Generalised graph Laplacians and canonical Feynman integrals with kinematics

To any graph with external half-edges and internal masses, we associate canonical integrals which depend non-trivially on particle masses and momenta, and are always finite. They are generalised Feynman integrals which satisfy graphical relations obtained from contracting edges in graphs, and a coproduct involving both ultra-violet and infra-red subgraphs. Their integrands are defined by evaluating bi-invariant forms which represent stable classes in the cohomology of the general linear group on a generalised graph Laplacian matrix which depends on the external kinematics of a graph.


Introduction
In the paper [Bro21] we introduced canonical differential forms on moduli spaces of metric graphs, and showed how they provide a connection between the cohomology of the commutative graph complex, the algebraic K-theory of the integers, and Feynman integrals.In this paper, we extend this theory to the case of graphs with external momenta and masses, with an emphasis on physical aspects.
Let us first recall some background on Feynman integrals.Consider a connected graph G, with n external legs (or half-edges), h G loops, and e G internal edges.Every external leg represents an incoming particle with momentum q i P R d subject to overall momentum conservation ř n i"1 q i " 0. The number of spacetime dimensions d will be d " 2 or d " 4 in this paper.To each edge e one additionally associates one of a finite set of particle masses m e , which are arbitrary real numbers.
For scalar theories, the parametric Feynman integral is the projective integral (1.1) where the edges of G are numbered from 1 to e G , the form Ω G is defined by and σ G " tpα 1 : . . .: α e G q P P e G ´1pRq : α i ě 0u is the coordinate simplex in projective space.The quantity D is typically an even integer, or, in the setting of dimensional regularisation, D " 2k ´ε, for a small positive ε.
The integrand involves the 'second Symanzik' polynomial which is expressed in terms of the two more basic polynomials Ψ G , Φ G pqq, which are homogeneous in the α e , e P E G of degrees h G , h G `1 respectively.They are defined as sums over spanning forests in the graph G with 1 or 2 connected components (see §6.1).The integral (1.1) diverges in general and can be regularised in a variety of manners including, for example, Laurent expansion in the parameter ε.
For most quantum field theories of relevance for collider physics, one is led to consider a wider class of Feynman integrals, which in parametric form (see, e.g.[Gol19]) have the following general shape (omitting Γ-factors for simplicity): (1.2) where a, b P Z (or a, b P Z `εZ), and the numerator N G is a polynomial in the parameters α i with typically complicated coefficients.Faced with the considerable difficulty in computing integrals (1.1) or (1.2), a common recent theme of research is to seek alternative theoretical frameworks in which the corresponding amplitudes are simpler and more highly structured, with a long term view to unearthing mathematical properties which are valid for general quantum field theories.Notable examples in this direction include the amplituhedron programme [AHT14], which organises certain amplitudes in N " 4 SYM according to geometric principles; string perturbation theory, which studies scattering amplitudes defined on punctured Riemann surfaces; or integrable 'fishnet' models [Zam80,GK16] which reduce to a small number of Feynman graphs.
In this paper, we introduce a special class of geometrically-defined integrals of the form (1.2), which are always finite irrespective of the graph G.They generalize [Bro21] to incorporate masses and momenta, and satisfy symmetry properties including a family of graphically-encoded relations.The latter involve both contraction of internal edges and also the 'motic' coproduct of [Bro17] which has applications to the study of both UV and IR divergences [BBH20,AHHM22].
1.1.Canonical forms and their integrals.Our starting point is the moduli space M trop g of stable tropical curves of genus g, or more precisely the open locus `Mtrop g ˘w"0 Ă M trop g consisting of graphs whose vertices have weight zero.The quotient `Mtrop g ˘w"0 {R ą0 is isomorphic to the quotient of Culler-Vogtmann's Outer space O g [CV86] by the group of outer automorphisms OutpF g q of the free group F g on g elements.Points in this space are equivalence classes of connected metric graphs (with no external legs), where a metric on a graph is an assignment of a length ℓ e ą 0 to every internal edge, normalised so that the total length of all edges is 1.
The tropical Torelli map [BMV11] on M trop g restricts to a map (1.3) `Mtrop g ˘w"0 ÝÑ P g {GL g pZq where P g is the space of positive definite g ˆg symmetric matrices X, upon which P P GL g pZq acts via X Þ Ñ P T XP .The map (1.3) assigns to a connected metric graph G the GL g pZq-equivalence class of any choice of graph Laplacian matrix Λ G .The latter is a g ˆg symmetric matrix whose determinant det Λ G " Ψ G equals the first Symanzik, or Kirchhoff, graph polynomial which arises in the integrals (1.1).The graph Laplacian may be interpreted as a tropical version of the Riemann polarization form on a compact Riemann surface of genus g.Canonical forms on `Mtrop g ˘w"0 are defined as follows.The forms trppX ´1dX q 4k`1 q, for k ě 1, are bi-invariant under left and right multiplication by GL g pRq and were shown by Borel to generate the stable cohomology of the symmetric space P g {GL g pZq.Their pull-backs trppΛ ´1 G dΛ G q 4k`1 q along the map (1.3) defines a distinguished family of differential forms on the images of cells in `Mtrop g ˘w"0 {R ą0 .The integrals of these forms are called canonical since they do not depend on any choices, and were studied in [Bro21].In that paper it was shown that interesting examples of Feynman residues for vacuum diagrams (see [Sch10] for a survey) of relevance to quantum field theory can arise as canonical integrals.A natural question is whether general Feynman integrals with non-trivial kinematics are also amenable to a 'canonical' geometrical interpretation of this kind.This is addressed in this paper.
1.1.1.Canonical forms with kinematic dependence.We consider a generalised moduli space M trop g,n,M , first considered in [BM19], of weighted metric graphs which have n external legs labelled from 1, . . ., n, and where each internal edge e is assigned one of M possible particle masses.When M " 0, this space coincides with the moduli space M trop g,n of n-marked tropical curves of genus g.Denote the open locus of graphs whose vertex weights are all zero by: (1.4) Consider in the first instance external particle momenta q 1 , . . ., q n P R 2 in twodimensional Euclidean space, which we identify with the complex numbers, and any choice of M internal particle masses.For any choice of routing µ e of the external momenta through the internal edges e of G, we consider a generalised graph Laplacian matrix (see (5.4)) whose determinant satisfies (1.5) det r Λ G pµ, mq " Ξ G pq, mq .
The generalised Laplacian r Λ G pµ, mq is a Hermitian pg `1q ˆpg `1q matrix (note that g `1 is the degree of the polynomial Ξ G pq, mq).It is the tropical version of a regularised Hermitian polarization form on the cohomology of a compact Riemann surface with n punctures.From this perspective, the external particle momenta are interpreted as tropical versions of the residues of a differential form of the third kind.The case when the external momenta lie in R 4 may be treated by replacing complex momenta with momenta in the ring of quaternions Q -R 4 , leading to a quaternionic Hermitian graph Laplacian which satisfies where Det of a quaternionic matrix is the determinant of its complex adjoint.One of our key results ( §9) is that the generalised graph Laplacian admits an asymptotic decomposition into block matrices as one approaches the boundary of the moduli space pM trop g,n,M q w"0 .This provides a geometric interpretation of asymptotic factorisation formulae for graph polynomials which are important for the study of infra-red and ultra-violet singularities of Feynman integrals [BBH20,AHHM22].
Define canonical forms 1 of the 'first' and 'second' kinds for all k ě 1, by (1.7) when the external momenta lie in R 2 -C.In the case of quaternionic momenta in R 4 -Q, the definition of canonical forms of the first kind is unchanged, but the canonical forms of the second kind are now defined for all k ě 1 by: where the quaternionic trace Tr is the trace of the complex adjoint.A crucial property of canonical forms (theorem 9.3) is that they factorise as one approaches infinity along the boundary faces of Feynman polytopes.
When the external momenta lie in R 2 -C, we define a canonical form ω G to be any homogeneous polynomial in the forms (1.7).When the momenta lie in

G
. For any connected graph G, and any such form ω G of degree e G `1, we may consider the integral (1.9) Since it is independent of all choices which go into defining generalised graph Laplacian matrices, we call it a canonical integral.
We prove the following facts about the integrals (1.9): (1) They are always finite.
(3) They satisfy graphical relations involving both the contraction of internal edges and a generalisation of the Connes-Kreimer coproduct which encodes not only ultraviolet, but also infra-red phenomena.
Thus the integrals (1.9), which may be loosely interpreted as certain 'volumes' of cells on a moduli space of tropical curves, pick out a distinguished class of generalised Feynman integrals with special properties.The reader may wish to turn to sections 11 -13 for examples of canonical integrals and their relations.
Remark 1.1.The canonical forms (1.7), (1.8) arise from the graded exterior algebras of invariant differential forms I Hn on the symmetric spaces associated to the following sequences of classical non-compact Lie groups H n .The following table is extracted from [Bor74], page 265, and based on results of H. Cartan: 1 The reader familiar with Chern-Simons theory may recognise canonical forms as a special case: for example, the Chern-Simons 3-form trpdA ^A `2 3 A ^A ^Aq reduces to ´1 3 trppΛ ´1dΛq 3 q upon substituting A " Λ ´1dΛ.The higher degree forms are similar.
where Ertx i us denotes the graded exterior algebra generated by elements x i in odd degrees i, and the inverse limit of the spaces of forms I Hn is taken in the category of graded exterior algebras.The generators are given by β n X in the case of real and complex matrices, and by TrppX ´1dX q n q in the quaternionic case.They always vanish when n is even, and additionally for n " 3 pmod 4q in the case of R and Q.
1.2.Discussion and questions for further research.It would be very interesting to find a momentum or position space formulation for the integrals (1.9), and to relate them more closely to the theory of graphical functions [GPS17,BS21] in the case when all masses vanish.The generalised graph Laplacian is a tropical version of 'single-valued' period integrals on a punctured Riemann sphere, which are prevalent in closed string perturbation theory, and suggests a possible connection with string perturbation theory.In a different direction, it was shown in [Bro21] that the canonical integrals for graphs without kinematics are closely connected to the homology of the commutative, even, graph complex.It would be interesting to find such a relation for the integrals (1.9) based on corollary 10.4, for example.See also [BK20] for related interpretations of Feynman integrals.
We also expect the work in this paper to have applications to the motivic study of Feynman integrals.For example, the existence of the generalised graph Laplacian implies that the graph motives at a fixed loop order have a universal family, and the canonical forms provide universally-defined classes in the relative cohomology of graph hypersurface complements and the de Rham realisation of graph motives.
In particular, since canonical integrals are distinguished elements of period matrices associated to ordinary Feynman integrals, they should be related to the latter via differential equations and also via the motivic coaction.The Stokes' relations proven here should thus enable one to transfer information about differential equations, periods, and infra-red singularities between graphs with different topologies.
Finally, it would also be interesting to study integrals of the form in dimensional regularisation, where ω G is canonical.They too will satisfy relations via Stokes' formula, but of a slightly different form to those considered here.
1.3.Contents.In §2 we recall some notations and conventions as well as some background on quaternionic matrices.In §3 we describe a moduli space of marked metric graphs with additional edge colourings, which encode particle masses.This space was previously introduced by Berghoff and Mühlbauer in [BM19].After a brief discussion in §4 of momentum routings, and complex and quaternionic momenta, we define the generalised Laplacians in §5.In the case when momenta are quaternionic and all masses are zero, they coincide with matrices previously considered by Bloch and Kreimer in [BK10].The way we arrived at the definition is explained in §14; namely by taking the tropical analogue of the single-valued integration pairing on the cohomology of a punctured Riemann surface.In §6 we recall the definitions of Symanzik polynomials, and prove a formula for Φ G in terms of spanning forest polynomials, which is used to deduce (1.5) and (1.6).Starting in §8, we recall the definitions of bi-invariant differential forms.A key result is a formula ( §9) for the asymptotic behaviour at infinity of graph Laplacian matrices, which is used to prove the convergence of canonical integrals in §10, along with the Stokes relations.It could also be used to prove (1.5) and (1.6).Sections 11 and 13 provide examples of canonical integrals for some well-studied Feynman diagrams, and §12 illustrates the general Stokes relations by deducing a graphical 5-term functional equation for the massive box diagram in 4 spacetime dimensions.
Declarations.The author has no relevant financial or non-financial interests to disclose, nor any competing interests to declare that are relevant to the content of this article.
Acknowledgements.Many thanks to Marko Berghoff, Paul Fendley, Lionel Mason, Erik Panzer, and all the participants of the seminar at Oxford in 2022 on graph complexes.This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement no.724638).

Notations and conventions
Graphs in this paper are finite, and usually connected.They have a finite number n of external half-edges.The set of internal edges is denoted by EpGq, and the set of vertices by V pGq.We write e G " |EpGq| for the number of internal edges, and h G for the number of loops of G.When an internal edge is directed, we denote by speq its source, and tpeq its target.An edge with speq " tpeq is called a self-edge or tadpole.

It admits a representation
The ring Q has an anti-involution, denoted x Þ Ñ x, which is induced by Hermitian conjugation V Þ Ñ pV q T on M 2 pCq.The conjugate of x " x 1 1 `x2 i `x3 j `x4 k is x " x 1 1 ´x2 i ´x3 j ´x4 k, and the square of the quaternion norm 4 q1 coincides with the square of the Euclidean norm.
2.1.1.Quaternionic matrices.Matrices with entries in the ring of quaternions share many properties with rings of matrices over a commutative ring (see [Zha97] for a survey).A convenient method for studying them is via the complex adjoint representation, which is the ring homomorphism (2.1) χ : M n pQq ÝÑ M n pM 2 pCqq , induced by the representation χ defined above.One shows that a matrix M P M n pQq has a unique left inverse (which is necessarily also a right inverse) if and only if its image χ M is invertible.However, the determinant of a quaternionic matrix is not defined in general, and its trace is not a similarity invariant, meaning that trpP ´1M P q ‰ trpM q for general P, M .For this reason it is customary to define the determinant and trace of M P M n pQq to be DetpM q " det χ M and TrpM q " tr χ M respectively.They take values in the complex numbers.
Two particular representations of quaternionic matrices are convenient: If one identifies M n pM 2 pCqq with M 2n pCq, then the image of χ is the set of matrices tV P M 2n pCq : V " J n V J ´1 n u, where J n is the block diagonal matrix consisting of n copies of χ j along the diagonal.
(ii).If one identifies M n pM 2 pCqq with M 2 pM n pCqq, then the image of χ consists of block matrices of the form: since any quaternionic matrix may be uniquely written A`j B, with A, B P M n pCq.
Here, and later, we identify the complex numbers with R1 `Ri Ă Q.
Either representation may be used to compute the quaternionic trace or determinant.We shall use piq in the examples.(i).Let M P M n pQq be a quaternionic Hermitian matrix, i.e., M T " M , where M denotes its quaternionic conjugate.Then χ M is Hermitian and where F is a polynomial in the entries of χ M which is invariant under complex conjugation.
(ii).In the case when M P M n pCq Ă M n pQq has complex entries (2.4) DetpM q " detpχ M q " detpM q 2 .Although we shall not use it, the Moore determinant F " NrppM q is a canonical solution to (2.3) which satisfies NrppM q " detpM q for all M P M n pCq.

Moduli of marked metric graphs with masses and momenta
We consider a moduli space of metric graphs with external legs and masses.It was previously studied in [BM19].
3.1.Graphs with external legs and massive edges.
3.1.1.Combinatorial masses.Let M ě 0 be a fixed integer.For a connected graph G, the assignment of a mass to each internal edge e P EpGq is encoded by The labelling 0 denotes the zero mass.
3.1.2.Physical masses.We fix a finite set of non-zero masses m i P R ˆ, for i " 1, . . ., M , and set m 0 " 0 to be the zero mass, i.e., m : t0, 1 . . ., M u Ñ R.Only the mass squares m 2 i will play a role in the theory.The mass m i denotes the mass of a particle of type i, and there is no need for the m i to be distinct.When drawing Feynman diagrams, our convention is to depict massless edges with a single line, and massive edges with a doubled line.
Two cases are of particular interest, namely the 'massless case' M " 0, when all edges have mass zero; and the 'generic mass case', when M " e G , and M : EpGq " Ñ t1, . . ., e G u is a bijection.
3.1.3.Combinatorial momenta.Let n ě 0 be a fixed integer.If n ě 1, the data of n external legs or 'markings' is given by a map n : t1, . . ., nu ÝÑ V pGq which can be realised combinatorially by attaching a half-edge, labelled i, to the vertex npiq.These external half-edges will always be oriented towards their endpoint, i.e., all external momenta are incoming.The map n does not need to be injective or surjective.In the case when n " 0, or when the image of n reduces to a single vertex, the associated graph is called a vacuum diagram, or momentumless.
The external half-edge i represents an incoming particle carrying momentum q i .Given such a collection of momenta, we obtain a vector at every vertex (3.2) q " pq v q v P `Rd ˘V pGq where q v " which is defined to be 0 if vertex v has no external edges attached to it, and otherwise is the sum of all incoming momenta at v.

3.2.
A category of graphs.Let us fix integers g, n, M ě 0.
Definition 3.1.A marked, weighted metric graph with masses and momenta is a tuple pG, w, n, M q where G is a connected graph, and w : V pGq ÝÑ N Y t0u (3.3) n : t1, . . ., nu ÝÑ V pGq M : EpGq ÝÑ t0, . . ., M u are the data of: a weighting w for every vertex; n external half lines n; and an assignment M of a mass label to each internal edge of G. Its genus is where h G is the first Betti (loop) number of G.Such a graph is called stable if, for every vertex v P V pGq, one has the inequality 2wpvq ´2 `dpvq ě 0 where dpvq is the total degree at v (including both internal and external half-edges).
An isomorphism of graphs pG, w, n, M q is an isomorphism of the underlying graphs which respects the additional data (3.3).
For every edge e P EpGq, the contraction pG, w, n, M q{e " pG{e, w 1 , n 1 , M 1 q , of e is defined as follows.If e has distinct endpoints, then G{e is the graph with edge e removed and its endpoints tspeq, tpequ identified.As usual, the weight of the new vertex is the sum of weights wpspeqq `wptpeqq, and n 1 is the composite of n with the quotient map V pGq Ñ V pG 1 q " V pGq{tspeq " tpequ.The map M 1 is the restriction of M to the subset EpG{eq " EpGqzteu.
When e is a tadpole or self-edge, and its endpoints coincide v " speq " tpeq, the contraction G{e is simply the deletion of edge e.The weight of the vertex v in G{e is defined to be one more than its weight in G: w 1 pvq " wpvq `1.The map n is unchanged, and M 1 is defined by restriction of M to EpGqzteu as before.
Edge contraction preserves both the genus and the property of stability.
Definition 3.2.For fixed g, n, M , let I g,n,M denote the category whose objects pG, w, n, M q are connected stable graphs of genus g, and whose morphisms are generated by isomorphisms and edge contractions.
When M " 0, it is nothing other than the category of marked stable metric graphs of genus g with n external markings.The category I g,n,M has a final object given by the graph consisting of a single vertex of weight g, and n external legs.
3.3.Associated moduli space.Based on ideas in [ACP22, BBC `20], one can efficiently define the moduli space of marked metric graphs as follows.
Let T op denote the category whose objects are topological spaces and whose morphisms are continuous maps.Consider a functor s : I Ñ T op, where I is a finite (diagram) category.Its realisation X s is the topological space A morphism from s : I Ñ T op to s 1 : J Ñ T op is the data of a functor j : I Ñ J, together with a natural transformation from s to s 1 ˝j.Such a morphism induces a continuous map X s Ñ X s 1 between the associated topological realisations.Definition 3.3.Define a functor (3.4) s : I opp g,n,M ÝÑ T op as follows.To the object pG, w, n, M q it associates the closed cell CpGq " CpG, w, n, M q " R EpGq ě0 , given by the space of non-negative edge lengths ℓ e , for e P EpGq.It only depends on the underlying graph G.An isomorphism pG, w, n, M q -pG 1 , w 1 , n 1 , M 1 q induces a bijection EpGq -EpG 1 q, and a linear isomorphism on the corresponding cells.Contraction of an edge e P EpGq induces a natural inclusion of cells ι e : CpG{eq ãÑ CpGq upon identifying CpG{eq with the subset ℓ e " 0. Definition 3.4.Define the tropical moduli space to be its topological realisation CpGq .
Although the cases when g " 0 are topologically interesting, they will play no role in this paper: in the absence of internal edges the canonical differential forms we shall consider are identically zero and only interesting for g ą 0.
Similarly, one can consider the functor Ls : I opp g,n,M Ñ T op which assigns to every object pG, w, n, M q the closed simplex σ G " tpℓ e q e P R EpGq ě0 : where the edge lengths are normalised to sum to 1.If EpGq is empty, then σ G is defined to be the empty set.
It defines a functor for the same reasons as above.The tropical moduli space M trop g,n,M is a cone whose cone point p is the cell associated to the final object in I g,n,M .The link M trop g,n,M ztpu{R ą0 is isomorphic to the topological space Remark 3.5.In the case when M " 0, one retrieves the definition of the moduli space M trop g,n of curves with n marked points.In general, the data of masses M is not completely anodine, since assigning distinct masses to the edges of a graph will in general reduce the size of its automorphism group.
The functor s defined above factors through a category of rational polyhedral cones as in [ACP22, BBC `20], which can in turn be upgraded to a category built out of affine spaces equipped with certain linear embeddings.There is a variant for the links Ls involving projective spaces.These notions will be pursued elsewhere.

3.4.
Homology and graph complexes.The homology of the spaces M trop g,n,M can be expressed in terms of complexes of graphs in the category I g,n,M via a variant of the classical result which relates cellular and singular homology (see [BBC `20, Proposition 2.1], [ACP22, Theorem 4.2]).We refer the reader to [BM19] for specific results about the homology of M trop g,n,M when M ą 0.

Interpretations of momenta
4.1.Genericity.Let G be a connected graph with external momenta q i P R d , and internal masses m e .We shall say that momenta are generic if (4.1) ˜ÿ iPI q i ¸2 ‰ 0 for all strict subsets I of the set of external half edges [Bro17,(1.18)].It is automatic from our definitions that all non-trivial edge masses m e are non-zero.Note that the genericity conditions for complex-valued momenta, which will not be considered in this paper, are more general than the Euclidean condition above.

Momentum routing.
Choose an orientation on every edge of G, such that external edges are oriented inwards.
Definition 4.1.A momentum routing relative to the edge orientation is the data, for every internal edge e P EpGq, of a momentum vector such that momentum conservation holds at every vertex v P V pGq, i.e., (4.2) where the first sum is over all edges emanating from v, and the second is over all edges terminating at v. In the second sum we include external half edges, for which we write µ e " q e for the incoming external momentum along e.
To see that momentum routings exist, we can reformulate (4.2) as follows.Let us write R " R d , and consider the exact sequence: where Bpeq " tpeq ´speq.The data of external momenta defines a vector q P R V G where q " pq v q vPV G via (3.2).The condition of momentum conservation (3.1) is precisely the statement that its image under the map R V G Ñ R is zero.Therefore q P ImpBq.A choice of momentum routing is any element Any two choices of momentum routing differ by an element of H 1 pG; Rq.Note that one may always assume that µ e " 0 for every self-edge or tadpole e.
Remark 4.2.One can define a canonical momentum routing by demanding that µ P R E G be orthogonal to the subspace H 1 pG; Rq with respect to the inner product on R E G for which the edges form an orthonormal basis.However, the canonical routing is not respected by contraction of edges.
4.3.Complex momenta.Suppose that the number of spacetime dimensions is d " 2, and the external momentum associated with the k th external half edge is (4.3) Having fixed a choice of square root i of ´1, there is an identification which is compatible with the Euclidean norm.In other words, if q k " px k , y k q and q k " x k `iy k for k " 1, 2, then the Euclidean inner product is (4.4) 2 q 1 .q 2 " 2 px 1 x 2 `y1 y 2 q " 2 Repq 1 q 2 q " q 1 q 2 `q1 q 2 and ||q 1 || 2 " q 1 q 1 .We can always assume (4.3) holds when the total number of external momenta is at most 3, since by momentum conservation, the Feynman integral only depends on two external momenta, which lie in a Euclidean plane.
4.4.Quaternionic momenta.Suppose that G is a connected graph as above, but the number of Euclidean dimensions is d " 4. The external momenta satisfy (4.5) q k " pq p1q k , . . ., q p4q k q P R 4 .Let us denote the corresponding quaternions §2.1 by q k P Q, where k k , which we may identify with their image in M 2 pCq under χ (2.1).The inner product will be written interchangably in either notation via the identity: (4.6) q i q j `qj q i " p2q i .qj q1 " 2 4 ÿ k"1 q pkq i q pkq j .

Generalised Laplacians for graphs with external legs
We define a generalised graph Laplacian of a connected graph G with external momenta subject to momentum conservation, and internal particle masses.In the following, let C be a ring, which is not necessarily commutative, equipped with an anti-commutative involution ι : C Ñ C such that its invariant subspace R " tc P C : ιpcq " cu is commutative.
Examples include: (1) C " C the complex numbers, equipped with complex conjugation.
(2) C " Q the ring of quaternions, with quaternionic conjugation.
(3) C " Qrµ e , µ e , m i s an abstract ring of kinematic variables, where ι is the Q linear map acting trivially on m i , and such that ιpµ e q " µ e , ιpµ e q " µ e .
In the first instance, the reader may wish to consider only the case p1q where C " C.
The generalised graph Laplacian requires the data of: ‚ an external momentum vector q P C V G subject to momentum conservation ‚ a vector of internal masses m P C E G which is invariant under ι.It is defined in two stages: first we define the graph Laplacian for zero masses, and then modify it very slightly to take into account the internal masses.5.1.Definition of the generalised graph Laplacian.Choose an orientation on the internal edges of G. Consider the exact sequence Since the momentum vector q P C V G maps to zero, we obtain an extension where E G " tµ P C E G : Bpµq P Cqu and the element 1 P C in the right-hand term is identified with q P C V G .We shall call this the extension by momenta.
Consider the pairing satisfying It restricts to a Hermitian form (with respect to the involution ι) on E G , which we also denote by x , y 0 , and which may be interpreted as a C-linear map y Þ Ñ px Þ Ñ xy, xy 0 q which we call the 'graph Laplacian for zero masses': Here, Hom pC,ιq pA, Bq denotes anti-linear maps ϕ : A Ñ B satisfying ϕpλxq " ιpλqϕpxq for all λ P C. Equivalently, (5.2) may be viewed as a C-linear map: The subspace H 1 pG; Cq Ă E G contains the subspace H 1 pG; Zq, which is invariant under the involution ι .Restricting the Hermitian form x , y 0 to the space H 1 pG; Zq therefore defines a symmetric bilinear form which is nothing other than the usual graph Laplacian Λ G , which depends neither on masses nor momenta.
The masses of G are encoded by a single element where m e " ιpm e q.The norm of this element with respect to (5.2) is We define the mass-correction Hermitian inner product x , y m on E G to be the unique inner product vanishing identically on H 1 pG; Cq: Thus Ă M G is zero except for the last column, which is the vector of masses m e .The generalised graph Laplacian matrix is defined to be where X Þ Ñ X ˚denotes Hermitian conjugation, i.e., X ˚" ιpXq T .Since Ă M G has entries which are invariant under ι, we have Ă M G only has a single non-zero entry in the bottom right-hand corner: In general, the matrix r Λ G has the following block-matrix form: (5.4) , where P P GLpH 1 pG; Zqq is a change of basis matrix with integer entries.In particular, one has detpP q 2 " detp r P q 2 " 1.

5.2.3.
Change of momentum routing.Two momentum routings µ, µ 1 differ by an element of H 1 pG; Cq.Therefore changing momentum routing (or changing a choice of splitting of (5.1)) is equivalent to modifying the matrix r Λ G by where r S is an invertible block matrix of the form (5.9) r S " ¨s1 I . . .
, where I denotes the h G ˆhG identity matrix and s 1 , . . ., s h P C.

Group of indeterminacy. The graph Laplacian matrix r
Λ G is therefore ambiguous up to the action of the semi-direct product (5.10)C g ⋊ GL g pZq where P P GL g pZq acts on S P C g via S Þ Ñ P SP ´1.Note that it only involves the additive structure on C, which is commutative, and not the multiplicative structure, which may not be.The group (5.10) can be viewed as the group of automorphisms of the extension E G (5.1) which respects the integral structure H 1 pG; Zq Ă H 1 pG; Cq.
Recall that C g ⋊ GL g pZq is the set pS, P q P C g ˆGL g pZq, which can be identified with the set of matrices (5.11) SP " ¨s1 P . . .
5.3.Complex adjoint of the generalised graph Laplacian.Let G be a connected graph as before, and suppose that the external momenta lie in C " Q, the ring of quaternions.The generalised graph Laplacian r Λ G pµ, mq is a quaternionic Hermitian form of rank g `1, and may be represented by a pg `1q ˆpg `1q Hermitian matrix with entries in Q.Its image under the complex adjoint map (2.1) is a Hermitian complex matrix of rank 2g `2.
where the quaternionic momenta are µ e " λ e `j ν e , for λ e , ν e P C, where is the complex adjoint of the ordinary graph Laplacian Λ G " pΛ ij q ij , and `m2 e `λe λ e `νe ν e ˘αe .
In the massless case m e " 0, these matrices were considered in [BK10].
The reader may prefer to use the equivalent matrix representation §2.1.1 (ii).

Symanzik and spanning forest polynomials
We recall the definition of Symanzik polynomials and relate the polynomial ϕ G pqq to a sum of spanning forest polynomials relative to a choice of momentum routing.For further background on Symanzik polynomials and some of their basic properties, we refer the reader to [Bro17, §1], for example.6.1.Definition of Symanzik polynomials.We recall the definition of the graph polynomials which arise in Feynman integrals in parametric form, since conventions can vary slightly in the literature.Let d ě 0 be any non-negative integer.In this section, G is any connected graph with external edges and external momenta q i P R d .Every internal edge e is assigned a mass m e P R.
6.1.1.1st Symanzik polynomial.Recall that the first Symanzik polynomial (also known as the 'Kirchhoff' or graph polynomial), is defined by where the sum is over all spanning trees of G.It does not depend on the external half-edges of G.It is equal to the determinant of the ordinary graph Laplacian: It is homogeneous of degree h G and is not identically zero.
6.1.2.2nd Symanzik polynomial.The second Symanzik polynomial depends on external momenta.It is the homogeneous polynomial of degree h G `1 defined by where the sum is over all spanning 2-trees of G (or spanning forests with exactly two connected components), and q Ti denotes the total momentum entering T i .By momentum conservation q T1 `qT2 " 0.
where the sum is over spanning forests F " T 1 Y . . .Y T r of G with exactly r connected components T i such that V pT i q X P " P i for 1 ď i ď r .
In other words, each tree T i contains the vertices in P i and no other vertices of P .
Trees consisting of a single vertex are permitted.We set ϕ P G " 0 if the sets P i are not all disjoint.6.3.2.An identity for the 2nd Symanzik polynomial.Proposition 6.3.Let G be as in the previous paragraph.For any choice of momentum routing, the second Symanzik polynomial satisfies: where we recall that ϕ P1,P2 G is zero if P 1 X P 2 is non-empty.Note that the second summand is symmetric in e and f and therefore each term is counted twice.
Proof.Let T " T 1 Y T 2 be a spanning forest in G with two connected components T 1 , T 2 .By momentum conservation (4.2), the momentum flowing into T 1 equals (6.7) q T1 " ´qT2 " where each sum is over all internal edges e P EpGq such that e Y T has no loops, and is hence a spanning tree of G.In particular, e has one endpoint in V pT 1 q and the other in V pT 2 q.
Suppose that e is an edge with distinct endpoints.It follows from (6.7) that the coefficient of µ e .µe in ´qT1 .qT2 is 1 if e Y T 1 Y T 2 is a tree, and 0 otherwise.By (6.3), the coefficient of µ e .µe in Φ G pqq is therefore equal to ÿ where the sum is over the set of spanning forests T with two connected components such that T Ye is a tree.The map T Þ Ñ T Ye is a bijection between the latter and the set of spanning trees which contain e, which in turn are in one-to-one correspondence with the set of spanning trees in G{e.We conclude that the coefficient of µ e .µe in the formula (6.3) is precisely α e Ψ G{e (which also equals α e Ψ G{{e ).
In the case when e is a loop, Ψ G{{e vanishes.By (6.7), µ e does not contribute to q T1 and therefore does not appear in Φ G pqq.This establishes the first line of (6.6).Now consider two distinct edges e, f and assume first of all that they have in total four distinct endpoints.Formula (6.7) implies that µ e .µf does not arise in ´qT1 .qT2 unless both T Y e and T Y f are trees.In this situation the coefficient is 1 if speq, spf q lie in the same connected component of T , and ´1 if they lie in different components.In the former case, this means precisely that T " T 1 Y T 2 is a spanning forest with 2 components such that the vertices of one component V pT 1 q contains tspeq, spf qu and the other, V pT 2 q, contains ttpeq, tpf qu.In the latter case, V pT 1 q contains tspeq, tpf qu and V pT 2 q contains tspf q, tpequ, or vice-versa.From (6.3), it follows that the coefficient of µ e .µf in Φ G pqq is exactly (6.8) 2 α e α f ´ϕtspeq,spfqu,ttpeq,tpfqu It remains to check that (6.8) remains true in the cases when tspeq, spf q, tpeq, tpf qu has 3 or fewer elements.The reader may verify this in every situation (e.g., speq " spf q, speq " tpf q, etc), using a similar argument and the fact that ϕ P1,P2 G vanishes if P 1 and P 2 are not disjoint.Note that if e or f is a loop (i.e., speq " tpeq) then µ e does not occur in (6.7), and indeed, both of the spanning forest polynomials in (6.8) are defined to be zero in this case.□ 6.3.3.Spanning forests and minors of graph Laplacians.The difference of spanning forest polynomials occurring in proposition 6.3 can be interpreted as a 'Dodgson polynomial' Ψ e,f G .We will make frequent use of the following well-known lemma.Lemma 6.4.Let G be a connected graph with directed edges.Suppose that e 1 , . . ., e n are distinct edges such that Gzte 1 , . . ., e n u is connected.Then h " h G ě n and we may choose cycles c 1 , . . ., c h P Z E G representing a basis for H 1 pG; Zq such that: In other words, for each 1 ď i ď n, the edge e i appears in cycle c i with coefficient 1 and occurs in no other cycle.
For a matrix M , let M i,j denote the matrix with row i and column j removed.Lemma 6.5.Let G be a connected graph with directed edges.
(i) Let e P EpGq be an edge such that Gzteu is connected (i.e., e is not a bridge).Choose cycles c 1 , . . ., c h as in the previous lemma and let Λ G be the graph Laplacian matrix with respect to this basis.Then we have G " Ψ Gze .(ii).Let e 1 , e 2 P EpGq be two edges such that Gzte 1 , e 2 u is connected, and choose cycles c 1 , . . ., c h as in the previous lemma.Then where s i is the source of the directed edge e i , and t i its target, for i " 1, 2.
Proof.piq.The variable α e only occurs in the matrix Λ G in a single place, namely the top left entry pΛ G q 1,1 .A row expansion shows that the coefficient of α 1 in detpΛ G q equals detpΛ 1,1 G q. On the other hand, the coefficient of α 1 in Ψ G is precisely Ψ Gze1 by the contraction-deletion identity Ψ G " α 1 Ψ Gze `ΨG{{e .piiq.By the same argument as in piq, we have detpΛ i,i G q " Ψ Gzei for i " 1, 2 and detpΛ 12,12 G q " Ψ Gzte1,e2u .Now the 'Dodgson identity' G which satisfies the identical identity (but is defined using the graph matrix, and not the Laplacian).An expression for the latter in terms of the requisite forest polynomials follows from [BY11, §2].□

Determinants of generalised graph Laplacians
Let G be a connected graph as in §5 with internal particle masses and external momenta lying in 2-dimensional Euclidean space R 2 -C.
Theorem 7.1.Let r Λ G pµ, mq be a generalised graph Laplacian for any choice of momentum routing µ e P C E G .It satisfies Now let G be as above with external momenta in R 4 -Q.
Theorem 7.2.Let r Λ G pµ, mq be a generalised quaternionic graph Laplacian with respect to a choice of momentum routing µ P Q E G .It satisfies Remark 7.3.The right-hand side of (7.1) or (7.2) only depends on external kinematics.It follows that the determinant of the generalised graph Laplacian with generic momentum variables µ e only depends on the quantities where the sum is over all internal edges incident to a vertex v P V pGq.
We first prove theorem 7.1 and use it to deduce theorem 7.2.It is instructive to prove theorem 7.1 by direct computation of the determinant of the left-hand side, making use of proposition 6.3.An alternative approach could exploit the asymptotic factorisation identities proven in §9, which were shown in [Bro17,§4] to determine Ξ G pq, mq essentially uniquely.

˜N ÿ
i"1 Since detpΛ G q " Ψ G , it is enough by definition of Ξ G pq, mq to prove (7.1) in the case when all masses are zero.Thus, theorem 7.1 is equivalent to: Now recall that r Λ G pµ, 0q takes the form: .
By §5.2.2, its determinant does not depend on the choices of cycles c i .To verify (7.3), we use proposition 6.3 and compare the coefficients of µ e1 µ e2 on both sides of (7.3).First we treat the case e 1 " e 2 " e.
Lemma 7.4.Let e be an edge of G, then the coefficients of µ e µ e on the left and right hand sides of (7.3) agree.
Proof.Consider the coefficient of µ e µ e in det r Λ G pµ, 0q.If e is a bridge, then we can find cycles c 1 , . . ., c h P Z E G representing a basis of H 1 pG; Zq which do not involve e at all.In this case, µ e only appears in r Λ G pµ, 0q in the bottom right hand corner, and the coefficient of µ e µ e is detpΛ G qα e " Ψ G α e " Ψ G{{e α e .
If e is not a bridge, then choose cycles c 1 , . . ., c h as in lemma 6.4.There are two possible types of terms in an expansion of the determinant which can involve µ e µ e , namely those involving the top right and bottom left corners c 1 pα, µq and c 1 pα, µq, and those involving the bottom right hand corner as before.In total they contribute α e detpΛ G q ´α2 e detpΛ 1,1 G q " α e `ΨG ´αe Ψ Gze ˘" α e Ψ G{{e .The minus sign on the left comes from p´1q h p´1q h´1 " ´1; the first equality is a consequence of lemma 6.5 (i); and the second equality is the contraction-deletion identity: Ψ G " α e Ψ Gze `ΨG{{e .In all cases, we conclude that the coefficient of µ e µ e in det r Λ G pµ, 0q is α e Ψ G{{e , in agreement with (6.6).□ Before proceeding to the case e 1 , e 2 distinct, we consider some boundary cases.
Lemma 7.5.Let e be a tadpole in G. Then det r Λ G pµ, 0q " α e det r Λ Gze pµ, 0q Proof.By lemma 6.4, choose the first cycle in a basis c 1 , . . ., c h of H 1 pGq to be c 1 " e, and all other c i to be independent of e.
The result follows by subtracting µ e times the first row from the final row, and performing an expansion of the determinant along the first column.□ Next we verify that the theorem is true in the massless case for 'banana' graphs, i.e., graphs with exactly two vertices joined by ě 3 internal edges.
Example 7.6.Let G have two vertices v 1 , v 2 , and n ě 3 edges.Orient them so that they all have source v 1 and target v 2 , except for e n , which has source v 2 and target v 1 .Consider the basis of cycles: c 1 " e 1 `en , c 2 " e 2 `en , . . ., c h " e n´1 `en where h " n ´1.With these conventions, the graph Laplacian is It follows from the explicit description of r Λ G pµ, mq in the form (5.4) that the coefficient of .
By subtracting the first row from all other rows, we see that det Λ 1,2 G " α 3 . . .α n , and therefore the coefficient of µ 1 µ 2 in detpΛ G pµ, 0qq is α 1 . . .α n .By symmetry, the same holds for any µ i µ j , where i ‰ j.This agrees with the coefficient of µ i µ j in Φ G pqq, by its definition (there is only one spanning two-tree).The case when i " j follows from lemma 7.4, and it follows that det r Λ G pµ, 0q " Φ G pqq as claimed.
7.2.Proof of theorem 7.1.Now let e 1 , e 2 P EpGq be two distinct edges and consider the coefficient of µ e1 µ e2 in detp r Λ G pµ, 0qq.Suppose first of all that one of the edges, say e 1 , is a bridge.By the argument given above, no terms in the matrix r Λ G pµ, 0q depend on µ e1 except for the bottom right-hand corner, and so the coefficient of µ e1 µ e2 in detp r Λ G pµ, 0qq is zero.Now since e 1 is a bridge, it follows that e 2 lies in just one of the two connected components of Gze 1 , and therefore: ϕ tspe1q,spe2qu,ttpe1q,tpe2qu G " ϕ tspe1q,tpe2qu,ttpe1q,spe2qu G " 0 .Now suppose that neither e 1 nor e 2 is a bridge, yet Gzte 1 , e 2 u has two connected components.In particular, e 2 is a bridge in Gzte 1 u.In this case we may choose the cycles c 1 , . . ., c h such that c 2 , . . ., c h are cycles in Gzte 1 u which do not depend on e 2 .Thus we can assume that only cycle c 1 involves either e 1 or e 2 , and furthermore we can direct the edges e 1 , e 2 in such a way that their coefficient in c 1 are both 1.In this case, the coefficient of µ e1 µ e2 in detp r Λ G pµ, 0qq is Since the sources of e 1 , e 2 lie in different components of Gzte 1 , e 2 u, we have ϕ tspe1q,spe2qu,ttpe1q,tpe2qu G " 0 .
There is a one-to-one correspondence between the set of spanning forests with two components T 1 Y T 2 such that tspe 1 q, tpe 2 qu Ă V pT 1 q and ttpe 1 q, spe 2 qu Ă V pT 2 q, and the set spanning trees T " T 1 Y T 2 Y e 2 of Gzte 1 u.This implies that ϕ tspe1q,tpe2qu,tspe1q,tpe2qu G " α e1 α e2 Ψ Gzte1u and again the coefficient of µ e1 µ e2 in detp r Λ G pµ, 0qq and Φ G pqq are in agreement.Finally, consider the case when Gzte 1 , e 2 u is connected.Choose cycles c 1 , . . ., c h as in lemma 6.4.By expanding det r Λ G pµ, 0q, its coefficient of µ e1 µ e2 is by lemma 6.5.By proposition 6.3, the coefficient of µ e1 µ e2 in det r Λ G pµ, 0q is equal to that of µ e1 .µe2 in Φ G pqq up to a possible sign.It remains to determine the sign.
If the coefficient of µ e1 µ e2 vanishes in det r Λ G pq, 0q then there is nothing to prove.Otherwise, it is enough to select a single non-zero monomial m in the coefficient of µ e1 µ e2 in det r Λ G pµ, 0q, and compare with the corresponding coefficient in Φ G pqq.Such a monomial corresponds to a unique spanning forest F " T 1 Y T 2 via the formula m " ś eRF α e .The sign is determined by setting all α e to 0 for e P F, which corresponds to contracting all edges in F. Since r Λ G{F pµ, 0q " r Λ G pµ, 0q ˇˇα e"0, ePF , it therefore suffices to prove the identity for the graph G 1 " G{F.Because F has two components, G 1 has exactly two vertices.We can assume that G 1 zte 1 , e 2 u is connected since we have already considered the disconnected case.By lemma 7.5, we can remove all tadpoles from G 1 , and thus assume that G 1 is a banana graph with two vertices.The result then follows from example 7.6.
7.3.Quaternionic case (proof of theorem 7.2).It follows from proposition 2.1 that there exists a well-defined polynomial F in the α i , m i and the components λ e , ν e of the momenta µ e " λ e `jν e , and their conjugates, such that Detp r Λ G pµ, mqq " detpχ r Λ G pµ,mq q " F 2 .By inspection of (5.12), the middle term is at most quadratic in the mass squares m 2 e , and therefore F may be written in the form More precisely, by expanding detpχ r Λ G pµ,mq q using the representation (5.12), one sees that the quadratic terms in the mass squares all arise from the term X 2 det χ Λ G " pX detpΛ G qq 2 using proposition 2.1 (ii).By comparing terms in m 4 e , it follows from this that and therefore Detp r Λ G pµ, 0qq " F 2 0 .We are thus reduced to proving the case when all masses are zero.
By inspection of the matrix (5.12), the determinant is of degree at most four in the momentum components λ e , ν e , λ e , ν e , and therefore F is of degree at most two in them.It suffices to check that for any two edges e, f , the coefficient of λ e λ f (or ν e ν f , etc) in F coincides with that of Φ G pqq.Since F does not depend on the choice of momentum routing, we can assume that the momenta through edges e, f are purely complex, since they lie in a copy of R 2 -C inside R 4 -Q.In other words, we may assume that ν e " ν f " 0 is zero.Since the coefficient of λ e λ f in F does not depend on the other momentum variables, we can assume that the latter are complex as well.Now if ν e " 0 for all edges e, the complex adjoint Laplacian χ r Λ G pµ,mq is simply the image under χ of the generalised Laplacian with complex momenta.Thus theorem 7.2 follows from theorem 7.1 and proposition 2.1 (ii).

Invariant forms associated to graphs
We define invariant differential forms associated to generalised graph Laplacians.8.1.Invariant forms.Let us define a graded exterior algebra in which each element β 2k`1 has degree 2k `1.It is a graded-commutative Hopf algebra with respect to the coproduct for which the generators β 2k`1 are primitive: We shall call a primitive element odd or even depending on whether k is odd or even.The Hopf subalgebra generated by the even forms β 4k`1 was denoted by Ω can Ă B can in [Bro21].Let R " À ně0 R 0 be a graded-commutative, unitary, differential graded algebra with differential d : R n Ñ R n`1 of degree 1. Definition 8.1.For any invertible matrix X P GL h pR 0 q, let us denote by By taking exterior products, we can define β X for any β P B can .
We briefly recall properties of these forms.Self-contained proofs of these facts, which are classical, may be found in [Bro21].
(1) The forms β 2k`1 X are closed: In particular, the odd primitive forms vanish on the space of symmetric matrices.
(3) The forms β 2k`1 X are bi-invariant: for any matrices A, B P GL h pR 0 q which are constant, i.e., dA " dB " 0.
(4) The primitive forms are additive: with respect to direct sums.It follows for general β P B can that where we write ∆ can β " ř β 1 b β 2 .
(5) Let λ P `R0 ˘ˆbe invertible.Then 8.2.Canonical forms associated to graphs.The canonical forms introduced in [Bro21], which do not depend on kinematics, will henceforth be called 'of the first kind' since their denominators involve the first Symanzik polynomial.
Definition 8.2.Let ω P B can .For any connected graph G with internal particle masses and external momenta in R 2 -C, define: (i).The first canonical form to be where Λ G is any choice of graph Laplacian matrix.
(ii).Define the second canonical form to be where r Λ G pµ, mq is any choice of generalised graph Laplacian matrix.It depends a priori on the external momenta µ e and particle masses m e .
(iii).Define a mixed canonical form, or simply canonical form to be any polynomial in the canonical forms ω G , ϖ G of the first or second kinds.
The case when momenta lie in R 4 is discussed below.Since the graph Laplacian Λ G is symmetric, odd canonical forms of the first kind are identically zero.This is not the case for the forms of the second kind.A mixed canonical form is abstractly represented, therefore, by an element of (8.4) The following theorem summarizes some basic properties of these forms.
Theorem 8.3.The primitive canonical graph forms are well-defined, projectively invariant closed differential forms of the shape: where N 1 , N 2 are polynomials with rational coefficients.Let pΞ G pq, mqq Ă P E G denote the generalised graph hypersurface.For any canonical form ω P 0 can of degree n, ω G is a closed, projectively invariant form of degree n: G q which, under the assumption of generic momenta ( §4.1), is smooth on the open coordinate simplex σ G Ă P E G pRq defined by α 1 , . . ., α e G ą 0.
Proof.The fact that the forms are closed and well-defined follows from bi-invariance §8.1, as does projective invariance.It is clear from the definition that their denominators only involve a power of the determinant: the order of this power follows from [Bro21][Theorem 5.2], and the expression for the determinant as a graph polynomial follows from theorem 7.1.The numerator N 2 is a priori only a polynomial in the momentum routing µ e , but again by bi-invariance, it is independent of the choice of momentum routing, and hence only depends on external momenta q v , q v .The fact that ω is smooth on the open simplex follows from the assumption that kinematics are generic, and hence Ξ G pq, mq and Ψ G are strictly positive on σ G .□ Remark 8.4.For any Hermitian matrix X, one has (8.5) As a consequence, ϖ 2k`1 G is real if k is even, and imaginary if k is odd.In the case when k is even, ϖ G will be a rational function of the inner products q i .qj of external momenta.In the case when k is odd, ϖ 2k`1 G may also depend on the quantities (8.6) λ ij " q i q j ´qj q i " ˘2b pq i .qj q 2 ´q2 i q 2 j Remark 8.5.If G has only two external momenta q 1 " ´q2 , r Λ G pµ, mq is equivalent to a symmetric matrix because we can find a momentum routing so that the µ e lie on the real line spanned by q 1 .In this case, ϖ 2k`1 G vanishes when k is odd.8.3.Further properties of canonical forms.The following properties of canonical forms are almost immediate from the definitions.Lemma 8.6.(i).Let π be an automorphism of G.It acts on R 0 by permutation of the edge variables x e .For any canonical form of mixed type ω P 0 can , π ˚ωG " ω G (ii).Let γ Ă G be any subset of edges such that γ neither contains a loop, nor is mass-momentum spanning.Then where L γ Ă P E G denotes the linear subspace defined by α e " 0 for e P γ.
Proof.The proof for a general canonical form are essentially the same as for the first canonical form ([Bro21, §6.2]).□ 8.4.Additional form in degree 1.The bi-invariant form is excluded because it is poorly behaved.In particular, the forms , where B denotes the Euler vector field.Nonetheless, the combination is projectively invariant since B 2 " 0, which provides an additional form in degree one.However, it has poles at infinity and so products of canonical forms involving ø 1 G often give rise to divergent integrals, and will be excluded on this account.See however, §11.1, for an example of a convergent integral associated to ø 1 G .
Remark 8.7.The significance of the form ø 1 G is that it appears in the connection which defines the twisted cohomology underlying dimensional regularisation.For example, one may consider the Mellin transform where the form ω G in the integrand is to be viewed as a cohomology class with respect to the twisted connection ∇pωq " ´sø 1 G ^ω `dω.
8.5.Quaternionic forms.Let G be a connected graph with external momenta in R 4 , which we identify with quaternions via §4.4.
Definition 8.8.For any ω P Ω can , define the quaternionic canonical form of the second kind associated to G to be where χ r Λ G pµ,mq is the image under (2.1) of any choice of quaternionic Laplacian.
Remark 8.9.Since the quaternionic graph Laplacian is Hermitian, its complex adjoint X " χ r Λ G pµ,mq satisfies the following identity, where J n was defined in §2.2, and is therefore similar to its transpose.For any such matrix β 2k`1 X " 0 if k is odd.For this reason only the even canonical forms are relevant here.
The quaternionic forms of the second kind have very similar properties to the complex case.We briefly summarise these properties in the following theorem.
Theorem 8.10.Every quaternionic canonical form of degree n of the second kind is a well-defined, closed, projectively-invariant differential form which is smooth on the open simplex σ G when momenta are generic (4.1).It is invariant with respect to automorphisms of the graph G and compatible with contraction of subgraphs which have no loops and are not mass-momentum spanning.
Remark 8.11.The primitive quaternionic forms may be defined directly using the quaternionic trace instead of passing via complex adjoints, using the identity: This uses the fact that M P M n pQq is invertible if and only if χ M P M 2n pCq is invertible §2.2.

Canonical forms at infinity
This section consists of technical results concerning the asymptotic behaviour of canonical forms in the limit when the generalised graph Laplacians become singular.9.1.Asymptotic behaviour of generalised Laplacian matrices.Let I Ă E G denote any subset of the set of edges of G. Consider the map s I : Qrα e , e P E G s ÝÑ Qrz, α e , e P E G s (9.1) s I pα e q " zα e if e P I , s I pα e q " α e otherwise .
Suppose that G has internal masses and external momenta in R 2 -C, for the time being.Recall that E G was the extension defined in (5.1).
Lemma 9.1.i).Let γ Ă E G be a subgraph.Then, writing matrices in block matrix form with respect to a choice of splitting the rescaled generalised graph Laplacian satisfies: ii).Now let γ Ă E G be a mass-momentum spanning subgraph.If we write matrices in block matrix form with respect to a splitting then the rescaled generalised graph Laplacian satisfies: ˆA zB zC z r Λ γ pµ, mq ˙where A " Λ G{γ pmod zq for some matrices A, B, C with entries in Crz, α e , e P E G s.
Proof.i).The short exact sequence 0 ÝÑ H 1 pγ; Zq ÝÑ H 1 pG; Zq ÝÑ H 1 pG{γ; Zq ÝÑ 0 defines a filtration 0 Ă H 1 pγ; Zq Ă H 1 pG; Zq.We may choose representatives c 1 , . . ., c g P Z E G for a basis of H 1 pG; Zq with the property that c 1 , . . ., c h Ă Z Eγ represent a basis for H 1 pγ; Zq.A choice of momentum routing µ defines a splitting E G " H 1 pG; Cq ' C of (5.1), and hence we may write The statement (9.2) follows from the general form (5.4) of the matrix r Λ G pµ, mq using the fact that s γ c i pα, µq " zc i pα, µq for 1 ď i ď h, and the fact that ii).Now let γ be mass-momentum spanning.As above, a choice of momentum routing µ defines a splitting E G -H 1 pG; Cq ' C, and we may write Let c 1 , . . ., c g be a system of representative cycles for a basis of H 1 pG; Zq, adapted to the filtration H 1 pγ; Zq Ă H 1 pG; Zq.Consider the form (5.4) of the matrix r Λ G pµ, mq with respect to this basis.Since γ is momentum spanning, one has by definition s γ c i pα, µq " zc i pα, µq for all 1 ď i ď g.Since γ is mass and momentum spanning, one has s γ X G " zX γ , where X ‚ is defined by formula (5.5).The statement therefore follows from s γ r Λ G pµ, mq ˇˇz"0 " Λ G{γ .□ Remark 9.2.By taking the determinant and using theorem 7.1, the previous lemma gives an interpretation of the 'asymptotic' factorisation formulae stated in [Bro10, Theorem 2.7] for generic kinematics.This observation leads to another proof of theorem 7.1 since the above factorisation formulae determine Ξ G pq, mq essentially uniquely [Bro10, Proposition 4.8].
Note that in the case when γ is mass-momentum spanning, both equations (9.2) and (9.3) are simultaneously valid.Since in this case G{γ is scaleless, r Λ G{γ pµ, mq has vanishing determinant and (9.2) is not relevant.9.2.Reminders on motives for graphs with generic kinematics.We assume generic kinematics (4.1) throughout this section.Let us denote by the union of graph hypersurfaces.By blowing up linear subspaces as in [BEK06] corresponding to motic subgraphs (the notion of a motic subgraph is defined in [Bro17, Definition 3.1]), one obtains a space [Bro10, Definition 6.3] (9.4) π G : P G ÝÑ P E G which contains a distinguished simple normal crossing divisor D whose image under π G is the union of coordinate hyperplanes.The irreducible components of D are of two types: exceptional divisors D γ corresponding to γ a motic subgraph of G, and divisors D e where e is an edge of E G , which are strict transforms of the coordinate hyperplanes α e " 0. We denote the strict transform of X G by Y G Ă P G .Every exceptional irreducible component D γ of D admits a canonical isomorphism D γ -P γ ˆP G{γ which induces an isomorphism (9.5) 9.3.Canonical forms along exceptional divisors.For any canonical form η, denote its pull-back to P G zpY G Y Eq along (9.4) by where E denotes the exceptional divisor of π G .It could a priori have poles along E, but the following theorem shows that in fact it does not.
Theorem 9.3.Let η G P 0 n can be any canonical form.Then r η G has no poles along D and extends to a smooth form on P G zY G , i.e., Its restriction to an irreducible boundary component D e of D satisfies: if D e is the strict transform of the hyperplane L e " V pα e q corresponding to a single edge e of G.The restriction of η G to exceptional divisors depend on its kind.For simplicity, assume that η G P tω G , ϖ G u is primitive (the general case follows by multiplying primitive forms together), and that γ Ă G is motic and e γ ą 1.
(i).Suppose that η G " ω G is of the first kind.Then if γ is a core (bridgeless but not necessarily connected) subgraph, and r ω G ˇˇDγ vanishes if D γ is the exceptional divisor corresponding to a m.m. subgraph which is not core.
(ii).Suppose that η G " ϖ G is of the second kind.Then Proof.The argument in [Bro21, Theorem 7.4] proves the case when η is of the first kind and γ is a core subgraph.The general result follows in an identical manner from a local description of the blow-ups in affine coordinates, combined with the asymptotic formulae of lemma 9.1.The essential part of the argument is the calculation (loc.cit.) of the behaviour of the bi-invariant form β n X where X is a matrix which is asymptotically block-diagonal as in (9.2) or (9.3).□ Remark 9.4.An interesting feature of equations (9.7) is that the right-hand side involves canonical forms of both the first and second kinds.In particular, if ω is primitive of degree n " 3 pmod 4q then the corresponding form of the first kind ω G vanishes, and the right-hand sides of (9.7) reduce to a single term.
The case of a non-primitive form is easily deduced from theorem 9.3 by taking products.For example, for a general form ω, equations (9.6), (9.7) take the form Remark 9.5.The differential form ø 1 G in general has poles along boundary divisors.Let G be as above, and consider a core subgraph γ Ă E G .Then by the partial factorisations for the first Symanzik polynomial (or by lemma 9.1 i)), one has:

It follows that
G{γ `Opzq and so ø 1 G has a simple pole along the exceptional boundary divisor D γ (which is given by z " 0 in local coordinates) and its residue depends only on the loop number of γ.A similar argument shows that ø 1 G also has poles along exceptional divisors D γ , when γ is mass-momentum spanning.Differential forms of the type ø 1 G ^η, where η is canonical, will therefore in general have poles along the divisor D, and will not be considered here.They naturally arise, however, when considering Stokes' formula for canonical forms in dimensional-regularisation.9.4.Quaternionic case.The above arguments similarly apply to the case of a canonical form of the second kind ϖ Q G associated to a quaternionic generalised Laplacian, by replacing r Λ G pµ, mq with the complex adjoint matrix χ r Λ G pµ,mq .Theorem 9.3 holds verbatim (with the only difference that odd canonical forms of the second kind vanish in the quaternionic setting).In particular, ϖ Q G has no poles along D and extends to a smooth differential form on P G zY G .Its restriction to irreducible components of D are given by (9.8).

Canonical integrals and relations from Stokes theorem
Throughout this section we assume generic (Euclidean) kinematics (4.1).10.1.Canonical integrals.Let G be a connected graph with an orientation e 1 . . .^eN P `Ź Z EpGq ˘ˆ.It induces an orientation on the simplex σ G .10.1.1.Complex momenta.Suppose that external momenta lie in R 2 -C.
Theorem 10.1.Let ω P 0 can be a canonical form of degree N `1 and let G be a connected graph with N edges.Then under assumption (4.1), the integral (10.1) is finite and defines an analytic function of the external kinematics.
Proof.This is a corollary of theorem 9.3.Let r σ G " π ´1 G pσ G q denote the closure in the analytic topology of the pull-back of the open simplex σ G to P G .One has The right-most integral converges because of the fact ( [Bro17]) that r σ G , which is compact, does not meet Y G , which is where the poles of r ω G are located.□ Reversing the orientation on G reverses the sign of the canonical integral (10.1).Furthermore, if τ : G " Ñ G 1 is an isomorphism of oriented graphs (with external legs and masses, etc), then since τ ˚ωG " ω G 1 , we have: Thus the integrals (10.1) are invariants of isomorphism classes of oriented graphs.10.1.2.Quaternionic case.The previous theorem, and comments which follow, hold verbatim for graphs with momenta in 4-dimensional Euclidean space with The only difference is that in this case we must assume ω P Ω can is even, since odd quaternionic canonical forms vanish by remark 8.9.10.2.Stokes' formula.In [Bro17], it was shown that one can extend the usual Connes-Kreimer coproduct to an enlarged coproduct ∆pGq " where the sum is over all motic subgraphs of G.It has two types of terms: if γ Ă G is a core subgraph which is not m.m., then γ is considered to be scaleless, and encodes ultraviolet divergences of the graph.If γ Ă G is m.m. then G{γ is scaleless, and γ encodes certain kinds of infrared divergences of G. 10.2.1.Complex momenta.
Theorem 10.2.Let ω P 0 can be a canonical form of degree k.Write its coproduct in the form ∆ can ω " Suppose that G is a connected oriented graph with external kinematics as above with k `2 edges.Then where the rightmost sum is over motic subgraphs γ ⊊ G not including G itself (which are not necessarily connected) such that e γ " deg ω 1 i `1 ą 1, and the orientations on σ G{e , σ γ ˆσG{γ are induced from those on σ G .
Proof.Identical to the proof of [Bro21, Theorem 8.5], using theorem 9.3.□ Note that in the case when ω P B can is purely of the second kind, one has where we have decomposed the set of motic subgraphs γ of G, such that 1 ă e γ ă e G into two types according to whether γ is m.m. or not.Note that the Stokes' formulae above remain valid by analytic continuation to a larger region of kinematic space wherever the integrals are finite.10.2.2.Quaternionic case.The statement is identical on replacing ω with ω Q .
Remark 10.3.In [Bro21, Theorem 8.5], the Stokes' formula is not quite correct as stated in the case when a graph G has a tadpole γ.In that case, the corresponding term was counted twice: once in the term G{e, where e " γ, and again in the form γ ˆG{γ.The condition imposed above, that e γ ą 1, fixes this problem.
Similarly, if γ is an m.m. subgraph which has only a single edge, then it only contributes a single term in the formula (10.2), which is counted in the left-most summation over all edges, and not in the right-hand double sum.10.3.Canonical forms of compact type.Let h ě 1.Let us say that a canonical form ω P B can is of compact type relative to h if it is divisible by the primitive form ω 2h´1 of degree 2h ´1, i.e., ω " ω 2h´1 ^η for some η P B can .It follows from the property (6) stated in §8.1, that ω vanishes on all matrices of rank ă h.
Corollary 10.4.Let G be as in theorem 10.2 and let ϖ P B can be of the second kind of degree e G ´2, and of compact type relative to h G `1. Thus ϖ " ϖ 2h G `1 ^η for some η P B can of the second kind.If G is core, then formula (10.3) reduces to Proof.Let γ Ă G be a strict motic subgraph with at least two edges.Suppose that γ is not m.m. and therefore core.We need to check that a term of the form G{γ vanishes in the right-hand side of (10.3).Either ω 1 or ω 2 is of compact type relative to h G `1.If the former, then This inequality holds since h γ `hG{γ `1 " h G `1, and h γ ą 0 because γ is core.Now suppose that γ is m.m..We need to check that any term of the form

Examples of canonical amplitudes (complex momenta)
For a graph G with edges numbered 1, . . ., N , we write We consider canonical differential forms in low degrees and their integrals.For now, we consider only the case when external momenta lie in R 2 -C.
11.1.Digression: the divergent form ø 1 .The integrals involving the differential form ø 1 diverge, and for that reason are not considered to be 'canonical' forms according to our definitions.However, this form provides some interesting first examples, which may on occasion lead to convergent integrals.Graphs which pair with ø 1 have two edges.Consider the moduli space M trop 1,2,2 of metric graphs with 1 loop, 2 external legs, and 2 distinct non-zero masses.It has two metric graphs of maximal dimension (see figure 1).The one on the right has a non-trivial subdivergence and so the integral of ø 1 is logarithmically divergent.Therefore, consider the graph G on the left.It is important that its edge masses m 1 , m 2 be distinct, for otherwise, this graph has a symmetry which induces an odd permutation of its set of edges, implying that its canonical integral vanishes.
Orient and number the edges as shown in the figure .A representative for a generator of the homology H 1 pG; Zq is c 1 " e 1 `e2 .Momentum conservation implies that q 2 " ´q1 P R 2 -C.Any momentum routing is given by a solution to: The generalised graph Laplacian matrix with respect to these choices is: Ṫhe relevant graph polynomials are given by 2 α 2 qpα 1 `α2 q , where we write q 2 1 for the Euclidean norm, which equals q 1 q 1 , where q 1 is q 1 viewed in C. Using (8.7), the exceptional form of degree 1 is Thus the associated integral is where the last equality follows by computing the projective integral on the affine chart α 2 " 1.The integral reduces to ẇhich is an agreement with the renormalised amplitude for a massive bubble.As expected, it vanishes when m 1 " m 2 for the symmetry reasons mentioned above.11.2.Canonical amplitudes associated to ϖ 3 .We now turn to bona fide canonical amplitudes, whose integrals are always finite by theorem 10.1.The canonical form of smallest degree is ϖ 3 P B can .It pairs with connected graphs which have 4 edges, and vanishes for graphs with fewer than 1 loop.Thus there are 3 cases:  2) is the unique graph which gives a cell of maximal dimension in M trop 1,4,4 .Choose any complex numbers µ e for each edge e such that momentum conservation (4.2) holds: µ i`1 ´µi " q i , where q i P R 2 , corresponding to q i P C, is the external momentum at vertex i subject 1 2 3 4 q 1 q 2 q 3 q 4 Figure 2. Massive box graph to momentum conservation q 1 `. . .`q4 " 0, and where we write µ 5 " µ 1 .The generalised graph Laplacian with respect to the cycle c 1 " e 1 `. . .`e4 is Ṫhe associated graph polynomials are: G is proportional to the Feynman differential form: where the numerator has the symmetric form: It can be re-expressed as a function of the external momenta: N G " 2i Im `q2 q 3 m 2 1 ´q3 q 4 m 2 2 `q4 q 1 m 2 3 ´q1 q 2 m 2 4 `q1 q 3 q 2 q 4 ˘.
In conclusion, we find that which is proportional to the Feynman integral for the massive box graph (viewed in D " 4 spacetime dimensions).11.2.2.Examples from M trop 2,3,4 .Figure 3 depicts cells of dimension 4 on M trop 2,3,4 .The graph on the right is an example of a 1-vertex join.There is a similar graph in which the central vertex has no incoming momentum, and the left-most vertex has two incoming vertices.In both cases, the graph Laplacian has the form and one verifies that ϖ 3 vanishes for this matrix.Therefore the only non-zero contribution is from the dunce's cap, depicted on the left of figure 3. It is one of the first examples of a graph with a non-trivial subdivergence (given by the subgraph spanned by edges 3, 4), so the associated Feynman integral diverges.By contrast, the canonical integral is necessarily finite and it is instructive to see why.Since there are only 3 external legs, we can always assume that the momenta are complex numbers q 1 , q 2 , q 3 subject to momentum conservation q 1 `q2 `q3 " 0. For the basis of cycles given by c 1 " e 1 `e3 `e2 , c 2 " e 4 ´e3 , and the orientations of edges as shown in figure 3, the generalised graph Laplacian takes the form: where (4.2) takes the form: A computation shows that the canonical form of the second kind in degree 3 is , where we note that the factor q 1 q 2 ´q2 q 1 " q 3 q 1 ´q1 q 3 is indeed symmetric, by momentum conservation.The numerator vanishes at α 3 " α 4 " 0, to compensate the pole arising from the vanishing of Ξ G pq, mq along this locus.Note also that if m 3 " m 4 then the integral of ϖ 3 G will vanish, due to the fact that interchanging edges 3 and 4 induces an odd permutation on the edges of G.In conclusion, for a suitable choice of orientation of G one has ż It is interesting to compare this to the renormalised amplitude associated to the graph G, which was discussed in [BK13].
11.2.3.The space M trop 3,2,4 .The associated graphs have at most two external momenta.By momentum conservation, they lie on a real line and so we may choose a real momentum routing for which the generalised Laplacian r Λ G pq, mq will be symmetric.Therefore ϖ 3 G vanishes, and there are no non-zero canonical integrals.
11.3.Canonical amplitudes in degree 5.In degree 5 there exist canonical forms ω 5 and ϖ 5 of both kinds.There is a unique vacuum diagram, namely the wheel with 3 spokes and no external legs, which pairs non-trivially with the former.Its amplitude is 6ζp3q, and is in detail in [Bro21].
Henceforth we shall focus on the form ϖ 5 .Since β 5 vanishes on matrices of rank ď 2, the form ϖ 5 G is zero unless h G ě 2. There are several graphs G with 6 edges and two or more loops which could potentially pair non-trivially with ϖ G , but we shall only focus on a couple of examples for reasons of space.There is a single potentially non-zero canonical integral on M trop 2,5,6 , given by the box-triangle graph depicted in figure 5.A computer calculation shows that where N 2 is a certain polynomial in the m 2 i and q i , q i , and the polynomial N 1 factorises into a product of two terms N 1 " f pα 1 , α 2 , α 3 , qqgpα 4 , α 5 , α 6 , q, mq where f is a homogeneous linear form in α 1 , α 2 , α 3 which does not depend on the masses and g is of degree 2 in the α i .The integral corresponding to the term N 2 is the usual Feynman integral in 4 space time dimensions.It would be interesting to know if the canonical integral admits a natural interpretation in terms of Feynman integrals via integration by parts identities, or if it may be obtained from the usual Feynman integral by application of a natural differential operator in the external kinematics.
q 1 q 4 q 5 q 2 q 3 Consider the massive pentagon diagram with arbitrary masses m 1 , . . ., m 5 P R and external particle momenta in q 1 , . . ., q 5 P R 2 -C subject to momentum conservation.Contracting each of its edges leads to five massive box diagrams.q 5 m 1 m 5 q 1 q 4 m 4 m 2 q 3 q 2 m 3 I Gi pϖ 3 , q, mq " 0 where G 1 , . . ., G 5 are the five diagrams pictured in figure 7.This relation can easily be checked in the massless case using Panzer's Hyperint [Pan15].
Since the canonical integrals for massive boxes are proportional to their Feynman integrals, we deduce a 5-term relation for massive box Feynman integrals.It is known [DD98] that the latter are expressible as a linear combination of a number of dilogarithms.The graphical relation above is presumably equivalent to Abel's 5-term equation for the dilogarithm.This functional equation should, in addition, be compatible with the motivic coaction which was computed in [Tap21].q 1 `q5 m 5 q 3 `q4 q 5 m 5 q 4 `q5 q 1 m 1 q 3 q 2 m 3 q 2 q 1 m 2 q 3 q 2 m 3 m 4 m 2 m 3 m 1 m 4 m 2 m 4 m 1 q 4 q 5 m 5 m 3 q 3 q 1 `q2 m 5 m 2 q 5 q 1 m 1 m 4 q 4 q 2 `q3 i ẇhere µ i P Q is a quaternionic momentum routing.Its complex adjoint may be represented by χ r Λ G pµ,mq " where we write µ i " λ i `jν i .We find by computer calculation that where N G is the determinant of the following matrix: where we write ||µ i || 2 " λ i λ i `νi ν i .Thus the canonical integral associated to the massive hexagon is proportional to the usual Feynman integral (viewed in 6 spacetime dimensions).Stokes' formula applied to a massive quaternionic heptagon diagram leads to a 7-term functional equation for these integrals.

Tropical single-valued integration on curves
In this section, which is not required for the paper, we provide some motivation for the definition of the generalised graph Laplacian by analogy with smooth projective algebraic curves with a finite number of punctures.
14.1.Periods on smooth algebraic curves.Let X be a smooth compact curve over a field k Ă C, and let Σ Ă XpCq denote a finite set of points.Consider the Gysin (residue) sequence in Rham cohomology 0 ÝÑ H 1 dR pXq ÝÑ H 1 dR pXzΣq Res ÝÑ H 0 dR pΣqp´1q ÝÑ H 2 dR pXq and suppose that we are given an element ('external momentum'): α q P ker `H0 dR pΣqp´1q ÝÑ H 2 dR pXq ˘bk C , which may be interpreted as a divisor of degree 0 on X with coefficients in C supported on Σ.By pulling back the above sequence along the copy of Cp´1q Ă H 0 dR pΣqp´1q b k C spanned by α q , we obtain a simple extension over C: (14.1) 0 ÝÑ H 1 dR pXq b k C ÝÑ E X ÝÑ Cp´1q ÝÑ 0 .Since F 1 Cp´1q " Cp´1q, it splits if and only if the sequence 0 ÝÑ F 1 H 1 dR pXq b k C ÝÑ F 1 E X ÝÑ Cp´1q ÝÑ 0 splits.Such a splitting is given by an element ('momentum routing') (14.2) µ P F 1 E X whose image in Cp´1q is α q .In other words, Res µ " α q .Since H 0 pX, Ω 1 X plog Σqq " Ñ gr 1 F pH 1 dR pXzΣqq, we can assume that µ is represented by a differential of the third kind with only logarithmic poles along Σ.It is uniquely defined only up to adding a linear combination of holomorphic forms: where ω 1 , . . ., ω g P H 0 pX, Ω 1 X q -F 1 H 1 dR pXq and λ i P C. We wish to consider a Hermitian form Q on F 1 E X defined formally by: Alas, the integrals on the right are not always convergent.There are three cases: (1) η 1 " ω i , and η 2 " ω j are holomorphic.Then the restiction of Q to H 1 dR pXq is given by the convergent integrals from which one retrieves the Riemann polarisation form.If k is real, then this actually defines a symmetric quadratic form on H 1 dR pXq.(2) η 1 " ω i is holomorphic, and η 2 " µ is of the third kind (or vice-versa).
Then the singular integral is convergent since it defines a single-valued period of H 1 dR pXzΣq.To see this, note that F 1 H 1 pX, Σq " Ñ F 1 H 1 pXq is an isomorphism and since ω i vanishes along Σ it may canonically be viewed as an element of H 1 pX, Σq.The integral above is an instance of the single-valued pairing [BD21, §6.4]: s : H 1 dR pX, Σq b H 1 dR pXzΣq ÝÑ C (3) Only in the case η 1 " η 2 " µ does one have an ill-defined integral: It is singular in a neighbourhood of a point σ P Σ, since the integrand may be expressed in local polar coordinates z " ρe iθ based at σ in the form µ ^µ " λ dρ ρ dθ `regular terms , for some λ P C, which is not 'polar-smooth' in the terminology of [BD21].One can regularise these integrals around each point of Σ by a number of methods (see also [BD21,Remark 3.22]).The method used is unimportant for this discussion but the value of the integral will in general depend upon it.In any case we may simply put 1 2πi where x m is any real and positive number of our choosing.The possibility of choosing x m relates to the presence of masses in Feynman graphs.
Thus, with respect to the basis ω 1 , . . ., ω g , µ of E X , the form Q is represented by a Hermitian matrix Q " ¨. . .
This is precisely the form of the generalised Laplacian matrix (compare (5.4)).
14.2.Tropical analogy.Let G be a connected graph (with no external half edges in the first instance).Choose an orientation on its edges.According to [MZ08], a differential form on G is a linear combination ω " where λ e P Z such that for all vertices v one has ř speq"v λ e " ř tpeq"v λ e .Thus the space of differential forms Ω 1 G on G lies in an exact sequence 0 ÝÑ Ω 1 G ÝÑ Z E G ÝÑ Z V G ÝÑ Z ÝÑ 0 and may be identified with H 1 pG; Zq.The inner product on Z E G is given by xde i , de j y " x i δ ij , and restricts to a positive definite quadratic form on Ω 1 G .
Now suppose that G has external half-edges, and that we are given an element q P kerpZ V G Ñ Zq b C. As in §5.1, we deduce an extension r Ω 1 G satisfying 0 ÝÑ Ω 1 G b C ÝÑ r Ω 1 G ÝÑ q C ÝÑ 0 and the above inner product (extended C-linearly) restricts to a Hermitian form which is a tropical analogue of (14.3).The generalised graph Laplacian matrix r Λ G pq, 0q for zero masses is precisely the matrix of this Hermitian form with respect to a splitting of r Ω 1 G , which is given by any differential form ν " ÿ e µ e de which satisfies the 'residue' equation ř speq"v λ e ´řtpeq"v λ e " q v for all vertices v.The presence of masses corresponds, as above, to modifying only the value of xν, νy in the bottom-right hand corner, and from this perspective, we see that the generalised graph Laplacian (5.4) is a tropical analogue of the regularised singlevalued integration pairing discussed in §14.1.

7. 1 .
Preliminary reductions.It follows from the presentation (5.4) of r Λ G pµ, mq in block matrix form that det r Λ G pµ, mq " mq " ˆzΛ γ zB zC D ẇhere B, C, D are matrices with entries in Crz, α e , e P E G s and D " r Λ G{γ pµ, mq pmod zq .

Figure 3 .
Figure 3. Left: The dunce's cap has a non-trivial divergent subgraph given by the subgraph spanned by edges 3, 4. Right: A double bubble graph, upon which the canonical form vanishes.

Figure 4 .
Figure 4. Some graphs with 6 edges for which ϖ 5 G vanishes

Figure 7 .
Figure 7. Five-term relation from pentagon diagram Changing the orientation of an edge does not modify the matrix r Λ G .To see this, observe that switching the orientation of any number of edges amounts to multiplying the matrix r H G on the left by a diagonal matrix E with entries in t1, ´1u: reversing the orientation of edge e results in a change of sign for the associated momentum µ e .The claim follows since E ˚DG E " D ePEpGq p e µ e α e , and cpα, ιpµqq for its image under ι.5.2.1.Change of edge orientations.G and since ĂM G does not depend on the edge orientations.5.2.2.Change of basis.The matrix r Λ G depends on the choice of basis for the homology H 1 pG; Zq.Changing basis modifies the matrix r §1.4]A subgraph γ Ă E G is momentum-spanning if all non-zero external momenta q v of G meet vertices vwhich lie in a single connected component of γ.It is called mass-spanning if it contains all massive edges of G.It is called mass-momentum spanning (or m.m. for short) if both hold.§1.7) which implies that the coefficients of every monomial in the variables α i in Ξ G are non-negative (this would not be true in Minkowski space).If γ is not m.m. then G{γ has a scale and one shows that Ξ G{γ pq, mq ą 0 in the region α e ą 0, for all e P E G{γ " E G zE γ .6.3.Spanning forest and Dodgson polynomials.The second Symanzik can be expressed using spanning forest polynomials, which were defined in [BY11].6.3.1.Spanning forest polynomials.Definition 6.2.Let P " P 1 Y . . .Y P r be any partition of a subset of vertices of G into disjoint sets.The associated spanning forest polynomial is defined by 6.1.3.Dependence on masses.The '2nd Symanzik' polynomial which occurs in the (Euclidean) parametric representation of Feynman integrals is (6.4) Ξ G pq, mq " Φ G pqq `˜ÿ e "0,ePγ " Ξ G{γ pq, mq is identically zero if and only if γ is an m.m. subgraph.In this paper we work in the Euclidean region (loc.cit.