analogous statement holds for singular cohomology.
3) Künneth formula. For any two topological spaces M 1 , M 2 , for each n ≥ 0, there is a natural isomorphism An analogous statement holds for singular cohomology.
commutes. Hence, f induces also a group homomorphism between the corresponding singular homology groups for each n ≥ 0.

5)
Pull-back. Let f : M 1 → M 2 be a continuous map between two topological spaces M 1 , M 2 . Then, f induces a morphism of cochain complexes called pull-back, sending ω 2 ∈ C n (M 2 ) to ω 1 = ω 2 • f * ∈ C n (M 1 ). Equivalently, the following diagram commutes. Hence, f induces also a group homomorphism between the corresponding singular cohomology groups for each n ≥ 0.

Relative Singular Homology
Let M be a topological space and ι : N → M the canonical inclusion of a topological subspace N ⊆ M . Denote their homological chain complexes. The morphism of complexes ι * : C • (N ) → C • (M ), obtained via push-forward, is injective. Thus, for each n ≥ 1, we define the double chain complex (43) and the differential ∂ n : C n (M, N ) → C n−1 (M, N ) acting as where the connecting morphisms are the push-forward maps ι * : H n (N, Q) → H n (M, Q) induced by the inclusion ι : N → M . Consider an element of the relative homology group H s n (M, N, Q). This is represented by a pair (σ N , σ M ) of singular chains σ N ∈ C n−1 (N ) and σ M ∈ C n (M ) satisfying Note that, since ι * is injective, the latter condition implies the former. Thus, relative homology classes are represented by chains in M whose boundary is contained in N . Relative cohomology groups H n s (M, N, Q) are defined similarly. Example 6. Let M = C * be the punctured complex plane and let N = {p, q} ⊂ M be the subspace consisting of the two distinct points p, q ∈ C * . Let σ 2 : ∆ 1 st → M be any continuous map such that σ 2 (0, 1) = p and σ 2 (1, 0) = q, such as the oriented segment starting at p and ending at q. Then Consequently, σ 2 defines a relative chain. It follows from the long exact sequence (45) that the only non-trivial relative homology group is H s 1 (M, N, Q). A basis of this group is given by the chain σ 2 and the chain σ 1 , introduced in Example 4, consisting of a counterclockwise circle containing the origin. Such a basis is graphically represented in Fig. 9.

De Rham Cohomology
Let M be a differentiable manifold of dimension n. A differential p-form on M is written in local coordinates as Let Ω p (M ) denote the R-vector space of differential p-forms on M and define the space of differential forms on M as The exterior derivative d : Ω(M ) → Ω(M ) is the unique R-linear map which sends p-forms into (p + 1)-forms and satisfies the following axioms: c) Let α be a p-form on M and β any differential form in Ω(M ). Denote α ∧ β their exterior product. Then, The associated cochain complex is and is called the (smooth) de Rham cohomology of M . A differential p-form ω is closed if dω = 0 and it is exact if there exists a differential (p − 1)-form η such that ω = dη. A classical theorem 11 by De Rham [21] asserts that the singular cohomology H • s (M, R) can be computed using differential forms 12 .
which sends the class of a differential form ω to the integration functional is an isomorphism.

Algebraic de Rham Cohomology
Assume X is an affine variety over Q of dimension n and write X = SpecR where R is the ring of regular functions on X, i.e. R = O(X). The algebraic 13 p-forms on X are the smooth differential p-forms on X with R-coefficients.
where f i1,...,ip (x 1 , ..., x n ) are regular functions on X. The space of algebraic p-forms is denoted Ω p alg−dR (X). Define the space of algebraic forms on X as The exterior derivative d : Ω alg−dR (X) → Ω alg−dR (X), defined as in Section 2.2, canonically yields a cochain complex called de Rham complex, whose associated cohomology, denoted H • alg−dR (X, Q) and called the algebraic de Rham cohomology of X, was first introduced by Grothendieck [34].
The only non-vanishing spaces of algebraic forms are Consequently, the following two groups are the only non-trivial cohomology groups of X. 11 De Rham's theorem was first presented in his PhD thesis, published in 1931, when cohomology groups had not been introduced yet. He did not state the theorem in the way it is described today, but gave an equivalent statement involving Betti numbers and integration of closed differential forms over cycles. 12 We refer to Bott and Tu [8] for a comprehensive investigation of differential forms in algebraic topology. 13 The algebraic substitute for the smooth differential form is rigorously defined through the notions of Kähler differential and exterior power. Also, the proper construction of the algebraic de Rham cohomology requires the notions of sheaf cohomology and hypercohomology that we do not use here. For details on these topics we refer to Kashiwara and Schapira [39].
The following fundamental theorem is proven by Grothendieck [34].
Theorem 5. Let X be a smooth affine variety defined over Q of dimension n. The following comparison isomorphism holds comp : for 0 ≤ k ≤ n.

Combining Grothendieck's and De Rham's theorems, an important remark follows.
Remark. Let X be a smooth affine variety over Q and let M be the differentiable manifold obtained as the space of complex points of X. Then, the smooth de Rham cohomology of M , equivalent to its singular cohomology, is isomorphic to the algebraic de Rham cohomology of X, i.e. the former can be computed considering algebraic forms only . Thus, a purely algebraic definition of cohomology is obtained.

Relative de Rham Cohomology
Let X be a smooth affine Q-variety. Denote Ω 0 (X) → Ω 1 (X) → Ω 2 (X) → ... its de Rham complex. Let D ⊆ X be a simple normal crossing divisor 14 and let D i , for i = 1, ..., r, be its smooth irreducible components. For simplicity, assume that each D i is defined over Q. For each I ⊆ {0, ..., r}, set and define The associated double cochain complex of Q-vector spaces K p,q = Ω q (D p ) is graphically represented as . . .
where the vertical differentials d ver are (−1) p d for each p ≥ 0 and the horizontal differentials d hor are linear combinations, with coefficients equal to ±1, of restriction maps d IJ : Ω q (D I ) → Ω q (D J ). Note that, thanks to the factor (−1) p in the definition of d ver , the vertical and horizontal differentials anticommute. Let (Ω • (X, D), δ) denote the total cochain complex associated to K p,q , that is For each n ≥ 0, the space Ω n (X, D) corresponds to the direct sum of the spaces on the n-th diagonal of the double cochain complex K p,q represented in (61). The total complex is in fact explicitly written down as The relative algebraic de Rham cohomology H • alg−dR (X, D) is the cohomology of the total cochain complex Ω • (X, D), that is where the only non-trivial horizontal differential is the evaluation map The corresponding total cochain complex is where the only non-trivial differential is explicitly written. The non-trivial relative algebraic de Rham cohomology groups are A basis of H 1 alg−dR (X, D) is given by the classes dx x , 0, 0 and dx z−1 , 0, 0 .

Pure Hodge Structures
As a consequence of Theorem 5, the Betti cohomology of an algebraic variety is endowed with a richer structure than the singular cohomology of a generic topological space. Recall the following definition.
Let M be a compact Kähler 15 manifold of dimension d. For each pair of integers p, q, let be the subspace of smooth de Rham cohomology classes that can be represented by a C ∞ -closed differential (p + q)form of type (p, q), i.e. that can be locally expressed as where the sum runs over the index subsets I = {i 1 , ..., i p } and J = {j 1 , ..., j q } of {1, ..., d} and f I,J are C ∞ -functions.
The following theorem by Hodge [37] marks the beginning of what is currently known as Hodge theory.
Theorem 6. Let M be as above. The following direct sum decomposition holds for n ≤ d.
We note that complex conjugation acts on the right-hand side of (71) through the action on the complex coefficients of the left-hand side, that is, where σ ∈ H n (M, Q) and ω ∈ C. Thus, the complex conjugate of H p,q (M ) is precisely H q,p (M ). This property is often called Hodge symmetry.
satisfying H p,q = H q,p . Then, H is called a pure Hodge structure of weight n and the given direct sum decomposition of its complexification H C is called Hodge decomposition.
An equivalent definition of pure Hodge structure is obtained by observing that the data encoded in the Hodge decomposition is equivalent to a finite decreasing filtration F • of H C , called Hodge filtration, such that, for all integers p, q with p + q = n + 1, we have The relation between the two equivalent descriptions is given by Let X be a smooth projective variety defined over Q and take M = X(C) to be the space of complex points of X. Then, by De Rham's and Grothendieck's theorems, the smooth de Rham cohomology of M is isomorphic to the algebraic de Rham cohomology of X after complexification, i.e.
As a consequence of Theorem 6, the Hodge decomposition can be easily referred to the algebraic de Rham cohomology of X and analogously the Hodge filtration F • is defined on H n alg−dR (X, Q). To keep track of these additional structures, we define the triple (1) f alg−dR is filtered with respect to the Hodge filtration, i.e.
(2) The following diagram commutes The definition implies that, if H and H have different weights, then every morphism of Hodge structures between them is zero. The following variant of Theorem 6 implies that pure Hodge structures are functorial for morphisms of algebraic varieties. Example 9. Let K be a subfield of C. For each n ∈ Z, we define where the filtration yields K = F −n K ⊇ F −n+1 K = 0 and the isomorphism comp : C → C is given by multiplication by (2πi) −n . Q(n) is a one-dimensional pure Hodge structure of weight −2n over K and is called a Tate-Hodge structure. As an example, where F • is the trivial filtration concentrated in degree 1.

Mixed Hodge Structures
The cohomology in degree n of a smooth projective complex variety X carries along a pure Hodge structure of weight n. However, this is no longer true when X fails to be smooth or projective. The generalisation of the notion of pure Hodge structure to any quasi-projective complex variety is due to Deligne [22], [23], [24], who proved that the cohomology of quasi-projective varieties over Q are iterated extensions of pure Hodge structures.
Theorem 8. Let X be a quasi-projective variety over Q.
(1) There exist a finite increasing filtration, called weight filtration and a finite decreasing filtration, called Hodge filtration such that F • induces a pure Hodge structure of weight m on each graded piece (2) If f : X → Y is a morphism of quasi-projective varieties, the induced maps on cohomology f * : H n (Y, Q) → H n (X, Q) and f * C : H n (Y, C) → H n (X, C) are filtered morphisms with respect to the two filtrations, i.e.
(3) If X is smooth, then Gr W m H n (X, Q) = 0 for all m < n. If X is projective, then Gr W m H n (X, Q) = 0 for all m > n.
The following definition generalises the notion of pure Hodge structure.
where W B • and W alg−dR • are the increasing filtrations associated to the Betti and algebraic de Rham cohomologies, respectively. Require that the comparison isomorphism is filtered with respect to the weight filtration, that is, and that for each integer m is a pure Hodge structure over Q of weight m. Then, H n (X) is called a mixed Hodge structure over Q.
We denote by MHS(Q) the category of mixed Hodge structures over Q. Deligne [23] proved that MHS(Q) is an abelian category. Moreover, MHS(Q) is naturally endowed with two forgetful functors sending the mixed Hodge structure H into the Q-vector spaces H B and H alg−dR , respectively. These functors are called the Betti and de Rham functors.

Numeric Periods
The following elementary definition was introduced by Kontsevich and Zagier [42].

Definition 10.
A numeric period is a complex number whose real and imaginary parts are values of absolutely convergent integrals of the formˆσ where the integrand f is a rational function with rational coefficients and the domain of integration σ ⊆ R n is defined by finite unions and intersections of domains of the form {g(x 1 , ..., x n ) ≥ 0} with g a rational function with rational coefficients.
If rational functions and coefficients are replaced in Definition 10 by algebraic functions and coefficients, the same set of numbers is obtained. On the other hand, algebraic functions in the integrand can be substituted by rational functions by enlarging the set of variables. Note that, because the integral of any real-valued function is equivalent to the volume subtended by its graph, any period admits a representation as the volume of a domain defined by polynomial inequalities with rational coefficients. Thus, the integrand can always be assumed to be the constant function 1. However, this extremely simplified framework does not prove to be particularly useful. Quite the opposite, in what follows, we mostly work with an even more general description of periods than the one given in Definition 10. We denote by P the set of periods. BeingQ ⊂ P ⊂ C, periods are generically transcendental numbers and nonetheless they contain only a finite amount of information, which is captured by the integrand and domain of integration of its integral representation as in (90). Indeed, just likeQ, P is countable. Many famous numbers belong to the class of periods. Here are some examples: (a) Algebraic numbers are periods, e.g.
(c) The transcendental number π is a period, as given by and alternatively by where γ 0 is a closed path encircling the origin in the complex plane.
(d) Values of the Gamma function at rational arguments satisfy (e) The elliptic integral representing the perimeter of an ellipse with radii a and b, is a period. Note that it is not an algebraic function of π for a = b, a, b ∈ Q >0 , (f) Values of the Riemann zeta function at integer arguments s ≥ 2 are periods, e.g.
(g) Multiple zeta values are periods by means of Kontsevich's integral representation (22).
(h) Convergent Feynman integrals, as in (16), are periods. Because the integral representation of a period is not unique, it is possible that a certain integral of a transcendental function admits a representation as a period as well. For example, log(2) is a period, and yet it can be written as the following integral of a transcendental function Indeed, there seems to be no general principle able to predict if a certain infinite sum or integral of a transcendental function is a period according to Definition 10 or able to determine whether two periods, given by explicit integrals, are equal or different. A number inQ also admits apparently different expressions, but those same techniques that work for checking the equality of algebraic numbers do not in general work for periods. In fact, two different periods may be numerically very close and yet be distinct 16 . However, the following conjecture is presented by Kontsevich and Zagier [42]. We note that even a proof of Conjecture 1 would not solve the additional problem of finding an algorithm to determine whether or not two given numbers in P are equal, or whether a given real number, known numerically to some accuracy, is equal within that accuracy to some period. Another fundamental open problem in the theory of periods is to explicitly exhibit one number which does not belong to P. Such numbers must exist, because P is a countable subset of C, but the concrete identification of one of such numbers has only been proposed conjecturally. Precisely, the basis of natural logarithms e and the Euler-Mascheroni constant γ are conjecturally not periods.
Before moving to a more sophisticated definition of periods written in the language of algebraic geometry, which is essential to subsequent developments, we mention the fruitful interplay between the theory of periods and the theory of linear differential equations. When the integrands or the domains of integration depend on some set of parameters, the integrals, as functions of these parameters, usually satisfy linear differential equations with algebraic coefficients. The solutions of these differential equations generate periods when evaluated at algebraic arguments. The differential equations occurring in this way are called Picard-Fuchs differential equations. The relation between periods and Picard-Fuchs equations has proved to be particularly productive in the case of elliptic curves, hypergeometric functions, modular forms and L-functions.

Algebra of Motivic Periods
The theory of periods can be alternatively developed within the formalism of algebraic geometry. We refer to Huber and Müller-Stach [38].
Definition 11. Let X be a smooth quasi-projective variety defined overQ, Y ⊂ X a subvariety, ω a close algebraic differential n-form on X vanishing on Y and γ a singular n-chain on the complex manifold X(C) with boundary contained in Y (C). The integral´γ ω ∈ C is a numeric period.
The equivalence of Definitions 11 and 10 follows from the observation that the algebraic chain γ can be deformed to a semi-algebraic chain and then broken up into small pieces, which can be bijectively projected onto open domains in R n with algebraic boundary. Without loss of generality, we work with coefficients in Q instead ofQ. We note that, like Definition 10, Definition 11 also contains redundancy. The integral´γ ω can be formally destructured into the quadruple (X, Y, ω, γ) and different quadrupoles can give the same resulting number. To get rid of this redundancy, the various forms of topological invariance of the integral must be suitably accounted for. Following Stokes' theorem, the integral is insensitive to the individual cycle and form, being instead determined by the homology and cohomology classes of these. Let us associate to ω its cohomology class in the n-th algebraic de Rham cohomology group of X relative to Y and to γ its homology class in the n-th Betti homology group of X relative to Y . Then, the first step towards a unique algebraic description of periods consists of the following substitutions into the quadrupole (X, Y, ω, γ). The problem of the coexistence of distinct, but similarly behaved, cohomologies associated to the same variety, which seems to imply an arbitrary choice here and in many other situations, has been tackled by Grothendieck 17 [20] with the introduction of the theory of motives. He suggested that there should exist a universal cohomology theory taking values in a Q-category of motives M. Thus, the notion of a motive is proposed to capture the intrinsic cohomological essence of a variety. Without delving into the category-theoretic details of the theory of motives, we give now an intuitive idea of its origin and fundamental features as necessary to review its application to the theory of periods. A more rigorous discussion of motives is presented in Section 4.5. We recall from Grothendieck's Theorem 5 that there is a comparison isomorphism Neglecting the presence of filtrations, the Hodge structure of X relative to Y is expressed as In the same way that the cohomology class of a differential form singles out its cohomological behaviour, the Hodge structure of an algebraic variety intuitively selects the content shared by its different coexisting cohomologies and filters out everything else. It is, therefore, the first approximate realisation of Grothendieck's idea of a motive. We define the motivic version of the period´γ ω as the triple where m in the superscript stands for motivic. We call a period in this guise a motivic period. This has proved to be the most profitable reformulation of the original notion of a period. However, a second source of redundancy in the description of periods via the integral formulation in Definition 11, corresponding to the same transformation rules in Conjecture 1, has yet to be factored out. 2) Change of variables. If f : where f * is the pull-back of f and f * is its push-forward.
3) Stokes' formula. Assume for simplicity that X is a smooth affine algebraic variety defined over Q of dimension d and D ⊆ X is a simple normal crossing divisor. Denote byD the normalisation 18 of D. We observe that the space of motivic periods P m is naturally endowed with an algebra structure. Indeed, new periods are obtained by taking sums and products of known ones.

Period Map
We call period map the evaluation homomorphism Following the construction in Section 3.2, the period map is explicitly surjective, while injectivity is, on the other hand, not proven. Indeed, a numeric period has a unique motivic realisation only conjecturally. Conjecture 1 is equivalent to the period conjecture below.
Conjecture 2. The period map per : P m → P is an isomorphism.
Let us briefly discuss the key idea underlying the period conjecture. A Q-morphism f : (X 1 , Y 1 ) → (X 2 , Y 2 ) between two pairs of algebraic varieties induces a change of coordinates between the corresponding algebraic de Rham cohomologies by pull-back, that is The same morphism f acts on the spaces of complex points underlying the given algebraic varieties and induces a change of coordinates between the corresponding singular homologies by push-forward, that is By means of such changes of coordinates, one can easily derive two distinct integral representations of the same numeric period. For example, taking [ The corresponding two motivic representations of the same numeric period could a priori be different motivic periods. However, they are identified with each other by change of variables. Indeed, the period conjecture corresponds to the statement that, whenever different motivic representations of the same period arise, they can always be interrelated by the three equivalence relations in Definition 12.
18D is locally the disjoint union of the irreducible components of D. for each pair of indices (i, j) with i, j = 1, ..., n. Define the period matrix of H as the n × n-matrix with complex entries (p ij ) i,j=1,...,n given by The period matrix expresses in a different guise the same information contained in the period map, once it has been restricted to a specific Hodge structure.

Motivic 2πi
The numeric period 2πi is given by the contour integral where γ 0 is a counterclockwise cycle encircling the origin in the punctured complex plane C * . As observed in Example 5, the complex manifold C * is isomorphic to the topological space of complex points G m (C) underlying the Q-algebraic variety G m . As shown in Examples 4 and 7, we have that which is alternatively represented by the pairing A second integral representation of 2πi is given by where dz∧dz (1+zz) 2 is a closed smooth algebraic 2-form over the closed manifold P 1 (C). Because P 1 (C) is compact and Kähler, Theorem 6 applies, giving the Hodge decomposition where the forms in H p,q alg−dR contain p copies of the holomorphic differential dz and q copies of the anti-holomorphic differential dz. Therefore, dz∧dz (1+zz) 2 ∈ H 1,1 alg−dR (P 1 ) and integral (119) corresponds to the following motivic period Remark. The two apparently different motivic periods in (117) and (121) are the same, thus preserving the period conjecture. To show this, define which satisfy the relations By the Mayer-Vietoris theorem applied to the singular homology groups, the following long exact sequence holds Similarly, one can prove that the whole Hodge structures H 1 (G m ) and H 2 (P 1 ) are isomorphic and that the change of coordinates occurring between them relates the cohomology classes dz∧dz (1+zz) 2 and dx x and the homology classes [γ 0 ] and P 1 (C) via pull-back and push-forward maps, respectively.

Motivic log(z)
Recall the integral representation of log(z), z ∈ Q\{1}, given by As in the case of 2πi, this is an integral over the punctured complex plane C * = G m (C). However, contrary to the case of 2πi, where the integration path γ 0 is closed, integral (126) is performed on an open path, precisely any continuous oriented path γ 1 ⊂ C * , starting at 1 and ending at z, which is contractible to the oriented segment from 1 to z. The integration path being open requires the framework of relative homology. Let G m be the ambient variety, C * the underlying topological space and {1, z} with z ∈ Q\{1} a simple normal crossing divisor in C * . As shown in Examples 6 and 8, we have {1, z}), comp), a motivic version of log(z) is which is alternatively represented by the pairing

Elementary Relations
Many relations among numeric periods are simply recast in the formalism of motivic periods. In fact, Hodge structures conjecturally capture all algebraic relations between periods.

Singularities and the Blow Up
Multiple zeta values and convergent Feynman integrals are periods by means of the integral representations (22) and (16), respectively. In both cases, singularities of the integrand can be contained in the domain of integration, a feature that does not occur in the examples of 2πi and log(z). Whenever singularities are present, they have to be taken care of with particular attention.
Example 13. The period ζ(2) is given by the following integral over the complex manifold C 2 . The domain of integration is the simplex and the integrand is the differential 2-form Observing that C 2 is isomorphic to the topological space of complex points A 2 (C), underlying the affine 19 Q-algebraic variety A 2 = SpecQ[x 1 , x 2 ], we may try to build ζ(2) m as we did for the examples in Section 3.4. Consider the lines l 0 = {x 1 = 0} and l 1 = {x 2 = 1} in the affine plane A 2 . Since L = l 0 ∪ l 1 is the locus of singular points of ω, the latter is an algebraic 2-form on X = A 2 \L. Thus, [ω] is a class in the second algebraic de Rham cohomology group of X and, consequently, we may want to consider the integral (133) as a period of X relative to some divisor containing the boundary of σ. In an attempt to do so, define the simple normal crossing divisor containing the boundary of σ. Note that D is not in X because D ∩ L = ∅. However, the divisor D\(D ∩ L) ⊂ X does no longer contain ∂σ. The problem arises from the fact that σ itself is not contained in X, intersecting the singular locus L in two points Removing the singular points p, q from D and considering the second relative Hodge structure H 2 (X, D\(D ∩ L)) does not solve the mentioned technical issue, because [σ] is not a class in H B 2 (X, D\(D ∩ L)). See Fig. 11. The example of ζ(2) shows how direct removal of singular points explicitly fails and motivates a more articulated geometric construction, called blow up, which proves to be successful in the case of ζ(2) and many more examples. Graphically, we may illustrate the procedure as the removal of a whole region of space centred at the singularity and the corresponding reshaping of the integration domain. See Fig. 12 for a qualitative representation of how the blow up of the two singular points p, q ∈ A 2 acts on σ in the case of ζ(2).

Motivic Multiple Zeta Values
Consider ζ(2) again. The blow up of the affine plane A 2 along the singular points p, q is defined as the closed subvariety given by the equations where [α i : β i ], i = 1, 2, are homogeneous coordinates on the two copies of P 1 . The projection of Y onto the first factor in A 2 × P 1 × P 1 is the proper surjective map The inverse of the projection ι = π −1 , mapping the affine plane A 2 into its blow up Y , replaces the singular points p, q ∈ A 2 by corresponding projective lines E p , E q ⊂ Y , called exceptional divisors. Precisely, we have Moreover, the restriction of ι to the complement in A 2 of the singular points p, q ι| A 2 \{p,q} : is an isomorphism. For any closed subset C ⊂ A 2 , the image ι(C) is called total transform of C. The strict transform of C, denotedĈ, is the closed subset of Y obtained by first removing the points p, q if they are in C, then applying ι, and finally taking the Zariski closure, that isĈ It follows that the strict transforms of l 0 , l 1 are the affine lines and their total transforms are We observe that L 0 , E p and L 1 , E q intersect in only one point each. Precisely Moreover, we have Similarly, we define the strict transformσ of the domain of integration. Observing that the closed points of E p can be interpreted as lines passing through p, and analogously that the closed points of E q can be interpreted as lines passing through q, we obtainσ which, combined with (146), imply thatσ See Fig. 13 for a graphical representation of the blow up. As the map ι is applied to the ambient variety, giving the reshaped domainσ, the differential form ω is replaced by its pull-back π * (ω), denoted byω. Let us now show that the pull-backω is only singular 20 on the strict transform L = L 0 ∪ L 1 . We use local coordinates on the blow up Y . In particular, consider a patch of Y around the point L 0 ∩ E p as shown in Fig. 14. Here, a local system of coordinates is explicitly given by where L 0 and E p have equations t = 0 and s = 0, respectively. Applying this change of variables toω, we havê It follows thatω is singular along the strict transform L 0 , while it is smooth along the exceptional divisor E p , because it has no pole at s = 0. Analogously, we find thatω is singular along L 1 , but not along E q . Then, the singular locus ofω is L. Observe that the complement Y \L is the closed affine subvariety of A 2 × A 1 × A 1 given by the equations where t, s are affine coordinates on the two copies of A 1 . Therefore, the differential formω determines a class in H 2 alg−dR (Y \L). Moreover, it follows from (149) that, moving from the original affine plane A 2 to the blow up Y , the singular locus of the differential formω and the domain of integrationσ are disjoint. As usual, we may want to consider the integral (133) as a period of Y \L relative to some divisor containing the boundary ofσ. The blow up construction is thus successful for the period ζ(2) if [σ] turns out to be a class in the given relative Betti homology group. To see this, recall that ∂σ is contained in the union D of the affine lines Thus, we naturally consider the normal crossing divisor M ⊂ Y defined by where M i =m i denotes the strict transform of m i for i = 1, 2, 3. Note that L ∩ M is the union of the points L 0 ∩ E p and L 1 ∩E q expressed in (146). Therefore,σ is contained in Y \L and ∂σ is contained in M \(M ∩L) ⊂ Y \L, implying by the equivalence relation under change of variables in P m . We observe that the whole period matrix of H is

Motivic Feynman Integrals
In an attempt to overcome singularity issues, the blow up procedure can be similarly applied to generic MZVs and other families of periods, such as convergent Feynman integrals. For an exposition of the general computation of the Hodge structure of a blow up we refer to Voisin [61].
Let G be a primitive log-divergent Feynman graph, E G the collection of its edges and n G = |E G |, as in Section 1.4. Recall that x e denotes the Schwinger parameter associated to e ∈ E G , and Ψ G , I G , and X G denote the first graph polynomial, the Feynman integral, and the graph hypersurface, as given in (10), (16), and (17), respectively. Denote by ω G and σ the integrand and the domain of integration of I G . Since ω G is a top-degree algebraic differential form on P n G −1 \X G , and ∂σ is contained in the union D of the coordinate hyperplanes {x e = 0, e ∈ E G }, we may intuitively try to build the motive I m G on the relative Hodge structure However, this naïve attempt fails whenever the hypersurface X G intersects the integration cycle σ non-trivially, implying the presence of non-negligible singularities. Whenever singularities are present, σ does not define an element in the corresponding naïve relative Betti homology group. To successfully build the motive I m G in the presence of singularities, the blow up technique is applied.
A linear subvariety L ⊂ 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 Since the coefficients of Ψ G are positive, the locus of problematic singularities is where the union is taken over all coordinate linear spaces L ⊂ X G .
Remark. The coordinate linear spaces L ⊂ X G are in one-to-one correspondence with the subgraphs γ ⊂ G such that l γ > 0. It follows where the union is taken over all subgraphs γ ⊂ G with l γ > 0. Here, L γ is the linear subvariety of P n G −1 defined by the equations {x e = 0, e ∈ E γ }.
The following theorem is proven, and an explicit algorithmic construction of the blow ups is given, by Bloch, Esnault and Kreimer [5].
Theorem 9. Let G be a primitive log-divergent Feynman graph such that every proper subgraph of G is primitive. There exists a tower π : P = P r → P r−1 → ... → P 1 → P 0 = P n G −1 (164) such that, for each i = 1, ..., r, P i is obtained by blowing up P i−1 along the strict transform of a coordinate linear space L i ⊂ X G , and the following conditions hold: (1) The pulled-back differentialω G = π * ω G has no poles along the exceptional divisors associated to the blow ups.
(2) Let B be the total transform of D in P , i.e.
Then, B ⊂ 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 σ in P does not meet Y G , that is,σ ∩ Y G (C) = ∅.
As a consequence of Theorem 9, the motive I m G associated to any subdivergence-free primitive log-divergent Feynman graph G can be written explicitly. Being ∂σ ⊂ B\(B ∩ Y G ), the domain of integration defines the class called Betti framing, while the integrand defines the class called de Rham framing. Brown and Doryn [14] present a method for explicit computation of the framings on the cohomology of Feynman graph hypersurfaces. Then, the Hodge structure H = H n G −1 (P \Y G , B\(B ∩ Y G )) is called the graph Hodge structure 21 , and the motivic Feynman integral I m G is given by Indeed, the pairing of the classes [ω G ] and [σ] yields the periodσω by the equivalence relation under change of variables in P m .
Example 14. Adopting the following notation P log = Q I G | G is a primitive log-divergent Feynman graph we observe that the sequence of inclusions P φ 4 ⊂ P log ⊂ P is preserved after promoting numeric periods to periods of motives, that is, P m φ 4 ⊂ P m log ⊂ P m holds.

Tannakian Formalism
We briefly introduce the fundamentals of the theory of Tannakian categories, following the more detailed and comprehensive exposition by Deligne et al [26]. The concept of a Tannakian category was first introduced by Saavedra Rivano [49] to encode the properties of the category Rep K (G) of the finite-dimensional K-linear representations of an affine group scheme G over a field K. Let us recall some preliminary notions in category theory. In the following, K is a given field.

Definition 14.
A K-linear category C is an additive category such that, for each pair of objects X, Y ∈ Ob(C), the group Hom C (X, Y ) is a K-vector space and the composition maps are K-bilinear.
Definition 15. Let C be a K-linear category endowed with a K-bilinear functor ⊗ : C × C → C.
(a.2) For all X, Y, Z, T ∈ Ob(C), the following diagram commutes such that the following two conditions hold: 2) For all X, Y ∈ Ob(C), the following composition is the identity (c) An associativity constraint and a commutativity constraint are compatible if, for all X, Y, Z ∈ Ob(C), the following diagram commutes (d) A pair (U, u) consisting of an object U ∈ Ob(C) and an isomorphism u : U → U ⊗ U is an identity object if the functor X → U ⊗ X is an equivalence of categories.
Definition 16. A K-linear tensor category is a tuple (C, ⊗, φ, ψ) consisting of a K-linear category C, a K-bilinear functor ⊗ : C × C → C, and compatible associativity and commutativity constraints φ, ψ such that C contains an identity object.
Equivalently, L is invertible if and only if there exists an object L ∈ Ob(C) such that L ⊗ L 1. Then, L is also invertible.
Definition 18. Let (C, ⊗) be a K-linear tensor category, where we omit the constraints φ, ψ for simplicity, and let X, Y ∈ Ob(C). Assume that there exists an object Z ∈ Ob(C) such that, for all T ∈ Ob(C), the functors T → Hom(T, Z) and T → Hom(T ⊗ X, Y ) admit a functorial isomorphism In this case, the functor T → Hom(T ⊗ 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 Hom(X, Y ) and it is unique up to isomorphism.
Definition 19. The dual of an object X ∈ Ob(C) is defined as X ∨ = Hom(X, 1). If X ∨ and (X ∨ ) ∨ exist, then there is a natural morphism X → (X ∨ ) ∨ , and the object X is reflexive if such a morphism is an isomorphism.
Definition 20. A K-linear tensor category (C, ⊗) is rigid if the following conditions hold: (1) For all X, Y ∈ Ob(C), Hom(X, Y ) exists.

Definition 21.
A Tannakian category over the field K is a rigid abelian K-linear tensor category T such that End(1) = K, and there exists an exact faithful K-linear tensor functor ω : T → Vec K , where Vec K is the category of finite-dimensional vector spaces over K. Any such functor is called a fibre functor.
Example 15. The category Vec K of finite-dimensional K-vector spaces, together with the identity functor, is a Tannakian category over K.
Example 16. The category GrVec K of finite-dimensional graded K-vector spaces, together with the forgetful functor ω : GrVec K → Vec K , sending (V, (V n ) n∈Z ) to V , is a Tannakian category over K.
Example 17. The category Rep K (G) of finite-dimensional K-linear representations of an abstract group G, together with the functor ω : Rep K (G) → Vec K that forgets the action of G, is a Tannakian category over K.
Let us fix a Tannakian category T over K and a fibre functor ω of T . Let R be a K-algebra. We denote by Aut ⊗ (ω)(R) the collection of families (λ X ) X∈Ob(T ) of R-linear automorphisms which are compatible with the tensor structure and functorial. Here, compatibility with the tensor structure and functoriality mean 22 that: (1) For all X 1 , X 2 ∈ Ob(T ), the following diagram commutes (2) The following diagram commutes (3) For all X, Y ∈ Ob(T ) and for every morphism α ∈ Hom(X, Y ), the following diagram commutes Denote Aut ⊗ (ω) = Aut ⊗ (ω)(K) the group of K-linear automorphisms of the fibre functor ω. Deligne et al [26] proved that all Tannakian categories are categories of finite-dimensional linear representations of a pro-algebraic group.
Theorem 10. Let T be a Tannakian category over K with a fibre functor ω.
(1) The functor R → Aut ⊗ (ω)(R) is representable by an affine group scheme over K, which is denoted as Aut ⊗ (ω) or G ω , and is called the Tannaka group of the pair (T , ω).
(2) For every X ∈ Ob(T ), the group Aut ⊗ (ω) acts naturally on ω(X) and the functor sending X to the vector space ω(X) with this action of Aut ⊗ (ω), is an equivalence of categories.
Theorem 11. Let T be a Tannakian category over K with two fibre functors ω and ω . The functor R → Isom ⊗ (ω, ω )(R) is representable by an affine scheme over K, which is denoted as Isom ⊗ (ω, ω ), and is a right torsor under Aut ⊗ (ω) and a left torsor under Aut ⊗ (ω ).

Motivic Galois Theory
Grothendieck's idea of a universal cohomology theory taking values in a Q-category of motives M is intimately connected to the theory of Hodge structures. Recall the rigorous notions of pure and mixed Hodge structures over Q, given in Sections 2.3 and 2.4. On the one hand, the cohomology of a smooth projective Q-variety is fundamentally described by a pure Hodge structure. On the other hand, applying the resolution of singularities by Hironaka [36], the cohomology of a singular quasi-projective Q-variety can be expressed in terms of cohomologies of smooth projective varieties, and since cohomologies of different degrees get mixed in this expression, it is fundamentally described by a mixed Hodge structure. Thus, enhancing the naïve description in Section 3.2, pure Hodge structures represent suitable candidates to actualise the idea of motives of smooth projective varieties proposed by Grothendieck. Similarly, mixed Hodge structures potentially represent motives of singular or quasi-projective varieties. Specifically looking at the application of Hodge theory to the theory of motivic periods, we identify the category of motives with the category of mixed Hodge structures over Q. For a thorough introduction to the theory of motives we refer to Voevodsky [60], André [4], Deligne and Goncharov [25], and Murre et al [46].
Recall that MHS(Q) is the category of mixed Hodge structures over Q, and ω B , ω dR are its two forgetful functors arising from the Betti and de Rham cohomologies, respectively. All the defining properties of a Tannakian category, encoded in Definition 21, apply. Indeed, MHS(Q) is a Tannakian category over Q, and both functors ω B and ω dR are fibre functors, thus justifying the use of the Tannakian machinery in the context of motives. The pro-algebraic group Aut ⊗ (ω dR ) is denoted G dR and called motivic Galois group. G dR (M ) is a group in GL(ω dR (M )) for every motive M ∈ Ob(M). Following Theorem 10, the category of motives is equivalent to the category of finite-dimensional Q-linear representations of the motivic Galois group, that is Remark. We observe that the motivic Galois group can alternatively be realised via Betti cohomology as G B = Aut ⊗ (ω B ), and the corresponding category of finite-dimensional Q-linear representations is still the same category of motives M.
In Tannakian formalism, the space of motivic periods P m is expressed as with implicit factorisation modulo bilinearity and functoriality. Thus, an alternative but equivalent description of motivic periods is obtained. Indeed, P m is isomorphic to the space of regular functions on the affine Q-scheme The isomorphism is made explicit by where σ • λ M gives Then, following Theorem 10, the motivic Galois group G dR has a natural action on Isom ⊗ (ω dR , ω B ) denoted by which induces a dual coaction on the corresponding spaces of regular functions where {e i } is a basis of ω dR (M ) and {e ∨ i } is the dual basis, called Galois coaction. We denote P dR = O(G dR ) the dual of the motivic Galois group and call it the space of de Rham periods.
Remark. 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. On the other hand, motivic periods have a well-defined map to numbers, while de Rham periods do not, although we can associate symbols to them. Thus, the Galois coaction turns the finite-dimensional Q-vector space P m into a comodule over the Hopf algebra P dR . A detailed discussion is presented by Brown [13], [12].
Example 18. Consider the motivic logarithm log(z) m for z ∈ Q\{1}. Following Section 3.4.2, we have where we write dx x = dx x , 0, 0 for simplicity. Thus, the corresponding motive is Direct application of the prescription in (190) gives the explicit decomposition Here, 1 m and log(z) m are called Galois conjugates of log(z) m .
Example 19. As for log(z) m , the Galois coaction of the motivic multiple zeta values ζ(s) m can be computed explicitly. In particular, for n ≥ 1, we have Thus, the Galois coaction is trivial on ζ(2) m , while ζ(2n + 1) m has the non-trivial Galois conjugate 1 m . Moreover

Coaction Conjecture
We look at the example of scalar massless φ 4 quantum field theory and consider the Galois coaction restricted to P m φ 4 . This is a priori valued in the whole space P dR ⊗ P m . However, after computing every known φ 4 -amplitude with loop order at most 7 and explicitly verifying that in each case the Galois coaction preserves the space P m φ 4 , Panzer and Schnetz 23 [47] proposed the following conjecture, known as the coaction conjecture.
Such a conjecture implies the existence of a fundamental hidden symmetry underlying the class of φ 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 φ 4 -periods from periods of all graphs, and thus explain this surprising evidence.
A first advancement in this direction has already been made. Suitably enlarging the space of amplitudes under consideration, the coaction conjecture is proven by Brown [12]. Define the finite-dimensional Q-vector space P m φ 4 associated to a φ 4 -graph G to be the space of motivic versions of all integrals of the form where k ≥ 1 is an integer, and P is any polynomial in Q[{x e }] such that I G converges.

Weights and the Small Graph Principle
We apply the notions of Hodge and weight filtrations, introduced in Sections 2.3 and 2.4, to the theory of motivic periods. For M ∈ Ob(M), the Q-vector space ω dR (M ) is equipped with a decreasing Hodge filtration F and an increasing weight filtration W dR , while the Q-vector space ω B (M ) is provided with a weight filtration W B only. Mixed Hodge structures, contrary to pure ones, do not have a well-defined weight. However, the graded quotients with respect to the weight filtration do possess a pure Hodge structure of definite weight, as described in Definition 8. These properties are used to define a notion of weight for motivic periods.
Definition 22. The weight filtration on ω dR (M ) induces a weight filtration on the space of motivic periods by Denote W = W dR for simplicity. A given motivic period [M, ω, σ] m is said to have weight at most n if it belongs to W n P m , and it has weight n if it belongs to the graded quotient Gr W n P m = W n P m /W n−1 P m .
Remark. We observe that the weight of motivic periods can alternatively, but equivalently, be defined from the Betti side via the weight filtration induced on P m by W B .
Example 20. Consider M = H 1 (G m , {1, z}) again. Its weight filtration in de Rham realisation is Observing that 0, 1 ∈ W 0 and 2πi, log(z) ∈ W 2 , the weight of each entry of the period matrix of M is determined. Indeed, 0, 1 are periods of weight zero, while 2πi, log(z) have weight 2.
Example 21. The weight filtration can be used to systematically study P m φ 4 weight by weight. For example, direct computation in low weight shows that The following conjecture, known as small graph principle, is due to Brown [12].
where the product runs over a subset {γ 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 Feynman graph are associated to its sub-quotient graphs with at most k + 1 edges. Thus, when interested in periods of weight at most k, it suggests to look at graphs with at most k + 1 edges. It follows that the topology of a given graph constrains the Galois theory of its amplitudes. The following theorem is proven by Brown [12].
Theorem 13. Let G be a primitive log-divergent Feynman graph. If G has a single vertex or a single loop, then M G = Q(0). Example 22. Because log(z) m has weight 2, the small graph principle suggests that any log(z) m appearing in the right-hand side of the coaction formula for a given φ 4 -period comes from graphs with at most three edges. Theorem 13 implies that all two-edge graphs are trivial, i.e. the associated motive is the Hodge-Tate motive Q(0), which does not have log(z) m in its period matrix. Writing down all possible graphs with three edges, we get the graphs shown in Fig. 15 along with the associated graph polynomials in the Schwinger parameters. The two outer graphs (a) and (d) are also trivial by Theorem 13, while the two middle graphs (b) and (c) satisfy M G = Q(0) ⊕ Q(−1). However, log(z) cannot be obtained as an integral with a denominator equal to either of their graph polynomial. It follows that log(z) m cannot be a Galois conjugate of any φ 4 -period. By the coaction conjecture and Equation (194), we conclude that log(z) m / ∈ P m φ 4 . Example 23. Direct computation by Panzer and Schnetz [47] shows that all φ 4 -periods of loop order up to 6 are Q-linear combinations of multiple zeta values. Following the small graph principle, we order the set of MZVs by weight 1 ζ(2) ζ(3) ζ(2) 2 ζ(5) ζ(3) 2 ζ(7) ζ(3, 5) · · · ζ(2)ζ(3) ζ(2) 3 ζ(2)ζ(5) ζ(2)ζ(3) 2 ζ(2) 2 ζ(3) . . .
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. Indeed, whenever it is shown that a given period is not a φ 4 -period, we automatically deduce that all periods that have the given one among their Galois conjugates cannot appear in P φ 4 either.
Remark. Structures even more fundamental that 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 [40] provide details on related notions in category theory.

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´σ ω in P. The general prescription for its investigation via the theory of motivic periods can be summarised as follows.
(2) Use all the known information about the mixed Hodge structure 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 numeric periods with their motivic version requests the validity of the period conjecture. Moreover, even disregarding the conjectural status of current results, 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 φ 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 H of motivic periods, a space A of de Rham periods and a corresponding coaction ∆ : H −→ P dR ⊗ P m . A potential coaction principle would then state that ∆(H) ⊆ A ⊗ H. Being A a Hopf algebra, we could canonically introduce the group C of homomorphisms from A to any commutative ring. It would follow that the coaction principle can be recast in terms of the group action C × H −→ H, 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 φ 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.
(1) 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 φ 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 [15] investigate a rigorous theory of motives associated to certain hypergeometric integrals.
(2) 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 [33] 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.
(3) In the same research direction, a high-precision computation of the 4-loop contribution to the electron anomalous magnetic moment g − 2 by Laporta [44] shows the presence of polylogarithmic parts with fourth and sixth roots of unity. This result is conjecturally recast in motivic formalism by Schnetz [54], giving a more compact expression which explicitly reveals a Galois structure. In this work, the 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 N = 4 super Yang-Mills theory. In each of these theories, after suitably defining the space of integrals or amplitudes 24 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 [52], [59] and subsequent developments for superstring perturbation theory and to the work of Caron-Huot et al [19], [18] for the planar limit of N = 4 super Yang-Mills theory.