AAn Introduction to Motivic Feynman Integrals
Claudia Rella
Department of Mathematics, University of Geneva7 Route de Drize, 1227 Carouge, Switzerland
Abstract
This article gives a short step-by-step introduction to the representation of parametric Feynman integrals inscalar perturbative quantum field theory as periods of motives. The application of motivic Galois theory to thealgebro-geometric and categorical structures underlying Feynman graphs is reviewed up to the current state ofresearch. The example of primitive log-divergent Feynman graphs in scalar massless φ quantum field theory isanalysed in detail. a r X i v : . [ m a t h - ph ] A ug ontents Introduction 31 Scalar Feynman Graphs 3 φ Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.5 Multiple Zeta Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 πi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.4.2 Motivic log( z ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.4.3 Elementary Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Conclusions 36References 38 ntroduction In Section 1, we describe the graph-theoretic framework for the investigation of the algebraic information containedin the topology of scalar Feynman diagrams. Perturbative quantum field theories possess an inherent algebraicstructure, which underlies the combinatorics of recursion governing renormalisation theory, and are thus deeplyconnected to the theory of graphs.In Section 2, we broadly review preliminary notions in algebraic geometry and algebraic topology. An algebraicvariety over Q is endowed with two distinct rational structures via algebraic de Rham cohomology and Betti co-homology, which are compatible only after complexification. The coexistence of these two cohomologies and theirpeculiar compatibility are linked to a specific class of complex numbers, known as numeric periods. The cohomologyof an algebraic variety is equipped with two filtrations, and the mixed Hodge structure arising from their interactionconstitutes the bridge between the theory of numeric periods and the theory of motives.In Section 3, we introduce the set of periods, lying between ¯ Q and C , among which are the numbers that come fromevaluating parametric Feynman integrals, and we briefly review their remarkable properties. Suitable cohomologicalstructures 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 fromthe presence of singularities are tackled by blow up. We adopt the category-theoretic Tannakian formalism wheremotivic periods, and motivic Feynman integrals in particular, reveal their most intriguing properties. We presentan overview of the current progress of research towards the general understanding of the structure of scatteringamplitudes via the theory of motivic periods, giving particular attention to recent results in massless scalar φ quantum field theory. A quantum field theory encodes in its Lagrangian every admissible interaction among particles, but it does it ina way that makes decoding this information a difficult task. Fixed the initial and final states, an interaction processis associated to a probability amplitude, called its
Feynman amplitude , which is determined by the set of kinematicand interaction terms in the Lagrangian. However, individual Lagrangian terms correspond to propagators andinteraction vertices which can be linked together in infinitely many distinct ways to connect the same pair of initialand final states. Each of these admissible realisations of the same interaction process has to be accounted for in aninfinite sum of contributions to the probability amplitude. We associate to each of these possibilities a graphicalrepresentation, called its
Feynman diagram , whose individual contribution to the probability amplitude is explicitlywritten in the form of a
Feynman integral by applying the formal correspondence between Lagrangian terms andgraphical components, which is established by convention through the set of Feynman rules of the theory. It is onlythe sum of the contributing Feynman integrals to a given process that has a physical meaning and not the individualintegrals, 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 insome small parameter of the theory, typically a suitable coupling constant. Thus, the Feynman amplitude can beexpanded in a formal power series, which has been shown to be divergent by Dyson [29]. The divergence does not,however, undermine the accuracy of predictions that can be made with the theory. Indeed, although a Feynmanamplitude receives contributions to any order in perturbation theory, practical calculations are made by truncatingthe sum at a certain order and directly evaluating only the remaining finitely many terms. Moreover, the explicitcalculation 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 froma non-1PI diagram at some given order can be expressed as a combination of lower-order 1PI contributions, whichhave 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 adiagram is associated to an unconstrained momentum and integrals over unconstrained loop momenta are the originof singularities in Feynman integrals. We distinguish two classes of singularities. The ultraviolet (UV) divergencesarise in the limit of infinite loop momentum, a regime that is far beyond the energy scale that we have currentlyexperimental access to and where we expect new physical phenomena to become relevant and corresponding new Serone et al [55] characterised the conditions under which some class of asymptotic perturbative series are Borel resummable, leadingto exact results without introducing non-perturbative effects in the form of trans-series. 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 renormalisationtheory . For a renormalizable theory, a suitable adjustment of the Lagrangian parameters allows to systematicallyre-express the predictions of the theory in terms of renormalized physical couplings, so that they decouple fromUV 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 loopmomentum. They cannot be removed by renormalisation and introduce numerous subtleties in the evaluation ofFeynman integrals which we are not dealing with in the present text. For a detailed and comprehensive presentationof 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 pertur-bative quantum field theory. Smirnov [57] summarised more than fifty years of advancements in the field, providingan overview of the most powerful, successful and well-established methods for evaluating Feynman integrals in asystematic 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 theproblem 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, includingunitarity methods, BCFW recursion relations, and the methods of leading singularities and maximal cuts. Overlap-ping divergences can be treated iteratively, thus revealing in the first place the recursive nature of renormalisationtheory. However, this combinatorics of subdivergences is only the first hint to a more fundamental algebraic structureinherent in all renormalizable quantum field theories and deeply connected to the theory of graphs . We consider a scalar quantum field theory in an even number D of space-time dimensions with Euclidean metric and allow different propagators to have different mass. A Feynman diagram is a connected directed graph whereeach edge represents a propagator and is assigned a momentum and a mass and each vertex stands for a tree-levelinteraction. External half-edges, also known as external legs, represent incoming or outgoing particles, while internaledges are the internal propagators of the diagram. We define the loop number to be the number of independent cyclesof the graph. We adopt the convention for which all external legs have arrows pointing inwards, and consequentlydistinguish 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 isnegative otherwise. We fix momenta on external lines and for each internal loop we choose an arbitrary orientationof the edges which is consistent with momentum conservation at each vertex of the graph and globally. Momentumconservation leaves one unconstrained free momentum variable for each loop. Thus, the loop number is equal to thenumber 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 Feynmanintegral, up to numerical prefactors, is I G = ( µ ) n − lD/ ˆ l (cid:89) r =1 d D k r iπ D/ n (cid:89) j =1 − q j + m j (1)where µ is a scale introduced to make the expression dimensionless, k , ..., k l are the independent loop momenta, m , ..., m n are the masses of the internal lines and q , ..., q n are the momenta flowing through them. These can beexpressed as q j = l (cid:88) i =1 λ ji k i + m (cid:88) i =1 σ ji p i (2) A first discussion about the appearance of transcendental numbers in Feynman integrals and its relation to the topology of Feynmangraphs is presented by Kreimer [43] in the framework of knot theory and link diagrams. It is common practice to compute amplitudes in Euclidean space. Moving to Minkowski space involves performing an extension byanalytic continuation known as Wick rotation. p , ..., p m are the external momenta and λ ji , σ ji ∈ {− , , } are constants depending on the particular graphstructure.Feynman [31] introduced the well-known manipulation consisting of defining a set of parameters x , ..., x n , called Feynman parameters , one for each internal edge of the graph, and applying the formula n (cid:89) j =1 P j = Γ( n ) ˆ { x j ≥ } d n x δ − n (cid:88) j =1 x j (cid:16)(cid:80) nj =1 x j P j (cid:17) n (3)with the choice P j = − q j + m j for j = 1 , ..., n . Here, Γ is the Euler Gamma function and δ is the Dirac Deltadistribution. Indeed, we can write n (cid:88) j =1 x j ( − q j + m j ) = − l (cid:88) r =1 l (cid:88) s =1 k r · ( M rs k s ) + l (cid:88) r =1 k r · Q r + J (4)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 } nj =1 . Applying Feynman parametrisationto (1), the l -dimensional integral over the loop momenta becomes an ( n − I G = Γ (cid:18) n − lD (cid:19) ˆ { x j ≥ } d n x δ − n (cid:88) j =1 x j U n − ( l +1) D/ F n − lD/ (5)which is characterised by the polynomials U = det( M ) and F = det( M )( J + QM − Q ) /µ , called first and secondSymanzik polynomials of the Feynman graph, respectively. Notice that the dimension D of space-time, entering theexponents in the integrand of (5), acts as regularisation. We use dimensional regularisation with D = 4 − (cid:15) and (cid:15) small parameter. A detailed description of Feynman parametrisation can be found in Srednicki [58]. Example . Consider the generic one-loop diagram with m = n external legs. Its Symanzik polynomials are U = n (cid:88) j =1 x j F = U n (cid:88) j =1 m j µ x j + n (cid:88) i,j =1 i The parametric Feynman integral in (5) can be written in a slightly different notation, which turns out to beparticularly useful henceforth. Neglecting prefactors and assuming D = 4, it is equivalent to the projective integral I G ( { p j , m e } ) = ˆ σ ΩΨ G (cid:18) Ψ G Ξ G ( { p j , m e } ) (cid:19) n G − l G (13)where σ is the real projective simplex given by σ = { [ x : ... : x n G ] ∈ P n G − ( R ) | x e ≥ , e = 1 , ..., n G } (14)and Ω is the top-degree differential form on P n G − expressed in local coordinates asΩ = n G (cid:88) e =1 ( − e x e dx ∧ ... ∧ (cid:100) dx e ∧ ... ∧ dx n G (15)One can check that the integrand is homogeneous of degree zero, so that the integral in projective space is well-definedand 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 thedomain of integration.Graphs satisfying the condition n G = 2 l G are called logarithmically divergent and constitute a particularlyinteresting class of graphs. In fact, their Feynman integral simplifies to I G = ˆ σ ΩΨ G (16)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 . Fora logarithmically divergent graph G , we define the graph hypersurface as the zero set of its first Symanzik polynomial X G = { [ x : ... : x n G ] ∈ P n G − | Ψ G ( x , ..., x n G ) = 0 } (17)which describes the singularities of its Feynman integral I G . The following theorem on the convergence of logarith-mically 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 ∅ (cid:54) = γ ⊂ G satisfies the condition n γ > l γ .A logarithmically divergent graph G such that I G is convergent is called primitive log-divergent , or simply prim-itive . We give particular attention to the class of primitive log-divergent graphs in scalar massless φ quantum fieldtheory. They are called φ -graphs , and have vertices with valency at most four. Feynman amplitudes in φ theoryhave been computed to much higher loop orders than most other quantum field theories thanks to the work ofBroadhurst and Kreimer [10], [11], and Schnetz [53]. Some of the simplest φ graphs are shown in Fig. 6 along withthe values of the associated Feynman integrals. (a) I G = 6 ζ (3) (b) I G = 20 ζ (5) (c) I G = 36 ζ (3) (d) I G = 32 P , Figure 6: Examples of φ graphs with 3, 4, 5 and 6 loops.Here, ζ is the Riemann zeta function, and P , = − ζ (3 , − ζ (5) ζ (3) + ζ (8). Among other contexts, the feature of no-scaling also occurs in the evaluation of Feynman diagrams concerning the anomalous magneticmoment of the electron, as presented by Laporta and Remiddi [45]. .5 Multiple Zeta Values The Riemann zeta function is defined, on the half-plane of complex numbers s ∈ C with Re( s ) > 1, by theabsolutely convergent series ζ ( s ) = ∞ (cid:88) n =1 n s (18)and extended to a meromorphic function on the whole complex plane with a single pole at s = 1. The firsttentative attempts to find polynomial relations among zeta values by multiplying terms of the form (18) have led toa generalisation of the notion of Riemann zeta value. Multiple zeta values, or MZVs, are the real numbers ζ ( s , ..., s l ) = (cid:88) n >n >...>n l ≥ n s · ... · n s l l (19)associated to tuples of integers s = ( s , ..., s l ), called multi-indices . To guarantee the convergence of the infiniteseries, only multi-indices such that s i ≥ i = 1 , ..., l and s ≥ admissible multi-indices. The integers wt( s ) = s + ... + s l and l are called weight and length of the multi-index s , respectively.Following the early observations that products of two zeta values are Q -linear combinations of zeta and doublezeta values, and that products of more than two zeta values are analogously expressed in terms of multiple zeta valuesof higher length, linear relations among MZVs have been the object of a more and more extensive investigation bymany mathematicians, including Brown, Cartier, Deligne, Drinfeld, ´Ecalle, Goncharov, Hain, Hoffman, Kontsevich,Terasoma, Zagier, Broadhurst and Kreimer. Indeed, the Q -linear relations among multiple zeta values directlyprovide insights on the widely sought-after algebraic relations among Riemann zeta values.The Q -vector space generated by multiple zeta values forms an algebra under the so-called stuffle product . Analyticmethods, like partial fraction expansions, provide only a few of the known relations among MZVs. Many more areobtained, although conjecturally, by performing extensive numerical experiments, as described by Bl¨umlein et al [6].However, enormous progress followed the analytic discovery of a crucial feature of multiple zeta values, that is, besidetheir representation as infinite series, they admit an alternative representation as iterated integrals over simplices ofweight-dimension. Let ∆ p = { ( t , ..., t p ) ∈ R p | ≥ t ≥ t ≥ ... ≥ t p ≥ } and define the following measures on theopen interval (0 , ω ( t ) = dtt , ω ( t ) = dt − t (20)If s is an admissible multi-index, write r i = s + ... + s i for each i = 1 , ..., l and set r = 0. Define the measure ω s on the interior of the simplex ∆ wt( s ) by ω s = l (cid:89) i =1 ω ( t r i − +1 ) · · · ω ( t r i − ) (cid:124) (cid:123)(cid:122) (cid:125) s i − ω ( t r i ) (21)The theorem below is due to Kontsevich. Theorem 2. Let s = ( s , ..., s l ) be an admissible multi-index. The multiple zeta value ζ ( s ) can be obtained by theconvergent improper integral ζ ( s ) = ζ ( s , ..., s l ) = ˆ ∆ wt( s ) ω s (22)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 contem-porarily endow the Q -vector space of MZVs of two distinct algebraic structures, expressed by the stuffle and shuffleproducts, to be the most productive source of information about these numbers. Relations among MZVs are also andmost interestingly derived by such a comparison. For a more detailed discussion of the classical theory of multiplezeta values we refer to the survey article by Fres´an and Gil [32].We observe the remarkable fact that Q -linear combinations of multiple zeta values are ubiquitous in the evaluationof 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 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 I Z l = 4 (2 l − l !( l − (cid:18) − − ( − l l − (cid:19) ζ (2 l − a) l = 5 (b) l = 6 Figure 7: Examples of zig-zag graphs with 5 and 6 loops.Another example is given by the anomalous magnetic moment of the electron in quantum electrodynamics. Thetree level Feynman diagram representing a slow-moving electron emitting a photon is depicted in Fig. 8 along withits one-loop correction. (a) Tree-level contribution (b) One-loop contribution Figure 8: Up to one-loop Feynman diagrams contributing to the anomalous magnetic moment of the electron.The two-loop correction comes from the contributions of seven distinct two-loop diagrams. The total two-loopFeynman amplitude has been evaluated by Petermann [48], giving + ζ (2) − ζ (2) log(2) + ζ (3), which involvesthe 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 longtime that all primitive amplitudes of the form (16) in massless φ theory should be Q -linear combinations of MZVs.Only recently this conjectural statement was proved false. Explicit examples of φ -amplitudes at high loop ordersnot expressible in terms of multiple zeta values have been found by Panzer and Schnetz [47]. In the same work,explicit computation of all φ -amplitudes with loop order up to 7 suggests that not all MZVs appear among them.For example, no φ -graph is known to evaluate to ζ (2) or ζ (2) . Remarkably, the integral representation of MZVspartially clarify the presence of these numbers in perturbative calculations in quantum field theory. Indeed, bothexpressions (16) and (22) are suitably interpreted as periods of algebraic varieties. 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∆ nst = { ( t , ..., t n ) ∈ R n +1 | n (cid:88) i =0 t i = 1 , t i ≥ , i = 0 , ..., n } (23)For each i = 0 , ..., n , the face map δ ni : ∆ n − st → ∆ nst is defined by δ ni ( t , ..., t n − ) = ( t , ..., t i − , , t i , ..., t n − ) (24)A singular n-chain in M is a continuous map σ : ∆ nst → M . For each n ≥ 0, let C n ( M ) = (cid:77) σ Z σ (25) If M is a differentiable manifold, we can assume the singular chains to be piecewise smooth, or smooth, without altering the homologygroups. 10e the free abelian group generated by singular n -chains. Elements of C n ( M ) are finite Z -linear combinations of thecontinuous maps σ : ∆ nst → M . For each n ≥ 1, the boundary map ∂ n : C n ( M ) → C n − ( M ) is defined by ∂ n ( σ ) = n (cid:88) i =0 ( − i ( σ ◦ δ ni ) (26)where the alternating signs in the sum guarantee that boundary maps satisfy the condition ∂ n − ◦ ∂ n = 0. The pair( C • ( M ) , ∂ • ) is called a homological chain complex and is graphically represented as ... ∂ n +1 −−−→ C n ( M ) ∂ n −−→ C n − ( M ) ∂ n − −−−→ ... ∂ −−→ C ( M ) ∂ −−→ C ( M ) (27) Definition 1. The singular homology of the topological space M is the homology of the complex ( C • ( M ) , ∂ • ), thatis H s n ( M, Z ) = (cid:40) C ( M ) / Im( ∂ ) n = 0Ker( ∂ n ) / Im( ∂ n +1 ) n ≥ n , chains in the kernel of the boundary map ∂ n are called (closed) cycles and chains in the image of theboundary map ∂ n +1 are called (exact) boundaries . Example . Let M = C ∗ be the punctured complex plane. The singular chains σ : ∆ st → C ∗ , (cid:55)→ σ : ∆ st → C ∗ , ( t, − t ) (cid:55)→ e πit (29)generate the singular homology groups H s0 ( C ∗ , Z ) and H s1 ( C ∗ , Z ), respectively. These are both free groups of rankone. All the other homology groups vanish.For each n ≥ 0, the free abelian group of singular n-cochains is defined by C n ( M ) = Hom( C n ( M ) , Z ) (30)Analogously, applying vector duality, the coboundary maps d n : C n ( M ) → C n +1 ( M ), which satisfy the condition d n +1 ◦ d n = 0, are introduced. This gives a cohomological chain complex ( C • ( M ) , d • ), graphically represented as ... d n +1 ←−−− C n +1 ( M ) d n ←−− C n ( M ) d n − ←−−− ... d ←−− C ( M ) d ←−− C ( M ) (31) Definition 2. The singular cohomology of the topological space M is the cohomology of the complex ( C • ( M ) , d • ),that is H n s ( M, Z ) = (cid:40) Ker( d ) n = 0Ker( d n ) / Im( d n − ) n ≥ Z , extend to other coefficientrings. For our purposes, we almost exclusively work with rational coefficients. This allows us to identify singularcohomology groups with the vector duals of the corresponding singular homology groups, that is H n s ( M, Q ) (cid:39) Hom( H s n ( M, Q ) , Q ) (33)Thus, classes in a cohomology group can be interpreted as classes of linear functionals on the corresponding homologygroup. The singular cohomology of a topological space given by the complex points of an algebraic variety definedover 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 isthe singular cohomology of the underlying topological space of complex points X ( C ) equipped with the analytictopology, that is H nB ( X ) = H n s ( X ( C ) , K ) (34) Example . Let G m = Spec Q [ x, /x ] be the multiplicative group. G m is an algebraic variety over Q and its underlyingtopological space of complex points is G m ( C ) = C ∗ . For each n ≥ 0, the n -th Betti cohomology group of G m is H nB ( G m ) = H n s ( C ∗ , Q ). This isomorphism is true for real or complex coefficients as well, while it does not hold for integer coefficients. .1.1 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 and M are homotopically equivalent topological spaces, then H s n ( M , Q ) (cid:39) H s n ( M , Q ) for each n ≥ 0. An analogous statement holds for singular cohomology.2) Mayer-Vietoris sequences . For any two open subspaces U, V ⊆ M such that M = U ∪ V , there is a long exactsequence of the following form ... H sn ( U ∩ V, Q ) H sn ( U, Q ) ⊕ H sn ( V, Q ) H sn ( M, Q ) H sn − ( U ∩ V, Q ) ... (35)An analogous statement holds for singular cohomology.3) K¨unneth formula . For any two topological spaces M , M , for each n ≥ 0, there is a natural isomorphism H s n ( M × M , Q ) (cid:39) (cid:77) i + j = n H s i ( M , Q ) ⊗ H s j ( M , Q ) (36)An analogous statement holds for singular cohomology.4) Push-forward . Let f : M → M be a continuous map between two topological spaces M , M . Then, f inducesa morphism of chain complexes f ∗ : C • ( M ) → C • ( M ) (37)called push-forward , sending σ ∈ C n ( M ) to σ = f ◦ σ ∈ C n ( M ). Equivalently, the following diagram∆ nst M M σ σ f (38)commutes. Hence, f induces also a group homomorphism between the corresponding singular homology groups f ∗ : H s n ( M , Q ) → H s n ( M , Q ) (39)for each n ≥ Pull-back. Let f : M → M be a continuous map between two topological spaces M , M . Then, f induces amorphism of cochain complexes f ∗ : C • ( M ) → C • ( M ) (40)called pull-back , sending ω ∈ C n ( M ) to ω = ω ◦ f ∗ ∈ C n ( M ). Equivalently, the following diagram H s n ( M , Q ) Q H s n ( M , Q ) ω f ∗ ω (41)commutes. Hence, f induces also a group homomorphism between the corresponding singular cohomologygroups f ∗ : H n s ( M , Q ) → H n s ( M , Q ) (42)for each n ≥ 0. 12 .1.2 Relative Singular Homology Let M be a topological space and ι : N (cid:44) → M the canonical inclusion of a topological subspace N ⊆ M . Denote( C • ( N ) , ∂ N • ) and ( C • ( M ) , ∂ M • ) 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 C n ( M, N ) = C n − ( N ) ⊕ C n ( M ) (43)and the differential ∂ n : C n ( M, N ) → C n − ( M, N ) acting as ∂ n ( σ N , σ M ) = ( − ∂ Nn − ( σ N ) , − ι ∗ ( σ N ) + ∂ Mn ( σ M )) (44)where ( σ N , σ M ) ∈ C n ( M, N ). Definition 4. The relative homology of the pair of topological spaces ( M, N ) is the homology of the double chaincomplex ( C • ( M, N ) , ∂ • ). For n ≥ 1, we denote the relative singular homology groups as H s n ( M, N, Q ).Relative homology satisfies the following long exact sequence ... H s n ( M, Q ) H s n ( M, N, Q ) H s n − ( N, Q ) H s n − ( M, Q ) H s n − ( M, N, Q ) ... (45)where the connecting morphisms are the push-forward maps ι ∗ : H n ( N, Q ) → H n ( M, Q ) induced by the inclusion ι : N (cid:44) → 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 − ( N ) and σ M ∈ C n ( M ) satisfying ∂ Nn − σ N = 0 , ∂ Mn σ M = − ι ∗ σ N (46)Note that, since ι ∗ is injective, the latter condition implies the former. Thus, relative homology classes are representedby chains in M whose boundary is contained in N . Relative cohomology groups H n s ( M, N, Q ) are defined similarly. Example . Let M = C ∗ be the punctured complex plane and let N = { p, q } ⊂ M be the subspace consisting of thetwo distinct points p, q ∈ C ∗ . Let σ : ∆ st → M be any continuous map such that σ (0 , 1) = p and σ (1 , 0) = q , suchas the oriented segment starting at p and ending at q . Then ∂ M σ = p − q ∈ C ( N ) (47)Consequently, σ defines a relative chain. It follows from the long exact sequence (45) that the only non-trivialrelative homology group is H s1 ( M, N, Q ). A basis of this group is given by the chain σ and the chain σ , introducedin Example 4, consisting of a counterclockwise circle containing the origin. Such a basis is graphically representedin Fig. 9. Figure 9: Basis of H s1 ( C ∗ , { p, q } , Q ). Let M be a differentiable manifold of dimension n . A differential p -form on M is written in local coordinates as ω = (cid:88) ≤ i ≤ ... ≤ i p ≤ n f i ,...,i p ( x , ..., x n ) dx i ∧ ... ∧ dx i p (48)where f i ,...,i p ( x , ..., x n ) are C ∞ -functions. Let Ω p ( M ) denote the R -vector space of differential p -forms on M anddefine the space of differential forms on M as Ω( M ) = n (cid:77) p =0 Ω p ( M ) (49)The exterior derivative d : Ω( M ) → Ω( M ) is the unique R -linear map which sends p -forms into ( p + 1)-forms andsatisfies the following axioms: 13) If f is a smooth function, df = (cid:80) ni =1 ∂f∂x i dx i is the ordinary differential of f .b) d ◦ d = 0.c) Let α be a p -form on M and β any differential form in Ω( M ). Denote α ∧ β their exterior product. Then, d ( α ∧ β ) = dα ∧ β + ( − p α ∧ dβ .The associated cochain complex is 0 → Ω ( M ) d −→ Ω ( M ) d −→ ... d −→ Ω n ( M ) → H • dR ( M, R ) 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 − η such that ω = dη . A classical theorem by De Rham [21] asserts that the singular cohomology H • s ( M, R ) can be computed using differential forms . Theorem 4. Let 0 ≤ k ≤ n . The map H kdR ( M, R ) −→ H k s ( M, R ) (cid:39) Hom( H s k ( M, R ) , R )[ ω ] (cid:55)−→ ˆ ω (51)which sends the class of a differential form ω to the integration functional ˆ ω : H s k ( M, R ) −→ R [ γ ] (cid:55)−→ ˆ γ ω (52)is an isomorphism. Assume X is an affine variety over Q of dimension n and write X = Spec R where R is the ring of regular functionson X , i.e. R = O ( X ). The algebraic p -forms on X are the smooth differential p -forms on X with R -coefficients.In local coordinates ω = (cid:88) ≤ i ≤ ... ≤ i p ≤ n f i ,...,i p ( x , ..., x n ) dx i ∧ ... ∧ dx i p (53)where f i ,...,i p ( x , ..., x n ) are regular functions on X . The space of algebraic p -forms is denoted Ω palg − dR ( X ). Definethe space of algebraic forms on X as Ω alg − dR ( X ) = n (cid:77) p =0 Ω palg − dR ( X ) (54)The exterior derivative d : Ω alg − dR ( X ) → Ω alg − dR ( X ), defined as in Section 2.2, canonically yields a cochain complex0 → Ω alg − dR ( X ) (cid:39) R d −→ Ω alg − dR ( X ) d −→ ... d −→ Ω nalg − dR ( X ) → de Rham complex , whose associated cohomology, denoted H • alg − dR ( X, Q ) and called the algebraic de Rhamcohomology of X , was first introduced by Grothendieck [34]. Example . Consider X = G m = Spec Q [ x, /x ]. The only non-vanishing spaces of algebraic forms areΩ alg − dR ( G m ) = Q [ x, /x ]Ω alg − dR ( G m ) = Q [ x, /x ] · dx (56)Consequently, the following two groups H alg − dR ( G m , Q ) = Q H alg − dR ( G m , Q ) = Q [ x, /x ] · dxd Q [ x, /x ] = Q (cid:20) dxx (cid:21) (57)are the only non-trivial cohomology groups of X . 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 integrationof closed differential forms over cycles. We refer to Bott and Tu [8] for a comprehensive investigation of differential forms in algebraic topology. The algebraic substitute for the smooth differential form is rigorously defined through the notions of K¨ahler differential and exteriorpower. Also, the proper construction of the algebraic de Rham cohomology requires the notions of sheaf cohomology and hypercohomologythat we do not use here. For details on these topics we refer to Kashiwara and Schapira [39]. Theorem 5. Let X be a smooth affine variety defined over Q of dimension n . The following comparison isomorphism holds comp : H kalg − dR ( X, Q ) ⊗ Q C ∼ −−→ H kB ( X, Q ) ⊗ Q C (58)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 spaceof complex points of X . Then, the smooth de Rham cohomology of M , equivalent to its singular cohomology, isisomorphic to the algebraic de Rham cohomology of X , i.e. the former can be computed considering algebraic formsonly . Thus, a purely algebraic definition of cohomology is obtained. Let X be a smooth affine Q -variety. Denote Ω ( X ) → Ω ( X ) → Ω ( X ) → ... its de Rham complex. Let D ⊆ X be a simple normal crossing divisor 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 ⊆ { , ..., r } , set D I = (cid:92) i ∈ I D i (59)and define D p = (cid:40) X p = 0 (cid:96) | I | = p D I p ≥ Q -vector spaces K p,q = Ω q ( D p ) is graphically represented as ... ... ... Ω ( X ) (cid:76) i Ω ( D i ) (cid:76) i 1, of restriction maps d IJ : Ω q ( D I ) → Ω q ( D J ). Note that, thanks to the factor( − p in the definition of d ver , the vertical and horizontal differentials anticommute. Let (Ω • ( X, D ) , δ ) denote thetotal cochain complex associated to K p,q , that isΩ • ( X, D ) = (cid:77) p + q = • K p,q , δ = ( d ver , d hor ) (62)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 doublecochain complex K p,q represented in (61). The total complex is in fact explicitly written down asΩ ( X, D ) (cid:39) Ω ( X ) δ −−→ Ω ( X, D ) (cid:39) Ω ( X ) ⊕ (cid:77) i Ω ( D i ) δ −−→ ... (63)The relative algebraic de Rham cohomology H • alg − dR ( X, D ) is the cohomology of the total cochain complex Ω • ( X, D ),that is H nalg − dR ( X, D ) = (cid:40) Ker( δ ) n = 0Ker( δ n ) / Im( δ n − ) n ≥ D looks locally like a collection of coordinate hypersurfaces. xample . Let X = G m = Spec Q [ x, /x ] and D = { , z } with z ∈ Q , z (cid:54) = 1. The corresponding double de Rhamcomplex is 0 Q (cid:2) x, x (cid:3) dx Q (cid:2) x, x (cid:3) Q ⊕ Q dd − d (65)where the only non-trivial horizontal differential is the evaluation map Q (cid:20) x, x (cid:21) −→ Q ⊕ Q f (cid:55)−→ ( f (1) , f ( z )) (66)The corresponding total cochain complex is Q (cid:20) x, x (cid:21) δ −−→ Q (cid:20) x, x (cid:21) dx ⊕ Q ⊕ Q f ( x ) (cid:55)−→ ( f (cid:48) ( x ) dx, f (1) , f ( z )) (67)where the only non-trivial differential is explicitly written. The non-trivial relative algebraic de Rham cohomologygroups are H alg − dR ( X, D ) = Ker( δ ) = 0 H alg − dR ( X, D ) = coKer( δ ) = Q (cid:2) x, x (cid:3) dx ⊕ Q ⊕ Q Im( δ ) (68)A basis of H alg − dR ( X, D ) is given by the classes (cid:2)(cid:0) dxx , , (cid:1)(cid:3) and (cid:104)(cid:16) dxz − , , (cid:17)(cid:105) . As a consequence of Theorem 5, the Betti cohomology of an algebraic variety is endowed with a richer structurethan the singular cohomology of a generic topological space. Recall the following definition. Definition 5. Let K be a field and ( V, F ), ( V (cid:48) , F ) be filtered K -vector spaces. A morphism f : V → V (cid:48) is called filtered if f ( F p V ) ⊆ F p V (cid:48) for each p ≥ M be a compact K¨ahler manifold of dimension d . For each pair of integers p, q , let H p,q ( M ) ⊆ H p + q ( M, C ) = H p + qdR ( M, C ) (69)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 (cid:88) I,J f I,J ( z , ..., z d ) dz i ∧ ... ∧ dz i p ∧ d ¯ z j ∧ ... ∧ d ¯ z j q (70)where the sum runs over the index subsets I = { i , ..., i p } and J = { j , ..., j q } of { , ..., 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 H n ( M, Q ) ⊗ Q C = (cid:77) p + q = n H p,q ( M ) (71)for n ≤ d . Recall that a K¨ahler manifold is a manifold with a complex structure, a Riemannian structure, and a symplectic structure which aremutually compatible. 16e note that complex conjugation acts on the right-hand side of (71) through the action on the complex coefficientsof the left-hand side, that is, σ ⊗ ω = σ ⊗ ¯ ω (72)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 isoften called Hodge symmetry . Definition 6. Let H be a finite-dimensional Q -vector space and let H C = H ⊗ Q C . Assume that H C possesses abigrading H C = (cid:77) p + q = n H p,q (73)satisfying H p,q = H q,p . Then, H is called a pure Hodge structure of weight n and the given direct sum decompositionof 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 Hodgedecomposition 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 F p H C ∩ F q H C = ∅ , F p H C ⊕ F q H C = H C (74)The relation between the two equivalent descriptions is given by H p,q = F p H C ∩ F q H C F p H C = (cid:77) i ≥ p H i,n − i (75)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 thealgebraic de Rham cohomology of X after complexification, i.e. H n ( M, C ) = H nalg − dR ( X, Q ) ⊗ Q C (76)As a consequence of Theorem 6, the Hodge decomposition can be easily referred to the algebraic de Rham cohomologyof X and analogously the Hodge filtration F • is defined on H nalg − dR ( X, Q ). To keep track of these additionalstructures, we define the triple H n ( X ) = ( H nB ( X, Q ) , ( H nalg − dR ( X, Q ) , F • ) , comp) (77)and call it a pure Hodge structure of weight n over Q . The comparison isomorphism induces a corresponding Hodgefiltration on the Betti cohomology which is still denoted by F • . Definition 7. Let H and H (cid:48) be two pure Hodge structures over Q , where we are omitting the weight for simplicity.A morphism between them f : H → H (cid:48) is a pair f = ( f B , f alg − dR ) consisting of two Q -linear maps f B : H B → H (cid:48) B and f alg − dR : H alg − dR → H (cid:48) alg − dR such that the following two conditions hold(1) f alg − dR is filtered with respect to the Hodge filtration, i.e. f alg − dR ( F • H alg − dR ) ⊆ F • H (cid:48) alg − dR (78)(2) The following diagram commutes H alg − dR ⊗ Q C H B ⊗ Q C H (cid:48) alg − dR ⊗ Q C H (cid:48) B ⊗ Q C comp f alg − dR ⊗ Q Id C f B ⊗ Q Id C comp (cid:48) (79)The definition implies that, if H and H (cid:48) have different weights, then every morphism of Hodge structures betweenthem is zero. The following variant of Theorem 6 implies that pure Hodge structures are functorial for morphismsof algebraic varieties. Theorem 7. Let X, Y be smooth projective varieties defined over Q and let H n ( X ) , H n ( Y ) be the correspondingpure Hodge structures of weight n . For any morphism f : X → Y of smooth projective varieties, the induced mapon cohomology f ∗ : H n ( Y ) → H n ( X ) is a morphism of pure Hodge structures.17 xample . Let K be a subfield of C . For each n ∈ Z , we define Q ( n ) = ( Q , ( K , F • ) , comp) (80)where the filtration yields K = F − n K ⊇ F − n +1 K = 0 and the isomorphism comp : C → C is given by multiplicationby (2 πi ) − n . Q ( n ) is a one-dimensional pure Hodge structure of weight − n over K and is called a Tate-Hodgestructure . As an example, Q ( − 1) is isomorphic to H ( G m ) = ( H B ( G m ) , ( H alg − dR ( G m ) , F • ) , comp) where F • is thetrivial filtration concentrated in degree 1. The cohomology in degree n of a smooth projective complex variety X carries along a pure Hodge structure ofweight n . However, this is no longer true when X fails to be smooth or projective. The generalisation of the notionof pure Hodge structure to any quasi-projective complex variety is due to Deligne [22], [23], [24], who proved thatthe 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 W − = 0 ⊆ W ⊆ W ⊆ ... ⊆ W n = H n ( X, Q ) (81)and a finite decreasing filtration, called Hodge filtration F = H n ( X, C ) ⊇ F ⊇ ... ⊇ F n ⊇ F n +1 = 0 (82)such that F • induces a pure Hodge structure of weight m on each graded pieceGr Wm H n ( X, Q ) = W m /W m − (83)(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. f ∗ ( W m H n ( Y, Q )) ⊆ W m H n ( X, Q ) f ∗ C ( F p H n ( Y, C )) ⊆ F p H n ( X, C ) (84)(3) If X is smooth, then Gr Wm H n ( X, Q ) = 0 for all m < n . If X is projective, then Gr Wm H n ( X, Q ) = 0 for all m > n .The following definition generalises the notion of pure Hodge structure. Definition 8. Let X be a quasi-projective variety over Q . Define the triple H n ( X ) = (( H nB ( X, Q ) , W B • ) , ( H nalg − dR ( X, Q ) , F • , W alg − dR • ) , comp) (85)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,comp( W alg − dR • ⊗ Q C ) = W B • ⊗ Q C (86)and that for each integer m Gr Wm H n = (Gr Wm H nB , (Gr Wm H nalg − dR , F • ) , comp) (87)is a pure Hodge structure over Q of weight m . Then, H n ( X ) is called a mixed Hodge structure over Q . Definition 9. Given two mixed Hodge structures H and H (cid:48) over Q , a morphism f : H → H (cid:48) between them is apair f = ( f B , f alg − dR ) consisting of two Q -linear maps f B : H B → H (cid:48) B and f alg − dR : H alg − dR → H (cid:48) alg − dR such that f B is filtered with respect to the weight filtration, while f alg − dR is filtered with respect to the weight and Hodgefiltrations, and both maps are compatible with the comparison isomorphism. In other words, we have f B ( W B • H B ) ⊆ W B • H (cid:48) B f alg − dR ( F • H alg − dR ) ⊆ F • H (cid:48) alg − dR f alg − dR ( W alg − dR • H alg − dR ) ⊆ W alg − dR • H (cid:48) alg − dR f alg − dR ◦ comp (cid:48) = comp ◦ ( f B ⊗ Id C ) (88)18e denote by MHS ( Q ) the category of mixed Hodge structures over Q . Deligne [23] proved that MHS ( Q ) is anabelian category. Moreover, MHS ( Q ) is naturally endowed with two forgetful functors ω B : MHS ( Q ) → Vec Q , ω dR : MHS ( Q ) → Vec Q (89)sending the mixed Hodge structure H into the Q -vector spaces H B and H alg − dR , respectively. These functors arecalled the Betti and de Rham functors . 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 absolutelyconvergent integrals of the form ˆ σ f ( x , ..., x n ) dx · ... · dx n (90)where the integrand f is a rational function with rational coefficients and the domain of integration σ ⊆ R n is definedby finite unions and intersections of domains of the form { g ( x , ..., x n ) ≥ } with g a rational function with rationalcoefficients.If rational functions and coefficients are replaced in Definition 10 by algebraic functions and coefficients, thesame set of numbers is obtained. On the other hand, algebraic functions in the integrand can be substituted byrational functions by enlarging the set of variables. Note that, because the integral of any real-valued function isequivalent to the volume subtended by its graph, any period admits a representation as the volume of a domaindefined by polynomial inequalities with rational coefficients. Thus, the integrand can always be assumed to be theconstant function 1. However, this extremely simplified framework does not prove to be particularly useful. Quitethe opposite, in what follows, we mostly work with an even more general description of periods than the one given inDefinition 10. We denote by P the set of periods. Being ¯ Q ⊂ P ⊂ C , periods are generically transcendental numbersand nonetheless they contain only a finite amount of information, which is captured by the integrand and domainof integration of its integral representation as in (90). Indeed, just like ¯ Q , P is countable. Many famous numbersbelong to the class of periods. Here are some examples:(a) Algebraic numbers are periods, e.g. √ ˆ x ≤ dx (91)(b) Logarithms of algebraic numbers are periods, e.g.log 2 = ˆ dxx (92)(c) The transcendental number π is a period, as given by π = ˆ − dx √ − x = + ∞ ˆ −∞ dx x = ˆ x + y ≤ dxdy (93)and alternatively by 2 πi = ˛ γ dzz (94)where γ is a closed path encircling the origin in the complex plane.(d) Values of the Gamma function at rational arguments satisfyΓ (cid:18) pq (cid:19) q ∈ P , p, q ∈ N (95)19e) The elliptic integral 2 b ˆ − b (cid:114) a x b − b x dx (96)representing the perimeter of an ellipse with radii a and b , is a period. Note that it is not an algebraic functionof π for a (cid:54) = b , a, b ∈ Q > ,(f) Values of the Riemann zeta function at integer arguments s ≥ ζ (3) = 1 + 12 + 13 + ... = ∞ (cid:88) n =1 n = ˚ The theory of periods can be alternatively developed within the formalism of algebraic geometry. We refer toHuber and M¨uller-Stach [38]. Definition 11. Let X be a smooth quasi-projective variety defined over ¯ Q , Y ⊂ X a subvariety, ω a close algebraicdifferential n -form on X vanishing on Y and γ a singular n -chain on the complex manifold X ( C ) with boundarycontained in Y ( C ). The integral ´ γ ω ∈ C is a numeric period. An example of two distinct periods which agree numerically to more than 80 digits is given by Shanks [56]. γ can be deformedto a semi-algebraic chain and then broken up into small pieces, which can be bijectively projected onto open domainsin R n with algebraic boundary. Without loss of generality, we work with coefficients in Q instead of ¯ Q . We notethat, like Definition 10, Definition 11 also contains redundancy. The integral ´ γ ω can be formally destructured intothe quadruple ( X, Y, ω, γ ) (99)and different quadrupoles can give the same resulting number. To get rid of this redundancy, the various formsof topological invariance of the integral must be suitably accounted for. Following Stokes’ theorem, the integral isinsensitive to the individual cycle and form, being instead determined by the homology and cohomology classes ofthese. 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 aunique algebraic description of periods consists of the following substitutions ω −→ [ ω ] ∈ H nalg − dR ( X, Y ) γ −→ [ γ ] ∈ H Bn ( X, Y ) (100)into the quadrupole ( X, Y, ω, γ ). The problem of the coexistence of distinct, but similarly behaved, cohomologiesassociated to the same variety, which seems to imply an arbitrary choice here and in many other situations, has beentackled by Grothendieck [20] with the introduction of the theory of motives . He suggested that there should exist auniversal cohomology theory taking values in a Q -category of motives M . Thus, the notion of a motive is proposedto capture the intrinsic cohomological essence of a variety. Without delving into the category-theoretic details ofthe theory of motives, we give now an intuitive idea of its origin and fundamental features as necessary to reviewits application to the theory of periods. A more rigorous discussion of motives is presented in Section 4.5. We recallfrom Grothendieck’s Theorem 5 that there is a comparison isomorphismcomp : H nalg − dR ( X, Y ) ⊗ Q C ∼ −−→ H nB ( X, Y ) ⊗ Q C (101)induced by the pairing H nalg − dR ( X, Y ) × H singn ( X ( C ) , Y ( C )) −→ C ([ ω ] , [ γ ]) (cid:55)−→ ˆ γ ω (102)Neglecting the presence of filtrations, the Hodge structure of X relative to Y is expressed as H n ( X, Y ) = ( H nB ( X, Y ) , H nalg − dR ( X, Y ) , comp) (103)In the same way that the cohomology class of a differential form singles out its cohomological behaviour, the Hodgestructure of an algebraic variety intuitively selects the content shared by its different coexisting cohomologies andfilters out everything else. It is, therefore, the first approximate realisation of Grothendieck’s idea of a motive. Wedefine the motivic version of the period ´ γ ω as the triple[ H n ( X, Y ) , [ ω ] , [ γ ]] m (104)where m in the superscript stands for motivic. We call a period in this guise a motivic period . This has proved to bethe most profitable reformulation of the original notion of a period. However, a second source of redundancy in thedescription of periods via the integral formulation in Definition 11, corresponding to the same transformation rulesin Conjecture 1, has yet to be factored out. Definition 12. The space P m of motivic periods is defined as the Q -vector space generated by the symbols[ H • ( X, Y ) , [ ω ] , [ γ ]] m after factorisation modulo the following three equivalence relations:1) Bilinearity . [ H • ( X, Y ) , [ ω ] , [ γ ]] m is bilinear in [ ω ] and [ γ ].2) Change of variables . If f : ( X , Y ) → ( X , Y ) is a Q -morphism of pairs of algebraic varieties, γ ∈ H B • ( X , Y )and ω ∈ H • alg − dR ( X , Y ), then[ H • ( X , Y ) , f ∗ [ ω ] , [ γ ]] m = [ H • ( X , Y ) , [ ω ] , f ∗ [ γ ]] m (105)where f ∗ is the pull-back of f and f ∗ is its push-forward. Grothendieck proposed the notion of a motive in a letter to Serre in 1964. He himself did not author any publication on motives,although he mentioned them frequently in his correspondence. The first formal expositions of the theory of motives are due to Demazure[27] and Kleiman [41], who based their work on Grothendieck’s lectures on the topic. 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 by ˜ D the normalisation of D . The variety ˜ D contains a simple normal crossing divisor ˜ D coming from double points in D . If [ ω ] ∈ H d − alg − dR ( X, D ) and[ γ ] ∈ H Bd ( X, D ), then [ H d ( X, D ) , δ [ ω ] , [ γ ]] m = [ H d − ( ˜ D, ˜ D ) , [ ω | ˜ D ] , ∂ [ γ ]] m (106)where δ : H d − alg − dR ( X, D ) → H dalg − dR ( X, D ) is the coboundary operator acting on the algebraic de Rhamcohomology and ∂ : H Bd ( X, D ) → H Bd − ( ˜ D, ˜ D ) is the boundary operator acting on the Betti homology.We observe that the space of motivic periods P m is naturally endowed with an algebra structure. Indeed, newperiods are obtained by taking sums and products of known ones. We call period map the evaluation homomorphismper : P m −→ P [ H n ( X, Y ) , [ ω ] , [ γ ]] m (cid:55)−→ ˆ γ ω (107)Following the construction in Section 3.2, the period map is explicitly surjective, while injectivity is, on the otherhand, not proven. Indeed, a numeric period has a unique motivic realisation only conjecturally. Conjecture 1 isequivalent to the period conjecture below. Conjecture . 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 , Y ) → ( X , Y )between two pairs of algebraic varieties induces a change of coordinates between the corresponding algebraic deRham cohomologies by pull-back, that is ( X , Y ) H • alg − dR ( X , Y )( X , Y ) H • alg − dR ( X , Y ) f f ∗ (108)The same morphism f acts on the spaces of complex points underlying the given algebraic varieties and induces achange of coordinates between the corresponding singular homologies by push-forward, that is( X ( C ) , Y ( C )) H s • ( X ( C ) , Y ( C ))( X ( C ) , Y ( C )) H s • ( X ( C ) , Y ( C )) f f ∗ (109)By means of such changes of coordinates, one can easily derive two distinct integral representations of the samenumeric period. For example, taking [ γ ] ∈ H s • ( X ( C ) , Y ( C )) and [ ω ] ∈ H • alg − dR ( X , Y ), we have ˆ f ∗ [ γ ] [ ω ] = ˆ [ γ ] f ∗ [ ω ] (110)The corresponding two motivic representations of the same numeric period[ H • ( X , Y ) , f ∗ [ ω ] , [ γ ]] m , [ H • ( X , Y ) , [ ω ] , f ∗ [ γ ]] m (111)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 thesame period arise, they can always be interrelated by the three equivalence relations in Definition 12. ˜ D is locally the disjoint union of the irreducible components of D . efinition 13. Let X be a smooth quasi-projective Q -variety, Y ⊂ X a subvariety and H = H • ( X, Y ) the Hodgestructure of X relative to Y . Assume that { [ ω i ] } ni =1 is a basis of the algebraic de Rham cohomology H • alg − dR ( X, Y )and that { [ γ j ] } nj =1 is a basis of the Betti homology H B • ( X, Y ). Denote per | H the period map restricted to the motivicperiods in P m that are built on the given Hodge structure H . Observe that per | H is entirely determined by thevalues that it takes when evaluated at [ H, [ ω i ] , [ γ j ]] m , that isper | H ([ H, [ ω i ] , [ γ j ]] m ) = ˆ γ i ω j (112)for each pair of indices ( i, j ) with i, j = 1 , ..., n . Define the period matrix of H as the n × n -matrix with complexentries ( p ij ) i,j =1 ,...,n given by p ij = ˆ γ i ω j (113)The period matrix expresses in a different guise the same information contained in the period map, once it hasbeen restricted to a specific Hodge structure. Example . Let H = H ( G m , { , z } ). As shown in Examples 6 and 8, a basis of the Betti homology H B ( G m , { , z } )is given by [ γ ] and [ γ ], and a basis of the algebraic de Rham cohomology H alg − dR ( G m , { , z } ) is given by [ ω ] = (cid:2)(cid:0) dxx , , (cid:1)(cid:3) and [ ω ] = (cid:104)(cid:16) dxz − , , (cid:17)(cid:105) . The period matrix of H is then (cid:18) πi log( z )0 1 (cid:19) (114) πi The numeric period 2 πi is given by the contour integral2 πi = ˛ γ dxx (115)where γ is a counterclockwise cycle encircling the origin in the punctured complex plane C ∗ . As observed inExample 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 H B ( G m ) = Q [ γ ] H alg − dR ( G m ) = Q (cid:20) dxx (cid:21) (116)Setting H ( G m ) = ( H B ( G m ) , H alg − dR ( G m ) , comp), a motivic version of 2 πi is(2 πi ) m = (cid:20) H ( G m ) , (cid:20) dxx (cid:21) , [ γ ] (cid:21) m (117)which is alternatively represented by the pairing H alg − dR ( G m ) × H B ( G m ) −→ C (cid:18)(cid:20) dxx (cid:21) , [ γ ] (cid:19) (cid:55)−→ ˛ γ dxx = 2 πi (118)A second integral representation of 2 πi is given by2 πi = ˆ P ( C ) dz ∧ d ¯ z (1 + z ¯ z ) (119)where dz ∧ d ¯ z (1+ z ¯ z ) is a closed smooth algebraic 2-form over the closed manifold P ( C ). Because P ( C ) is compact andK¨ahler, Theorem 6 applies, giving the Hodge decomposition H alg − dR ( P ) ⊗ Q C = (cid:77) p + q =2 H p,qalg − dR ( P ) (120)23here the forms in H p,qalg − dR contain p copies of the holomorphic differential dz and q copies of the anti-holomorphicdifferential d ¯ z . Therefore, (cid:104) dz ∧ d ¯ z (1+ z ¯ z ) (cid:105) ∈ H , alg − dR ( P ) and integral (119) corresponds to the following motivic period(2 πi ) m = (cid:20) H ( P ) , (cid:20) dz ∧ d ¯ z (1 + z ¯ z ) (cid:21) , (cid:2) P ( C ) (cid:3)(cid:21) m (121) Remark. The two apparently different motivic periods in (117) and (121) are the same, thus preserving the periodconjecture. To show this, define A = P ( C ) \{∞} ∼ = C ⊂ P ( C ) , B = P ( C ) \{ } ∼ = C ⊂ P ( C ) (122)which satisfy the relations A ∩ B (cid:39) C ∗ (cid:39) G m ( C ) , A ∪ B = P ( C ) (123)By the Mayer-Vietoris theorem applied to the singular homology groups, the following long exact sequence holds0 H s ( A ∪ B ) H s ( A ) ⊕ H s ( B ) H s ( A ) ⊕ H s ( B ) (cid:124) (cid:123)(cid:122) (cid:125) (cid:39) H s ( A ∪ B ) H s ( A ∩ B ) H s ( A ∩ B ) H s ( A ∪ B ) H s ( A ) ⊕ H s ( B ) (cid:124) (cid:123)(cid:122) (cid:125) (cid:39) (124)Here, the step H s ( A ∩ B ) → H s ( A ∪ B ) is an isomorphism, giving H s ( G m ( C )) (cid:39) H s ( P ( C )) (125)Similarly, one can prove that the whole Hodge structures H ( G m ) and H ( P ) are isomorphic and that the changeof coordinates occurring between them relates the cohomology classes (cid:104) dz ∧ d ¯ z (1+ z ¯ z ) (cid:105) and (cid:2) dxx (cid:3) and the homology classes[ γ ] and (cid:2) P ( C ) (cid:3) via pull-back and push-forward maps, respectively. log( z )Recall the integral representation of log( z ), z ∈ Q \{ } , given bylog( z ) = ˆ z dxx (126)As in the case of 2 πi , this is an integral over the punctured complex plane C ∗ = G m ( C ). However, contrary tothe case of 2 πi , where the integration path γ is closed, integral (126) is performed on an open path, precisely anycontinuous oriented path γ ⊂ C ∗ , starting at 1 and ending at z , which is contractible to the oriented segment from 1to 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 { , z } with z ∈ Q \{ } a simple normal crossing divisor in C ∗ . As shown inExamples 6 and 8, we have H B ( G m , { , z } ) = Q [ γ , γ ] H alg − dR ( G m , { , z } ) = Q (cid:20)(cid:18) dxx , , (cid:19) , (cid:18) dxz − , , (cid:19)(cid:21) (127)Setting as usual H ( G m , { , z } ) = ( H B ( G m , { , z } ) , H alg − dR ( G m , { , z } ) , comp), a motivic version of log( z ) islog( z ) m = (cid:20) H ( G m , { , z } ) , (cid:20)(cid:18) dxx , , (cid:19)(cid:21) , [ γ ] (cid:21) m (128)which is alternatively represented by the pairing H alg − dR ( G m , { , z } ) × H B ( G m , { , z } ) −→ C (cid:18)(cid:20)(cid:18) dxx , , (cid:19)(cid:21) , [ γ ] (cid:19) (cid:55)−→ ˆ γ dxx = log( z ) (129)24 .4.3 Elementary Relations Many relations among numeric periods are simply recast in the formalism of motivic periods. In fact, Hodgestructures conjecturally capture all algebraic relations between periods. Example . For a, b ∈ Q \{ } , we have log( ab ) m = log( a ) m + log( b ) m (130) Example . Consider H = H ( G m , { , z } ) again, and let γ be the union of the paths γ and γ in the puncturedcomplex plane, as shown in Fig. 10. Figure 10: The paths γ , γ and γ in C ∗ .The numeric period obtained by integrating ω along γ is ˆ γ ω = ˆ γ ω + ˆ γ ω = 2 πi + log( z ) (131)and in the formalism of motives we have(2 πi + log( z )) m = [ H, [ ω ] , [ γ ]] m = [ H, [ ω ] , [ γ ∪ γ ]] m = [ H, [ ω ] , [ γ ]] m + [ H, [ ω ] , [ γ ]] m = (2 πi ) m + log( z ) m (132)where we have used that (cid:2) H ( G m , { , z } ) , [ ω ] , [ γ ] (cid:3) m = (cid:2) H ( G m ) , [ ω ] , [ γ ] (cid:3) m . 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, afeature that does not occur in the examples of 2 πi and log( z ). Whenever singularities are present, they have to betaken care of with particular attention. Example . The period ζ (2) is given by the following integral ζ (2) = ˆ ≥ x ≥ x ≥ dx x dx − x (133)over the complex manifold C . The domain of integration is the simplex σ = { ( x , x ) ∈ C | ≥ x ≥ x ≥ } (134)and the integrand is the differential 2-form ω = dx x dx − x (135)25bserving that C is isomorphic to the topological space of complex points A ( C ), underlying the affine Q -algebraicvariety A = Spec Q [ x , x ], we may try to build ζ (2) m as we did for the examples in Section 3.4. Consider the lines l = { x = 0 } and l = { x = 1 } in the affine plane A . Since L = l ∪ l is the locus of singular points of ω , thelatter is an algebraic 2-form on X = A \ 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 containingthe boundary of σ . In an attempt to do so, define the simple normal crossing divisor D = { x = x } ∪ { x = 1 } ∪ { x = 0 } ⊂ A (136)containing the boundary of σ . Note that D is not in X because D ∩ L (cid:54) = ∅ . 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 thesingular locus L in two points p = (0 , 0) = σ ∩ l = D ∩ l q = (1 , 1) = σ ∩ l = D ∩ l (137)Removing the singular points p, q from D and considering the second relative Hodge structure H ( X, D \ ( D ∩ L ))does not solve the mentioned technical issue, because [ σ ] is not a class in H B ( X, D \ ( D ∩ L )). See Fig. 11.Figure 11: Construction of ζ (2) m in the affine plane A .The example of ζ (2) shows how direct removal of singular points explicitly fails and motivates a more articulatedgeometric 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 andthe corresponding reshaping of the integration domain. See Fig. 12 for a qualitative representation of how the blowup of the two singular points p, q ∈ A acts on σ in the case of ζ (2). (a) Before the blow up (b) After the blow up Figure 12: Qualitative illustration of the blow up of the singular points of ζ (2). Consider ζ (2) again. The blow up of the affine plane A along the singular points p, q is defined as the closedsubvariety Y = Blow p,q ( A ) ⊂ A × P × P (138) For any positive integer n , the n -dimensional affine variety over Q is defined as A n = Spec Q [ x , ..., x n ]. For any field extension K ⊇ Q ,the space of K -points of A n is A n ( K ) = K n . The multiplicative group G m = Spec Q [ x, x ] satisfies G m = Spec Q [ x , x ] / (1 − x x ) = A \{ } ⊂ A , that is, G m is an hyperbola in A . x α = x β ( x − α = ( x − β (139)where [ α i : β i ], i = 1 , 2, are homogeneous coordinates on the two copies of P . The projection of Y onto the firstfactor in A × P × P is the proper surjective map π : Y −→ A ( x , x ) × [ α : β ] × [ α : β ] (cid:55)−→ ( x , x ) (140)The inverse of the projection ι = π − , mapping the affine plane A into its blow up Y , replaces the singular points p, q ∈ A by corresponding projective lines E p , E q ⊂ Y , called exceptional divisors . Precisely, we have ι ( p ) = ι (0 , 0) = (0 , × P × [1 : 1] = E p ι ( q ) = ι (1 , 1) = (1 , × [1 : 1] × P = E q (141)Moreover, the restriction of ι to the complement in A of the singular points p, qι | A \{ p,q } : A \{ p, q } −→ Y \ ( E p ∪ E q )( x , x ) (cid:55)−→ ( x , x ) × [1 : 1] × [1 : 1] (142)is an isomorphism. For any closed subset C ⊂ A , the image ι ( C ) is called total transform of C . The strict transform of C , denoted ˆ C , 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 isˆ C = ι ( C \{ p, q } ) ⊆ ι ( C ) (143)It follows that the strict transforms of l , l are the affine lines L = ˆ l = { (0 , x ) × [1 : 0] × [1 − x : 1] | x ∈ A } L = ˆ l = { ( x , × [1 : x ] × [0 : 1] | x ∈ A } (144)and their total transforms are ι ( l ) = L ∪ E p ι ( l ) = L ∪ E q (145)We observe that L , E p and L , E q intersect in only one point each. Precisely L ∩ E p = { (0 , × [1 : 0] × [1 : 1] } L ∩ E q = { (1 , × [1 : 1] × [0 : 1] } (146)Moreover, we have L ∩ E p = ∅ = L ∩ E q L ∩ L = { (0 , × [1 : 0] × [0 : 1] } L ∩ E q = ∅ (147)Similarly, we define the strict transform ˆ σ of the domain of integration. Observing that the closed points of E p canbe interpreted as lines passing through p , and analogously that the closed points of E q can be interpreted as linespassing through q , we obtain ˆ σ ∩ E p = { (0 , × [ t : 1] × [1 : 1] | ≤ t ≤ } ˆ σ ∩ E q = { (1 , × [1 : 1] × [1 : t ] | ≤ t ≤ } (148)which, combined with (146), imply that ˆ σ ∩ L = ∅ , ˆ σ ∩ L = ∅ (149)See Fig. 13 for a graphical representation of the blow up.27igure 13: The strict transform of σ in the blow up Y .As the map ι is applied to the ambient variety, giving the reshaped domain ˆ σ , the differential form ω is replacedby its pull-back π ∗ ( ω ), denoted by ˆ ω . Let us now show that the pull-back ˆ ω is only singular on the strict transform L = L ∪ L . We use local coordinates on the blow up Y . In particular, consider a patch of Y around the point L ∩ E p as shown in Fig. 14.Figure 14: Local patch of Y around the intersection of L and E p .Here, a local system of coordinates is explicitly given by t = x x = β α , s = x (150)where L and E p have equations t = 0 and s = 0, respectively. Applying this change of variables to ˆ ω , we haveˆ ω = d ( st ) st ∧ ds − s = dss ∧ ds − s + dtt ∧ ds − s = dtt ∧ ds − s (151)It follows that ˆ ω is singular along the strict transform L , while it is smooth along the exceptional divisor E p , becauseit has no pole at s = 0. Analogously, we find that ˆ ω is singular along L , but not along E q . Then, the singular locusof ˆ ω is L . Observe that the complement Y \ L is the closed affine subvariety of A × A × A given by the equations x t = x x − x − s (152)where t, s are affine coordinates on the two copies of A . Therefore, the differential form ˆ ω determines a class in H alg − dR ( Y \ L ). Moreover, it follows from (149) that, moving from the original affine plane A to the blow up Y ,the singular locus of the differential form ˆ ω and the domain of integration ˆ σ are disjoint. As usual, we may want toconsider the integral (133) as a period of Y \ L relative to some divisor containing the boundary of ˆ σ . The blow upconstruction is thus successful for the period ζ (2) if [ˆ σ ] turns out to be a class in the given relative Betti homologygroup. To see this, recall that ∂σ is contained in the union D of the affine lines m = { x = x } , m = { x = 1 } , m = { x = 0 } (153)Thus, we naturally consider the normal crossing divisor M ⊂ Y defined by M = ι ( D ) = ι ( m ∪ m ∪ m ) = E p ∪ E q ∪ M ∪ M ∪ M (154) In principle, ˆ ω might have singularities along the total transform of l ∪ l , i.e. L ∪ L ∪ E p ∪ E q . However, in the case of ζ (2),it turns out that ˆ ω has no singularities along the exceptional divisors. More generally, this condition determines whether the blow upprescription turns out to be successful or not for a given period. M i = ˆ m i denotes the strict transform of m i for i = 1 , , 3. Note that L ∩ M is the union of the points L ∩ E p and L ∩ E q expressed in (146). Therefore, ˆ σ is contained in Y \ L and ∂ ˆ σ is contained in M \ ( M ∩ L ) ⊂ Y \ L , implying[ˆ σ ] ∈ H B ( Y \ L, M \ ( M ∩ L )) (155)Besides, the restriction of ˆ ω to every irreducible component M i , i = 1 , , 3, of M gives zero, implying[ˆ ω ] ∈ H alg − dR ( Y \ L, M \ ( M ∩ L )) (156)Setting the Hodge structure H = H ( Y \ L, M \ ( M ∩ L )), the resulting motivic version of ζ (2) is ζ (2) m = [ H, [ˆ ω ] , [ˆ σ ]] m (157)Indeed, the pairing of [ˆ σ ] and [ˆ ω ] yields ˆ ˆ σ ˆ ω = ˆ ˆ σ π ∗ ( ω ) = ˆ π ∗ (ˆ σ ) ω = ˆ σ ω = ζ (2) (158)by the equivalence relation under change of variables in P m . We observe that the whole period matrix of H is (cid:18) (2 πi ) ζ (2)0 1 (cid:19) (159) In an attempt to overcome singularity issues, the blow up procedure can be similarly applied to generic MZVsand other families of periods, such as convergent Feynman integrals. For an exposition of the general computationof 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 Section1.4. Recall that x e denotes the Schwinger parameter associated to e ∈ E G , and Ψ G , I G , and X G denote the firstgraph 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 differentialform on P n G − \ X G , and ∂σ is contained in the union D of the coordinate hyperplanes { x e = 0 , e ∈ E G } , we mayintuitively try to build the motive I m G on the relative Hodge structure H n G − ( P n G − \ X G , D \ ( D ∩ X G )) (160)However, this na¨ıve 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 elementin the corresponding na¨ıve relative Betti homology group. To successfully build the motive I m G in the presence ofsingularities, the blow up technique is applied.A linear subvariety L ⊂ P n G − defined by the vanishing of a subset of the set of Schwinger parameters is calleda coordinate linear space , while its subspace of real points with non-negative coordinates is denoted by L ( R ≥ ) = { [ x e ] e ∈ E G ∈ L | x e ∈ R ≥ } (161)Since the coefficients of Ψ G are positive, the locus of problematic singularities is σ ∩ X G ( C ) = (cid:91) L ⊂ X G L ( R ≥ ) (162)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 suchthat l γ > 0. It follows σ ∩ X G ( C ) = (cid:91) γ ⊂ G L γ ( R ≥ ) (163)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 − definedby 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]. 29 heorem 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 − → ... → P → P = P n G − (164)such that, for each i = 1 , ..., r , P i is obtained by blowing up P i − along the strict transform of a coordinate linearspace L i ⊂ X G , and the following conditions hold:(1) The pulled-back differential ˆ ω 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. B = ι ( D ) = π − (cid:32) (cid:91) e ∈ E G { x e = 0 } (cid:33) (165)Then, B ⊂ P is a normal crossing divisor such that none of the non-empty intersections of its irreduciblecomponents 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, ˆ σ ∩ Y G ( C ) = ∅ .As a consequence of Theorem 9, the motive I m G associated to any subdivergence-free primitive log-divergentFeynman graph G can be written explicitly. Being ∂ ˆ σ ⊂ B \ ( B ∩ Y G ), the domain of integration defines the class[ˆ σ ] ∈ H Bn G − ( P \ Y G , B \ ( B ∩ Y G )) (166)called Betti framing , while the integrand defines the class[ˆ ω G ] ∈ H n G − alg − dR ( P \ Y G , B \ ( B ∩ Y G )) (167)called de Rham framing . Brown and Doryn [14] present a method for explicit computation of the framings on thecohomology of Feynman graph hypersurfaces. Then, the Hodge structure H = H n G − ( P \ Y G , B \ ( B ∩ Y G )) is calledthe graph Hodge structure , and the motivic Feynman integral I m G is given by I m G = [ H, [ˆ ω G ] , [ˆ σ ]] m (168)Indeed, the pairing of the classes [ˆ ω G ] and [ˆ σ ] yields the period ˆ ˆ σ ˆ ω G = ˆ ˆ σ π ∗ ( ω G ) = ˆ π ∗ (ˆ σ ) ω G = ˆ σ ω G = I G (169)by the equivalence relation under change of variables in P m . Example . Adopting the following notation P log = Q (cid:104) I G | G is a primitive log-divergent Feynman graph (cid:105)P φ = Q (cid:104) I G | G is a primitive log-divergent Feynman graph in φ theory (cid:105) (170)we observe that the sequence of inclusions P φ ⊂ P log ⊂ P is preserved after promoting numeric periods to periodsof motives, that is, P m φ ⊂ P m log ⊂ P m holds. We briefly introduce the fundamentals of the theory of Tannakian categories , following the more detailed andcomprehensive exposition by Deligne et al [26]. The concept of a Tannakian category was first introduced by SaavedraRivano [49] to encode the properties of the category Rep K ( G ) of the finite-dimensional K -linear representations ofan 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 ), thegroup Hom C ( X, Y ) is a K -vector space and the composition maps are K -bilinear. The graph Hodge structure is also explicitly known in the general case of renormalised amplitudes of single-scale graphs due to thework of Brown and Kreimer [16], who pave the way for the rigorous investigation of divergent Feynman graphs and their renormalisedamplitudes from an algebro-geometric perspective. efinition 15. Let C be a K -linear category endowed with a K -bilinear functor ⊗ : C × C → C .(a) An associativity constraint for ( C , ⊗ ) is a natural transformation φ = φ · , · , · : · ⊗ ( · ⊗ · ) −→ ( · ⊗ · ) ⊗ · (171)such that the following two conditions hold:(a.1) For all X, Y, Z ∈ Ob( C ), the map φ X,Y,Z is an isomorphism.(a.2) For all X, Y, Z, T ∈ Ob( C ), the following diagram commutes X ⊗ ( Y ⊗ ( Z ⊗ T )) X ⊗ (( Y ⊗ Z ) ⊗ T ) ( X ⊗ Y ) ⊗ ( Z ⊗ T )( X ⊗ ( Y ⊗ Z )) ⊗ T (( X ⊗ Y ) ⊗ Z ) ⊗ T Id ⊗ φ Y,Z,T φ X,Y,Z ⊗ T φ X,Y ⊗ Z,T φ X ⊗ Y,Z,T φ X,Y,Z ⊗ Id (172)(b) A commutativity constraint for ( C , ⊗ ) is a natural transformation ψ = ψ · , ∗ : · ⊗ ∗ −→ ∗ ⊗ · (173)such that the following two conditions hold:(b.1) For all X, Y ∈ Ob( C ), the map ψ X,Y is an isomorphism.(b.2) For all X, Y ∈ Ob( C ), the following composition is the identity ψ Y,X ◦ ψ X,Y : X ⊗ Y −→ X ⊗ Y (174)(c) An associativity constraint and a commutativity constraint are compatible if, for all X, Y, Z ∈ Ob( C ), thefollowing diagram commutes X ⊗ ( Y ⊗ Z ) ( X ⊗ Y ) ⊗ ZX ⊗ ( Z ⊗ Y ) Z ⊗ ( X ⊗ Y )( X ⊗ Z ) ⊗ Y ( Z ⊗ X ) ⊗ Y φ X,Y,Z Id ⊗ ψ Y,Z ψ X ⊗ Y,Z φ X,Z,Y φ X,Z,Y ψ X,Z ⊗ Id (175)(d) A pair ( U, u ) consisting of an object U ∈ Ob( C ) and an isomorphism u : U → U ⊗ U is an identity object if thefunctor X (cid:55)→ 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 -bilinearfunctor ⊗ : C × C → C , and compatible associativity and commutativity constraints φ , ψ such that C contains anidentity object. Definition 17. An object L ∈ Ob( C ) is invertible if the functor X (cid:55)→ L ⊗ X is an equivalence of categories.Equivalently, L is invertible if and only if there exists an object L (cid:48) ∈ Ob( C ) such that L ⊗ L (cid:48) (cid:39) . Then, L (cid:48) is alsoinvertible. Definition 18. Let ( C , ⊗ ) be a K -linear tensor category, where we omit the constraints φ , ψ for simplicity, andlet X, Y ∈ Ob( C ). Assume that there exists an object Z ∈ Ob( C ) such that, for all T ∈ Ob( C ), the functors T (cid:55)→ Hom( T, Z ) and T (cid:55)→ Hom( T ⊗ X, Y ) admit a functorial isomorphismHom( T, Z ) ∼ −−→ Hom( T ⊗ X, Y ) (176)In this case, the functor T (cid:55)→ 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, ). If X ∨ and ( X ∨ ) ∨ exist, thenthere is a natural morphism X (cid:55)→ ( X ∨ ) ∨ , and the object X is reflexive if such a morphism is an isomorphism.31 efinition 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.(2) For all X , X , Y , Y ∈ Ob( C ), the natural morphismHom( X , Y ) ⊗ Hom( X , Y ) −→ Hom( X ⊗ X , Y ⊗ Y ) (177)is an isomorphism.(3) All objects are reflexive. Definition 21. A Tannakian category over the field K is a rigid abelian K -linear tensor category T such thatEnd( ) = K , and there exists an exact faithful K -linear tensor functor ω : T → Vec K , where Vec K is the category offinite-dimensional vector spaces over K . Any such functor is called a fibre functor . Example . The category Vec K of finite-dimensional K -vector spaces, together with the identity functor, is a Tan-nakian category over K . Example . 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 . The category Rep K ( G ) of finite-dimensional K -linear representations of an abstract group G , togetherwith 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 byAut ⊗ ( ω )( R ) the collection of families ( λ X ) X ∈ Ob( T ) of R -linear automorphisms λ X : ω ( X ) ⊗ K R −→ ω ( X ) ⊗ K R (178)which are compatible with the tensor structure and functorial. Here, compatibility with the tensor structure andfunctoriality mean that:(1) For all X , X ∈ Ob( T ), the following diagram commutes ω ( X ⊗ X ) ⊗ R ω ( X ⊗ X ) ⊗ Rω ( X ) ⊗ ω ( X ) ⊗ R ω ( X ) ⊗ ω ( X ) ⊗ R ( ω ( X ) ⊗ R ) ⊗ R ( ω ( X ) ⊗ R ) ( ω ( X ) ⊗ R ) ⊗ R ( ω ( X ) ⊗ R ) λ X ⊗ X λ X ⊗ R λ X (179)(2) The following diagram commutes ω ( ) ⊗ R ω ( ) ⊗ RR R λ Id (180)(3) For all X, Y ∈ Ob( T ) and for every morphism α ∈ Hom( X, Y ), the following diagram commutes ω ( X ) ⊗ R ω ( X ) ⊗ Rω ( Y ) ⊗ R ω ( Y ) ⊗ R λ X ω ( α ) ⊗ Id ω ( α ) ⊗ Id λ Y (181)Denote Aut ⊗ ( ω ) = Aut ⊗ ( ω )( K ) the group of K -linear automorphisms of the fibre functor ω . Deligne et al [26] provedthat 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 (cid:55)→ 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 ). In the given diagrams, all unlabelled tensor products are over K and all unlabelled arrows are the natural isomorphisms. X ∈ Ob( T ), the group Aut ⊗ ( ω ) acts naturally on ω ( X ) and the functor T Rep K ( G ω ) X ω ( X ) G ω (182)sending X to the vector space ω ( X ) with this action of Aut ⊗ ( ω ), is an equivalence of categories.Given a second fibre functor ω (cid:48) , we analogously define Isom ⊗ ( ω, ω (cid:48) )( R ) to be the collection of families ( τ X ) X ∈ Ob( T ) of R -linear isomorphisms τ X : ω ( X ) ⊗ K R −→ ω (cid:48) ( X ) ⊗ K R (183)which are compatible with the tensor structure and functorial. Again, we denote Isom ⊗ ( ω, ω (cid:48) ) = Isom ⊗ ( ω, ω (cid:48) )( K ).Deligne et al [26] proved the following result. Theorem 11. Let T be a Tannakian category over K with two fibre functors ω and ω (cid:48) . The functor R (cid:55)→ Isom ⊗ ( ω, ω (cid:48) )( R ) is representable by an affine scheme over K , which is denoted as Isom ⊗ ( ω, ω (cid:48) ), and is a righttorsor under Aut ⊗ ( ω ) and a left torsor under Aut ⊗ ( ω (cid:48) ). Grothendieck’s idea of a universal cohomology theory taking values in a Q -category of motives M is intimatelyconnected 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 fundamentallydescribed 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 smoothprojective varieties, and since cohomologies of different degrees get mixed in this expression, it is fundamentallydescribed by a mixed Hodge structure. Thus, enhancing the na¨ıve description in Section 3.2, pure Hodge structuresrepresent 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. Specificallylooking at the application of Hodge theory to the theory of motivic periods, we identify the category of motives withthe category of mixed Hodge structures over Q . For a thorough introduction to the theory of motives we refer toVoevodsky [60], Andr´e [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 functorsarising 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 arefibre functors, thus justifying the use of the Tannakian machinery in the context of motives. The pro-algebraic groupAut ⊗ ( 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 M (cid:39) Rep Q ( G dR ) (184) 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 categoryof motives M .In Tannakian formalism, the space of motivic periods P m is expressed as P m = Q (cid:104) [ M, ω, σ ] m | M ∈ Ob( M ) , ω ∈ ω dR ( M ) , σ ∈ ω B ( M ) ∨ (cid:105) (185)with implicit factorisation modulo bilinearity and functoriality. Thus, an alternative but equivalent description ofmotivic periods is obtained. Indeed, P m is isomorphic to the space of regular functions on the affine Q -schemeIsom ⊗ ( ω dR , ω B ), that is P m (cid:39) O (Isom ⊗ ( ω dR , ω B )) (186)The isomorphism is made explicit by P m −→ O (Isom ⊗ ( ω dR , ω B ))[ M, ω, σ ] m (cid:55)−→ (cid:0) ( λ X ) X ∈ Ob( M ) (cid:55)→ σ ◦ λ M (cid:1) (187)where σ ◦ λ M gives ω dR ( M ) ω B ( M ) Q ω λ M ( ω ) σ ( λ M ( ω )) λ M σ (188)33hen, following Theorem 10, the motivic Galois group G dR has a natural action on Isom ⊗ ( ω dR , ω B ) denoted by ∇ : G dR ⊗ Isom ⊗ ( ω dR , ω B ) −→ Isom ⊗ ( ω dR , ω B ) (189)which induces a dual coaction on the corresponding spaces of regular functions∆ : P m O ( G dR ) ⊗ P m [ M, ω, σ ] m (cid:80) ni =1 [ M, ω, e ∨ i ] dR ⊗ [ M, e i , σ ] m (190)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 ) thedual 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 isnot, thus making the coaction intrinsically asymmetric. On the other hand, motivic periods have a well-defined mapto numbers, while de Rham periods do not, although we can associate symbols to them. Thus, the Galois coactionturns the finite-dimensional Q -vector space P m into a comodule over the Hopf algebra P dR . A detailed discussion ispresented by Brown [13], [12]. Example . Consider the motivic logarithm log( z ) m for z ∈ Q \{ } . Following Section 3.4.2, we havelog( z ) m = (cid:20) H ( G m , { , z } ) , (cid:20) dxx (cid:21) , [ γ ] (cid:21) m (191)where we write (cid:2) dxx (cid:3) = (cid:2)(cid:0) dxx , , (cid:1)(cid:3) for simplicity. Thus, the corresponding motive is M = H ( G m , { , z } ), while thecomplete period matrix of M is (cid:18) πi log( z )0 1 (cid:19) (192)Direct application of the prescription in (190) gives the explicit decomposition∆ (cid:20) M, (cid:20) dxx (cid:21) , [ γ ] (cid:21) m = (cid:34) M, (cid:20) dxx (cid:21) , (cid:20) dxz − (cid:21) ∨ (cid:35) dR ⊗ (cid:20) M, (cid:20) dxz − (cid:21) , [ γ ] (cid:21) m + (cid:34) M, (cid:20) dxx (cid:21) , (cid:20) dxx (cid:21) ∨ (cid:35) dR ⊗ (cid:20) M, (cid:20) dxx (cid:21) , [ γ ] (cid:21) m (193)which is equivalent to ∆ log( z ) m = log( z ) dR ⊗ m + (2 πi ) dR ⊗ log( z ) m (194)Here, 1 m and log( z ) m are called Galois conjugates of log( z ) m . Example . 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 ∆ ζ (2) m = 1 dR ⊗ ζ (2) m ∆ ζ (2 n + 1) m = ζ (2 n + 1) dR ⊗ m + 1 dR ⊗ ζ (2 n + 1) m (195)Thus, the Galois coaction is trivial on ζ (2) m , while ζ (2 n + 1) m has the non-trivial Galois conjugate 1 m . Moreover∆( ζ (2) m ζ (2 n + 1) m ) = ζ (2 n + 1) dR ⊗ ζ (2) m + 1 dR ⊗ ζ (2) m ζ (2 n + 1) m (196) We look at the example of scalar massless φ quantum field theory and consider the Galois coaction restricted to P m φ . This is a priori valued in the whole space P dR ⊗ P m . However, after computing every known φ -amplitude withloop order at most 7 and explicitly verifying that in each case the Galois coaction preserves the space P m φ , Panzerand Schnetz [47] proposed the following conjecture, known as the coaction conjecture . Conjecture . Galois conjugates of φ -periods are still φ -periods, i.e.∆( P m φ ) ⊆ P dR ⊗ P m φ (197) Panzer and Schnetz [47] explicitly computed the first examples of φ -amplitudes which are not MZVs. Such numbers are polyloga-rithms at 2nd and 6th roots of unity. The coaction conjecture is verified for them as well. φ -periods thatwe do not yet properly understand. Indeed, the unexpected observations by Panzer and Schnetz, and the resultingconjecture, have greatly stimulated research, motivating the search for a mathematical mechanism able to distinguish φ -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 underconsideration, the coaction conjecture is proven by Brown [12]. Define the finite-dimensional Q -vector space P m˜ φ associated to a φ -graph G to be the space of motivic versions of all integrals of the form I G = ˆ σ P ( { x e } ) ΩΨ kG (198)where k ≥ P is any polynomial in Q [ { x e } ] such that I G converges. Theorem 12. P m˜ φ is stable under the Galois coaction, i.e. ∆( P m˜ φ ) ⊆ P dR ⊗ P m˜ φ . We apply the notions of Hodge and weight filtrations, introduced in Sections 2.3 and 2.4, to the theory of motivicperiods. For M ∈ Ob( M ), the Q -vector space ω dR ( M ) is equipped with a decreasing Hodge filtration F and anincreasing 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 quotientswith respect to the weight filtration do possess a pure Hodge structure of definite weight, as described in Definition8. 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 W dR • P m = Q (cid:104) [ M, ω, σ ] m | ω ∈ W dR • ω dR ( M ) (cid:105) (199)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 Wn P m = W n P m /W n − P m . Remark. We observe that the weight of motivic periods can alternatively, but equivalently, be defined from the Bettiside via the weight filtration induced on P m by W B . Example . Consider M = H ( G m , { , z } ) again. Its weight filtration in de Rham realisation is W − = 0 ⊆ W = W = Q (0) ⊆ W = H ( G m , { , z } ) (200)Observing that 0 , ∈ W and 2 πi, log( z ) ∈ W , the weight of each entry of the period matrix of M is determined.Indeed, 0 , πi, log( z ) have weight 2. Example . The weight filtration can be used to systematically study P m φ weight by weight. For example, directcomputation in low weight shows that W P m φ = W P m φ = W P m φ = Q (0) (201)The following conjecture, known as small graph principle , is due to Brown [12]. Conjecture . Let G be a primitive log-divergent Feynman graph in scalar massless φ theory. Denote by [ M G , ω G , σ ] m the explicit form of its motivic Feynman integral I m G . The elements in the right-hand side of the coaction formulafor ∆[ M G , ω G , σ ] m can be expressed in the form (cid:89) i [ M γ i , ω γ i , σ ] m (202)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 aprimitive Feynman graph are associated to its sub-quotient graphs with at most k + 1 edges. Thus, when interestedin periods of weight at most k , it suggests to look at graphs with at most k + 1 edges. It follows that the topologyof 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). 35 xample . Because log( z ) m has weight 2, the small graph principle suggests that any log( z ) m appearing in theright-hand side of the coaction formula for a given φ -period comes from graphs with at most three edges. Theorem13 implies that all two-edge graphs are trivial, i.e. the associated motive is the Hodge-Tate motive Q (0), which doesnot have log( z ) m in its period matrix. Writing down all possible graphs with three edges, we get the graphs shownin Fig. 15 along with the associated graph polynomials in the Schwinger parameters. (a) x + x + x (b) x x + x x + x x (c) x ( x + x ) (d) x x x Figure 15: Feynman graphs with 3 edges and their first graph polynomials.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 ( − z ) cannot be obtained as an integral with a denominator equal to either of theirgraph polynomial. It follows that log( z ) m cannot be a Galois conjugate of any φ -period. By the coaction conjectureand Equation (194), we conclude that log( z ) m / ∈ P m φ . Example . Direct computation by Panzer and Schnetz [47] shows that all φ -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 byweight 1 ζ (2) ζ (3) ζ (2) ζ (5) ζ (3) ζ (7) ζ (3 , · · · ζ (2) ζ (3) ζ (2) ζ (2) ζ (5) ζ (2) ζ (3) ζ (2) ζ (3) ... (203)As a consequence of the coaction conjecture and Equation (196), ζ (2) m / ∈ P m φ implies that all elements which arelinear in ζ (2) cannot be φ -periods. Analogously, ( ζ (2) ) m / ∈ P m φ implies that all MZVs quadratic in ζ (2) are not φ -periods. The set of MZVs that can appear as φ -periods is then reduced to1 ζ (3) ζ (5) ζ (3) ζ (7) ζ (3 , · · · ζ (2) ... (204)From similar considerations, other highly non-trivial constraints at all loop orders in perturbation theory can bederived using the Galois coaction and weight filtrations. Indeed, whenever it is shown that a given period is not a φ -period, we automatically deduce that all periods that have the given one among their Galois conjugates cannotappear in P φ either. Remark. Structures even more fundamental that those captured by the coaction conjecture and the small graphprinciple underly the space of motivic periods of Feynman graphs. Although not being sufficiently explored in theliterature, the notion of operad in the category of motives imposes strong constraints on the admissible periods andit should be the object of further investigation. The operad structure underlying the space of motivic Feynmanintegrals 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 Feyn-man integrals, the theory of motivic periods has revealed unexpected features, placing restrictions on the set ofnumbers which can occur as amplitudes and paving the way for a more comprehensive understanding of their generalstructure. Indeed, the coaction conjecture gives new constraints at each loop order, which in turn propagate to allhigher loop orders because of the recursive structure inherent in perturbative quantum field theories. At the sametime, the small graph principle makes finite computations at low-loop into all-order results.36ssume to deal with a Feynman integral of the form ´ σ ω in P . The general prescription for its investigation viathe theory of motivic periods can be summarised as follows.(1) Associate the integral representation ´ σ ω to a motivic representation [ H, ω, σ ] m , deriving explicitly the corre-sponding algebraic varieties and cohomology classes.(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 tothe 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 motivicversion requests the validity of the period conjecture. Moreover, even disregarding the conjectural status of currentresults, the present state of understanding of motivic amplitudes is still far from building a theory. Although thegiven general prescription for the investigation of motivic Feynman integrals has been particularly fruitful for masslessscalar φ 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, aspace A of de Rham periods and a corresponding coaction ∆ : H −→ P dR ⊗ P m . A potential coaction principlewould then state that ∆( H ) ⊆ A ⊗ H . Being A a Hopf algebra, we could canonically introduce the group C ofhomomorphisms from A to any commutative ring. It would follow that the coaction principle can be recast in termsof the group action C × H −→ H , that is, the space of amplitudes is stable under the action of the group C , oftenreferred to as cosmic Galois group . This speculative construction, that broadly reproduces the general prescriptionsummarised above, motivates a programme of research leading towards a systematic study of scattering amplitudesvia the representation theory of groups.Although practically harder than the φ -case, like-minded attempts are already on the way to gather informationabout 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 evidencesuggesting that scalar Feynman integrals of small graphs with non-trivial masses and momenta satisfy similarproperties to φ -periods. A diagrammatic coaction for specific families of integrals appearing in the evaluationof scalar Feynman diagrams, such as multiple polylogarithms and generalised hypergeometric functions, isproposed and a connection between this diagrammatic coaction and graphical operations on Feynman diagramsis conjectured. At one-loop order, a fully explicit and very compact representation of the coaction in terms ofone-loop integrals and their cuts is found. Moreover, Brown and Dupont [15] investigate a rigorous theory ofmotives associated to certain hypergeometric integrals.(2) A subsequent generalisation arises transitioning from scalar quantum field theories to gauge theories. Theproblem of dealing with much more involved parametric integrands which are not explicitly expressed in termsof the Symanzik polynomials of the associated Feynman graphs has only recently been tackled. A combinatoricand graph-theoretic approach to Schwinger parametric Feynman integrals in quantum electrodynamics byGolz [33] has revealed that the parametric integrands can be explicitly written in terms of new types ofgraph polynomials related to specific subgraphs. The tensor structure of quantum electrodynamics is givena diagrammatic interpretation. The resulting significant simplification of the integrands paves the way for asystematic motivic description of gauge theories.(3) In the same research direction, a high-precision computation of the 4-loop contribution to the electron anoma-lous magnetic moment g − Q -vector spaces of Galois conjugatesof the g − N = 4 super Yang-Mills theory. In eachof these theories, after suitably defining the space of integrals or amplitudes under consideration, a version of the In various modern approaches to N = 4 SYM, including the bootstrap method, on-shell techniques, and the amplituhedron, theamplitude 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. N = 4 super Yang-Mills theory. Acknowledgements This work is partially supported by the Italian Department of Education, Research and University (Torno Subito13474/19.09.2018 POR-Lazio-FSE/2014-2020) and the Swiss National Centre of Competence in Research SwissMAP(Excellence Fellowship Master Class in Mathematical Physics 2019-2020). I thank Francis Brown and Lionel Masonfor useful discussions. References [1] S. Abreu, R. Britto, C. Duhr, and E. Gardi. Diagrammatic Hopf algebra of cut Feynman integrals: the one-loop case. J. HighEnergy Phys. , (12):090, front matter+72, 2017.[2] S. Abreu, R. Britto, C. Duhr, E. Gardi, and J. Matthew. Coaction for Feynman integrals and diagrams. PoS , LL2018:047, 2018.[3] Samuel Abreu, Ruth Britto, Claude Duhr, Einan Gardi, and James Matthew. From positive geometries to a coaction on hypergeo-metric functions. JHEP , 02:122, 2020.[4] Y. Andr´e. Une introduction aux motifs (motifs purs, motifs mixtes, p´eriodes) , volume 17 of Panoramas et Synth`eses [Panoramasand Syntheses] . 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