Schwinger, ltd: Loop-tree duality in the parametric representation

We derive a variant of the loop-tree duality for Feynman integrals in the Schwinger parametric representation. This is achieved by decomposing the integration domain into a disjoint union of cells, one for each spanning tree of the graph under consideration. Each of these cells is the total space of a fiber bundle with contractible fibers over a cube. Loop-tree duality emerges then as the result of first decomposing the integration domain, then integrating along the fibers of each fiber bundle. As a byproduct we obtain a new proof that the moduli space of graphs is homotopy equivalent to its spine. In addition, we outline a potential application to Kontsevich's graph (co-)homology.

1. Introduction 1.1. Background. Loop-tree duality expresses the momentum space integral of a Feynman diagram G as a linear combination of (simpler) integrals indexed by the set of spanning trees of the underlying graph. This simplification is achieved by iterated applications of Cauchy's residue theorem. We sketch the idea, in order to motivate the alternative approach presented down below. For the general story see the review article [Agu+21] as well as the references therein. We follow the arguments of [Run+19].
Consider the Feynman integral of a one-loop graph on n edges in a scalar, massive theory, (1) where each m i > 0 and each L i is Z-linear in the loop momentum k ∈ R 1,D−1 and the external momenta p 1 , . . . , p n ∈ R 1,D−1 . The precise form of the L i depends on a choice of momentum flow, but the integral is independent of this by momentum conservation.
Let us abbreviate the integrand by We can do the integration over the energy component k 0 in eq. (1) by using the residue theorem, where Γ is a curve that runs first along the real axis from −∞ to ∞, then closes along a semi-circle in the upper/lower half-plane through the point ±i∞ (either choose one or average over both choices). For generic values of p this will collect n different residues, one for each pole ρ i = ρ i ( k, p) : k 0 = (L i (k, p)) 0 ± L 2 i (k, p) + m 2 i − iε of the propagators L 2 i (k, p) − m 2 i + iε, i = 1, . . . , n. See also the derivation in [BK20, §7] using divided differences to systematically extract the poles of each propagator.

This transforms I G into a linear combination of integrals
(2) indexed by the spanning trees of G (each pole belongs to a propagator/edge, its complement is a spanning tree of G).
It seems natural to generalize this to the multi-loop case, (3) using induction on the number of loops = h 1 (G). However, the location of each pole depends not only on k and p, but also on ε. It is therefore not immediately clear how to compute iterated residues of this form. We refer to [Agu+21] for a thorough discussion and general solution of this problem.
1.2. Parametric space and sector decompositions. One may wonder whether a version of this loop-tree duality exists for Feynman integrals in Schwinger variables.
In this case there seems to be no straightforward analytical method to relate a parametric Feynman integral to a linear combination of integrals indexed by the spanning trees of G. Of course one could use the Schwinger trick to transform an integral of the form (2) into a parametric version. However, if one wishes to work exclusively in parametric space, then a more combinatorial approach can be based on Hepp's sector decomposition [Hep66;Spe75]. For a detailed account we refer to [Hei08]; see also [KY21,§4.3] as well as [Sch18] which discusses general sector decompositions from the viewpoint of toric geometry. This approach leads to formulae of the sought-after form, but there seems to be no general account of loop-tree duality for parametric Feynman integrals in the literature, at least not under this name.
1.3. An alternative approach. We propose here a different, more geometric solution.
The idea was sketched in [BK20]. It is based on a study of the combinatorial geometry of the integration domain. Recall that a parametric Feynman integral arises from integrating a differential form ω G (defined in section 4.1) over a simplex in projective space, (4) We will see below that (up to a subset of measure zero) the integration domain ∆ G decomposes into a disjoint union of cells D T , indexed by the set of spanning trees of G, such that each cell is the total space of a fiber bundle π T over a cube C T = (0, 1) |E T | .
The construction of the map π T can be sketched as follows. The set of all subgraphs of G is partially ordered by inclusion. The order complex of this poset 1 is isomorphic to the We call the subcomplex formed by the vertices w γ , where γ = G/F for F ⊂ G a forest, the spine of ∆ G . The other vertices represent the faces of ∆ G where the set {e ∈ E G | x e = 0} determines a subgraph In the context of moduli spaces of graphs these faces are called faces at infinity (see section 2.3). Each spanning tree T ⊂ G represents a maximal cell C T in the spine of ∆ G . It is a t = |E T | dimensional cube, the union of all simplices This defines a smooth fiber bundle that extends to a piecewise smooth fiber bundle on a certain closure of D T (which includes C T ). In fact, it can be extended even further to obtain a fibration π G of ∆ G over its spine. Furthermore, these maps fit together to form a fibration π of the moduli space of graphs over its spine; see remark 2.7.
With this at hand we are able to rewrite I G as where (π T ) * denotes the push-forward along π T , that is, integration along the fibers of this bundle.
Here it is important to note that both steps are only well-defined if I G converges absolutely. Thus, to be precise, we should either start with a renormalized integrand or work with some regularization of ω G and later worry about the analytic continuation to the points of physical interest. We discuss this in detail in section 4.1.
1 The order complex of P is the simplicial complex whose vertices are the elements of P and p 0 , . . . , p n ∈ P form an n-simplex iff p 0 ≤ . . . ≤ p n .
[1 : 0 : 0] [0 : 1 : 0] [0 : 0 : 1] Figure 1. The sunrise graph G and the simplex ∆ G ⊂ P G = P(C 3 ). The red part is the spine of ∆ G , the simplicial/cubical complex on the central vertex w G and the three vertices w G/e i . The blue lines indicate the fibers over the cube C T for T = e 2 , the black vertices, corresponding to graphs G/{e i , e j }, are the faces at infinity.
In summary, we can express a parametric Feynman integral as a linear combination of (simpler) integrals of the push-forwards of ω G over the cubes C T . This turns eq. (4) into the sum running over all spanning trees of G.
This parametric loop-tree duality is arguably much "cleaner" than its momentum space counterpart. Note, however, that the technical details and intricacies have not simply disappeared; they are hidden in the geometry of the decomposition of ∆ G and the structure of the fiber bundles π T . We demonstrate this by a couple of examples throughout the paper, in particular we use the sunrise diagram ( fig. 1) as a running example.
When comparing the two variants, it is also important to note that in momentum space we really reduce the number of integration steps; the residue theorem takes care of all k 0 -integrations. On the other hand, in the parametric version the fiber integration is a non-trivial task, as exemplified by our discussion of the 'wheel with three spokes' in 1.4. Organization of the paper. We start section 2 by introducing some notation and conventions. Then we discuss decompositions of the graph simplex ∆ G and how this gives rise to the fibrations π T . An important aspect is understanding the structure of the fibers which takes up most of this section. After this is established, we discuss an example in detail. The section finishes with some comments on the naturality of the construction and other possible fibrations.
In section 3 we quickly recall the definition of and elementary facts about fiber integration.
Section 4 applies the content of the previous two sections to Feynman

Simplex fibrations
2.1. Notation and basic definitions. The following conventions will be used throughout the paper.
• We write x = (x 1 , . . . , x d ) for a vector x ∈ k d with k ∈ {R, C} and [x] = [x 1 : . . . : x d ] for the corresponding point in P(k d ) -we sometimes omit the brackets [·] if the meaning is clear from the context. We set R + = (0, ∞).
• For a graph G we let E G and V G denote its set of (internal) edges and vertices, respectively. We set e G = |E G |, v G = |V G |, and we write h G = dim H 1 (G; Q) for the rank or loop number of G.
• All graphs we consider here will be finite and connected.
• A graph is core or 1-particle irreducible (1PI) if removing any edge reduces its first Betti number by one.
• A graph may have external edges (legs) 2 . These are defined in the usual way correspond to single non-paired half-edges. A tadpole (self-loop) is an edge formed by two half-edges that are connected to the same vertex.
• A tree is a graph T such that h T = 0. A forest is a disjoint union of trees.
We write G/γ for the graph obtained from G by collapsing each connected component of γ to a vertex.
• A spanning tree of G is a subgraph T ⊂ G such that T is a tree and V T = V G . A spanning forest of G is a subgraph F ⊂ G such that F is a forest and V F = V G .
• We sometimes abuse notation by identifying edge sets and subgraphs in the obvious way.
• If no confusion is possible, we write x for a singleton {x}.
2.2. The graph simplex and its spine. Let G be a graph without tadpoles (the case with tadpoles is discussed in remark 2.9). Set For every spanning forest F ⊂ G we define a subset C F of ∆ G by (5) It can be parametrized by an open cube of dimension e F : To ease notation, suppose that the edges of G are ordered such that the edges of F are labeled by 1, . . . , f . Then C F is parametrized by the map 2.
3. An excursion into the moduli space of graphs and its spine. We sketch the geometric background that motivated the construction of the fibration presented in this paper. This section is not essential and may be skipped on a first read.
We follow the exposition in [CGP21;CGP22] which studies the moduli space of tropical curves which are certain weighted, marked metric graphs. A slight adjustment produces the moduli spaces of graphs (as a subset of the latter) which is classically defined as the quotient of Outer space by the action of Out(F n ) [CV86]. 3 We discuss here the case without legs/external edges, but it easily generalizes to the case where such special (half-)edges are allowed.
Let 2 ≤ g ∈ N. Define a category Γ g by • ob(Γ g ) is the set of all graphs of rank g that have no bridges/separating edges (1PI) and all vertices at least three-valent.
• hom(Γ g ) are given by forest collapses or isomorphisms of graphs.
denote the space of metrics on G that are allowed to vanish on forests in G. Given a This defines a functor from Γ g to the category of topological spaces. The moduli space of rank g graphs MG g is defined as the colimit of this functor.
If there were no graph isomorphisms in hom(Γ g ), the space MG g would be a union of cones, with some of their faces deleted, identified along their common boundaries.
Isomorphisms act however non-trivially on this space, so that MG g is an orbifold. For instance, if G is the sunrise graph ( fig. 1), then the corresponding cone in MG 2 is the quotient of R 3 ≥0 by the action of S 3 permuting the coordinates x 1 , x 2 , x 3 . It is often convenient to normalize the metrics on graphs. For this define a function λ : MG g → R ≥0 by measuring the volume of a metric, λ( ) = e∈E G (e). Consider the subset M G g ⊂ MG g defined by M G g = λ −1 (1). Note that this turns the cones σ(G) into simplices. In the following we will work exclusively with this space and henceforth refer to M G g as the moduli space of (rank g) graphs.
If we ignore isomorphisms for a moment, we can think of M G g as a semi-simplicial complex (aka ∆-complex [Hat02]) with some of its faces deleted. If σ G is a cell in M G g , parametrizing the space of metrics of unit volume on G, then If on the other hand a metric vanishes on a subgraph γ with h γ > 0, then this face is not in M G g ; it is called a face at infinity.
One may compactify M G g by various methods of which two are important for us.
Firstly, we can add all faces at infinity to form a (semi)simplicial completion M G g of M G g [CGP21]. Secondly, we can truncate small "neigborhoods of infinity" in each cell 3 It is actually much easier to define the moduli space of graphs via the detour through either tropical curves or Outer space. Interested readers are therefore encouraged to consult the given references. σ G to obtain polytopes J G that assemble to a compact space M G g , homotopy equivalent to M G g . This is equivalent (homeomorphic) to the iterated blow-up of M G g along the faces at infinity. For the details we refer to [BSV18]; see also [Bro21;Ber20].
In the context of Outer space it is well-known [CV86] that M G g deformation retracts onto its spine S g , a subset which has the structure of a cubical complex [HV98]. As an abstract complex it is defined as the order complex of the poset (ob The geometric realization of this complex can be embedded as a subcomplex S g of the barycentric subdivision β(M G g ) of the semi-simplicial completion of M G g . The deformation retract r : M G g → S g is then described by collapsing all the cells that have vertices at infinity, that is, in M G g \ M G g .
Above (section 2.2) we have seen how to embed S g into M G g . In remark 2.7 we sketch how the maps π T can be assembled to define a fibration π : M G g → S g , producing another proof that both spaces are homotopy equivalent.

2.4.
A fibration over the spine. Let T be a spanning tree of G, that is, a maximal forest T ⊂ G. We want to define a subset D T of ∆ G (or σ G , as defined in section 2.3) such that there is a surjective map π T : D T → C T . Moreover, we want the union of the sets D T where T ranges over all spanning trees of G to form a partition of ∆ G (up to a set of measure zero).
Putting the cart before the horse, we start by describing the fibers of this sought-after map. In fact, the fibers are the very thing we are interested in because our eventual goal is to integrate forms along the fibers of this map. In any case, we need to establish that there is a map with the asserted properties.
As mentioned in the introduction, the idea is to define π T by bundling together all fibers that connect C T to certain subsets of the faces at infinity in ∆ G , that is, to the faces where {e ∈ E G | x e = 0} defines a subgraph γ with h γ > 0. These subsets of faces will be indexed by certain core subgraphs of G. To make this precise consider the set of all proper core subgraphs of G that share at least one edge with T and whose union with T does not cover the whole graph. This set is partially ordered by the inclusion relation. We let C max (T ) denote its maximal elements.
Lemma 2.1. Let G be a connected graph. The cardinality of C max (T ) is equal to h G , the rank of G. In particular, |C max (T )| is independent of the choice of spanning tree T of G.
Here we are tacitly working with isomorphism classes of graphs tadpoles/self-loops connected to a single vertex). There is a bijection The injection ← is given by expanding T again and chopping off edges from the subgraph I ∪ T to make it core. For the other direction → recall that every γ in C max (T ) shares at least one edge with T and its union with T is not the full graph G. Thus, after collapsing T the subgraph γ becomes a rose with at most h G − 1 petals. If the number of petals is less than h G − 1, then γ was not maximal.
We now describe the fibers of the map π T . Given a point x ∈ C T we let the fiber of π T over x be the set where v γ is the map that sends all x e with e ∈ E γ to zero, In order to define the map π T we need the following Proof. First of all note that Cone  So let x = x ∈ C T and suppose first that h G = 2. This means C If not, then using the parametrization by eq. (6) we can find a non-empty subset I ⊂ E T \ E γ such that Points in the intersection of C x and C x satisfy
Since I is disjoint from E γ , the first item reads x e = λx e . Thus, λ = 1. With (4) we find then µ = µ . Finally, from (2) and (3) we infer x e = x e for all e ∈ E G , a contradiction.
The general case h G > 2 follows by the same reasoning as above, using the following claim: For any subset A ⊂ C max (T ) with |A| = h G − 1 (1) there is an edge e that does not belong to T , but to every element of A.
(2) for any γ ∈ A there is an e(γ) in the complement of T and γ which belongs to This is a corollary of lemma 2.1. We can think of an element γ ∈ C max (T ) as a subset I ⊂ {1, . . . , h G } of cardinality h G −1, or equivalently, as a one-element subset of the latter.
That is, we can specify an element γ ∈ C max (T ) uniquely by an edge in the complement of T that does not belong to γ, but to any other γ ∈ C max (T ). By the same argument, A is determined by choosing h G − 1 "forbidden edges" out of {1, . . . , h G }. Thus, there is always one left which has to belong to every γ ∈ A.
With this we can solve the system of equations for C x ∩ C x . (1) leads to λ = λ and (2) leads to µ i = µ i for every i = 1, . . . , h G−1 . Hence, x = x , a contradiction. The previous lemma allows to finally define the map π T . We set D T = x∈C T π T −1 (x) and define π T by This is well-defined by lemma 2.3.
The closure of D T in ∆ G has a convenient description in terms of β(∆ G ), the first barycentric subdivision of ∆ G 5 . The vertices of the latter complex are in one-to-one correspondence with proper subgraphs γ G. We can parametrize them by Then D T is the interior of the subcomplex of β(∆ G ) spanned by the vertex set By definition D T contains all rays from points in C T (including the vertices of type w F ) to the vertices of type w γ with γ ∈ C max (T ). Furthermore, if γ 1 ∩ γ 2 = ∅, then the closure of Cone x (v γ 1 (x), v γ 2 (x)) contains the vertex w γ 1 ∩γ 2 : (9) w γ 1 ∩γ 2 = lim µ→∞ [x + µv γ 1 (x) + µv γ 2 (x)] for any x with x e = 1 if e ∈ E γ 1 ∩γ 2 .
Additionally, D T contains all the vertices w γ∪F for F ⊂ T and γ ∈ C max (T ): Both eqs. (9) and (10) have solutions x in (the closure of) C T .
Proposition 2.4. Up to a set of measure zero (codimension greater than one) the sets Proof. The first assertion is true because every point x in ∆ G lies in a maximal simplex σ = (w ∅ , w e 1 , w {e 1 ,e 2 } , . . .) of β(∆ G ). Let F be the forest in G, defined by Then we can use eqs. (9) and (10) to infer that x ∈ D T for any tree T ⊂ G such that The second assertion follows from the fact that D T and D T intersect in the closure of Proposition 2.6. The map π T : D T → C T , defined in eq. (8), is a smooth fiber bundle whose fibers are diffeomorphic to h G copies of R e G −e T −1 = R h G −1 .
Proof. Recall that D T is a disjoint union of h G open cones over the points in C T . The map π T simply maps all points in a cone to its basepoint which is well-defined by lemma 2.3, and globally trivial (by construction, and also because the base is contractible). Moreover, it is obviously smooth.
Note that π T is well-defined on D T minus the points at infinity in ∆ G . We could thus extend the family {π T : D T → C T } T ∈T (G) to a single continuous -in fact, piecewise smooth -map π G on ∆ G (but not on ∆ G ). In other words, it extends to ∆ G , the closure of ∆ G in the compactification M G g (Figure 4 indicates the map π G on the compactified cell ∆ G ).
In both cases the resulting map is no longer a fiber bundle, but only a fibration 6 . The fibers over cubes C F for F a spanning k-forest with k > 1 are unions of faces of the (closure of the) cones (7), hence also contractible, but not necessarily homeomorphic to the other fibers. For an example see fig. 1 (or fig. 4): None of the three maps π T is defined at the three corners of ∆ G . Each map can be extended to the lines {x i = x j , x k > x i : i, j, k∈ {1, 2, 3}}, but the fiber over the center w ∅ = [1 : 1 : 1] is not homeomorphic to R.
At this point one might wonder why we do not work with ∆ G instead of ∆ G . This has certain advantages, for instance making the fibers of each π T compact. This in turn would streamline the application of fiber integration in section 4.2. The price to pay is that the map π T , in particular its fibers, are more difficult to describe in this setting. Remark 2.7. The fact that there is a fibration π G from ∆ G minus its faces at infinity to F ⊂G C F , where the union is over all spanning forests of G, implies that the two spaces are homotopy equivalent. Moreover, the construction of π G is functorial with respect to forest collapses and isomorphisms: (1) If G and G are isomorphic as graphs, then π G and π G are isomorphic as bundles.
(2) Each π T extends to the faces {x i = 0 | i ∈ I, I ⊂ E T } by π T /I = (π T ) |{x i =0|i∈I} . If there is I ⊂ T, T for two trees T ⊂ G and T ⊂ G , such that G/I = G /I, then by (1) the respective bundles π T and π T agree when restricted to the corresponding common face of ∆ G and ∆ G .
Thus, we can combine the above observations to deduce the existence of a homotopy equivalence π between the moduli space of graphs and its spine, a well-known fact [CV86].
See also [BK] for another proof in a similar setting but without providing an explicit map.
We finish this section with an example. If we let ϕ denote the affine chart (x 1 : x 3 ) we have in these coordinates so that π T : D T → C T is locally given by the map π loc Note that z < w, so it really maps into the unit cube. The fiber over a point (u, 1) is By the obvious symmetry of G we have an analogous description for the other two spanning trees of G. Note that this example generalizes in a straightforward way to the "n-edge banana" graphs, consisting of two vertices connected by n edges. For each n the spine is a onedimensional "star", formed by n rays connecting the center of ∆ G ⊂ P(C n ) to the barycenter of its n codimension one facets. The fiber over each point in a 1-cube C T is the union of h G = n − 1 cones, each homeomorphic to R n−2 (the closure PL homeomorphic to a simplex of dimension n − 2). See fig. 5.
Remark 2.9. If the graph G has tadpoles the construction has to be adjusted. Consider, for instance, a graph G on three edges formed by a 2-edge banana with a tadpole (fig. 6).
Then it is easy to see that the definitions in eqs. (7) and (8) are not sufficient; the union of the fibers π T −1 (x) does not cover ∆ G . This case can be repaired by adding the tadpole e 3 to C max (T ), but it is unclear how to do this in general. Firstly, the spine S G can obviously be embedded in ∆ G in many ways. However, our choice is in some sense canonical. Consider for instance the sunrise graph ( fig. 1). The space of edge lengths without normalization is a 3-dimensional cube R 3 + . Working on the closure we can replace it by the unit cube [0, 1] 3 . The graph simplex ∆ G is the link of the origin. If we embed it as the intersection of [0, 1] 3 with the unit sphere (e.g. with respect to the 1-norm || · || 1 to obtain an euclidean simplex), then the spine C G emerges naturally as the projection of the codimension one "back faces" that connect to the point (1, 1, 1), see fig. 8. This construction generalizes in a straightforward way to every graph simplex and also to the full moduli space of graphs (or tropical curves); see [BK].
There are also many ways of expressing ∆ G as the total space of a fibration over its spine. Consider again the sunrise graph in fig. 1. In this case it is easy to imagine different bundles that cover ∆ G up to a set of measure zero. It is however not clear how to make these ideas work in general, except using the geometric approach presented here, projecting the spine into the faces at infinity.

A reminder on fiber integration
We recall the notion of push-forward or integration along the fibers of a bundle, following [Nic07]. See also [BT95].
Fiber integration is typically defined for bundles with compact fibers or as a map on compactly supported differential forms (or differential forms with compact support in the fiber direction). In contrast, the bundles that we consider have non-compact fibers. In addition, the Feynman forms ω G that we want to integrate are not compactly supported, neither in the total space nor in the fiber. We are saved, though, by Fubini's theorem and the fact that the total integral σ G ω G is absolutely convergent (when suitably regularized or renormalized). Alternatively, we could work on the compactified cell ∆ G in order to have the fibers of π T compact.
In any case we use the following proposition as a definition for fiber integration.
Let Ω cpt (X) denote the space of compactly supported forms on a smooth manifold X.
We cite proposition 3.4.47 from [Nic07]: Proposition 3.1. Let p : E → B be an orientable fiber bundle with fiber F , k = dim F .
Then there exists a linear operator It is uniquely defined by its action on forms supported on local trivializations where The operator p * is called push-forward along p or integration along the fibers of p.

Feynman integrals
We apply the results of the previous two sections to parametric Feynman integrals. We start with a quick review on parametric Feynman integrals. A thorough discussion can be found in [Pan15b].

Feynman integrals in Schwinger parameters. Let G be a (massive scalar)
Feynman diagram, that is, a finite connected graph with n labeled edges (masses m e and powers a e of propagators) and k labeled legs (momenta p i of in/out-going particles).
The Feynman integral of G is (up to a prefactor of constants and Γ-functions) = (a 1 , . . . , a n ) and Here ψ G and ϕ G are the two graph polynomials x e , p(T 1 ) = −p(T 2 ) denoting the sum of all momenta flowing into the T 1 -component of the 2-forest F . The complex number is the superficial degree of divergence of G. As the name suggests, it determines the region of parameters (p, m, a, D) where I G converges absolutely.
We henceforth assume that the parameters (a, D) lie in the region determined by this criterion. Otherwise we would have to renormalize the integrand [BK11] in order to make sense of the integral. Since we are mainly interested in structural properties we omit a discussion of this procedure 7 . The main message is that the integral I G converges absolutely, so that we can apply the aforementioned formalism of fiber integration to this case.

4.2.
Loop-tree duality. To arrive at a parametric variant of loop-tree duality we decompose the integration domain of I G as ∆ G = T ∈T (G) D T , using proposition 2.4.
We can thus rewrite I G as The individual integrals I G,T can be further simplified, The second identity is valid only outside a set of measure zero X T ⊂ D T where the fiber bundle π T is not smooth (see proposition 2.6). To be precise we should definê It is important to note that there is no map (π T ) * : Ω • (D T ) → Ω •−h G +1 (C T ); the pushforward is only defined on those forms for which the fiber integral is finite 8 , in particular those forms ω ∈ Ω • (D T ) for which D T ω converges absolutely.
Putting everything together, we have proved as in eq. (11) can be written as a sum of integrals over unit cubes C T ∼ = (0, 1) e T , indexed by the set T (G) of spanning trees of G, where ω G,T = (π T ) * ω G ∈ Ω e T (C T ) andπ T the smooth fiber bundle defined above in eq. (12). Thus in this case eq. (13) takes the form

4.3.
Examples. Recall the discussion in example 2.8. For T = e 1 we have in the affine chart with x 3 = 1 This is smooth on the complement of C T in D T , soD T = D T − C T . In these coordinates the integral over the sector D T is given by with f G determined by the Feynman rules in section 4.1. Then the fiber integral is and eq. (13) takes thus the form See also [Kre22,Appendix E] which discusses the renormalized parametric sunrise integral along similar lines.
To be more concrete let us consider a simple toy model. If z dw (zw + 1) 2 (z + 1) 2 dz, and thus which may be confirmed by direct integration. As another example we consider the wheel with three spokes (fig. 7). Its period is the integral The spanning trees of G come in two different shapes, either paths on three edges, for instance {e 1 , e 2 , e 3 }, or "claws" 9 , for instance {e 1 , e 2 , e 6 }. In both cases the set C max (T ) consists of the core subgraphs obtained from adding to T two out of the three edges from its complement. In the affine chart x 6 = 1 where C T = {(x 1 , x 2 , x 3 , 1, 1) | x i ∈ (0, 1)} we find du v 4 f 6 (vx 1 , vx 2 , vx 3 , vu, v) du v 4 f 6 (vx 1 , vx 2 , vx 3 , v, vu) dx 1 dx 2 dx 3 .
We can explicitly compute the forms ω G,T by means of the contraction-deletion identity Write ψ I J for ψ G where G is the graph G with edges e i , i ∈ I, removed and e j , j ∈ J, contracted. Then It follows that We may therefore iterate the process of fiber integration.
Let us look at the 4-edge banana graph B 4 in fig. 5. The closure in ∆ B 4 of the fiber over x ∈ C e 4 is (PL) homeomorphic to a 2-simplex which we identify with the space of metrics on the graph G − T = G − e 4 , a 3-edge banana B 3 . Moreover, the three lines from x to the corners at infinity on which π e 4 is not smooth correspond precisely to the three lines where π B 3 fails to be a fiber bundle (see fig. 4). This means we can express the integral along the fibers of π B 4 itself as a (sum of) fiber integrals of the form I B 3 , up to a set of measure zero.
For a general banana graph we can iterate this process until we are left with onedimensional fibers, that is, until G − T 1 − T 2 − . . . − T k is the 3-edge banana graph. Here, no further simplification is possible.
Note that this is very reminiscent of the recursive structure used in [Kre22] to discuss the analytic structure of Feynman integrals of banana graphs.
It would be interesting to study if and how this generalizes to graphs with banana subgraphs (that is, graphs with multi-edges).

5.2.
Periods. The loop-tree duality formula eq. (13) arises solely from a decomposition of the integration domain subject to the combinatorics of the graph G.
If the integrand is sufficiently nice, for instance in the case of periods, or generalized periods in the sense of [Bro10], it may be possible to further simplify the forms ω G,T by exploiting the structure of the graph polynomial ψ G (cf. [Bro10, §2, §3]). The example of the wheel with three spokes shows that we can always do one fiber integration using the contraction-deletion identity for ψ G .
A natural question to ask then is for which forms ω G the pushforwards ω G,T can be computed explicitly and how does it depend on the combinatorics of G and T . where we integrate over the full cell σ(G) in MG g , that is, without normalizing the volume of G. The decomposition of ∆ G that we constructed above can be "lifted" to a decomposition of σ(G): The spine becomes a fan based at the origin 0 ∈ R n , the maps π T assemble to one-dimensional families of fiber bundles.
In this representation the problem of ultraviolet and infrared renormalization splits in an interesting way. While the divergences at x e → 0 are still located in the fibers of π T , Figure 8. The cell σ(G) of the sunrise graph. The gray simplex is ∆ G , its spine indicated by the red dashed lines . The three red cones form the associated fan in R 3 + . To find the fiber over a point x in one of these cones, translate ∆ G , so that x ∈ τ.∆ G = { 3 i=1 y i = τ }, then take the corresponding fiber in τ.∆ G (as in fig. 1). the divergences at x e → ∞ (see e.g. [AHM22]) lie now on the respective components of the spine. This means that ultraviolet renormalization is needed to define the (lifted) push-forwards (π T ) * while the infrared divergences appear then at the level of the forms ω G,T . 5.4. Differential forms on graph complexes. The parametric loop-tree duality formalism applies of course also to other differential forms on ∆ G , not just those given by Feynman rules. Moreover, one may also try to apply it to forms on the (full) moduli space of graphs or tropical curves, for instance the canonical forms constructed in [Bro21;Bro22].
Recall the definition of M G g in section 2.3. A smooth differential form of degree k on M G g is a family ω = ω G ∈ Ω k (∆ G ) | G ∈ ob(Γ g ) such that (1) if ϕ : G ∼ → G is an isomorphism and ∆ ϕ the induced map ∆ G ∼ → ∆ G , then if e ∈ E G is not a tadpole and ι e is the inclusion ∆ G/e → ∆ G , then ω G extends to the face ι e (∆ G/e ) and satisfies ι * e ω G = ω G/e . Examples can be found in [Bro21;Bro22]. Note that Feynman rules do not define differential forms on M G g ; they define a form ω G on ∆ G for each G, but these do not assemble to a differential form in the above sense. Instead one may interpret the family {ω G } G as a distribution on the cell complex M G g [Ber20].
Given a differential form ω on M G g such that ∆ G ω G converges absolutely for every graph G, we can associate to it a differential form η on the maximal cubes of the spine S g . On a cube C G T ⊂ ∆ G (we add a superscript label to cubes and maps in order to keep track of the cell we are working in) where T ∈ T (G) use the push-forward along the corresponding restriction π G T of π : M G g → S g to set Now the question is if and how η can be extended to the full spine S g . A cube C G T has two kinds of faces; they correspond either to collapsing an edge e in T , or removing it from T .
In the first case, if we send x e to zero, we get a cube C G/e T /e in ∆ G/e . By functoriality of both ω and π (item (2) above and remark 2.7) we have ι * e η G T = η G/e T /e .
In the second case the push-forward along π is not defined, since over C G T −e the map π is not a fiber bundle. If we simply set where τ e is the inclusion C G T −e → C G T defined by sending x e to 1, then it is not clear whether this is well-defined. In fact, examples show that it is not. However, if η G T and η G T are closed, and for T − e = T − e the differences τ * e η G T − τ * e η G T = dα are exact, then (under some mild technical assumptions) the triple (α, η G T , η G T ) defines a cohomology class on C T ∪ C T ∪ C T −e . 10 These ideas and their application to the study of graph complexes will be pursued in future work.