Mixed Hodge Structures and Renormalization in Physics
aa r X i v : . [ h e p - t h ] S e p MIXED HODGE STRUCTURES AND RENORMALIZATION IN PHYSICS
SPENCER BLOCH AND DIRK KREIMER Introduction monodromy as the chain varies.Even methods like minimal subtraction in the context of dimensional or analytic regularization implicitlymodify the integrand through the definition of a measure R d D k via analytic continuation. Still, as a regulatordimensional regularization is close to our approach in so far as it leaves the rational integrand assigned to agraph unchanged. Minimal subtraction as a renormalization scheme differs though from the renormalizationschemes which we consider -momentum subtractions essentially- by a finite renormalization. Many of thenice algebro-geometric structures developed below are not transparent in that scheme.The advantages of the limiting Hodge method are firstly that it is linked to a very central and powerfulprogram in mathematics: the study of Hodge structures and their variations. As a consequence, one gainsa number of tools, like weight, Hodge, and monodromy filtrations to study and classify the Feynman ampli-tudes. Secondly, the method depends on the integration chain, and hence on the graph, but it is in somesense independent of the integrand. For this reason it should adapt naturally e.g. to gauge theories wherethe numerator of the integrand is complicated.An important point is to analyse the nature of the poles. Limiting mixed Hodge structures demand thatthe divergent subintegrals have at worst log poles. This does not imply that we can not apply our approach toperturbative amplitudes which have worse than logarithmic degree of divergence. It only means that we haveto correctly isolate the polynomials in masses and external momenta which accompany those divergences suchthat the corresponding integrands have singularities provided by log-poles. This is essentially automatic fromthe notion of a residue available by our very methods. As a very pleasant byproduct, we learn that physicalrenormalization schemes -on-shell subtractions, momentum subtractions, Weinberg’s scheme,- belong to aclass of schemes for which this is indeed automatic.Moreover, for technical reasons, it is convenient to work with projective rather than affine integrals. Oneof the central physics results in this paper is that the renormalization problem can be reduced to the studyof logarithmically divergent, projective integrals. This is again familiar from analytic regulators. The factthat it can be achieved here by leaving the integrand completely intact will hopefully some fine day allow tounderstand the nature of the periods assigned to renormalized values in quantum field theory.A remark for Mathematicians: our focus in this paper has been renormalization, which is a problemarising in physics. We suspect, however, that similar methods will apply more generally for example toperiod integrals whenever the domain of integration is contained in R + n and the integrand is a rationalfunction with polar locus defined by a polynomial with non-negative real coefficients. The toric methodsand the monodromy computations should go through in that generality. Acknowledgments.
Both authors thank Francis Brown, H´el`ene Esnault and Karen Yeats for helpful dis-cussions. This work was partially supported by NSF grants DMS-0603781 and DMS-0653004. S.B. thanksthe IHES for hospitality January-March 2006 and January-March 2008. D.K. thanks Chicago University forhospitality in February 2007.1.2.
Physics Introduction.
This paper studies the renormalization problem in the context of parametricrepresentations, with an emphasis on algebro-geometric properties. We will not study the nature of the eriods one obtains from renormalizable quantum field theories in an even dimension of space-time. Instead,we provide the combinatorics of renormalization such that a future motivic analysis of renormalized ampli-tudes is feasible along the lines of [2]. Our result will in particular put renormalization in the frameworkof a limiting mixed Hodge structure, which hopefully provides a good starting point for an analysis of theperiods in renormalized amplitudes. That these amplitudes are provided by numbers which are periods (inthe sense of [11]) is an immediate consequence of the properties of parametric representations, and will alsoemerge naturally below (see Thm.(7.3)).The main result of this paper is a careful study of the singularities of the first Kirchhoff–Symanzikpolynomial, which carries all the short-distance singularities of the theory. The study of this polynomial canproceed via an analysis with the help of projective integrals. Along the way, we will also give useful formulasfor parametric representations involving affine integrals, and clarify the role of the second Kirchhoff–Symanzikpolynomial for affine and projective integrals.Our methods are general, but in concrete examples we restrict ourselves to φ theory. Parametric rep-resentations are used which result from free-field propagators for propagation in flat space-time. In suchcircumstances, the advantages of analytic regularizations are also available in our study of parametric rep-resentations as we will see. In particular, our use of projective integrals below combines such advantageswith the possibility to discuss renormalization on the level of the pairing between integration chains and deRham classes.In examples, special emphasis is given to the study of particular renormalization schemes, the momentumscheme (MOM-scheme, Weinberg’s scheme, on-shell subtractions).Also, we often consider Green functions as functions of a single kinematical scale q >
0. Green functionsare defined throughout as the scalar coefficient functions (structure functions) for the radiative correctionsto tree-level amplitudes r . They are to be regarded as scalar quantities of the form 1 + O ( ~ ). Renormalizedamplitudes are then, in finite order in perturbation theory, polynomial corrections in L = ln q /µ ( µ >
0) without constant term, providing the quantum corrections to the tree-level amplitudes appearing asmonomials in a renormalizable Lagrangian [15]:(1.1) φ R (Γ) = aug(Γ) X j =1 p j (Γ) L j . Correspondingly, Green functions become triangular series in two variables(1.2) G r ( α, L ) = 1 + ∞ X j =1 γ rj ( α ) L j = 1 + ∞ X j =1 c rj ( L ) α j . The series γ rj ( α ) are related by the renormalization group which leaves only the γ r ( α ) undetermined, whilethe polynomials c rj ( L ) are bounded in degree by j . The series γ r fulfill ordinary differential equations drivenby the primitive graphs of the theory [16].The limiting Hodge structure A (Γ) which we consider for each Feynman graph Γ provides contribution ofa graph Γ to the coefficients of γ r in the limit. This limit is a period matrix (a column vector here) whichhas, from top to bottom, the periods provided by a renormalized graph Γ as entries. The first entry is thecontribution to γ r of a graph with res(Γ) = r and the k -th is a rational multiple of the contribution to γ rk . Insection 9.1 we determine the rational weights which connect these periods to the coefficients p j (Γ) attributedto the renormalization of a graph Γ.We include a discussion of the structure of renormalization which comes from an analysis of the secondKirchhoff–Symanzik polynomial. While this polynomial does not provide short-distance singularities in itsown right, it leads to integrals of the form(1.3) Z ω ln( f )for a renormalized Feynman amplitude, with ω a de Rham class determined by the first Kirchhoff–Symanzikpolynomial, and f -congruent to one along any remaining exceptional divisor- determined by the second.We do not actually do the monodromy calculation for integrals (1.3) involving a logarithm, but it will besimilar to the calculation for (1.5) which we do. A full discussion of the Hodge structure of a Green functionseems feasible but will be postponed to future work. X Figure 1.
Picture of X and L Math Introduction.
Let P n − be the projective space of lines in C n which we view as an algebraicvariety with homogeneous coordinates A , . . . , A n . Let ψ ( A , . . . , A n ) be a homogeneous polynomial of somedegree d , and let X ⊂ P n − be the hypersurface defined by ψ = 0. We assume the coefficients of ψ are allreal and ≥
0. Let σ = { [ a , . . . , a n ] | a i ≥ , ∀ i } be the topological ( n − P n − , where[ . . . ] refers to homogeneous coordinates. We will also use the notation σ = P n − ( R ≥ ). Our assumptionabout coefficients implies(1.4) σ ∩ X = [ L ⊂ X L ( R ≥ ) , where L runs through all coordinate coordinate linear spaces L : A i = · · · = A i p = 0 contained in X (see(see Fig.1)). The genesis of the renormalization problem in physics is the need to assign values to integrals(1.5) Z σ ω where ω is an algebraic ( n − P n − with poles along X . The problem is an important one for physicalapplications, and there is an extensive literature (see, for example, [10, 22, 21]) focusing on practical formulaeto reinterpret (1.5) in some consistent way as a polynomial in log t . (Here t parametrizes a deformation ofthe integration chain. As a first approximation, one can think of R ∞ t ω when ω has a logarithmic pole at t = 0.)A similar problem arises in pure mathematics in the study of degenerating varieties, e.g. a family of ellipticcurves degenerating to a rational curve with a node. In the classical setup, one is given a family f : X → D ,where D is a disk with parameter t . The map f is proper (so the fibres X t are compact). X is assumed tobe non-singular, as are the fibres X t , t = 0. X may be singular, though one commonly invokes resolution ofsingularities to assume X ⊂ X is a normal crossing divisor. Choose a basis σ ,t , . . . , σ r,t for the homologyof the fibre H p ( X t , Q ) in some fixed degree p . By standard results in differential topology, the fibre space islocally topologically trivial over D ∗ = D − { } , and we may choose the classes σ i,t to be locally constant.If we fix a smooth fibre t = 0, the monodromy transformation m : H p ( X t ) → H p ( X t ) is obtained bywinding around t = 0. An important theorem ([7], III,2) says this transformation is quasi-unipotent, i.e.after possibly introducing a root t ′ = t /n (which has the effect of replacing m by m n ), m − id is nilpotent.The matrix(1.6) N := log m = − h ( id − m ) + ( id − m ) / . . . i s thus also nilpotent. This is the mathematical equivalent of locality in physics. It insures that ourrenormalization of (1.5) will be a polynomial in log t rather than an infinite series. We take a cohomology class[ ω t ] ∈ H p ( X t , C ) which varies algebraically. For example, in a family of elliptic curves y = x ( x − x − t ),the holomorphic 1-form ω t = dx/y is such a class. Note ω t is single-valued over all of D ∗ . It is not locallyconstant. The expression(1.7) exp (cid:16) − ( N log t ) / πi (cid:17) R σ ,t ω t ... R σ r,t ω t . is then single-valued and analytic on D ∗ . Suppose ω t chosen such that the entries of the column vector in(1.7) grow at worst like powers of | log | t || as | t | →
0. A standard result in complex analysis then impliesthat (1.7) is analytic at t = 0. We can write this(1.8) R σ ,t ω t ... R σ r,t ω t ∼ exp (cid:16) ( N log t ) / πi (cid:17) a ... a r . Here the a j are constants which are periods of a limiting Hodge structure . The exponential on the rightexpands as a matrix whose entries are polynomials in log t , and the equivalence relation ∼ means that thedifference between the two sides is a column vector of (multi-valued) analytic functions vanishing at t = 0.We would like to apply this program to the integral (1.5). Let ∆ : Q n A j = 0 be the coordinate divisorin P n − . Note that the chain σ has boundary in ∆, so as a first attempt to interpret (1.5) as a period , wemight consider the pairing(1.9) H n − ( P n − − X, ∆ − X ∩ ∆) × H n − ( P n − − X, ∆ − X ∩ ∆) → C The form ω is an algebraic ( n − σ meets X (1.4), so we do not get a class in homology. Instead we consider a family of coordinatedivisors ∆ t : Q n A j,t = 0 with ∆ = ∆. (For details, see section 6.) For t = ε > σ ε which is what the physicists would call a cutoff. We have ∂σ ε ⊂ ∆ ε and σ ε ∩ X = ∅ , so R σ ε ω is defined.One knows on abstract grounds that the monodromy of H n − ( P n − − X, ∆ t − X ∩ ∆ t )is quasi-unipotent as above ([7], III, § σ ε in the specific case of Feynman amplitudes in physics. More precisely, X will be a graphhypersurface X Γ associated to a graph Γ (section 5). We will write down chains τ εγ , one for each flag of core ( one particle irreducible in physics) subgraphs γ = { Γ ( · · · Γ p ( γ ) ( Γ } , representing linearly independenthomology classes in H n − ( P n − − X, ∆ ε − X ∩ ∆ ε ). (The combinatorics here is similar to that found in [1],[18].) We will show the monodromy in our case is given by(1.10) m ( σ ε ) = σ ε + X γ ( − p ( γ ) τ εγ . We will then exhibit a nilpotent matrix N such that(1.11) m ( σ ε )... m ( τ εγ )... = exp( N ) σ ε ... τ εγ ... . With this in hand, renormalization is automatic for any physical theory for which Γ and its subgraphs areat worst logarithmically divergent after taking out suitable polynomials in masses and momenta. Namely, uch a physical theory gives a differential form ω Γ as in (1.5) and we may repeat the above argument:(1.12) exp( − ( N log t ) / πi ) R σ t ω ... R τ tγ ω ... is single-valued on the punctured disk. The hypothesis of log divergence at worst for subgraphs of Γ willimply that the integrals will grow at worst like a power of log as | t | → Z σ t ω Γ = r X k =0 b k (log t ) k + O ( t ) , where O ( t ) denotes a (multi-valued) analytic function vanishing at t = 0. The renormalization schemesconsidered here can be characterized by the condition b = 0.Of course, the requirement that a physical theory have at worst log divergences is a very strong constraint.The difficult computations in section 7 show how general divergences encountered in physics can be reducedto log divergences. Remarks 1.1.
The renormalization scheme outlined above, and worked out in detail in the following sec-tions, has a number of properties, some of which may seem strange to the physicist.(i) It does not work in renormalization schemes which demand counter-terms which are not defined by sub-tractions at fixed values of masses and momenta of the theory. So conditions on the regulator for example, asin minimal subtraction where one defines the counterterm by projection onto a pole part, are not considered.In such schemes, and for graphs which are worse than log divergent, a topological procedure of the sort givenhere can not work. It is necessary instead to modify the integrand ω Γ in a non-canonical way.(ii) On the other hand, our approach is very canonical. It depends on the choice of a parameter t , as anyrenormalization scheme must. Somewhat more subtle is the dependence on the monodromy associated to thechoice of a family ∆ t of coordinate divisors deforming the given ∆ = ∆ . We have taken the most evidentsuch monodromy, moving all the vertices of the simplex. Note that this choice is stable in the sense that asmall deformation leaves the monodromy unchanged.(iii) It would seem that our answer is much more complicated than need be, because Γ will in generalcontain far more core subgraphs than divergent subgraphs. For example, in ϕ -theory, the “dunce’s cap”(see Fig.2) has only one divergent subgraph, given in the picture by edges 1 ,
2. It has 3 core subgraphs(3 , , , (3 , , , (1 , τ εγ aretubes, and the integral R τ εγ ω Γ is basically a residue which will vanish unless γ ⊂ Γ is a divergent subgraph.In (1.12), the column vector of integrals will consist mostly of 0’s and the final regularization (1.13) willinvolve only divergent subgraphs.(iv) An important property of the theory is the presence of a limiting mixed Hodge structure . The constantson the right hand side of (1.8) are periods of a mixed Hodge structure called the limiting MHS for thedegeneration. One may hope that the tendency for Feynman amplitudes to be multi-zeta numbers [4] willsome day be understood in terms of this Hodge structure. From the point of view of this paper, the vectorspace W ⊂ H n − ( P n − − X Γ , ∆ t − X Γ ∩ ∆ t ) spanned by σ t and the τ tγ is invariant under the monodromy.One may ask whether the image of W in the limiting MHS spans a sub-Hodge structure. If so, we wouldexpect that this HS would be linked to the multi-zeta numbers. Note that W is highly non-trivial even whenΓ has no subdivergences. This W is an essentially new invariant which comes out of the monodromy. Seesection (9.2) for a final discussion of our viewpoint.(v) There are a number of renormalization schemes in physics, some of which are not compatible with ourapproach. One general test is that our scheme depends only on the graph polynomials of Γ. For example,suppose Γ = Γ ∪ Γ where the Γ i meet at a single vertex. Then the renormalization polynomial in log t ourtheory yields for Γ will be the product of the renormalizations for the Γ i .Most of the mathematical work involved concerns the calculation of monodromy for a particular topologicalchain. It is perhaps worth taking a minute to discuss a toy model. Suppose one wants to calculate R ∞ ω , Figure 2.
Dunce’s cap. Here and in following figures, external half-edges are often notdrawn and are determined by the requirement that all vertices are four-valent.where ω = dz ( z − i ) z . The integral diverges, so instead we consider R ∞ t ω as a function of t = εe iθ for 0 ≤ θ ≤ π .If we take the path [ t, ∞ ] to be a great circle, then as t winds around 0, the path will get tangled in thesingularity of ω at z = i . Assuming we do not understand the singularities of our integral far from 0, thiscould be a problem. Instead we chose our path to follow the small circle from εe iθ to ε and then the positivereal axis from ε to ∞ . The variation of monodromy is the difference in the paths for θ = 0 and θ = 2 π .In this case, it is the circle {| t | = ε } . If we assume something (at worst superficial log divergence for thegiven graph and all subgraphs in the given physical theory) about the behavior of ω near the pole at 0, thenthe behavior of our integral for | t | << ω . A glance at fig.(10) suggests that our toy model is too simple. We have to work with two scales, ε << η <<
1. This is because in the more complicated situation, we have to deal with cylinders of smallradius η , but then we have further to slightly deform the boundaries of the cylinder (cf. fig.(12)).1.4. Leitfaden.
Section 2 is devoted to Hopf algebras of graphs and of trees. These have played a centralrole in the combinatorics of renormalization. In particular, the insight afforded by passing from graphsto trees is important. Since the combinatorics of core subgraphs is even more complicated than that ofdivergent subgraphs, it seemed worth going carefully through the construction. Section 3 studies the toricvariety we obtain from a graph Γ by blowing up certain coordinate linear spaces in the projective space withhomogeneous coordinates labeled by the edges of Γ. The orbits of the torus action are related to flags of coresubgraphs of the given graph. In section 4, we use the R -structure on our toric variety to construct certaintopological chains which will be used to explicit the monodromy. Section 5 recalls the basic properties ofthe graph polynomial ψ Γ ≡ ψ (Γ) and the graph hypersurface X Γ : ψ Γ = 0. The crucial point is corollary 5.3which says that the strict transform of X Γ on our toric blowup avoids points with coordinates ≥
0. Any chainwe construct which stays close to the locus of such points necessarily is away from X Γ and hence also awayfrom the polar locus of our integrand. Section 6 computes the monodromy of our chain. Section 7 considershow to reduce Feynman amplitude calculations as they arise in physics, including masses and momenta aswell as divergences which are worse than logarithmic, to the basic situation where limiting methods canapply. In section 8 we calculate the nilpotent matrix N which is the log of the monodromy transformation,and in section 9 we prove the main renormalization theorem in the log divergent case, to which we havereduced the theory. . Hopf algebras of trees and graphs
Graphs.
In this section we bring together material on graphs and the graph Hopf algebra which willbe used in the sequel. We also discuss Hopf algebras related to rooted trees and prove a result (proposition2.5) relating the Hopf algebra of core graphs to a suitable Hopf algebra of labeled trees. Strictly speakingthis is not used in the paper, but it provides the best way we know to understand flags of core subgraphs,and these play a central role in the monodromy computations. Trees labeled by divergent subgraphs have along history in renormalization theory [12], [13].A graph Γ is determined by giving a finite set HE (Γ) of half-edges, together with two further sets E (Γ)(edges) and V (Γ) (vertices) and surjective maps(2.1) p V : HE (Γ) → V ; p E : HE (Γ) → E. (Note we do not allow isolated vertices.) In combinatorics, one typically assumes all fibres p − E ( e ) consist ofexactly two half-edges ( e an internal edge ), while in physics the calculus of path integrals and correlationfunctions dictates that one admit external edges e ∈ E with p − E ( e ) = 1. If all internal edges of Γ areshrunk to 0, the resulting graph (with no internal edges) is called the residue res(Γ). In certain theories, thevertices are decomposed into different types V = ∐ V i , and the valence of the vertices in V i , p − V ( v ), is fixedindependent of v ∈ V i .We will typically work with labeled graphs which are triples (Γ , A, φ : A ∼ = E (Γ)). We refer to A as theset of edges.A graph is a topological space with Betti numbers | Γ | = h (Γ) = dim H (Γ , Q ) and h (Γ). We say Γ is connected if h = 1. Sometimes h is referred to as the loop number .A subgraph γ ⊂ Γ is determined (for us) by a subset E ( γ ) ⊂ E (Γ). We write Γ //γ for the quotientgraph obtained by contracting all edges of γ to points. If γ is not connected, Γ //γ is different from the naivequotient Γ /γ . If γ = Γ, we take Γ // Γ = ∅ to be the empty set. It will be convenient when we discuss Hopfalgebras below to have the empty set as a graph.Also, for γ = e a single edge, we have the contraction Γ //e = Γ /e . In this case we also consider the cut graph Γ − e obtained by removing e and also any remaining isolated vertex.A graph Γ is said to be core (1 P I in physics terminology) if for any edge e we have | Γ − e | < | Γ | .A cycle γ ⊂ Γ is a core subgraph such that | γ | = 1. If Γ is core, it can be written as a union of cycles (seee.g. the proof of lemma 7.4 in [2]).2.1.1. Self-energy graphs.
Special care has to be taken when the residue res( γ ) of a connected component γ of some subgraph consists of two half-edges connected to a vertex, | res( γ ) | = 2. Such graphs are calledself-energy graphs in physics. In such a situation, if the internal edges of γ contract to a point, we are leftwith two edges in Γ //γ , which are connected at this point u . It might happen that the theory provides morethan one two-point vertex. In fact, for a massive theory, there are two two-point vertices provided by thetheory corresponding to the two monomials in the Lagrangian quadratic in the fields, we call them of massand kinetic type. Γ //γ represents then a sum over two graphs by summing over the two types of vertices forthat point u . (see Fig.(3) for an example).The edges and vertices of various types have weights. We set the weight of an edge to be two, the weightof a vertex with valence greater than two is zero, the weight of a vertex of mass type is zero, the weight ofthe kinetic type is +2.Then, the superficial degree of divergence sdd(Γ) for a connected core graph Γ is(2.2) sdd(Γ) = 4 | Γ | − | Γ [1] | + 2 | Γ [0] , kin | , where Γ [0] , kin is the set of vertices of kinetic type, and Γ [1] the set of internal edges. Γ [0] , the set of interactionvertices (for which we assume we have only one type) does not show up as they have weight zero, nor doesΓ [0] , mass . By | · · · | we denote the cardinality of these sets.Note that a graph Γ //γ which has one two-point vertex labeled m (of mass type) which appears aftercontracting a self-energy subgraph γ has an improved power-counting as its edge number is 2 h (Γ //γ ) + 1. Ifthe two-point vertex is labeled by (cid:3) (kinetic type), it has not changed though: sdd(Γ //γ ) = sdd(Γ), as theweight of the two-point vertex compensates for the weight of the extra propagator. Quite often, in masslesstheories, one then omits the use of these two-point vertices altogether.
12 32 3456456 X u
Figure 3.
This vertex graph has a propagator correction given by edges 4 , ,
6. The non-trivial part of the coproduct then delivers on the left the subgraph with internal edges 4 , , , ,
3. There isa two-point vertex u between edges 2 ,
3. Choosing two labels u = m or u = (cid:3) allows todistinguish between mass and wave-function renormalization. We remind the reader thatthe corresponding monomials in the Lagrangian are m φ / φ (cid:3) φ/ Hopf algebras of graphs.
Let P be a class of graphs. We assume ∅ ∈ P and that Γ ∈ P and Γ ′ ∼ = Γimplies Γ ′ ∈ P . We say P is closed under extension if given γ ⊂ Γ we have(2.3) γ, Γ ∈ P ⇔ γ, Γ //γ ∈ P . Easy examples of such classes of graphs are P = core graphs, and P = log divergent graphs, where Γ islog divergent (in φ theory) if it is core and if further E (Γ i ) = 2 | Γ i | for every connected component Γ i ⊂ Γ.(Both examples are closed under extension by virtue of the identity | γ | + | Γ //γ | = | Γ | .) Examples which arisein physical theories are more subtle. Verification of (2.3) requires an analysis of which graphs can arise froma given Lagrangian. To verify P = { Γ | sdd(Γ) ≥ } satisfies (2.3) one must consider self-energy graphs andthe role of vertices of kinetic type as discussed above.In particular, in massless φ theory divergent graphs are closed under extension, and so is the class ofgraphs for which 4 | Γ |− | Γ [1] | +2 | Γ [0] , kin | +2 | Γ [0] , mass | ≥
0. Note that this may contain superficially convergentgraphs if there are sufficiently many two-point vertices of mass type. It pays to include them in the class ofgraphs to be considered, which enables one to discuss the effect of mass in the renormalization group flow.Associated to a class P which is closed under extension as above, we define a (commutative, but notcocommutative) Hopf algebra H = H P as follows. As a vector space, H is freely spanned by isomorphismclasses of graphs in P . (A number of variants are possible. One may work with oriented graphs, forexample. In this case, the theory of graph homology yields a (graded commutative) differential graded Hopfalgebra. One may also rigidify by working with disjoint unions of subgraphs of a given labeled graph.) H becomes a commutative algebra with 1 = [ ∅ ] and product given by disjoint union. Define a comultiplication∆ : H → H ⊗ H :(2.4) ∆(Γ) = X γ ⊂ Γ γ ∈P γ ⊗ Γ //γ. One checks that (2.3) implies that (2.4) is coassociative. Since H is graded by loop numbers and each H n is finite dimensional, the theory of Hopf algebras guarantees the existence of an antipode, so H is a Hopfalgebra.
25 64 3
Figure 4.
In Eq.(2.6), we give the coproduct for this wheel with three spokes in the coreHopf algebra.If P ′ ⊂ P with Hopf algebras H ′ , H ( e.g. take P to be core graphs, and P ′ ⊂ P divergent core graphs) thenthe map H ։ H ′ obtained by sending Γ
6∈ P ′ is a homomorphism of Hopf algebras. For example, thedivergent Hopf algebra carries the information needed for renormalization [13], while the core Hopf algebra H C determines the monodromy. In terms of groupschemes, one has Spec ( H log. div. ) ֒ → Spec ( H C ) is a closedsubgroupscheme, and renormalization can be viewed as a morphism from the affine line with coordinate L to Spec ( H log. div. ). Already here we use that for divergent graphs with sdd(Γ) >
0, we can evaluate themas polynomials in masses and external momenta with coefficients determined from log divergent graphs, seebelow.Let Γ i , i = 1 , v i ∈ Γ i be vertices. Let Γ = Γ ∪ Γ where the two graphs are joined by identifying v ∼ v . Then Γ is core (cf.proposition 3.2). Further, core subgraphs Γ ′ ⊂ Γ all arise as the image of Γ ′ ∐ Γ ′ → Γ for Γ ′ i ⊂ Γ i core.Thus(2.5) ∆(Γ) = X Γ ′ ⊗ Γ // Γ ′ = (cid:16) X Γ ′ ⊗ Γ // Γ ′ (cid:17)(cid:16) X Γ ′ ⊗ Γ // Γ ′ (cid:17) + X (Γ ′ − Γ ′ · Γ ′ ) ⊗ (Γ // Γ ′ · Γ // Γ ′ ) + X Γ ′ ⊗ (cid:16) Γ // Γ ′ − Γ // Γ ′ · Γ // Γ ′ (cid:17) . It follows that the vector space I ⊂ H C spanned by elements Γ − Γ · Γ as above satisfies ∆( I ) ⊂ I ⊗ H C + H C ⊗ I . Since I is an ideal, we see that H C := H C /I is a commutative Hopf algebra. Roughly speaking, H C is obtained from H C by identifying one vertex reducible graphs with products of the component pieces.Generalization to theories with more vertex and edge types are straightforward.Fig.(4) gives the wheel with three spokes. This graph, which in φ theory (external edges to be added suchthat each vertex is four-valent) has a residue 6 ζ (3) for conceptual reasons [2], has a coproduct (we omit edgelabels and identify terms which are identical under this omission, which gives the indicated multiplicities)∆ = ⊗ I (2.6) + I ⊗ +4 ⊗ +3 ⊗ ⊗ . or example, the three possible labelings for the four-edge cycle in the third line are 4523, 5631 and 6412.While the graph has a non-trivial coproduct in the core Hopf algebra, it is a primitive element in therenormalization Hopf algebra. It is tempting to hope that the core coproduct relates to the Hodge structureunderlying the period which appears in the residue of this graph.2.3. Rooted tree Hopf algebras [12] , [3] . We introduce the Hopf algebra of decorated non-planar rootedtrees H T using non-empty finite sets as decorations (decorations will be sets of edge labels of Feynman graphsbelow) to label the vertices of the rooted tree Hopf algebra H T ( ∅ ). Products in H T are disjoint unions oftrees (forests). We write the coproduct as(2.7) ∆( T ) = T ⊗ I + I ⊗ T + X admissible cuts C P C ( T ) ⊗ R C ( T ) . Edges are oriented away from the root and a vertex which has no outgoing edge we call a foot. An admissiblecut is a subset of edges of a tree such that no path from the root to any vertex of T traverses more than oneelement of that subset. Such a cut C separates T into at least 2 components. The component containingthe root is denoted R C ( T ), and the product of the other components is P C ( T ).A ladder is a tree without side branching. Decorated ladders generate a sub-Hopf algebra L T ⊂ H T . Ageneral element in L T is a sum of bamboo forests , that is disjoint unions of ladders. Decorated ladders havean associative shuffle product(2.8) L ⋆ L := X k ∈ shuffle( ℓ ,ℓ ) L ( k )where ℓ i denotes the ordered set of decorations for L i and shuffle( ℓ , ℓ ) is the set of all ordered sets obtainedby shuffling together ℓ and ℓ . Lemma 2.1.
Let
K ⊂ L T be the ideal generated by elements of the form L · L − L ⋆ L . Then ∆( K ) ⊂K ⊗ L T + L T ⊗ K .Proof. Write ∆( L i ) = P d i j =0 L ij ⊗ L d i − ji where d i is the length of L i and L ij (resp. L ji ) is the bottom (resp.top) subladder of length j . Then ∆( L )∆( L ) = X j,µ L j L µ ⊗ L d − j L d − µ (2.9) ∆( L ⋆ L ) = X k ∆( L ( k )) = X k,ν L ( k ) ν ⊗ L ( k ) d + d − ν . Consider pairs ( j, µ ) of indices in (2.9) and write j + µ = ν . Among the pairs k, ν we consider the subset K ( j, µ ) for which the first ν = j + µ elements of the ordered set consist of a shuffle of the decorations on theladders L j , L µ . It is clear that the remaining d + d − ν elements of k will then run through shuffles ofthe decorations of L d − j , L d − µ , so(2.10) ∆( L )∆( L ) − ∆( L ⋆ L ) = X j,µ (cid:16) ( L j L µ − X k ∈ K ( j,µ ) L ( k ) j + µ ) ⊗ L d − j L d − µ (cid:17) + X j,µ (cid:16) X k ∈ K ( j,µ ) L ( k ) j + µ ⊗ ( L d − j L d − µ − L ( k ) d + d − j − µ ) (cid:17) ∈ K ⊗ L T + L T ⊗ K . (cid:3) Remark 2.2.
Any bamboo forest is equivalent mod K to a sum of stalks. Indeed, one has e.g.(2.11) L · L · L ≡ ( L ⋆ L ) · L ≡ ( L ⋆ L ) ⋆ L ≡ L ⋆ L ⋆ L . For any decoration ℓ , one has an operator [3](2.12) B ℓ + : H T → H T which carries any forest to the tree obtained by connecting a single root vertex with decoration ℓ to all theroots of the forest. This operator is a Hochschild 1-cocycle, i.e.(2.13) ∆ B ℓ + = B ℓ + ⊗ I + (id ⊗ B ℓ + )∆ . et J ⊂ H T be the smallest ideal containing the ideal K as in lemma 2.1 and stable under all the operators B ℓ + . Generators of J as an abelian group are obtained by starting with elements of K and successivelyapplying B ℓ + for various ℓ and multiplying by elements of H T . It follows from (2.13) that ∆ J ⊂ J ⊗ H T + H T ⊗ J . Define(2.14) H T := H T / J . A flag in a core graph Γ is a chain(2.15) f := ∅ ( Γ ( · · · ( Γ n = Γof core subgraphs. Write F (Γ) for the collection of all maximal flags of Γ. One checks easily that for amaximal flag, n = | Γ | . Let us consider an example. ( ( , (2.16) ( ( , (2.17) ( ( , (2.18) ( ( , (2.19) ( ( , (2.20) ( ( , (2.21) ( ( , (2.22) ( ( , (2.23) ( ( , (2.24) ( ( , (2.25) ( ( , (2.26) ( ( , (2.27)are the twelve flags for the graph given in Fig.(5). We omitted the edge labels in the above flags. Note thatonly the first two , (2.16,2.17) are relevant for the renormalization Hopf algebra to be introduced below. Figure 5.
A graph with overlapping subdivergences. The renormalization Hopf algebragives ∆ ′ ⊗ ⊗
34 + 3456 ⊗
12. Note that each edge belongs to somesubgraph with sdd ≥ f we associate the ladder L ( f ) with n vertices decorated by Γ i − Γ i − . (More precisely, thefoot is decorated by Γ and the root by Γ − Γ n − .). Define(2.28) ρ L : H C → L T ; ρ L (Γ) := X f ∈ F (Γ) L ( f )Here the set of labels D will be the set of subsets of graph labels. Lemma 2.3.
The map ρ L is a homomorphism of Hopf algebras.Proof. For a flag f let f ( p ) be the bottom p vertices with the given labeling, and let f ( p ) be the top n − p vertices with the quotient labeling gotten by contracting the core graph associated to the bottom p vertices.For γ ⊂ Γ a core subgraph, define F (Γ , γ ) := { f ∈ F (Γ) | γ ∈ f } . There is a natural identification(2.29) F (Γ , γ ) = F ( γ ) × F (Γ //γ ) . We have(2.30) ( ρ L ⊗ ρ L ) ◦ ∆ C (Γ) = X γ ρ L ( γ ) ⊗ ρ L (Γ //γ ) = X γ X f ∈ F (Γ ,γ ) L ( f | γ | ) ⊗ L ( f | γ | ) . On the other hand(2.31) ∆ L ◦ ρ L (Γ) = X f ∈ F (Γ) n X i =1 L ( f ( i ) ) ⊗ L ( f ( i ) ) . The assertion of the lemma is that there is a 1 − { γ, max. flag of Γ containing γ } ↔ { max. flag of Γ , i ≤ n } . This is clear. (cid:3)
In fact, the tree structure associated to a maximal flag f of Γ is rather more intricate than just a ladder.Though we do not use this tree structure in the sequel, we present the construction in some detail to helpin understanding the difference between the core and renormalization Hopf algebra.We want to associate a forest T ( f ) to the flag f , and we proceed by induction on n = | Γ | . We can writeΓ = S Γ ( j ) in such a way that all the Γ ( j ) are core and one vertex irreducible, and such that | Γ | = P | Γ ( j ) | .This decomposition is unique. If it is nontrivial, we define T ( f ) = Q T ( f ( j ) ) where f ( j ) is the induced flagfrom f on Γ ( j ) . We now may assume Γ is one vertex irreducible. If the Γ i in our flag are all one vertexirreducible, we take T ( f ) = L ( f ) to be a ladder as above. Otherwise, let m < n be maximal such thatΓ m ( Γ is one vertex reducible. By induction, we have a forest T ( f | Γ m ). To define T ( f ), we glue the footof the ladder with decorations Γ m +1 − Γ m , . . . , Γ − Γ n − to all the roots of T ( f | Γ m ). (For an example, seefigs.(6) and (7).) Lemma 2.4.
Let
Γ = S Γ ( j ) where Γ and the Γ ( j ) are core. Assume | Γ | = P j | Γ ( j ) | . Then, viewing flags f ∈ F (Γ) as sets of core subgraphs, ordered by inclusion, there is a − correspondence between F (Γ) andshuffles of the F (Γ ( j ) ) .Proof. One checks easily that the Γ ( j ) can have no edges in common. Further, there is a 1 − ′ ⊂ Γ and collections of core subgraphs Γ ( j ) ′ ⊂ Γ ( j ) . Here, the dictionary is givenby Γ ′
7→ { Γ ′ ∩ Γ ( j ) } and { Γ ( j ) ′ } 7→ S Γ ( j ) ′ . The assertion of the lemma follows. (cid:3) Figure 6.
The core Hopf algebra on rooted trees. We indicate subgraphs by edge labelson the vertices of rooted trees. The dots indicate seven more such trees, corresponding toflags Γ i ( Γ j ( Γ k with Γ i a cycle on four edges. The last tree represents a sum of twoflags, 34 ( ( ( (
123 4 5 6 12 3 45 61a2b 345612 123456 123456 125634c or dd or c c or dd or c+ + + +12ab
Figure 7.
The two graphs differ in how the subdivergences are inserted. a, c ∈ ,
4, and b, d ∈ , c = a, b = d . So there are eight such legal trees, plus the two which areidentical between the two graphs. Note the permutation of labels at the feet of the treesin ρ (Γ): 1 a b ↔ ab . Keeping that order, we can uniquely reconstruct each graph fromthe knowledge of the labels at the feet: 1 a b, ,
56 and 12 ab, ,
56, which are the cycles ineach graph. Note that in the difference of the two graphs, only the difference of those eighttrees remains, corresponding to a primitive element in the renormalization Hopf algebra.The core Hopf algebra hence stores much more information than the renormalization Hopfalgebra, which we hope to use in the future to understand the periods assigned to Feynmangraphs by the Feynman rules.As a consequence of lemma 2.4 we may partition the flags F (Γ) associated to a core Γ as follows. Given f ∈ F (Γ), Let Γ m ⊂ Γ be maximal in the flag f such that Γ m is 1-vertex reducible. The flag f induces aflag f m on Γ m , and we know that it is a shuffle of flags f ( j ) m on Γ ( j ) m where Γ m = S Γ ( j ) m as in the lemma. Wesay two flags are equivalent, f ∼ f ′ , if f and f ′ agree at Γ m and above, and if they simply correspond totwo different shuffles of the flags f ( j ) m . We now have(2.33) T ( f ) ≡ X f ′ ∼ f L ( f ′ ) mod J . Indeed, T ( f ) is obtained by successive B ℓ + operations applied to the forest T ( f | Γ m ). The latter, by remark2.2, coincides with the righthand side of (2.33). We conclude Proposition 2.5.
With notation as above, there exist homomorphisms of Hopf algebras (2.34) H C ρ L −−−−→ L T y y H C ρ T −−−−→ H T ere ρ T (Γ) is the sum T ( f ) over equivalence classes of flags f as above. We will barely use H T in thefollowing, and introduced it for completeness and the benefit of the reader used to it.2.4. Renormalization Hopf algebras.
In a similar manner, one may define homomorphisms(2.35) ρ R : H R → H T for any one of the renormalization Hopf algebras obtained by imposing restrictions on external leg structure.For a graph Γ, let, as before, the residue of Γ, res(Γ), be the graph with no loops obtained by shrinking allits internal edges to a point. What remains are the external half edges connected to that point (cf. section2.1). Note that ”doubling” an edge by putting a two-point vertex in it does not change the residue.In φ theory for example, graphs have 2 m external legs, with m ≥
0. For a renormalizable theory, thereis a finite set of external leg structures R such that we obtain a renormalization Hopf algebra for that set.For example, for massive φ theory, there are three such structures: the four-point vertex, and twotwo-point vertices, of kinetic type and mass type.Let us now consider flags associated to core graphs. Such chains · · · Γ i ( Γ i +1 ( · · · ( Γ correspond todecorated ladders, and the coproduct on the level of such ladders is a sum over all possibilities to cut anedge in such a ladder, splitting the chain(2.36) [ · · · ( Γ i ] ⊗ [Γ i +1 // Γ i ( · · · ( Γ // Γ i ] . So let us call such an admissible cut renormalization-admissible, if all core graphs Γ i , Γ // Γ i obtained bythe cut have residues in R .The set of renormalization-admissible cuts is a subset of the admissible cuts of a core graph, and thecoproduct respects this. Hence the renormalization Hopf algebra H R is a quotient Hopf algebra of the coreHopf algebra.If we enlarge the set R to include other local field operators appearing for example in an operator productexpansion we get quotient Hopf algebras between the core and the renormalization Hopf algebra.2.5. External leg structures.
External edges are usually labeled by data which characterize the amplitudeunder consideration. Let σ be such data. For graphs Γ with a given residue res(Γ), there is a finite set τ ∈ { σ } res(Γ) of possible data τ . A choice of such data determines a labeling of the corresponding vertex towhich a subgraph shrinks. Let Γ //γ τ be that co-graph with the corresponding vertex labeling.One gets a Hopf algebra structure on pairs (Γ , σ ) by using the renormalization coproduct ∆(Γ) = Γ ′ ⊗ Γ ′′ by setting ∆(Γ , σ ) = P τ ∈{ σ } res(Γ ′ ) (Γ ′ , τ ) ⊗ (Γ ′′ τ , σ ). We regard the decomposition into external leg structuresas a partition of unity and write(2.37) X τ ∈{ σ } res(Γ) (Γ , τ ) = (Γ , I ) . In our applications we only need this for (sub)graphs γ with | res( γ ) | = 2, and the use of these notions willbecome clear in the applications below.3. Combinatorics of blow-ups
We consider P n − with fixed homogeneous coordinates A := { A , . . . , A n } . Suppose given a subset S ⊂ A . Assume A 6∈ S and that S has the property that whenever µ , µ ∈ S with µ ∪ µ = A , then µ ∪ µ ∈ S . For µ ∈ S we write L µ ⊂ P n − for the coordinate linear space defined by A i = 0 , i ∈ µ . Write L ( S ) := { L µ | µ ∈ S } . We see that(3.1) L µ i ∈ L ( S ); L µ ∩ L µ = ∅ ⇒ L µ ∩ L µ ∈ L ( S ) . We can stratify the set L ( S ) taking L ( S ) to be the set of all minimal elements (under inclusion) of L ( S ).More generally, L ( S ) i will be the set of minimal elements in L ( S ) − ` i − j =1 L ( S ) j . Proposition 3.1. (i) Elements in L ( S ) are all disjoint, so we may define P ( S ) to be the variety defined byblowing up elements in L ( S ) on P n − . We do not need to specify an order in which to perform the blowups.(ii) More generally, the strict transforms of elements in L ( S ) i +1 to the space P ( S ) i obtained by successivelyblowing the strict transform of L ( S ) j , j = 1 , . . . , i are disjoint, so we may inductively define P ( S ) to be thesuccessive blowup of the L ( S ) i . iii) Let E i ⊂ P ( S ) correspond to the blowup of L µ i , i = 1 , . . . , r . ( E i is the unique exceptional divisorwith image L µ i in P ( S ) .)Then E ∩ · · · ∩ E r = ∅ if and only if after possibly reordering, we have inclusions L µ ⊂ · · · ⊂ L µ r .(iv) The total exceptional divisor E ⊂ P ( S ) is a normal crossings divisor.(v) Let M ⊂ P n − be a coordinate linear space. Assume M L for any L ∈ L ( S ) . Then M ∩ L ( S ) := { M ∩ L | L ∈ L ( S ) } satisfies (3.1) . The strict transform of M in P ( S ) is obtained by blowing up elementsof M ∩ L ( S ) on M as in (i) and (ii) above.Proof. If L = L ∈ L ( S ) i and L ∩ L = ∅ , then L ∩ L ∈ L ( S ) j for some j < i . This means that when weget to the i -th step, L ∩ L has already been blown up, so the strict transforms of the L i are disjoint, proving(ii). For ( iii ), T E i = ∅ ⇐ L µ ⊂ · · · ⊂ L µ r follows from the above argument. Conversely, if we have strictinclusions among the L µ i , we may write (abusively) L µ i /L µ i − for the projective space with homogeneouscoordinates the homogeneous coordinates on L µ i vanishing on L µ i − . The exceptional divisor on the blowupof L µ i − ⊂ L µ i is identified with L µ i − × ( L µ i /L µ i − ). A straightforward calculation identifies nonemptyopen sets (open toric orbits in the sense to be discussed below) in T E i and(3.2) L µ × ( L µ /L µ ) × · · · × ( L µ r /L µ r − )The remaining parts of the proposition follow from the algorithm in [8]. (cid:3) For us, sets S as above will arise in the context of graphs. Recall in 2.1 we defined the notion of coregraph. Proposition 3.2.
Let Γ be a graph, and let Γ , Γ ⊂ Γ be core subgraphs. Then the union Γ ∪ Γ is a coresubgraph.Proof. Removing an edge increases the Euler-Poincar´e characteristic by 1. If h doesn’t drop, then either h increases (the graph disconnects when e is removed) or e has a unary vertex so removing e drops thenumber of vertices. Suppose e is an edge of Γ (assumed core). Then e cannot have a unary vertex. If, onthe other hand, removing e disconnects Γ ∪ Γ , then since the Γ i are core what must happen is that eachΓ i has precisely one vertex of e . But this would imply that Γ is not core, a contradiction. (cid:3) To a graph Γ we may associate the projective space P (Γ) with homogeneous coordinates A e , e ∈ E (Γ)labeled by the edges of Γ. Let Γ be a core graph. A coordinate linear space L ⊂ P (Γ) is a non-empty linearspace defined by some subset of the homogeneous coordinate functions, L : A e = · · · = A e p = 0. Define L (Γ) to be the set of coordinate linear spaces in P (Γ) such that the corresponding set of edges e i , . . . , e i p isthe edge set of a core subgraph Γ ′ ⊂ Γ. It follows from proposition 3.2 that L (Γ) satisfies condition (3.1), sothe iterated blowup(3.3) π : P (Γ) → P (Γ)as in proposition 3.1 is defined. Define(3.4) L = [ L ∈ L (Γ) L ⊂ P (Γ); E = [ E L = π − L . Lemma 3.3.
Suppose P (Γ) = P n − with coordinates A , . . . , A n . Let L ⊂ P (Γ) be defined by A = · · · = A p = 0 . Let π L : P L → P (Γ) be the blowup of L . Then the exceptional divisor E ⊂ P L is identifiedwith P p − × L . Further A , . . . , A p induce coordinates on the vertical fibres P p − and A p +1 , . . . , A n givehomogeneous coordinates on L .Proof. This is standard. One way to see it is to use the map P n − − L → P p − , [ a , . . . , a n ] [ a , . . . , a p ].(Here, and in the sequel, [ · · · ] denotes a point in homogeneous coordinates.) This extends to a map f on P L :(3.5) E ֒ → −−−−→ P L f −−−−→ P p − y π L | E y π L L ֒ → −−−−→ P n − . The resulting map π L | E × f : E ∼ = L × P p − . (cid:3) t will be helpful to better understand the geometry of P (Γ). Let G m = Spec Q [ t, t − ] be the standard onedimensional algebraic torus. Define T = G nm / G m where the quotient is taken with respect to the diagonalembedding. For all practical purposes, it suffices to consider complex points(3.6) T ( C ) = C × n / C × ∼ = C × n − . A toric variety P is an equivariant (partial) compactification of T . In other words, T ⊂ P is an open set,and we have an extension of the natural group map m (3.7) T × T ⊂ −−−−→ T × P m y ¯ m y T ⊂ −−−−→ P. For example, P (Γ) is a toric variety for a torus T (Γ). Canonically, we may write T (Γ) = ( Q e ∈ Edge(Γ) G m ) (cid:14) G m . More important for us:
Proposition 3.4. (i) P (Γ) is a toric variety for T = T (Γ) .(ii) The orbits of T on P (Γ) are in − correspondence with pairs ( F, Γ p ( · · · ( Γ ( Γ) . Here F ⊂ Γ isa (possibly empty) subforest (subgraph with h ( F ) = 0 ) and the Γ i are core subgraphs of Γ . We require thatthe image of F i := F ∩ Γ i in Γ i // Γ i +1 be a subforest for each i . (cf. (3.2) ). The orbit associated to sucha pair is canonically identified with the open orbit in the toric variety P (Γ p //F p ) × P ((Γ p − // Γ p ) //F p − ) ×· · · × P ((Γ // Γ ) //F ) .Proof. A general reference for toric varieties is [9]. The fact (i) that P (Γ) is a toric variety follows inductivelyfrom the fact that the blowup of an invariant ideal I in a toric variety is toric. Indeed, the torus acts on I and hence on the blowup Proj( I ).We recall some toric constructions. Let N = Z Edge(Γ) / Z , and let M = hom( N, Z ). We have canonically T = Spec Q [ M ] where Q [ M ] is the group ring of the lattice M . A fan (op. cit., 1.4, p. 20) F is a finite setof convex cones in N R = N ⊗ R satisfying certain simple axioms. To a cone C ⊂ N R one associates the dualcone (op. cit. p. 4)(3.8) C ∨ = { m ∈ M R | h m, c i ≥ , ∀ c ∈ C } (resp. the semigroup C ∨ Z = C ∨ ∩ M ). The toric variety V ( F ) associated to the fan F is then a union ofthe affine sets U ( C ) := Spec Q [ C ∨ Z ]. For example, our N has rank n −
1. There are n evident elements e determined by the n edges of Γ. Let C e = { P e ′ = e r e ′ e ′ | r e ′ ≥ } be the cone spanned by all edges except e . The spanning edges for C e form a basis for N which implies that U ( C e ) ∼ = A n − . Since all the coordinaterings lie in Q [ M ] (i.e. T (Γ) ⊂ U ( C e )), one is able to glue together the U ( C e ). The resulting toric varietyassociated to the fan { C e | e ∈ Edge( E ) } is canonically identified with P (Γ). Remark 3.5.
Our toric varieties will all be smooth (closures of orbits in smooth toric varieties are smooth),which is equivalent ([9], §
2) to the condition that cones in the fan are all generated by subsets of bases forthe lattice N . Faces of these cones are in 1 − − C in the fan (op. cit.3.1, p.51). The subgroup of N generated by C ∩ N corresponds to the subgroup of T which acts triviallyon the orbit. For example, in the case of projective space P n − , there are n cones C e of dimension n − n fixed points (0 , . . . , , . . . , ∈ P n − . For any S ( Edge(Γ), the cone C ( S ) spanned bythe edges of S corresponds to the orbit { ( . . . , x e , . . . ) | x e = 0 ⇔ e ∈ S } ⊂ P n − . Let L : A e = 0 , e ∈ Γ ′ ⊂ Γbe a coordinate linear space in P (Γ) associated to a subgraph Γ ′ ⊂ Γ. It follows from lemma 3.3 that theexceptional divisor E L ⊂ P L in the blowup of L can be identified with(3.9) E L = P (Γ ′ ) × P (Γ // Γ ′ ) . Let e (Γ ′ ) = P e ∈ Γ ′ e ⊂ N R , and write τ (Γ ′ ) = R ≥ · e (Γ ′ ). The subgroup Z · e (Γ ′ ) ⊂ N determines a1-parameter subgroup G (Γ ′ ) ⊂ T = Spec Q [ M ]. It follows from (3.9) that G (Γ ′ ) acts trivially on E L . Onehas τ (Γ ′ ) ⊂ C ′ ⊂ C e for all e Γ ′ , where C ′ is the cone generated by the edges of Γ ′ . For all e ′ ∈ Γ ′ we efine a subcone C e,e ′ ⊂ C e to be spanned by τ (Γ ′ ) together with all edges of Γ except e, e ′ . The fan for P L is then(3.10) { C e , e ∈ Γ ′ } ∪ { C e,e ′ , e Γ ′ , e ′ ∈ Γ ′ } . Note that C e , e Γ ′ is not a cone in the fan for P L . More generally, let F be the fan for P (Γ). Certainly, F will contain as cones the half-lines τ (Γ ′ ) for all core subgraphs Γ ′ ⊂ Γ as well as the R ≥ e, e ∈ Γ. butwe must make precise which subsets of this set of half-lines span higher dimensional cones in F . By generaltheory, the cones correspond to the nonempty orbits. In other words,(3.11) R ≥ e , . . . , R ≥ e p , R ≥ e (Γ ) , . . . , R ≥ e (Γ q )span a cone in F if and only if the intersection(3.12) E ∩ · · · ∩ E q ∩ D ∩ · · · ∩ D p = ∅ , where E i ⊂ P (Γ) is the exceptional divisor corresponding to L (Γ i ) and D j ⊂ P (Γ) is the strict transform ofthe coordinate divisor A e i = 0 in P (Γ). To understand (3.12), consider the simple case E ∩ D . We have acore subgraph Γ ⊂ Γ, and an edge e of Γ. We know by lemma 3.3 that E ∼ = P (Γ ) × P (Γ // Γ ). If e is anedge of Γ , then D ∩ E = P (Γ //e ) × P (Γ // Γ ). Otherwise D ∩ E = P (Γ ) × P ((Γ // Γ ) //e ) . One (degenerate) possibility is that e is an edge of Γ which forms a loop (tadpole). In this case, e is itselfa core subgraph of Γ, and the divisor D should be treated as one of the exceptional divisors E i . Thus, weomit this possibility. Another possibility is that e Γ , but that the image of e in Γ // Γ forms a loop.In this case, Γ := Γ ∪ e is a core subgraph, so the linear space L : A e = 0 , e ∈ Γ gets blown up inthe process of constructing P (Γ). But blowing L separates E and D , so the intersection of the stricttransforms of D and E in P (Γ) is empty. The general argument to show that (3.12) is empty if and onlyif the conditions of (ii) in the proposition are fulfilled is similar and is left for the reader. Note that the casewhere there are no divisors D i follows from proposition 3.1(iii). (cid:3) We are particularly interested in orbits corresponding to filtrations by core subgraphs Γ p ( · · · ( Γ ( Γ.Let V ⊂ P (Γ) be the closure of this orbit. We want to exhibit a toric neighborhood of V which retracts onto V as a vector bundle of rank p . As in the proof of proposition 3.4 we have e (Γ i ) := P e ∈ Γ i e . The cone C spanned by the e (Γ i ) lies in the fan F . For cones C ′ ∈ F we write C ′ > C if C is a subcone of C ′ . By thegeneral theory, this will happen if and only if C ⊂ C ′ is a subcone which appears on the boundary of C ′ .The orbit corresponding to C ′ will then appear in the closure of the orbit for C . Proposition 3.6.
With notation as above, Let F C ⊂ F be the subset of cones C ′ such that we have C ′ ≤ C ′′ ≥ C for some C ′′ ∈ F . Write P ⊂ P (Γ) for the open toric subvariety corresponding to the subfan F C ⊂ F . We have V ֒ → P ⊂ P (Γ) . Further there is a retraction π : P → V realizing P as a rank p vector bundle over V which is equivariant for the action of the torus T .Proof. One has the following functoriality for toric varieties [9], § φ : N ′ → N ′′ is a homomor-phism of lattices (finitely generated free abelian groups). Let F ′ , F ′′ be fans in N ′ R , N ′′ R . Suppose for eachcone σ ′ ∈ F ′ there exists a cone σ ′′ ∈ F ′′ such that φ ( σ ′ ) ⊂ σ ′′ . Then there is an induced map on toricvarieties V ( F ′ ) → V ( F ′′ ). Let N ′ = N = Z n / Z as above, and N ′′ = N ′ / ( Z e (Γ ) + · · · + Z e (Γ p )). One hasthe evident surjection φ : N ′ ։ N ′′ . We take as fan F ′ = F C ⊂ F . The closure V of the orbit correspondsto the fan F ′′ in N ′′ R given by the images of all cones C ′′ ≥ C (op. cit. § C ′′ is generated by e (Γ ) , . . . , e (Γ p ) , f , . . . , f q , and there are no linear relations among these elements (remark 3.5). A subcone C ′ ≤ C ′′ is generated by a subset e (Γ i ) , . . . , e (Γ i a ) , f , . . . , f b . The image is simply the cone in N ′′ R generatedby the images of the f ’s. If we have another cone C ′ ≤ C ′′ ≥ C in F ′ with the same image in F ′′ , it willhave generators say g , . . . , g b together with some of the e (Γ i )’s. Reordering the g ’s, we find that there arerelations(3.13) f i + X a ij e (Γ j ) = g i + X b ij e (Γ j )with a ij , b ij ≥
0. It follows that the cones in F spanned by f i , e (Γ ) , . . . , e (Γ p ) and g i , e (Γ ) , . . . , e (Γ p ) meetin a subset strictly larger that the cone spanned by the e (Γ j ). By the fan axioms, the intersection of twocones in a fan is a common face of both, so these two cones coincide, which implies f i = g i . In particular, or each cone in F ′′ , there is a unique minimal cone in F ′ lying over it. This is the hypothesis for [19], p. 58,proposition 1.33. One concludes that the map π : P → V induced by the map F ′ → F ′′ is an equivariantfibration, with fibre the toric bundle associated to the fan generated by the e (Γ i ) , ≤ i ≤ p . This toricvariety is just affine p -space, so we get an equivariant A p -fibration over V . Any such fibration is necessarilya vector bundle with structure group G pm . Indeed, this amounts to saying that any automorphism of thepolynomial ring k [ x , . . . , x p ] which intertwines the diagonal action of G pm is necessarily of the form x i c i x i with c i ∈ k × . (cid:3) Remark 3.7.
We will need to understand how these constructions are compatible. Let V be a closed orbitcorresponding to a cone C as above, and let V ⊂ V be a smaller closed orbit corresponding to a larger cone C > C . (The correspondence between cones and orbits is inclusion-reversing.) As above we have a toricvariety V ⊂ P ⊂ P (Γ) and a retraction π : P → V . The fan F ′ for P is given by the set of cones C ′ in F such that(3.14) C ′ ≤ C ′′ ≥ C ( > C ) . It follows that F ′ ⊂ F ′ = F C , so P ⊂ P is an open subvariety. Let V ⊂ V be the image of the composition P ⊂ P π −→ V . Then V is the open toric subvariety of V corresponding as above to the closed orbit V ⊂ V ,and we have a retraction V π V −−→ V . One gets commutative diagrams(3.15) P P ⊂ −−−−→ P π y π y π y V π V ←−−−− V ⊂ −−−−→ V and(3.16) P | V ⊂ −−−−→ P π y π y V V . Remark 3.8.
Using the toric structure, one can realize these vector bundles as direct sums of line bundlescorresponding to characters of the tori acting on the fibres.The inclusion on the top line of (3.16) correspondsto characters which act trivially on all of V . Remark 3.9. (compare proposition 3.4). Given a flag of core subgraphs(3.17) Γ p ( Γ p − ( · · · ( Γ ( Γ , let L i ⊂ P (Γ) be defined by the edge variables for edges in Γ i , so we have L ( · · · ( L p ( P (Γ). For L ⊂ P (Γ) a coordinate linear space, let T ( L ) ⊂ L be the subtorus where none of the coordinates vanish.Then the orbit associated to (3.17) is(3.18) T ( L ) × T ( L /L ) × · · · × T ( L p /L p − ) × T ( P n − /L p )(Here the notation L i +1 /L i is as in (3.2).)4. Topological Chains on Toric Varieties
One can define the notion of non-negative real points V ( R ≥ ) and positive real points V ( R > ). For atorus T = Spec Q [ N ∨ ] for some N ∼ = Z g we take T ( R > ) = { φ : Q [ N ∨ ] → R | φ ( n ) > , ∀ n ∈ N ∨ } . A toric variety V can be stratified as a disjoint union of tori V = ` T α . Define V ( R ≥ ) = a T α ( R > );(4.1) V ( R > ) = T ( R > ) , where T ⊂ V is the open orbit. Let V ⊂ P (Γ) be the closure of the orbit associated to a flag (3.17), and let T ( V ) ⊂ T = Spec Q [ N ∨ ] be the subtorus acting trivially on V . Let π V : P V → V be the vector bundle as in P H G L P H G L E ΕΕ Ε ΗΣΗ , Ε P H G L L Figure 8. P (Γ) and the real chain σ η,εP (Γ) .proposition 3.6. We write P V = L ⊕ · · · ⊕ L p as a direct sum of line bundles, where each L i is equivariantfor T ( V ). Let K ( V ) ∼ = ( S ) p ⊂ T ( V )( C ) be the maximal compact subgroup. Note that one has a canonicalidentification T ( V ) = G pm associated to the 1-parameter subgroups of T ( V ) generated by e (Γ i ) ∈ N . Inparticular, the identification K ( V ) = ( S ) p is canonical as well. For all closed orbits V we may fix metricson the L i which are compatible under inclusions (3.16) and are (necessarily) invariant under the action of K ( V ). We fix also a constant η >
0. We can then define S ηV ⊂ P V to be the product of the circle bundlesof radius η embedded in the L i . S ηV becomes a principal bundle over V with structure group K ( V ). Notethat S ηV ∩ P V ( R ≥ ) contains a unique point in every fibre of S ηV over a point of V ( R ). Let 0 < ε << η beanother constant. We need to define a chain σ η,εV ⊂ V ( R > ). We consider closures V ⊂ V of codimension1 orbits in V . For each such V we have an open P ( V ) ⊂ V and a retraction P ( V ) → V which is a linebundle with a metric. The fibres of P ( V ) ( R > ) have a canonical coordinate r >
0. If V corresponds to anintersection of V = E ∩ · · · ∩ E p with another exceptional divisor E p +1 , then we remove from each fibre of P ( V ) ( R > ) over V ( R > ) the locus where r < η . If, on the other hand V corresponds to an intersectionof V with one of the D i (i.e. with a strict transform of one of the coordinate divisors), then we remove thelocus r < ε . Repeating this process for each V (i.e. for each irreducible toric divisor in V ), we obtain acompact σ η,εV ⊂ V ( R > ) which stays away from the boundary components. (Here ”boundary components”are exceptional divisors together with strict transforms of coordinate divisors.) Example 4.1.
Consider the case V = P (Γ). Let π : P (Γ) → P (Γ), and let σ = { ( A , . . . , A n ) | A i ≥ } ⊂ P (Γ)( R )be the original integration chain. We have σ η,εP (Γ) ⊂ π − ( σ ) defined by excising away points within a distanceof η from an E i or ε from the strict transform D j of a coordinate divisor A j = 0. (cf. fig.(8)). It is a manifoldwith corners.Define τ η,εV to be the inverse image of σ η,εV in S ηV . The fibres of τ η,εV over σ η,εV are products ( S ) p with acanonical origin at the point where this fibre meets P V ( R ≥ ). For an angle 0 ≤ θ ≤ π , we can thus define τ η,ε,θV ⊂ τ η,εV to be swept out by the origin in each fibre under the action of [0 , θ ] p ⊂ K ( V ). The chains τ η,ε,θV have R -dimension n − P (Γ) and P (Γ). e e e + e fig. Figure 9.
Fan for Example 4.2.
Example 4.2.
Here is an example which is too simple to correspond to any graph, but is sufficient to clarifythe toric picture. Take(4.2) L : A = A = 0; L : A = 0in P with coordinates A , A , A . Take P π −→ P to be the blowup of L = (0 , , E ⊂ P be theexceptional divisor, and let E ⊂ P be the strict transform of L . Note that E is already a divisor so it isnot necessary to blow up again. Take V = E ∼ = P . The fan F for P is fig.(9). The cone C = R ≥ · ( e + e ),so the fan F ′ = F C ⊂ F is the subset of cones lying in the first quadrant. The toric variety P V is A with (0 ,
0) blown up. It projects down onto V as a line bundle. S ηV ⊂ P V ( C ) is then a circle bundle over V ( C ). V has two suborbits V = E ∩ E and V = E ∩ D , where D is the strict transform of the divisor A = 0 in P . We may interpret z := A /A as a coordinate on V , so V : z = 0 and V : z = ∞ . We have P ( V ) = V − { z = ∞} and P ( V ) = V − { z = 0 } . The real chain σ η,εV = { η ≤ z ≤ /ε } , and τ η,εV is the S -bundle of radius η over σ η,εV . On the other hand, V corresponds to the cone labeled C in fig.(9), andthe fan F C is just C itself. The toric variety P V ∼ = A is a rank 2 vector bundle over the point V . Wehave P V ⊂ P V . In this case σ η,εV is simply the point V , and τ η,εV ∼ = S × S ⊂ P V ( C ). In local coordinatesaround V given by eigenfunctions for the torus action we have τ η,ε,θV = { ( ηe iµ , z ) | η ≤ z ≤ /ε, ≤ µ ≤ θ } (4.3) τ η,ε,θV = { ( ηe iµ , ηe iν ) | ≤ µ, ν ≤ θ } τ η,ε,θV ∩ τ η,ε,θV = { ( ηe iµ , η ) | ≤ µ ≤ θ } . We want now to establish a basic formula for the boundary of the chains τ η,ε,θV . Here V runs through theclosures of orbits in P (Γ) associated to flags of core subgraphs (3.17). We include the big orbit V = P (Γ).We write | V | := codim( V /P (Γ)). We may express the boundary chains ∂τ η,ε,θV locally (in fact Zariski-locally)in coordinates which are eigenfunctions for the torus action. It is clear (cf. (4.3)) that boundary terms areobtained by setting a suitable one of these coordinates to be constant: either ηe iθ or η or ε . (The presenceof 1 /ε in the first line of (4.3) simply means that the appropriate coordinate near that point is 1 /z .) Proposition 4.3.
For a suitable orientation, the boundary (4.4) ∂ X V ( − | V | τ η,ε,θV Η ΤΗ , Ε , Θ VV Η , Ε , ΘΤ P H G L ΞΗ , Ε , Θ Figure 10.
The monodromy chain, with angular variable θ . will contain no chains with one coordinate constant = η .Proof. (Cf. fig.(10) ). For a given boundary term, we can choose local eigenfunction coordinates x , . . . , x n − such that be boundary term is given by x = η . We take the chains to be oriented in some consistant wayby this ordering of coordinates. (Note that these coordinates are defined on a Zariski open set. Theobstruction to choosing consistent orientations for various open sets is a class in the first Zariski cohomologyof P (Γ) with constant Z / Z -coefficients. Since this cohomology group vanishes, we can choose such consistentorientations.) If ∂τ η,ε,θV contains a term with x = η , there are two possibilities. Either x is a real coordinateon τ η,ε,θV or it is a circular coordinate. If x is a real coordinate, then the fact that x = η appears in theboundary means that locally x = 0 defines a codimension 1 orbit closure V ֒ → V . In ∂τ η,ε,θV , x willappear as a circular coordinate. Since | V | = | V | + 1, the same chain x = η will appear in ∂τ η,ε,θV and in ∂τ η,ε,θV and will cancel in (4.4). If, on the other hand, x = ηe iθ is a circular coordinate, then for suitableordering of coordinates, the chain will be an ( S ) p -bundle over a chain σ contained in the locus where certaincoordinates ≥
0. But then (4.4) will contain another chain which is an ( S ) p − -bundle over { x ≥ η } × σ ,and the boundary components involving x = η will occur with opposite signs and will cancel. (cid:3) The boundary chain (4.4) is an ( n − < ε < η . We want to construct an( n − ξ η,ε,θ which amounts to a scaling η → ε . To do this, we construct a vector field v on P (Γ). Let E = P E i be the exceptional divisor. v will be 0 outside a neighborhood N of E . Locally, at a point on N which is close to divisors E , . . . , E p we have coordinates x , . . . , x p which are eigenfunctions for the torusaction such that locally E i : x i = 0. Locally we will take v to be radial and inward-pointing in each x i . Weglue these local v ’s using a partition of unity. ”Flowing” the ( n − n − ξ η,ε,θ . If this is done with care, we can arrange(4.5) ∂ξ η,ε,θ ≡ ∂ X V ( − | V | τ η,ε,θV − ∂ X V ( − | V | τ ε,ε,θV . Here ≡ means that the two sides differ by a chain lying in an ε -neighborhood of the strict transform D ofthe coordinate divisor ∆ in P (Γ). Another important property of the chain ξ η,ε,θ is Lemma 4.4. ξ η,ε, π ≡ ξ η,ε, .Proof. The point is that ∂τ η,ε, πV ≡ V = P (Γ), and τ η,ε,θP (Γ) is independent of θ . (Seefig.(10)). (cid:3) Define the chain c η,ε,θ = P V ( − | V | τ η,ε,θV − ξ η,ε,θ . We have(4.6) ∂c η,ε,θ = ∂ X V ( − | V | τ ε,ε,θV . Note that c η,ε, = σ η,εP (Γ) , i.e. all chains involving at least one circular variable die at θ = 0. We define thevariation,(4.7) var ( c η,ε,θ ) = c η,ε, π − c η,ε, ≡ X V ( P (Γ) ( − | V | τ ε,εV . t is a sum of “( S ) p -tubes” over all E ∩ · · · ∩ E p ( P (Γ).5. The Graph Hypersurface
Associated to a graph Γ with n edges, one has the graph polynomial(5.1) ψ Γ ( A , . . . , A n ) = X T Y e T A e where T runs through spanning trees of Γ. This polynomial has degree h (Γ). For more detail, see [2] andthe references cited there. Let X = X Γ : ψ Γ = 0 be the graph hypersurface in P n − . For µ ⊂ Edge(Γ), let L µ ⊂ P (Γ) be defined by A e = 0 , e ∈ µ . Let Γ µ = S e ∈ µ e ⊂ Γ be the subgraph with edges in µ . Note thedictionary Γ µ ↔ L µ is inclusion reversing. Lemma 5.1. (i) L µ ⊂ X Γ ⊂ P (Γ) if and only if h (Γ µ ) > .(ii) If h (Γ µ ) > , there exists a unique ν ⊆ µ such that h (Γ ν ) = h (Γ µ ) and such that moreover Γ ν is acore graph.(iii) We have in (ii) that ν = S ξ where ξ runs through all minimal subsets of µ such that L ξ ⊂ X .(iv) L µ = L ν ∩ M , where M is a coordinate linear space not contained in X Γ .Proof. These assertions are straightforward from the results in [2], section 3. Note that (iv) justifies ourstrategy of only blowing up core subgraphs. (cid:3)
We have seen (remark 3.4) that our blowup P (Γ) is stratified as a union of tori indexed by pairs(5.2) ( F, { Γ p ( · · · ( Γ ( Γ //γ } )where F ⊂ Γ is a suitable subforest and the Γ i are core. Proposition 5.2. (i) As in proposition 3.4, the torus corresponding to (5.2) is (5.3) T (Γ p //F p ) × T ((Γ p − // Γ p ) //F p − ) × · · · × T ((Γ // Γ ) //F ) . Here T (Γ) := P (Γ) − ∆ , where ∆ : Q e ∈ Edge (Γ) A e = 0 .(ii) The strict transform Y of X Γ in P (Γ) meets the stratum (5.3) in a union of pullbacks (5.4) pr − ( X p ) ∪ pr − ( X p − // Γ p ) ∪ · · · ∪ pr − p ( X //γ ) // Γ ) . Here the pr i are the projections to the various subtori in (5.3) , and X denotes the restriction of the corre-sponding graph hypersurface to the open torus in the projective space.Proof. Let Γ ′ ⊂ Γ be a subgraph and let L : A e = 0 , e ∈ Edge(Γ ′ ). Assume h (Γ ′ ) >
0, so L ⊂ X Γ . Let P L → P (Γ) be the blowup of L . Let E L ⊂ P L be the exceptional divisor, and let Y L ⊂ P L be the stricttransform of X Γ . The basic geometric result (op. cit. prop. 3.5) is that E L = P (Γ ′ ) × P (Γ // Γ ′ ) and(5.5) Y L ∩ E L = (cid:16) X Γ ′ × P (Γ // Γ ′ ) (cid:17) ∪ (cid:16) P (Γ ′ ) × X Γ // Γ ′ (cid:17) . The assertions of the proposition follow by an induction argument. (cid:3)
Corollary 5.3.
The strict transform Y of X Γ in P (Γ) does not meet the non-negative points P (Γ)( R ≥ )(4.1) .Proof. It suffices by (4.1) to show that Y doesn’t meet the positive points in any stratum. By proposition5.2, it suffices to show that for any graph Γ, the graph hypersurface X Γ has no R -points with coordinatesall >
0. This is immediate because ψ Γ is a sum of monomials with non-negative coefficients. (cid:3) Remark 5.4.
The Feynman amplitude is obtained by calculating an integral over σ = P (Γ)( R ≥ ) withan integrand which has a pole along X Γ . Again using that ψ Γ is a sum of monomials with non-negativecoefficients, one sees from lemma 5.1 that(5.6) σ ∩ X Γ = [ µ L µ ( R ≥ )where L µ ↔ Γ µ with Γ µ ⊂ Γ a core subgraph. The iterated blowup P (Γ) → P (Γ) is exactly what is necessaryto separate the non-negative real points from the strict transform of X Γ . p p p + tq p + tq p + tq D t X Figure 11.
Moving ∆ t . Remark 5.5.
The points where ψ Γ = 0 have some remarkable properties. It is shown in [20] that for anyangular sector S with angle < π , ψ Γ ( a , . . . , a n ) = 0 at any complex projective point a such that the a i = 0and all the arg( a i ) lie in S . 6. Monodromy
Let p i = (0 , . . . , , , . . . , ∈ C n be the i -th coordinate vector. Define σ aff = { n X i =1 τ i p i | τ i ≥ , X τ i = 1 } ⊂ C n − { (0 , . . . , } → P n − . Fix a positive constant ε << q k = ( q k , . . . , q kn ) ∈ R n , ≤ k ≤ n with 1 − ε < q kj ≤ | q jk − q ℓ,m | ≤ ε . We assume the q k are algebraically generic. Write r k ( t ) = p k + tq k ∈ C n . Define (cf.fig.(11))(6.1) σ afft = { n X i =1 τ k r k ( t ) | τ k ≥ , X τ k = 1 } We write σ and e σ t for the images of these chains in P n − . Of course, σ = σ P n − as above, and we knowthat σ ∩ X Γ = S L ⊂L σ L . Here L is as in (3.4). Lemma 6.1.
Let
L ⊂ N L be a neighborhood of L in P n − and let σ ⊂ N σ be a neighborhood of σ . Thenthere exists ε > such that ε ≤ ε implies that for all ≤ θ ≤ π , we have e σ εe iθ ⊂ N σ and e σ εe iθ ∩ X Γ ⊂ N L .Proof. We have σ ∩ X Γ ⊂ L . By compacity, e σ εe iθ ⊂ N σ for ε <<
1. Again by compacity, if we shrink N σ wewill have N σ ∩ X Γ ⊂ N L . (cid:3) Remark 6.2.
Write H k,t for the projective span of the points r ( t ) , . . . , [ r k ( t ) , . . . , r n ( t ) , and let ∆ t = S nk =1 H k,t . Thus, ∆ = ∆ and we may consider the monodromy for ∆ εe iθ , ≤ θ ≤ π . Moreprecisely, renormalization in physics involves an integral over the chain σ . The integrand has poles along X Γ . Since σ ∩ X Γ = ∅ , the integral is possibly divergent. On the other hand, by corollary 5.3, the chain σ ε does not meet X Γ and so represents a singular homology class(6.2) [ σ ε ] ∈ H n − ( P n − − X Γ , ∆ ε − ∆ ε ∩ X Γ , Z ) . Ε EE I R ³ M D j D k Η Τ , Η , Ε E Figure 12.
The chain τ η,εE .(Since all q kj >
0, it follows that σ ε ⊂ σ , and points in σ ε have all coordinates > P n − − X Γ , ∆ εe iθ − ∆ εe iθ ∩ X Γ ) as a family over the circle and we continuously deform ourchain σ ε to a family of chains σ εe iθ on P n − − X Γ with boundary on ∆ εe iθ − ∆ εe iθ ∩ X Γ . (We will not be ableto take σ εe iθ = e σ εe iθ because this chain can meet X Γ .) The monodromy map m is an automorphism of (6.2)obtained by winding around the circle: m ( σ ε ) = σ εe πi . We will calculate m ( σ ε ) and see that it determinesin a natural way the renormalization expansion we want.Recall we have π : P (Γ) → P (Γ), and π − ( X Γ ) = Y Γ ∪ E , where Y = Y Γ is the strict transform of X Γ and E = S E i is the exceptional divisor. (The E i are closures of orbits associated to core subgraphs of Γ.) Wemay transfer our monodromy problem to P (Γ). ∆ εe iθ is in general position with respect to the blowups, sowe obtain a family of divisors ∆ ′ εe iθ = π ∗ ∆ εe iθ on P (Γ). Since π : P (Γ) − E − Y Γ ∼ = P (Γ) − X Γ , we have anisomorphism of topological pairs(6.3) (cid:16) P (Γ) − E − Y Γ , ∆ ′ εe iθ − ∆ ′ εe iθ ∩ ( E ∪ Y Γ ) (cid:17) ∼ = (cid:16) P (Γ) − X Γ , ∆ εe iθ − ∆ εe iθ ∩ X Γ (cid:17) . In section 4 we have defined chains τ η,ε,θV , ξ η,ε,θ , c η,ε,θ on P (Γ). These chains sit on (or, in the case of ξ ,within) various ( S ) p -bundles over P (Γ)( R ≥ ) where the S have radius η with respect to a chosen metric.From corollary 5.3 it follows that for 0 < η <<
1, none of these chains meets Y Γ . By construction, thesechains do not meet E , so they may be identified with chains on P (Γ) − X Γ . We claim that a small modificationof the chains c η,ε,θ will represent the monodromy chains σ εe iθ . The monodromy chains σ εe iθ should haveboundary on ∆ εe iθ . On the other hand, the chains c η,ε,θ were cut off so they had boundaries on tubes adistance ε from the toric divisors D j given by the strict transforms of the A j = 0 (see fig.(12)). We must“massage” these brutal cutoffs to get them into ∆ εe iθ . Our chains τ sit on tubes or products of tubes orproducts of tubes of radius η which we can think of as lying on P n − − L . Since ε << η , when we deform∆ → ∆ εe iθ the homotopy type of the circles, or product of circles where these divisors intersect the tubesdoesn’t change. This may seem strange because L ⊂ ∆ while ∆ εe iθ is in general position with respect to L , but the intersections with a hollow tubular neighborhood of L are canonically homotopic. Indeed, wemay take ∆ εe iθ to correspond to a point in a small contractible disk in the moduli space for coordinatesimplices around the point corresponding to ∆. The canonical path up to homotopy between the two pointsin moduli will induce the desired homotopy on the intersections. (See fig.(13). The two sets of four dots onthe circles are canonically homotopic.). In more detail, by corollary 5.3, the chains τ η,ε,θ are bounded awayfrom X Γ by a bound which is independent of ε as ε →
0. Outside of some tubular neighborhood N of X Γ we may find a space M disjoint from X Γ such that M contains open neighborhoods of both ∆ − N ∩ ∆ and∆ εe iθ − N ∩ ∆ εe iθ and such that we have deformation retractions M → ∆ − N ∩ ∆ and M → ∆ εe iθ − N ∩ ∆ εe iθ . Ε e i Θ D L L fig.8
Figure 13.
Homotopy invariance of ∆ t ∩ tube over L .Shrinking ε , we may assume our ε -cutoffs lie in M . We may then use the deformation retract to extendthe chain slightly to a chain ˜ τ η,ε,θV which bounds on ∆ εe iθ . It remains to consider the chains ξ η,ε,θ . Recallthese were obtained by flowing the chain ∂ P V ( − | V | τ η,ε,θV inward toward the exceptional divisor E , so η → ε (cf. fig.(10)). We are in a small neighborhood of E ( R ≥ ) hence by corollary 5.3 we are away from X Γ . The point to be checked is that the term ∂ P V ( − | V | τ ε,ε,θV is very close to ∆ εe iθ so by the samedeformation retraction argument as above we can extend the chain to bound on ∆ εe iθ . The subtlety is thatwe are ε -close to E as well, so we need the distance from ∆ εe iθ to be o ( ε ). Recall (6.1) we have the vertices r k ( εe iθ ) = [ q k εe iθ , . . . , q kk εe iθ , . . . , q kn εe iθ ] ∈ P n − . The coordinate divisor ∆ εe iθ is determined by theseprojective points. The projective point does not change if we scale the coordinates by e iθ , so the image in P n − of the affine simplex below, parametrized by τ , . . . , τ n ≥ , P τ j = 1, will have boundary in ∆ εe iθ :(6.4) e iθ τ (1 + εe iθ q , . . . , εe iθ q n ) + · · · + e iθ τ p ( εe iθ q p , . . . , εe iθ q pp , . . . , εe iθ q pn )+ τ p +1 ( εe iθ q p +1 , , . . . , εe iθ q p +1 ,p +1 , . . . , εe iθ q p +1 ,n ) + · · · + τ n ( εe iθ q n , . . . , εe iθ q nn + 1) . Consider for example ∂τ ε,ε,θV where V is the orbit closure corresponding to the blowup of A = · · · = A p = 0.Take in (6.4) τ , . . . , τ p ≤ ε so terms in τ j ε may be neglected for j ≤ p . Take u j := τ j /τ k where k > p ischosen so that (say) τ k ≥ /n . As a consequence, u , . . . , u p ≤ nε . The corresponding projective point canthen be written(6.5) h e iθ ( u + ε ) + O ( ε ) , . . . , e iθ ( u p + ε ) + O ( ε ) , u p +1 + e iθ ε + O ( ε ) , . . . , u n + e iθ ε + O ( ε ) i . The boundary is given by setting one or more of the u j = 0. Points in ∂τ ε,ε,θV can be approximated by points(6.5) which then deform into ∆ εe iθ . To see this, note that since V is a codimension 1 orbit closure, therewill locally be one coordinate on P (Γ) near V which takes the constant value εe iθ on ∂τ ε,ε,θV (cf. fig.(10)).On the other hand, (6.5) is in homogeneous coordinates on P (Γ). To transform to P (Γ) near a general pointof V , one fixes ℓ ≤ p and looks at ratios(6.6) e iθ ( u j + ε ) + O ( ε ) e iθ ( u ℓ + ε ) + O ( ε )for 1 ≤ j = ℓ ≤ p . Clearly, at the boundary u ℓ = 0 we will get p − u j /ε + O ( ε ) which are closeto R ≥ , and one coordinate (corresponding to the local defining equation for V ) of the form εe iθ + O ( ε ).The remaining coordinates on V are ratios of the u j + εe iθ + O ( ε ) , j ≥ p + 1. Since u k = 1, these ratiosare again close to R ≥ . The calculation for orbit closures V of codimension ≥ P (Γ) is similar and is leftfor the reader. We have proven roposition 6.3. With notation as above, the monodromy of the chain σ ε ∈ H n − ( P n − − X Γ , ∆ ε − X Γ ∩ ∆ ε ) is represented by the chains ˜ c η,ε,θ given by modifying the chains c η,ε,θ to have boundary in ∆ εe iθ . In particular,the monodromy m ( σ ε ) = ˜ c η,ε, π is given by (6.7) m ( σ ε ) = X V ( − | V | ˜ τ ε,εV where ˜ τ ε,εV is the chain τ ε,εV defined in section 4 with boundary extended to ∆ ε as above. It will be convenient to simplify the notation and write(6.8) τ εV := ˜ τ ε,εV . Parametric representations
In this section we list well-known representations of the Feynman rules and then prepare for a subsequentanalysis of short-distance singularities in terms of mixed Hodge structures.7.1.
Kirchhoff–Symanzik polynomials.
Let ψ (Γ) = X T Y e T A e , (7.1) φ (Γ) = X T ∪ T = T Q ( T ) · Q ( T ) Y e T ∪ T A e , (7.2)be the two homogenous Kirchhoff–Symanzik polynomials [10, 22]. Here, T is a spanning tree of the 1PIgraph Γ and T , T are disjoint trees which together cover all vertices of Γ. Also, Q ( T i ) is the sum of allexternal momenta attached to vertices covered by T i . Note that φ (Γ) can be written as(7.3) X kinetic invariants ( q i · q j ) R q i · q j . Here, q i are external momenta attached to T and q j to T , and R q i · q j are rational functions of the edgevariables only, and the sum is over independent such kinematical invariants where momentum conservationhas been taken into account. We extend the definition to the empty graph I by ψ ( I ) = 1, φ ( I ) = 0.Let | · | γ denote the degree of a polynomial with regard to variables of the graph γ . Lemma 7.1. i) deg φ = deg ψ + 1 .ii) (7.4) ψ (Γ) = ψ (Γ //γ ) ψ ( γ ) + ψ Γ ,γ with | ψ Γ ,γ | γ > | ψ ( γ ) | γ for all core graphs Γ and subgraphs γ .iii) (7.5) φ (Γ) = φ (Γ //γ ) ψ ( γ ) + φ Γ ,γ with | φ Γ ,γ | γ > | ψ ( γ ) | γ for all core graphs Γ and subgraphs γ . Proof: i) by definition, ii) has been proved in [2], iii) follows similarly by noting that the two-trees of φ areobtained from the spanning trees of ψ by removing an edge. If that edge belongs to Γ //γ , we get φ (Γ //γ ) ψ ( γ ).If it belongs to γ , we get a monomial m with | m | γ > | ψ ( γ ) | γ . (cid:3) Note that it might happen that φ (Γ //γ ) = 0, if the external momenta flows through subgraphs γ only. Insuch a case (which can lead to infrared divergences) one easily shows φ Γ ,γ = ψ (Γ //γ ) φ ( γ ). .2. Feynman rules. ¿From these polynomials one constructs the Feynman rules of a given theory. Forexample we have in φ theory for a vertex graph Γ, sdd(Γ) = 0,(7.6) Φ(Γ) = Z R k>ǫ e − P edges e A e m e − φ (Γ) ψ (Γ) ψ (Γ) dA · · · dA | Γ [1] | . We will write R >ǫ dA Γ to abbreviate the affine chain of integration.The integral is over the k -dimensional hypercube of positive real coordinates in R >ǫ with a small strip ofwidth 1 ≫ ǫ > C ∋ ι (Γ) := e − P e A e m e − φ (Γ) ψ (Γ) ψ (Γ) ,ι (Γ) = ι (cid:0) Γ)( { m } , { q i · q j } , { A } (cid:1) as a function of the set of internal masses { m } , the set of external momenta { q i · q j } (which can be considered as labels on external half-edges) and the set of graph coordinates { A } ,and ι takes values in C . We often omit the A dependence and abbreviate P = { m } , { q i · q j } for all theseexternal parameters of the integrand: ι = ι ( P ). The renormalization schemes we consider are determinedby the condition that the Green function shall vanish at a particular renormalization point R , so thatrenormalization becomes an iterated sequence of subtractions(7.8) ι − ( P, R ) := ι ( P ) − ι ( R ) . We let sdd(Γ) be the superficial degree of divergence of a graph Γ given as (see also Eq.(2.2) for a refinedversion)(7.9) sdd(Γ) := D | Γ | − X edges e w e − X vertices v w v , where | Γ | is the rank of the first Betti homology, D the dimension of spacetime which we keep as an integer, w e the weights of the propagator for edge e as prescribed by free field theory and w v the weight of the vertexas prescribed by the interaction Lagrangian. Note that we can set the width ǫ to zero, R >ǫ dA Γ → R > dA Γ if the integrand ι − (Γ) is evaluated on a graph Γ which has no divergent subgraphs.Throughout, we assume that all all masses and external momenta are in general position so that there areno zeroes in the φ -polynomial off the origin for positive values of the A variables. In particular, we assumethat the point P is chosen appropriately away from all mass-shell and kinematical singularities. We remindthe reader of the notation (Γ , σ ) (section (2.5)) where σ stores all the necessary detail on how to evaluatethe graph Γ.A special role is played by the evaluations (Γ , σ P =0 ). They set all internal masses and momenta to zero.Note that this leads immediately to infrared divergences: the Feynman integrands ι ( · )( P = 0) are missingthe exponential in the numerator, which provides a regulator at large values of the A variables, and hence aninfrared regulator. The ultraviolet singularities at small values of the A variables are taken into account bythe renormalization procedure itself, and hence by our limiting mixed Hodge structure. We will eliminate thecase P = 0 below using that ι − evaluates to zero if there is no dependence on masses or external momenta.7.3. General remarks on renormalization and QFT.
We now consider the renormalization Hopf alge-bra H R of 1PI Feynman graphs in section (2.4). We use the notation(7.10) ∆(Γ) = X γ γ ⊗ Γ //γ, for its coproduct. Also, ∆( I ) = I ⊗ I . Projection P into the augmentation ideal on the rhs is written as(7.11) (id ⊗ P )∆(Γ) = X ∅6 =Γ //γ γ ⊗ Γ //γ, so that for example the antipode S is(7.12) S (Γ) = − X ∅6 =Γ //γ S ( γ )Γ //γ =: − ¯Γ . urthermore, we introduce a forest notation for the antipode:(7.13) S (Γ) = X [for] ( − | [for] | Γ // [for] | [for] | Y j =1 γ [for] ,j , where the sum is over all forests [for] and the product is over all subgraphs which make up the forest. Here,a forest [for] is a possibly empty collection of proper superficially divergent 1PI subgraphs γ [for] ,j of Γ whichare mutually disjoint or nested. We call a forest [for] maximal if Γ // [for] is a primitive element of the Hopfalgebra. As edge sets(7.14) Γ = (Γ // [for]) ∪ ( ∪ j γ j ) . This is in one-to-one correspondence with the representation of the antipode as a sum over all cuts on rootedtrees ρ T (Γ) as detailed in section (2.3) above. The integer | [for] | is the number of edges removed in thisrepresentation.Let us first assume that the graph Γ and all its core subgraphs have a non-positive superficial degree ofdivergence, so they are convergent or provide log-pole: sdd ≤ ι (Γ)( P ) depends on P = { m } , { q i · q j } only through the argument of the exponential,we redefine the second Kirchhoff–Symanzik polynomial as follows:(7.15) φ (Γ)( { q i · q j } ) → ϕ (Γ)( P ) := φ (Γ)( { q i · q j } ) + ψ (Γ) X e A e m e . Then, the unrenormalized integrand is(7.16) ι (Γ)( P ) = exp − ϕ (Γ)( P ) ψ (Γ) ψ (Γ) . With this notation, the renormalized integrand is (in all sums and products over j here and in the following, j runs from 1 to | [for] | ) ι R (Γ)( P, R ) = X [for] ( − [for] exp − (cid:16) ϕ (Γ // [for])( P ) ψ (Γ // [for]) + P j ϕ ( γ j )( R ) ψ ( γ j ) (cid:17) ψ ( γ// [for]) Q j ψ ( γ j ) − X [for] ( − [for] exp − (cid:16) ϕ (Γ // [for])( R ) ψ (Γ // [for]) + P j ϕ ( γ j )( R ) ψ ( γ j ) (cid:17) ψ ( γ// [for]) Q j ψ ( γ j )(7.17) =: ¯ ι (Γ)( P, R ) + S ι (Γ)( R ) , where + S ι (Γ)( R ) = − ¯ ι (Γ)( R, R ) is the integrand for the counterterm, and ¯ ι (Γ)( P, R ), the integrand inthe first line, delivers upon integrating Bogoliubov’s ¯ R operation. Note that this formula (7.17) is just theevaluation(7.18) m ( S ιR ⊗ ι )∆(Γ) , which guarantees that the corresponding Feynman integral exists in the limit ǫ → ǫ to ∞ each edge variable. For the renormalizedFeynman integral Φ R (Γ)( P ) we can take the limit ǫ →
0, while for the ¯ R -operation(7.19) ¯Φ(Γ)( P, R ; ǫ ) = Z ǫ ¯ ι (Γ)( P, R ) , and the counterterm(7.20) S Φ R ; ǫ (Γ) = − ¯Φ(Γ)( R, R ; ǫ ) , the lower boundary remains as a dimension-full parameter in the integral. Note that the result (7.17) abovecan also be written in the P − R form, typical for renormalization schemes which subtract by constraints on hysical parameters:(7.21) ι R (Γ)( P, R ) = X ∅6 =Γ //γ [ ι (Γ //γ )( P ) − ι (Γ //γ )( R )] S ιR ; ǫ ( γ ) , and as(7.22) ¯ ι (Γ)( P, R ) = X ∅6 =Γ //γ S ιR ; ǫ ( γ ) ι (Γ //γ )( P ) ⇒ ι R (Γ)( P, R ) = X γ S ιR ; ǫ ( γ ) ι (Γ //γ )( P ) , using the notation (7.11,7.10). Similarly, for Feynman integrals,(7.23) ¯Φ(Γ)( P, R ; ǫ ) = X ∅6 =Γ //γ S Φ R ; ǫ ( γ )Φ(Γ //γ )( P ) , Φ R (Γ)( P ) = lim ǫ → X γ S Φ R ; ǫ ( γ )Φ(Γ //γ )( P ) . When it comes to actually calculating the integral (7.6) (or, in its renormalized form (7.17)), somethingrather remarkable happens. By lemma 7.1(i), the term in the exponential in these integrals is homogeneousof degree 1 in the edge variables A i . The assumption sdd(Γ) = 0 means dA/ψ is homogeneous of degree 0.Making the change of variable A i = ta i , we find(7.24) dA/ψ ( A ) = dt/t ∧ ( X ( − j − a j da ∧ · · · ∧ d da j ∧ · · · ) /ψ ( a ) = dt/t ∧ Ω /ψ . Note that Ω /ψ is naturally a meromorphic form on the projective space P (Γ) with homogeneous coordinatesthe a i . Writing σ = { a i ≥ } ⊂ P (Γ)( R ), we see that the renormalized integral can be rewritten up to aterm which is O ( ε ) as a sum of terms of the form(7.25) Z σ Ω /ψ j Z ∞ ε (cid:16) e ( − tf j ( a )) − e ( − tg j ( a )) (cid:17) dt/t = Z σ Ω /ψ j (cid:16) E ( εf j ( a )) − E ( εg j ( a )) (cid:17) , where(7.26) E ( z ) := Z ∞ e − tz dtt = − γ E − ln z + O ( z ); z → f j ( a ) , g j ( a ) are defined by taking the locus a i ≥ , P a i = 1.) As long as f j ( a ) , g j ( a ) >
0, we may allow ε → a . The Euler constant and log ε terms cancel. When the dustsettles, we are left with the projective representation for the renormalized Feynman integral(7.27) Φ R (Γ)( P ) = Z σ Ω Γ X [for] ( − [for] ln (cid:16) ϕ (Γ // [for])( P ) Q j ψ ( γ j )+ P j ϕ ( γ j )( R ) ψ (Γ // [for]) Q h = j ψ ( γ h ) ϕ (Γ // [for])( R ) Q j ψ ( γ j )+ P j ϕ ( γ j )( R ) ψ (Γ // [for]) Q h = j ψ ( γ h ) (cid:17) ψ (Γ // [for]) Q j ψ ( γ j ) . Note that the use of σ is justified as long as the integrand has all subdivergences subtracted, so is in the ¯ ι form, so that lower boundaries in the a i variables can be set to zero indeed.By (7.21), this can be equivalently written as(7.28) Φ R (Γ)( P ) = lim ǫ → X γ S Φ R ; ǫ ( γ ) Z >ǫ dA Γ //γ ι − (Γ //γ )( P, R ) , in any renormalization scheme which is described by kinematical subtractions P → R . Remark 7.2.
It will be our goal to replace the affine R dA by the projective R d Ω in the above. The presenceof lower boundaries, which can not be ignored as the integrand has divergent subgraphs, allows this only uponintroducing suitable chains τ ǫγ as discussed in previous sections. Next, we relax the case of log-divergence. .4. Reduction of graphs with ssd(Γ) > . We start with an example. To keep things simple but not toosimple, we consider the one-loop self-energy graph in φ theory, a scalar field theory with a cubic interactionin six dimensions of space-time. We have(7.29) Φ(Γ)( P ) = Z >ǫ dA Γ ι (Γ)( P ) = Z >ǫ dA Γ e − ϕ (Γ) ψ (Γ) ψ (Γ) ≡ Z ∞ ǫ dA dA e − m A A q A A A A ( A + A ) . We will renormalize by suitable subtractions at chosen values of masses and momenta in the ϕ -polynomial.We hence (with subdivergences taken care of by suitable bar-operations ι → ¯ ι in the general case) replace ι (Γ)( P ) by ι (Γ)( P ) − ι (Γ)(0), as this leaves ι − (Γ)( P, R ) invariant.Then the above can be written, with this subtraction, and by the familiar change of variables A i = ta i ,and by one partial integration in t ,Φ(Γ)( P ) = Z σ d Ω Z ∞ ǫ dtt [ m ( a + a ) + q a a ] e − t m a a q a a a a ( a + a ) − Z σ d Ω [ m ( a + a ) + q a a ]( a + a ) , (7.30)where we expanded the boundary term up to terms constant in ǫ , which gave the term in the second line. Wediscarded already the pure pole term ∼ /ǫ from Φ(Γ)( P = 0) = R >ǫ dA ( A + A ) = R ∞ ǫ dt/t R ∞ db / (1 + b ) .Note that graphs Γ with sdd > q say, φ (Γ) = φ (Γ)( q ), for which we write φ (Γ) q .The result in (7.30) leads us to define two top-degree forms. (Here Ω = a da − a da and we still write φ, ψ for the Kirchhoff–Symanzik polynomials regarded as dependent on either a i or A i variables below).(7.31) ω (cid:3) = ω (cid:3) (Γ) = Ω φ (Γ) ψ (Γ) = Ω a a ( a + a ) , and(7.32) ω m = ω m (Γ) = Ω ( a + a ) ψ (Γ) = Ω 1( a + a ) , so that Φ(Γ)( P ) = − m Z σ [ ω (cid:3) + ω m ] − ( q − m ) Z σ ω (cid:3) (7.33) + m Z σ [ ω (cid:3) + ω m ] Z ∞ ǫ dtt e − t ϕ (Γ)( P ) ψ (Γ) +( q − m ) Z σ ω (cid:3) Z ∞ ǫ dtt e − t ϕ (Γ)( P ) ψ (Γ) . There are corresponding affine integrands ι (cid:3) (Γ) = φ (Γ) ψ (Γ) e − ϕ (Γ)( P ) ψ (Γ) , (7.34) ι m (Γ) = ( a + a ) ψ (Γ) e − ϕ (Γ)( P ) ψ (Γ) . (7.35)The graph Γ is renormalized by a choice of a renormalization condition R (cid:3) for the coefficient of q − m (wave function renormalization), and by the choice of a condition R m for the mass renormalization. R isoften still used to denote the pair of those.(7.36) Φ(Γ)( P ) + m δ m + q z (cid:3) = Φ R (cid:3) ,R m (Γ)( P ) . The mass counterterm is then(7.37) m δ m = − m Z σ [ ω (cid:3) + ω m ] (cid:18) − Z ∞ ǫ dtt e − t ϕ (Γ)( Rm ψ (Γ) (cid:19) , nd the wave-function renormalization q z (cid:3) is(7.38) q z (cid:3) = − q Z σ ω (cid:3) (cid:18) − Z ∞ ǫ dtt e − t ϕ (Γ)( R (cid:3) ) ψ (Γ) (cid:19) . Note the term 1 in the () brackets does not involve exponentials.The corresponding renormalized contribution is(7.39) Φ R (Γ)( P ) = ( q − m ) Z σ ω (cid:3) ln ϕ ( P ) ϕ ( R (cid:3) ) + m Z σ [ ω (cid:3) + ω m ] ln ϕ (Γ)( P ) ϕ (Γ)( R m ) . The transition from the unrenormalized contribution to the renormalized one is particularly simple upondefining Feynman rules in accordance with external leg structures:Φ((Γ , σ (cid:3) )) = ( q − m ) Z >ǫ dA φ (Γ) e − ϕ (Γ)( P ) ψ (Γ) ψ (Γ) D/ , (7.40) Φ((Γ , σ m )) = m Z >ǫ dA [ φ (Γ) + ψ (Γ) P e A e ] e − ϕ (Γ)( P ) ψ (Γ) ψ (Γ) D/ , (7.41)(7.42)so that renormalization proceeds as before on log-divergent integrands.This example extends straightforwardly to the case of Γ having divergent subgraphs. Let us return to φ theory and define for a core graph Γ with sdd(Γ) = 2, (so that it is a self-energy graph and hence has onlytwo external legs, and thus a single kinematical invariant q ), and graph-polynomials ψ (Γ), φ (Γ) = φ q (Γ), ϕ (Γ) = ϕ (Γ)( P ) = φ q (Γ) + ψ (Γ) P e A e m e , the forms(7.43) ω (cid:3) (Γ) = Ω Γ φ (Γ) ψ (Γ) , (7.44) ω m (Γ) = Ω Γ φ (Γ) + ψ (Γ) P e A e ψ (Γ) . The corresponding complete affine integrands ι (cid:3) , ι m are immediate replacing a i by A i variables, and mul-tiplying by exponentials exp − ϕ (Γ)( P ) /ψ (Γ), with P → R for counterterms.One finds by a straightforward computationΦ R (cid:3) ((Γ , σ (cid:3) ))( P ) = X γ S Φ R ; ǫ ( γ ) Z ǫ ω (cid:3) (Γ //γ ) ln ϕ (Γ //γ )( P ) ϕ (Γ //γ )( R (cid:3) )(7.45) = Z Ω Γ X [for] ( − [for] ω (cid:3) (Γ // [for]) ln ϕ (Γ // [for])( P ) ϕ (Γ // [for])( R (cid:3) ) , (7.46)and Φ R m ((Γ , σ m ))( P ) = X γ S Φ R ; ǫ ( γ ) Z ǫ ω m (Γ //γ ) ln ϕ (Γ //γ )( P ) ϕ (Γ //γ )( R m )(7.47) = Z Ω Γ X [for] ( − [for] ω m (Γ // [for]) ln ϕ (Γ // [for])( P ) ϕ (Γ // [for])( R m ) . (7.48)We set(7.49) Φ R (Γ)( P ) ≡ Φ R ((Γ , I ))( P ) = φ R (cid:3) ((Γ , σ (cid:3) ))( P ) + Φ R m ((Γ , σ m ))( P ) , in the external leg structure notation of section (2.5). We can combine the results for graphs Γ for all degreesof divergence sdd(Γ) ≥ ω (Γ) = Ω /ψ (Γ) for a log divergent graph with the results above. Andthat’s that. Well, we have to hasten and say a word about the Feynman rules when the subgraphs γ havesdd( γ ) >
0, and hence also about S Φ R ; ǫ ( γ ) in that case.We use, with P the projection into the augmentation ideal, the notation(7.50) ¯Γ = Γ + m ( S ◦ P ⊗ P )∆ =: Γ + (Γ ′ ) − Γ ′′ . et us consider the quotient Hopf algebra given by quadratically divergent graphs: ∆ (Γ) = P γ, sdd( γ )=2 γ ⊗ Γ //γ . We write(7.51) ∆ (Γ) =: Γ ⊗ I + I ⊗ Γ + Γ ′ ⊗ Γ ′′ . We add 0 = +Γ ′ − Γ ′′ − Γ ′ − Γ ′′ , so¯Γ = Γ + Γ ′ − Γ ′′ − Γ ′ − Γ ′′ + Γ ′− Γ ′′ (7.52) = (cid:16) Γ + Γ ′ − Γ ′′ (cid:17) + (cid:16) Γ ′− − Γ ′ − (cid:17) Γ ′′ . (7.53)Here the sum is over all terms of the coproduct with the Γ ′ terms being present whenever Γ ′ is quadraticallydivergent.Evaluating the terms Γ ′ by 1 /ψ ( γ ′ ) = ι ( γ ′ )( P = 0) decomposes the bar-operation on the level ofintegrands as follows.(7.54) ¯ ι (Γ)( P ) = I z }| {(cid:16) ι (Γ)( P ) + ι (Γ ′ − )( P = 0) ι (Γ ′′ )( P ) (cid:17) + ι (Γ ′− )( R ) ι (Γ ′′ )( P ) , where ι (Γ ′− )( R ) ≡ S ιR ; ǫ (Γ ′ ) appears because a subtraction of a P = 0 term, from a quadratically divergentterm, precisely delivers those counterterms by our previous analysis. Note that they contain terms which donot have an exponential, as in the example (7.38,7.37). Often, as a two-point vertex of mass type improvesthe powercounting of the co-graph, we might keep self-energy subgraphs massless, in which case only termsinvolving R (cid:3) contribute.We are left to decompose the terms denoted I . We find by direct computation I = II z }| {h ω (Γ) + ω (Γ ′ − ) ω (Γ ′′ ) i e − ϕ (Γ)( P ) ψ (Γ) (7.55) − ω (Γ ′ − ) ω (Γ ′′ ) " e − ϕ (Γ)( P ) ψ (Γ) − e − ϕ (Γ // Γ ′ P ) ψ (Γ // Γ ′ III . (7.56)The terms denoted II gives us the final integrand ι (Γ)( P ) with a corresponding form ω II (Γ). ω II (Γ) = ω (Γ)if there are no subgraphs with sdd = 2. Note that II has the full Γ as an argument in the commonexponential,(7.57) II = ω II (Γ) exp( − ϕ (Γ) /ψ (Γ)) , which defines ω II . The rational coefficient ω II has log-poles only for all subgraphs including the ones withsdd = 2.The terms III is considered in t, a i variables. We can integrate t as before. As the rational part of theintegrand factorizes in Γ ′ and Γ ′′ variables, we similarly decompose the former into s, b i , i ∈ Γ ′ , variables.We note s only appears in the log (after the t integration) as a coefficient of φ Γ , Γ ′ , using Lemma (7.1).Partial integration in s eliminates the log and delivers a top-degree form for the b i integration. These termsprecisely compensate against the constant terms mentioned above, as φ Γ , Γ ′ = φ (Γ ′ ) φ (Γ − Γ ′ ), using thatres(Γ ′ ) = 2.We hence summarize Theorem 7.3. (7.58) Φ R (Γ)( P ) = lim ǫ → X γ S Φ R ; ǫ ( γ ) Z ǫ ω II (Γ //γ ) ln ϕ (Γ //γ )( P ) ϕ (Γ //γ )( R ) . It is understood that each counterterm is computed with a subtraction R as befits its argument γ , and forms Γ are chosen in accordance with the previous derivations. Here, ω II is constructed to have log-poles only. s a projective integral this reads Φ R (Γ)( P ) = Z Ω Γ X [for] ( − [for] ×× ln ϕ (Γ // [for])(P) Q j ψ ( γ j ) + P j ϕ ( γ j )(R) ψ (Γ // [for]) Q h =j ψ ( γ h ) ϕ (Γ // [for])(R) Q j ψ ( γ j ) + P j ϕ ( γ j )(R) ψ (Γ // [for]) Q h =j ψ ( γ h ) ! × ω (Γ // [for]) Y j ω ( γ j ) . (7.59) Remark 7.4.
Similar formulas can be obtained for the bar-operations and counterterms, with the samerational functions in the integrands, and exponentials exp( − ϕ (Γ //γ )( X ) /ψ (Γ //γ )) , with X = P or X = R as needed. Remark 7.5.
We have worked with choices of renormalizations for mass and wave functions, R → R (cid:3) , R m .One can actually also define P → P (cid:3) , P m , and for example set masses to zero in all exponentials ( ϕ ( · )( P ) → φ q ( · ) ), that’s essentially the Weinberg scheme if one then subtracts at q = µ . Remark 7.6.
This all is nicely reflected in properties of analytic regulators. For example in dimensionalregularization the identity R d D k [ k ] ρ = 0 , ∀ ρ , leads to Φ(Γ)( P = 0) = 0 immediately, where Φ now indicatesunrenormalized Feynman rules using that regulator. Remark 7.7.
We are working so far with constant lower boundaries. The chains introduced in previoussections have moving lower boundaries which respect the hierarchy in each flag. We will study that differencein section (9.1).
Specifics of the MOM-scheme.
We define the MOM-scheme by setting all masses to zero in radiativecorrections and keeping a single kinematical invariant q in the φ -polynomial, P = { } , { q i · q j ∼ q } ,(7.60) φ (Γ) = q R q (Γ) . Such a situation arises if we set masses to zero (possibly after factorization of a polynomial part from theamplitude as in the Weinberg scheme), and for vertices if we consider the case of zero momentum transfers,or evaluate at a symmetric point q i = q , where i denotes the external half-edges of Γ. If we want toemphasize the q dependence we write φ q . Trivially, φ q = q φ . In the MOM-scheme, subtractions aredone at q = µ , which defines R for all graphs. Counter-terms in the MOM-scheme become very simplewhen expressed in parametric integrals thanks to the homogeneity of the φ -polynomial. Note that we hencehave ϕ (Γ) = φ (Γ) as we have set all masses to zero.In a MOM-scheme, renormalized diagrams are polynomials in ln q /µ : Theorem 7.8.
For all Γ , (7.61) Φ MOM (Γ)( q /µ ) = aug(Γ) X j =1 c j (Γ) ln j q /µ . Here, aug(Γ) = max [for] | [for] | .Proof: Consider a sequence γ ( γ · · · γ aug(Γ) ( Γ. This is in one-to-one correspondence with some decoratedrooted tree appearing in ρ R (Γ) (2.35). Choose one edge e j ∈ γ j /γ j − in each decoration and de-homogenizewith respect to that edge. We get a sequence of lower boundaries ǫ, ǫ/A , ǫ/A /A , · · · . Use the affine rep-resentation and integrate to obtain the result. (cid:3) MOM scheme results from residues.
In such a scheme, it is particularly useful to take a derivativewith respect to ln q . We consider(7.62) p (Γ) := q ∂ q Φ MOM (Γ)( q /µ ) | q µ , where we evaluate at q = µ after taking the derivative. This number, which for a primitive element ofthe renormalization Hopf algebra is the residue of that graph in the sense of [2], is our main concern for ageneral graph. It will be obtained in the limit of the limiting mixed Hodge structure we construct. emark 7.9. It is not that this limit would not exist for general schemes. But the limit would be a compli-cated function of ratios of masses and kinematical invariants, which has a constant term given by the number p (Γ) and beyond that a dependence on these ratios which demands a much finer Hodge theoretic study thanwe can offer here. But first we need to remind ourselves how coefficients of higher powers of logarithms of complicated graphsrelated to coefficients of lower powers of sub- and co-graphs thanks to the renormalization group.7.5.2.
The counterterm S ΦMOM . For S ΦMOM (Γ) =: P aug(Γ) j =1 s j (Γ) ln j µ , we simply use the renormalizationgroup or the scattering type formula. In particular, we have(7.63) S ΦM OM (Γ) = aug(Γ) X j =1 j ! ( − j [ p ⊗ · · · ⊗ p ] | {z } j factors ∆ j − (Γ) . This is easily derived [6, 17] upon noting that p (Γ) = Φ( S ⋆ Y (Γ)).Note that this determines counter-terms by iteration: for a k -loop graph, knowledge of all the lower ordercounterterms suffices to determine all contributions to the k -loop counterterm but the lowest order coefficientof ln µ . But then, that coefficient is given by the formula(7.64) s (Γ) = p (Γ) ln µ , which itself only involves counter-terms of less than k loops, by the structure of the bar operation.7.5.3. p (Γ) from co-graphs. We can now summarize the consequences of the renormalization group and ourprojective representations for parametric representations of Feynman integrals. The interesting question isabout the logs which we had in numerators. Thm.(7.3) becomes
Theorem 7.10. (7.65) p (Γ) = lim ǫ → X γ S ΦMOM; ǫ ( γ ) q ∂ q Z ǫ ω II (Γ //γ ) ln φ q /µ (Γ //γ ) . This limit is p (Γ) = Z Ω Γ X [for] ( − [for] ×× q ∂ q ln φ q /µ (Γ // [for]) Y j ψ ( γ j ) + X j φ ( γ j ) ψ (Γ // [for]) Y h =j ψ ( γ h ) × ω (Γ // [for]) Y j ω ( γ j ) . (7.66) The derivative with respect to ln q can be taken inside the integral in (7.65) if and only if all edges carryingexternal momentum are in the complement C (Γ) of all edges belonging to divergent subgraphs. In that case, q ∂ q ln φ q (Γ //γ ) = 1 and no logs in the numerator appear. Remark 7.11.
Note that overlapping divergent graphs can force all edges to belong to divergent subgraphs,cf. Fig.(5).
Proof: If all edges carrying external momentum are in the complement to divergent subgraphs, we bring thecounter-terms under the integrand using the bar-operation. We can take the limit ǫ → q by assumption: each φ q (Γ //γ ) is a linear combination of terms A e ψ e (Γ //γ ), where e is in that complement C (Γ) of subgraph edges, and ψ e (Γ //γ ) = ψ (Γ //γ/e ). Applying then the Chen-Wu theorem [21] with respectto the elements of C (Γ) disentangles the q dependence from the limit in ǫ . (cid:3) Figure 14.
The Dunce’s cap, again. We label the edges 1 , , ,
4. Resolved in trees, wefind three trees in the core Hopf algebra. We label the vertices by edge labels of thegraph. The sets 123 and 124 correspond to a triangle graph as indicated, the sets 12 and34 are one-loop vertex graphs, and tadpoles appear in the coproduct on the rhs for edges3 or 4. The coproduct in the core Hopf algebra is, expressed in edge labels, ∆ ′ (1234) =123 ⊗ ⊗ ⊗
12. Only the last term appears in the renormalization Hopf algebra.
Remark 7.12.
Note that the discussion below with respect to the limiting Hodge structure assumes that wehave this situation of disentanglement of divergent subgraphs and edges carrying external momentum. Wehence have no logarithms in the numerator. But note that the general case does no harm to the ensuingdiscussion: by Lemma (7.1), any logarithms in the numerator are congruent to one along any exceptionaldivisor of X Γ // [for] . Furthermore, when external momentum interferes with subgraphs, all logs can be turnedto rational functions by a partial integration. The fact that the second Kirchhoff–Symanzik polynomial isa linear combination of ψ -polynomials, applied to graphs with an extra shrunken edge, in the MOM-caseestablishes these rational functions to have poles coming from our analysis of this ψ (Γ) polynomial. A fullmathematical discussion of this ” R ω ln f ” situation should be subject to future work. Examples. ¿From now on we measure q in units of µ so that subtractions are done at 1. Thissimplifies notation. Let us first consider the Dunce’s cap in detail, (14). We have the following data(path q (Γ) refers to the momentum path through the graph): ψ (Γ) = ( A + A )( A + A ) + A A , (7.67) ψ ( γ ) = A + A , ψ (Γ //γ ) = A + A , (7.68) path q (Γ) = e , (7.69) φ (Γ) = A ( A A + A A + A A ) = A ψ (Γ //e ) = A ψ (Γ) , (7.70) φ ( γ ) = A A , φ (Γ //γ ) = A A , (7.71) { [for] } = {∅ , (34) } . (7.72)(7.73) Φ(Γ) ǫ ( q ) = Z ∞ ǫ Y i =1 dA i exp − q φ (Γ) ψ (Γ) ψ (Γ) . ence we choose a function τ ( ǫ ) which goes to zero rapidly enough so that lim ǫ → τ ( ǫ ) /ǫ = 0 and compute¯Φ ǫ (Γ)( q , µ ) = Z ∞ ǫ dA dA Z ∞ τ ( ǫ ) dA dA exp − q φ (Γ) ψ (Γ) ψ (Γ) − exp h − q φ (Γ //γ ) ψ (Γ //γ ) i ψ (Γ //γ ) exp h − φ ( γ ) ψ ( γ ) i ψ ( γ ) (7.74) = Z ∞ q ǫ dA dA Z ∞ τ ( ǫ ) q dA dA exp − φ (Γ) ψ (Γ) ψ (Γ) − Z ∞ q ǫ dA dA Z ∞ τ ( ǫ ) dA dA exp h − φ (Γ //γ ) ψ (Γ //γ ) i ψ (Γ //γ ) exp h − φ ( γ ) ψ ( γ ) i ψ ( γ ) . (7.75)Let us now re-scale to variables A i → A B i for all variables i ∈ , ,
4. We get¯Φ ǫ (Γ)( q , µ ) = Z ∞ q ǫ dA A Z ∞ q ǫ dB Z ∞ q τ ( ǫ ) /A dB dB exp − A B B + B B + B B )(1+ B )( B + B )+ B B [(1 + B )( B + B ) + B B ] − (Z ∞ q ǫ dA A Z ∞ q ǫ dB Z ∞ τ ( ǫ ) /A dB dB exp − A B B (1 + B ) exp − A B B B + B ( B + B ) ) . (7.76)We re-scale once more B = B C . Also, we set the lower boundaries in the B and C integrations to zero.This is justified as A and B remain positive.¯Φ ǫ (Γ)( q , µ ) = Z ∞ q ǫ dA A Z ∞ dB Z ∞ q τ ( ǫ ) /A dB B Z ∞ dC exp − A B + B C + C B )(1+ B )(1+ C )+ B C [(1 + B )(1 + C ) + B C ] − Z ∞ q ǫ dA A Z ∞ dB Z ∞ τ ( ǫ ) /A dB B Z ∞ dC ( exp − A B B (1 + B ) exp − A B C C (1 + C ) ) . (7.77)Taking a derivative wrt ln q and using that lim ǫ → τ ( ǫ ) /ǫ = 0, delivers three remaining terms ∂ ln q ¯Φ ǫ (Γ) q =1 = Z ∞ dB Z ∞ τ ( ǫ ) /ǫ dB B Z ∞ dC (cid:26) B )(1 + C ) + B C ] (cid:27) − Z ∞ dB Z ∞ τ ( ǫ ) / ( q ǫ ) dB B Z ∞ dC ( B ) e − ǫq B C C (1 + C ) ) + Z ∞ q ǫ dA A Z ∞ dB Z ∞ dC ( e − A B B [(1 + B )(1 + C )] ) . (7.78)Integrating B in the second line and A in the third, we find ∂ ln q ¯Φ ǫ (Γ) q =1 = Z ∞ dB Z ∞ τ ( ǫ ) /ǫ dB B Z ∞ dC B )(1 + C ) + B C ] + ln τ ( ǫ ) /ǫ Z Ω γ ψ ( γ ) Z Ω Γ //γ ψ (Γ //γ ) . (7.79)Using the exponential integral, those B and A integrations also deliver finite contributions(7.80) − Z ∞ dB Z ∞ dC ( B ) ln C C (1 + C ) ) + Z ∞ dB Z ∞ dC ( ln B B [(1 + B )(1 + C )] ) = 0 . his cancellation of logs is no accident: while in this simple example it looks as if it originates from the factthat the co-graph and subgraph are identical, actually the cross-ratio(7.81) ln φ (Γ //γ ) ψ ( γ ) ψ (Γ //γ ) φ ( γ )vanishes identically when integrated against the de-homogenized product measure(7.82) Z dA Γ //γ dA γ ψ (Γ //γ ) ψ ( γ ) . This is precisely because C (Γ) = e has an empty intersection with γ [1] = e , e .But then, this cancelation of logs will break down if φ (Γ) is not as nicely disentangled from φ ( γ ) forall log-poles as it is here, and will be replaced by logs congruent to 1 along subdivergences in general, inaccordance with Thm.(7.10).Let us study this in some detail. Consider the graph on the upper left in Fig.(7), and consider the finiteln φ/ψ -type contributions of the exponential integral to in the vicinity of the exceptional divisor for thesubspace A = A = 0.Routing an external momentum through edges 1,6, we have the following graph polynomials: φ (Γ) = A [ A A ( A + A ) + A A ( A + A ) + A ( A + A )( A + A )](7.83) + A A [( A + A )( A + A ) + A A ] φ (Γ /
34) = A [ A A + A ( A + A )] + A A [( A + A )](7.84) φ (34) = A A (7.85) ψ (Γ /
34) = ( A + A )( A + A ) + A A (7.86) ψ (34) = A + A . (7.87)We have C (Γ) = e , e , and ∪ γ ( Γ , res( γ ) ≥ γ [1] = e , e , e , e . The intersection is e . We hence find, withsuitable de-homogenization,(7.88) ln X z }| { B B (1 + B ) + Y z }| { B B + B ( B + B ) (1+ B )( B + B )+ B B − ln C C [(1 + B )( B + B ) + B B ] [1 + C ] dB dC dB dB . Here, the term X denotes a term which would be absent if the momenta would only go through edge 1 andhence the above intersection would be empty, while Y indicates the terms from the momentum flow throughedge 1.This is of the form ln( f Γ //γ /f γ )[ ω Γ //γ ∧ ω γ ]. If the term X would be absent, a partial integration(7.89) Z ∞ ǫ ln xu + vxu + w ( xu + w ) ∼ Z ∞ ǫ xu + w ) would show the vanishing of this expression as above. The presence of X leaves us with a contribution whichcan be written, replacing ln C / (1 + C ) by ln Y /ψ (Γ / X z }| { B B (1 + B ) + Y z }| { B B + B ( B + B ) B B + B ( B + B ) | {z } Y [(1 + B )( B + B ) + B B ] [1 + C ] . As promised, it is congruent to one along the remaining log-pole at A = A = 0. It has to be: the forestwhere the subgraph 56 shrinks to a point looses the momentum flow through edge 6 and could not contributeany counterterm for a pole remaining in the terms discussed above.Note that in general higher powers of logarithms can appear in the numerator as subgraphs can havesubstructure. Lacking a handle to notate all the log-poles which do not cancel due to partial integrationidentities known beyond mankind we consider it understood that all terms from the asymptotic expansionof the exponential integral up to constant terms (higher order terms in ǫ are not needed as all poles arelogarithmic only) are kept without being shown explicitly in further notation. We emphasize though that ll those logarithm terms in the numerator are congruent to one along log-poles -and deserve study in theirown right elsewhere-, and hence thanks to Lemma (7.1) which guarantees indeed all necessary cancelations,we have in all cases:(7.91) p (Γ) = lim ǫ → ∂ ln q ¯Φ ǫ (Γ) q =1 . Remark 7.13.
There is freedom in the choice of τ , a natural choice comes from the rooted tree representation ρ (Γ) of the forest. Each forest is part of a legal tree t and any subgraph γ corresponds to a vertex v in thattree. If d v is the distance of v to the root of t , τ ( ǫ ) = ǫ d v +1 is a natural choice. N for physicists: The antipode as monodromy. Let us now come back to the core Hopf algebra andprepare for an analysis in terms of limiting mixed Hodge structures. This will be achieved in two steps: ananalysis of the structure of the antipode of the renormalization Hopf algebra, which will then allow to definea matrix N for the monodromy in question such that S (Γ) can be expressed in a particularly nice way. Infact, because of orientations, the N which arises in the monodromy calculation is the negative of the N computed in this section. We omit the minus sign to simplify the notation.Let us consider the antipode first. Thanks to the above lemma we can write for the antipode S (Γ)(8.1) S (Γ) = − | Γ | X j =0 ( − j X | C | = j X t P C ( t ) R C ( t ) . Here, we abuse notation in an obvious manner identifying Γ and ρ T (Γ), the latter being the indicated sumover trees, in accordance with Eq.(2.34).We also define R (Γ) = − S (Γ). Let us now label the edges of each t (Γ) once and for all by 1 , , · · · , | Γ | − | Γ | − C with | C | = 1, and(8.2) (cid:18) | Γ | − j (cid:19) cuts of cardinality | C | = j . We hence can define a vector v (Γ) with 2 | Γ |− entries in H , ordered accordingto a never decreasing cardinality of cuts:(8.3) v (Γ) = (Γ , X t P C ( t ) R C ( t ) | {z } “ | Γ |− ” entries of cardinality 1 , · · · , X t P C ( t ) R C ( t ) | {z } “ | Γ |− j ” entries of cardinality j , · · · ) T . Example: Dunce’s cap with edges 1 , , , ,
4, comare Fig.(14). The core coproductis(8.4) ∆ ′ c = 123 ⊗ ⊗ ⊗ . The vector v is then(8.5) v = (cid:18) (cid:19) . Let N (2) be the to-by-two matrix(8.6) N (2) = (cid:18) (cid:19) . Note that(8.7) (cid:20)(cid:18) (cid:19) − N (2) (cid:21) (cid:18) (cid:19) = (cid:18) R (1234)(123)(4) + (124)(3) + (12)(34) (cid:19) , with(8.8) R (1234) = 1234 − (123)(4) − (124)(3) − (34)(12) . n fact, it is our first task to find a nilpotent matrix N , N | Γ | = 0, such that(8.9) | Γ |− X j =0 ( − j N j /j ! = ( R (Γ) , X t R ( P C ( t )) R ( R C ( t )) | {z } “ | Γ |− ” entries of cardinality 1 , · · · , X t R ( P C ( t )) R ( R C ( t )) | {z } “ | Γ |− j ” entries of cardinality j , · · · ) , ) T . For P C ( t ) = Q i t i we here have abbreviated R ( P C ( t )) for Q i R ( t i ).8.2. The matrix N . Let M (0 ,
1) be the space of matrices with entries in the two point set { , } .Let now m + 1 be the number of loops m = | Γ | − m × m square matrix N ≡ N ( m ) , N m +1 = 0, in M (0 ,
1) as follows.Consider first the m + 1-th row of the Pascal triangle, for example for m = 3 it reads 1 , , ,
1. For thisexample, we will then construct blocks of sizes 1 ×
1, 1 ×
3, 3 ×
3, 3 × ×
1, all with entries either 0or 1.So this gives us in general m +2 blocks M ( m ) j , 0 ≤ j ≤ m +1, of matrices of size M ( m )0 : 1 × M ( m )1 : 1 × m , M ( m )2 : m × m ( m − / · · · , M ( m ) m : m × M ( m ) m +1 : 1 × M ( m ) j , 0 ≤ j < ( m + 2) /
2, fill the columns, from left to right, by never increasing sequencesof binary numbers (read from top to bottom) where each such number contains j entries 1 for the block M ( m ) j . Put M ( m )0 = (0) in the left upper corner and M ( m )1 to the left of it. For j ≥
2, put the block M ( m ) j below and to the right of the block M ( m ) j − , in N . All entries in N outside these blocks are zero. Determinethe entries of the blocks M ( m ) j , m + 1 ≥ j ≥ ( m + 2) /
2, by the requirement that N ⊥ = N , where N ⊥ isobtained from N by reflection along the diagonal which goes from the lower left to the upper right. We write M ( m ) i ⊥ = M ( m ) m +1 − i . For odd integer m , we have M ( m )( m +1) / ⊥ = M ( m )( m +1) / , by construction. Here are M (3) j and N, N , N for m = 3:(8.10) M (3)0 = (0) , M (3)1 = (1 , , , M (3)2 = , M (3)3 = , M (3)4 = (0) . (3) = | | | |
00 0 0 0 | |
00 0 0 0 | |
00 0 0 0 0 0 0 | | | , (8.11) N (3)2 = , (8.12) N (3)3 = . (8.13)We can now write, for 1 ≤ j ≤ m ,(8.14) N j = j ! n ( m ) j , where the matrix n ( m ) j ∈ M (0 , n − LN ( m ) o = m X j =0 ( − L ) j j ! N ( m ) j = m X j =0 ( − L ) j n ( m ) j . This is obvious from the set-up above. Furthermore, directly from construction, n ( m ) j , j ≥
1, has a blockstructure into blocks of size(8.16) (1 × m ) , · · · , · · · |{z} j − , · · · , ( m × , located in the uppermost right corner of size 2 m − j +1 × m − j +1 as in the above example.8.3. Math:The Matrix N . In this section we compute the matrix N which gives the log of the monodromy.Because of orientations, the answer we get is the negative of the physical N computed in the previous section.Our basic result gives the monodromy(8.17) m ( σ ) = X I ( − p τ I = σ + X I, p ≥ ( − p τ I . Here we have changed notation. I = { i , . . . , i p } refers to a flag Γ i ( · · · ( Γ i p ( Γ of core subgraphs. Moregenerally(8.18) m ( τ I ) = X J ⊃ I ( − q − p τ J . Here J = { j , . . . , j q } ⊃ I . to verify (8.18), consider e.g. the case corresponding to Γ ( Γ. We haveseen (lemma 3.3) that the blowup of P (Γ) along the linear space defined by the edge variables associated toedges of Γ yields as exceptional divisor E ∼ = P (Γ ) × P (Γ // Γ ). In fact, the strict transform of E in thefull blowup P (Γ) can be identified with P (Γ ) × P (Γ // Γ ). To see this, note that by proposition 3.4, the ntersection in P (Γ) of distinct exceptional divisors E ∩ · · · ∩ E p is non-empty if and only if after reordering,the corresponding core subgraphs of Γ form a flag. This means, for example, that E ∩ E I = ∅ if and only ifthe flag corresponding to I has a subflag of core subgraphs contained in Γ , and the remaining core subgraphsform a flag containing Γ . In this way, we blow up appropriate linear spaces in P (Γ ) and in P (Γ / Γ ). theresult is P (Γ ) × P (Γ // Γ ) ⊂ P (Γ). The chain τ is an S -bundle over the chain σ P (Γ ) × σ P (Γ // Γ ) (slightlymodified along the boundaries as above), and the monodromy map is the product of the monodromies oneach factor. (The monodromy takes place on P (Γ ) × P (Γ // Γ ). In the end, one takes the S -bundle over m ( σ P (Γ ) × σ P (Γ // Γ ) ).) But this yields exactly (8.18). The result for a general m ( τ I ) is precisely analogous. Tocompute N , suppose Γ has exactly k core subgraphs Γ ′ ( Γ. (This means that P (Γ) will have k exceptionaldivisors E i .) Consider the commutative ring(8.19) R := Q [ x , . . . , x k ] / ( x , . . . , x k , M , . . . , M r ) , where we think of the x i as corresponding to exceptional divisors E i on P (Γ), and the M j are monomialscorresponding to empty intersections of the E i . The notation means that we factor the polynomial ringin the x i by the ideal generated by the indicated elements. We may if we like drop the M j from theideal. This will simply mean the column vector on which N acts will have many entries equal to 0. Asa vector space, we can identify R with the free vector space on σ and the τ I by mapping σ τ I Q i ∈ I x i . With this identification, the monodromy map m is given (compare (8.18)) by multiplicationby (1 − x )(1 − x ) · · · (1 − x k ). But the map R → End vec. sp. ( R ) given by multiplication is a homomorphismof rings, so log( m ) is given by (note x i = 0)(8.20) log (cid:16) (1 − x ) · · · (1 − x k ) (cid:17) = − X x i . Thus N is the matrix for the map given by multiplication by − P x i . If we ignore the relations M j andjust write the matrix for the action on Q [ x , . . . , x k ] / ( x , . . . , x k ), it has size 2 k × k and is strictly uppertriangular. For k = 3, the matrix is − N (3) (8.11).9. Renormalization: the removal of log-poles
Recall we have defined sdd(Γ), the degree of superficial divergence of a graph with respect to a givenphysical theory, (2.2). The choice of the theory determines a differential form ω Γ associated to Γ. We willbe interested in the logarithmic divergent case, when sdd(Γ) ≥
0, but ω Γ has been chosen such that it onlyhas log-poles, see in particular section 7.4. The affine integral in this case will be overall logarithmicallydivergent, but this overall divergence can be eliminated by passing to the associated projective integral. If,for all core subgraphs Γ ′ ⊂ Γ, we have sdd(Γ ′ ) <
0, then the projective integral actually converges and weare done. If Γ ′ > ≤ ω II .Below, we spell all results out for the case ω II = Ω n − /ψ , and we set ψ Γ ≡ ψ (Γ). Lemma 9.1.
Let Γ ′ ( Γ be core graphs and assume sdd (Γ) = 0 . Let L ⊂ X Γ ⊂ P (Γ) be the coordinatelinear space defined by the edges occurring in Γ ′ . Let π : P L → P (Γ) be the blowup of L . Then π ∗ ω Γ has alogarithmic pole on E if and only if sdd (Γ ′ ) = 0 . Similarly, the pullback of ω Γ to the full core blowup P (Γ) (cf. formula (3.3) ) has a log pole of order along the exceptional divisor E Γ ′ associated to Γ ′ if and only ifsdd (Γ ′ ) = 0 .Proof. We give the proof for φ -theory. Let the loop number | Γ | = m so the graph has 2 m edges (2.2). Let(9.1) Ω m − = X ( − i A i dA ∧ · · · ∧ d dA i ∧ · · · ∧ dA m = A m m d ( A /A m ) ∧ · · · ∧ d ( A m − /A m ) . Then(9.2) ω Γ = Ω m − ψ . Suppose L : A = · · · = A p = 0. We can write the graph polynomial ([2], prop. 3.5)(9.3) ψ Γ = ψ Γ ′ ( A , . . . , A p ) ψ Γ // Γ ′ ( A p +1 , . . . , A m ) + R here the degree of R in A , . . . , A p is strictly greater than deg ψ Γ ′ = | Γ ′ | . Let a i = A i /A m , and let b i = a i /a p , i < p . Locally on P we can take b , . . . , b p − , a p , a p +1 , . . . , A m as local coordinates and write(9.4) ω Γ = ± a p − | Γ ′ | p da p a p ∧ db ∧ · · · ∧ da m − F . Here F is some polynomial in the a i ’s and the b j ’s which is not divisible by a p . The assertion for the blowupof L follows immediately. The assertion for P (Γ) is also clear because we can find a non-empty open set on P (Γ) meeting L such that the inverse images in P (Γ) and in P L are isomorphic. (cid:3) We want to state the basic renormalization result coming out of our monodromy method. For this, werestrict to the case(9.5) sdd(Γ ′ ) ≤ , ∀ Γ ′ ⊆ Γ , with an understanding that appropriate forms ω II (Γ ′ ) have been chosen so that the differential forms haslog-poles only. The following lemma applies then to φ -theory. A physicist wishing to apply our results toanother theory needs only check the lemma holds with ω Γ replaced by the integrand given by Feynman rules. Lemma 9.2.
Let τ εV be the chains on P (Γ) constructed above (section 4) (including the case τ εP (Γ) = σ ε ).Then, assuming (9.5) , we will have (9.6) (cid:12)(cid:12)(cid:12) Z τ εV ω Γ (cid:12)(cid:12)(cid:12) = O ( | log | ε || k ) , | ε | → for some k ≥ .Proof. We first consider the integral for the chain σ ε = τ εP (Γ) . Locally on the blowup P (Γ) the integrand willlook like (9.4) but there may be more than one log form; i.e. e ωda p /a p ∧· · ·∧ da p k /a p k . An easy estimate forsuch an integral over a compact chain satisfying a j ≥ ε gives C ( | log ε | ) k . The integrals over τ εV , V ( P (Γ)involve first integrating over one or more circles. Locally the chain is an ( S ) p -bundle over an intersection x = · · · = x p = 0 in local coordinates. We may compute the integral by first taking residues. V will be theclosure of a torus orbit in P (Γ) associated to a flag Γ p ( · · · ( Γ ( Γ (proposition 3.4). We may assume x i is a local equation for the exceptional divisor in P (Γ) associated to Γ i ⊂ Γ. By lemma 9.1, our integrandwill have a pole on x i = 0 if and only if sdd(Γ i ) = 0. (Note that the integrand has no singularities on τ εV ,so we may integrate in any order.) The situation is confusing because sdd(Γ i ) < ⇒ sdd(Γ // Γ i ) > i ) = 0 , ∀ i , the residue integral is(9.7) Z Q j τ εP (Γ j// Γ j +1) ω Γ p ∧ · · · ∧ ω Γ // Γ . Since sdd(Γ i // Γ i +1 ) = 0, we may simply write (9.7) as a product of integrals and argue as above. (cid:3) We want now to apply the argument sketched in the introduction to our situation. There is one mathe-matical point which must be dealt with first. We want to consider R σ t ω Γ as a function of t . Here we mustbe a bit careful. For t = εe iθ and | θ | << θ grows, our chain may meet X Γ . Topologically,we have (proposition 6.3) the chains ˜ c η,ε,θ which miss X Γ and which represent the correct homology class in H ∗ ( P (Γ) − X Γ , ∆ t − X Γ ∩ ∆ t ), but one must show our integral depends only on the class in homology relativeto ∆ t , i.e. ω Γ integrates to zero over any chain on ∆ t − X Γ ∩ ∆ t . Intuitively, this is because ω Γ | ∆ t = 0, but,because ∆ t has singularities it is best to be more precise. Quite generally, assume U is a smooth variety ofdimension r , and D ⊂ U is a normal crossings divisor (i.e. for any point u ∈ U there exist local coordinates x , . . . , x r near u , and p ≤ r such that D : x x · · · x p = 0 near u ). One has sheaves(9.8) Ω qU (log D )( − D ) ⊂ Ω qU ⊂ Ω qU (log D )where Ω qU is the sheaf of algebraic (or complex analytic; in fact, either will work here) q -forms on X ,and Ω qU (log D ) is obtained by adjoining locally wedges of differential forms dx i /x i , ≤ i ≤ p . Locally,Ω qU (log D )( − D ) := x x · · · x p Ω qU (log D ). All three sheaves are easily seen to be stable under exterior ifferential (for varying q ). The resulting complexes calculate the de Rham cohomology for ( U, D ) , U, ( U − D )respectively, [7]. Note that in the top degree r = dim U we have(9.9) Ω rU (log D )( − D ) = O U · x x · · · x p dx ∧ · · · ∧ dx p x x · · · x p dx p +1 ∧ · · · ∧ dx r = Ω rU . It follows that we get a maps(9.10) Ω rU [ − r ] → Ω ∗ U (log D )( − D ); Γ( U, Ω rU ) → H rDR ( U, D ) . In particular, taking U = P (Γ) − X Γ , we see that integrals R ch.rel.∆ t ω Γ are well-defined. Theorem 9.3.
We suppose given a graph Γ such that all core subgraphs Γ ′ ⊆ Γ have superficial divergencesdd (Γ ′ ) ≤ for a given physical theory. Let ω Γ be the form associated to the given theory. Let N be theupper-triangular matrix of size K × K described in the previous section, where K is the number of chains ofcore subgraphs Γ p ( · · · ( Γ . Then the lefthand side of the expression below is single-valued and analytic for t in a disk about so thelimit (9.11) lim | t |→ exp( − N log t πi ) R τ tP (Γ) ω Γ ... R τ tV ω Γ ... = a ... a k exists.Proof. The proof proceeds as outlined in section 1.3. N is chosen to be nilpotent and such that the lefthandside has no monodromy. The lemma 9.2 assures that terms have at worst log growth. Since they aresingle-valued on D ∗ , they extend to the origin. (cid:3) Remark 9.4.
It is time to compare what we are calculating here with what a physicist computes accordingto Thm.(7.3). The transition is understood upon noticing that in our constructions of chains, we pick up theresidue from each exceptional divisor by computing the monodromy. In physics we iterate those residues asiterated integrals. Below the top entry a this gives different rational weights to them in according with thescattering type formula of [6] . We discuss this below in section (9.1). Definition 9.5.
With notation as above, the renormalized value R σ ω Γ is the top entry in the column vectorexp(+ N log t πi ) a ... ! . Remark 9.6.
Note that the terms R τ tV ω Γ on the lefthand side of (9.11) may be calculated recursively. Asin lemma 9.2 above, V corresponds to a flag of core subgraphs of Γ. As in formula (9.7), the integral diesunless all the Γ i // Γ i +1 are log divergent. In this case, one gets(9.12) (2 πi ) p − Y Z τ tP (Γ i// Γ i +1) ω Γ i // Γ i +1 . If, in addition, the subquotients Γ i // Γ i +1 are primitive , i.e. they are log divergent but have no divergentsubgraphs, then the integrals in (9.12) will converge as | t | →
0. Upto a term which is O ( t ) and can beignored in the limit, they may be replaced by their limits as t →
0. These entries in (9.11) may then betaken to be constant.
Example 9.7.
Consider the dunce’s cap fig.(2). It has 3 core subgraphs, but only the 2-edged graph γ withedges 1 , N = (cid:18) −
10 0 (cid:19) . The constant entry in the column vector is(9.13) 2 πi Z σ γ Ω ψ γ Z σ Γ //γ Ω ψ //γ = 2 πi (cid:16) Z ∞ da ( a + 1) (cid:17) = 2 πi. t remains to connect R σ ω Γ to the physicists computation.9.1. lMHS vs Φ R . Let us understand how the period matrix p T = ( a , a , · · · , a r ) which we have con-structed connects to the coefficients c j (9.14) Φ MOM (Γ)( q /µ ) = r X j =1 c j (Γ) ln j q /µ . Going to variables t Γ , a , . . . , a | Γ [1] | , X a i = 1 ,t , b , . . . , b | Γ | , X b i = 1 ,. . . ,t p , z , . . . , b | Γ p [1] | , X z i = 1 , for a chain of core graphs Γ p ( · · · ( Γ ( Γ gives, for each such flag and constant lower boundaries ǫ , aniterated integral over(9.15) Z ∞ ǫ dt Z ∞ ǫ/t dt · · · Z ∞ ǫ/t/t ··· /t p − dt p . As the integral has a logarithmic pole along any t i integration, the difference between integrating against thechains, which only collect the coefficients of ln ǫ for each such integral, and the iteration above is a factorialfor each flag. A summation over all flags established the desired relation using tree factorials [14]:As the entries in the vector ( a , · · · ) T are in one-to-one correspondence with forests of Γ, identifying a withthe empty forest, we can write the top-entry defined in Defn.(9.5) as(9.16) X [for] (cid:18) ln t πi (cid:19) | [for] | a [for] , where(9.17) a [for] = p (Γ // [for]) Y j p ( γ j ) , using the notation of Eqs.(7.14,7.62). Then,(9.18) ∂ ln t Φ MOM (Γ)( t ) = X [for] aug(Γ) (cid:18) ln t [for] ∗ ! (cid:19) | [for] | a [for] . Here, [for] ∗ ! is a forest factorial defined as follows. Any forest [for] defines a tree T and a collection of edges C such that P C ( T ) and R C ( T ) denote the core sub- and co-graphs in question. The complement set T [1] /C defines a forest ∪ i t i say. We set [for] ∗ ! = Q i t i !, for standard tree factorials t i ! [14]. For example, comparingthe two graphs(9.19) Γ = , Γ = , we have the two vectors(9.20) p (cid:18) (cid:19) p (cid:18) (cid:19) p (cid:18) (cid:19) p (cid:18) (cid:19) p (cid:18) (cid:19) p (cid:18) (cid:19) p (cid:18) (cid:19) p (cid:18) (cid:19) nd(9.21) p (cid:18) (cid:19) p (cid:18) (cid:19) p (cid:18) (cid:19) p (cid:18) (cid:19) p (cid:18) (cid:19) p (cid:18) (cid:19) p (cid:18) (cid:19) p (cid:18) (cid:19) . Hence, we find the same ln t term upon computing Eq.(9.16) for the monodromy.On the other hand, the tree factorials deliver 1/2 for that term in the case of Γ , and 1 for Γ , while weget 2 in both cases for the term ∼ ln t . Indeed, the flag(9.22) ( ( corresponds to a tree with two edges. The term ∼ ln t comes from the cut C which corresponds to both ofthese edges. The complement is the empty cut, whose tree factorial is 3! simply. As we took a derivativewith respect to ln t , we get a factor of aug(Γ) = 3, which leaves us with a factor 3 /
3! = 1 / Limiting Mixed Hodge Structures.
In this final paragraph at the suggestion of the referee weoutline the structure of a limiting mixed Hodge structure associated to a variation of mixed Hodge structureand how it might apply to the Feynman graph amplitudes.Let Γ be a log divergent graph with n loops and 2 n edges. The graph hypersurface X Γ : ψ Γ = 0 is ahypersurface in P n − , and the Feynman integrand represents a cohomology class(9.23) h Ω ψ i ∈ H n − ( P n − − X Γ , C ) = H n − ( P n − − X Γ , Q ) ⊗ C = H C = H Q ⊗ C . The cohomology group has a mixed Hodge structure , which means there are defined two filtrations:(i) The weight filtration W ∗ H Q which is defined over Q and increasing. It looks like(9.24) 0 ⊂ W n H Q ⊂ W n +1 H Q ⊂ · · · ⊂ W n − H Q = H Q . Blowing up on X Γ so it becomes a normal crossings divisor D ∗ , there is a spectral sequence relating thegraded pieces W n − i /W n − i − to the Tate twist by − i of the cohomology in degree 2 n − − i of thecodimension i − D . (So, for example, gr W n is related to ⊕ j H n − ( D j )( −
1) where D = S D j .)(ii) The Hodge filtration F ∗ H C which is defined over C and decreasing:(9.25) (0) ⊂ F n − ⊂ F n − ⊂ · · · ⊂ F = H C . The filtrations are subject to the compatibility condition that the filtration(9.26) F p ( gr Wq ⊗ C ) := F p H C ∩ W q ⊗ C . F p H C ∩ W q − ⊗ C is the Hodge filtration of a pure Hodge structure of weight q . (This is simply the condition that F ∗ gr Wq ⊗ C be q -opposite to its complex conjugate, i.e. that gr Wq ⊗ C = F p ⊕ F q − p +1 for any p .)Let us say that a class ω ∈ H C has Hodge level p if ω ∈ F p H C − F p +1 H C . An important problem isto determine the Hodge level of the Feynman form (9.23). One may speculate that the Hodge level of theFeynman form equals the transcendental weight of the period. (The transcendental weight of a multizetanumber ζ ( n , . . . , n p ) is the sum of the n i .) For example, in [4] one finds many examples of Feynmanamplitudes of the form ∗ ζ ( N ) where * is rational. In all known cases N = 2 n −
3. To estimate the Hodgelevel, one may use the pole order filtration [7], 3.12. One blows up on X Γ ⊂ P n − to replace X by a normalcrossings divisor D = S ri =1 D i . Let ω on P n − − X Γ be a (2 n − I ⊂ { , . . . , r } be the indices i such that ω has a pole along D i . Write p i + 1 for the order of this pole, with p i ≥
0. Then the Hodge level f ω is ≥ n − − P p i . (For a more precise statement, see op. cit.) For example, if X Γ is smooth (thishappens only when n = 1) one would get p = 1 so the Hodge level would be ≥ n − Proposition 9.8.
For the Feynman form, at least of the p i ≥ . The pole order calculation thus suggeststhe Hodge level of the Feynman form above is ≤ n − .Proof. The situation for n = 1 is trivial, so we assume n ≥
2. The space of symmetric n × n -matrices hasdimension d := n ( n +1)2 . Let P d − be viewed as the projectivized space of such matrices, so a point correspondsto a matrix upto scale. The determinant of the universal matrix defines a hypersurface X ⊂ P d − . Moregenerally, we define X p ⊂ P d − to be the locus where the rank of the corresponding symmetric matrix is ≤ n − p . We have X = X , and it is easy to see that X p has codimension p ( p +1)2 in P d − . Points in X p willhave multiplicity ≥ p on X .There is an inclusion ρ : P n − ֒ → P d − such that X Γ = X ∩ P n − . Points of X ∩ P n − will havemultiplicity ≥ X Γ and codimension ≤ P n − . This means that in the local ring on P n − at ageneral point of X ∩ P n − , there will be functions x , x , x which form part of a system of coordinateson P n − such that a local defining equation ψ for X Γ lies in ( x , x , x ) . We may construct our normalcrossings divisor D as above by first blowing up X ∩ P n − in P n − . Subsequent blowups will not affectthe pole order, which may be computed at the generic point of the exceptional divisor E . We have(9.27) dx dx dx · · · ψ = x dx d ( x /x ) d ( x /x ) · · · x φ ( x , x /x , x /x , . . . ) . It follows that the Feynman form has a double pole on E as well as a double pole on the strict transform of X Γ in the blowup. (cid:3) Remark 9.9. (i) To give a complete proof that the Hodge level is ≤ n − H n − ( P n − − X Γ ).(iii) The data in [4] suggests that double zetas which occur will have transcendental weight 2 n −
4. Forexample, the bipartite graph Γ consisting of the 12 edges joining sets of 3 and 4 vertices has Feynmanamplitude a rational multiple of ζ (3 , X ∩ P n − has multiplicity ≥ ≤
6. If one could show that for the bipartite Γ that this codimension drops to 5, thenthe same argument as above would yield 3 poles with p i ≥
1, suggesting a Hodge level 2 n − P = P (Γ) → P n − . Let B ⊂ P be the complement of the big toric orbit in P . It is the union of the strict transform of the coordinate divisor ∆ ⊂ P n − and the exceptional divisors.Let Y ⊂ P be the strict transform of X Γ . The relevant cohomology group is the middle group in the sequence(9.28) H n − ( B − Y ∩ B ; Q ) → H n − ( P − Y, B − Y ∩ B ; Q ) → H n − ( P − Y, Q ) . If all the subgraphs Γ ′ ( Γ have sdd(Γ ′ ) <
0, then renormalization is unnecessary. The Feynman amplitudeas we have defined it is simply a period of the mixed Hodge structure (9.28). The weight filtration for thegroup on the left involves the cohomology of the strata of the normal crossings divisor B . For example, wehave an exact sequence(9.29) H ( B (1) − Y ∩ B (1) , Q ) → H ( B (0) − Y ∩ B (0) , Q ) → W H n − ( B − Y ∩ B (0) , Q ) . Here we write B ( i ) for the disjoint union of the components of the strata of dimension i . We know fromcorollary 5.3 that Y ∩ B (0) = ∅ , and a bit of thought about the combinatorics of B ( i ) , i = 0 , W H n − ( B − Y ∩ B, Q ) = Q (0). This gives a map of the trivial Hodge structure Q (0) to our period motive:(9.30) Q (0) → H n − ( P − Y, B − Y ∩ B ; Q ) . When the period is a rational multiple of ζ (2 n −
3) we expect that there is a map of Hodge structures Q (3 − n ) → H n − ( P n − − X Γ , Q ) and that the extension of Q (3 − n ) by Q (0) associated to ζ (2 n −
3) isa subquotient of (9.28).Finally the main focus of this paper has been the renormalization case when one or more proper subgraphsof Γ has sdd = 0. In this case, the Feynman form will have a pole along one or more divisor in B , so (9.28)is no longer the relevant Hodge structure. In this case, we work with the limiting mixed Hodge structure lim associated to H t := H n − ( P n − − X Γ , ∆ t − ∆ t ∩ X Γ ). Let D be a small disk around t = 0, andlet D ∗ = D − { } . Then H D ∗ = S t =0 H t becomes a local system on D ∗ . Let H D ∗ = H D ∗ ⊗ O D ∗ be thecorresponding analytic bundle. If we untwist by the monodromy, we get a trivial local system ( h = dim H t )(9.31) C hD ∗ ∼ = exp( − N log t ) H D ∗ ⊂ H D ∗ . Since this local system is trivial, it extends (trivially) across t = 0. It also has a canonical Q -structuredefined from the Q -structure at any point t = 0. The analytic bundle H D ∗ has a Hodge filtration F ∗ H D ∗ coming from the Hodge filtrations on the H t . (Note the Hodge filtration is not horizontal, so there is noHodge filtration on the local system H D ∗ .) From (9.31) we get a canonical trivialization of the analyticbundle H D ∗ ∼ = O hD ∗ and hence a canonical extension across t = 0. One can show [5], 2.1(i) that the Hodgefiltration extends across t = 0 as well.Thus, on the fibre H we have a Hodge filtration and a Q -structure. If you think in terms of periods,i.e. using the pairing H ∨ , Q × H → C , the above description of the Hodge filtration as a limit across t = 0coincides with the computation (9.11). What we have not given is the weight filtration. This monodromyweight or limiting weight filtration is more subtle, essentially being determined by the endomorphism N together with the given weight filtrations on the fibres H t . We hope that the computation of N in thispaper will help to understand this structure, but at the moment the weight structures on the H t are notwell enough understood to say more. For the general theory, the interested reader is referred to [5] and thereferences cited there. eferences [1] Bestvina, M., and Feighn, M., The topology at infinity of Out ( F n ), Inv. Math. , no. 3, (2000), pp. 651-692.[2] Bloch, S, Esnault, H., and Kreimer, D., On Motives Associated to Graph Polynomials, Comm. Math. Phys. (2006),181-225.[3] Bergbauer, C., and Kreimer, D., Hopf algebras in renormalization theory: locality and Dyson-Schwinger equations fromHochschild cohomology, IRMA Lect.Math.Theor.Phys. (2006) 133-164; hep-th/0506190.[4] Broadhurst, D.J., and Kreimer, D., Association of multiple zeta values with positive knots via Feynman diagrams up to 9loops, Phys. Lett. B 393 (1997) 403.[5] Cattani, E., and Kaplan, A., Degenerating variations of Hodge structure, in