Motives of melonic graphs
MMOTIVES OF MELONIC GRAPHS
PAOLO ALUFFI, MATILDE MARCOLLI, AND WALEED QAISAR
Abstract.
We investigate recursive relations for the Grothendieck classes of theaffine graph hypersurface complements of melonic graphs. We compute these classesexplicitly for several families of melonic graphs, focusing on the case of graphs withvalence-4 internal vertices, relevant to CTKT tensor models. The results hint ata complex and interesting structure, in terms of divisibility relations or nontrivialrelations between classes of graphs in different families. Using the recursive re-lations we prove that the Grothendieck classes of all melonic graphs are positiveas polynomials in the class of the moduli space M , . We also conjecture thatthe corresponding polynomials are log-concave, on the basis of hundreds of explicitcomputations. Contents
1. Introduction 21.1. Graph polynomials and CTKT models 21.2. CTKT models and melonic Feynman graphs 31.3. Families of melonic graphs 51.4. Summary of the paper 52. Melonic graphs 72.1. The construction of melonic graphs 72.2. Reduced melonic constructions 92.3. Melonic graphs and bipartite rooted trees 92.4. Valence-four melonic graphs 112.5. Melonic vacuum bubbles 113. Grothendieck classes of melonic graphs 123.1. The Grothendieck ring of varieties 123.2. Kirchhoff–Symanzik polynomials 123.3. Grothendieck classes of graph hypersurface complements 143.4. Recursion formulas for the Grothendieck classes 153.5. Positivity and log-concavity 184. Explicit computations, I 214.1. Recursion for the Γ n graphs 224.2. Relations of vacuum and non-vacuum graphs 245. Explicit computations, II 245.1. Rational generating functions for Γ n graphs 245.2. Graphs Γ vn with arbitrary valence 256. Vacua 27 a r X i v : . [ m a t h . AG ] J u l PAOLO ALUFFI, MATILDE MARCOLLI, AND WALEED QAISAR
Introduction
In this paper we obtain a recursive formula for the Grothendieck classes (virtualmotives) of the graph hypersurfaces associated to the melon-tadpole graphs. Thisprovides a recursively constructed family of mixed-Tate motives, which includes themotives associated to the leading melonic terms of certain bosonic tensor models.Our motivation in considering the behavior of the motives of melon and melon-tadpole graphs comes from the fact that several interesting physical models are dom-inated in the large N limit by melonic graphs. This is the case for SYK models(see [13] for a rigorous diagrammatic proof), as well as in group field theory (see forinstance [7]) and tensor models ([12], [16], [21], [23]), which include generalizationsof the SYK models (see for instance [18], [31]).1.1. Graph polynomials and CTKT models.
The graph polynomials that oneexpects to find, when representing amplitudes in Feynman parametric form in thesetting of group field theory and tensor models, are usually of the form describedin [19] or [30]. The Tanasa graph polynomials of [30] are generalizations of theBollob´as-Riordan polynomial that satisfy the deletion-contraction relation. Similarly,the Gurau polynomials of [19] also satisfy a deletion-contraction relation. The motivesof hypersurfaces associated to these polynomials may be, in principle, amenable tothe kind of algebro-geometric techniques discussed in [4], which we rely on in thispaper, but in a form more similar to the case of the Potts models we analyzedin [5]. However, the computation of the Grothendieck class we obtain here reliesessentially on the recursive form of the Grothendieck class for splitting an edge andfor replacing an edge by a number of parallel edges, obtained in [4]. These formulasdo not have a simple counterpart for the case of the Potts models and other graphpolynomials with deletion-contraction. This means that a more general computationof the polynomials of [19] or [30] probably requires a much more thorough analysisand would not be an immediate generalization of the argument presented here. Otherparametric realizations of tensor models, such as [17], do not even satisfy a deletion-contraction relation, hence they cannot be addressed via the method of [4] and ofthis paper.The case we focus on here, however, is simpler and it involves the usual graphhypersurfaces associated to the Kirchhoff–Symanzik polynomial of the graph, fora massless scalar theory. These are relevant to tensor models in the case of themelonic sector of the CTKT models. We briefly recall below the setting used in [9]
OTIVES OF MELONIC GRAPHS 3
Figure 1.
Tetrahedron, pillow, and double-trace contractions inCTKT models, [9].that motivates the computations we present in this paper. The case of the graphpolynomials of [19] or [30] will be left to a future investigation. Note that, if a similarargument can be applied to such polynomials, or to the massive melonic graphs, onedoes not expect to obtain a family of motives with the mixed-Tate property, sinceit is known that already for small graphs in such families the mixed-Tate propertyfails, [10], [28]. Thus, the mixed-Tate property is certainly specific to the case of themassless Kirchhoff–Symanzik polynomial.1.2.
CTKT models and melonic Feynman graphs.
We focus here on the mod-ified version of the O ( N ) model of Klebanov and Tarnopolsky [25] considered in [9],which generalizes the zero-dimensional version of [14]. These models are referred toin [9] as CTKT models and we will maintain the same terminology here.We recall the following setting from [9]. One considers a real rank three tensorfield φ a ( x ), with a = ( a , a , a ) that transforms under O ( N ) , with action functional(1.1) S [ φ ] = 12 (cid:90) φ a ( x ) ( − ∆) φ a ( x ) d vol( x ) + S int [ φ ] S int [ φ ] = m (cid:90) φ a ( x ) δ ab φ b ( x ) d vol( x )+ λ t N / (cid:90) δ t abcd φ a ( x ) φ b ( x ) φ c ( x ) φ d ( x ) d vol( x )+ (cid:90) (cid:18) λ p N δ p ab ; cd + λ d N δ d ab ; cd (cid:19) φ a ( x ) φ b ( x ) φ c ( x ) φ d ( x ) d vol( x )with ∆ = ∂ µ ∂ µ and with δ t abcd = δ a b δ c d δ a c δ b d δ a d δ b c δ ab = (cid:81) i δ a i ,b i ,δ p ab ; cd = (cid:80) i δ a i c i δ b i d i (cid:81) i (cid:54) = j δ a j b j δ c j d j , δ d ab ; cd = δ ab δ cd . The labels t, p, d distinguish the tetrahedron, pillow, and double-trace patterns ofcontraction. When edges are colored (red, green, or blue) according to the values ofthe tensor indices in { , , } these different quartic terms correspond to the graphsof Figure 1 (with three different choices of the pillow contraction depending on thecolor of the vertical edge). PAOLO ALUFFI, MATILDE MARCOLLI, AND WALEED QAISAR
Figure 2.
Melon-tadpole graphs in CTKT models, [9].When one computes the contributions to the expansion at leading order in 1 /N andall orders in the coupling constants, this is usually done using the 4-colored graphsexpansion of tensor models ([11], [20], [22]) with 3-colored graphs for the differentinteraction terms as mentioned above (bubbles) and another color for the propagatorsconnecting these 3-colored bubbles. However, as shown in [9], it is possible to alsoconsider an expansion in ordinary Feynman graphs, which are obtained by shrinkingall the bubbles to points. The free energy of the model is written in [9] in the formof a sum over connected vacuum 4-colored graphs with labelled tensor vertices, F = (cid:88) G N F − n t − n p − n d λ n t t n t ! 4 n t λ n p p n p ! 12 n p λ n d d n d ! 4 n d ( − n t + n p + n d +1 A ( G ) , with n t ( G ), n p ( G ), and n d ( G ) the number of tetrahedral, pillow, and double-tracebubbles, respectively, and F ( G ) the number of faces and with A ( G ) the amplitudeof G written in terms of edge propagators (see § G in this expansion with ordinary Feynman graphs by first replacingall the pillow and double trace bubbles with their minimal resolution in terms oftetrahedral bubbles (as in Figure 3 of [9]). An ordinary Feynman graph is thenobtained by replacing these bubble by vertices. The resulting graph corresponds toa term of order zero in 1 /N iff it is a melon-tadpole graph, that is, a graph obtainedby iterated insertion of melons or tadpoles into a melon or tadpole (Figure 2). Inthe absence of pillows and double traces one would obtain just melonic graphs. Theamplitudes A ( G ) of the resulting ordinary melon-tadpole Feynman graphs can thenbe computed in the Feynman parametric form, in terms of the Kirchhoff–Symanzikpolynomial, as in [9]. We will not discuss here the renormalization problem forthe resulting Feynman integrals, for which we refer the reader to [9]. We focushere instead on the algebro-geometric and motivic properties of these melon-tadpoleFeynman integrals.From the point of view of motivic structures in quantum field theory (see [27]for a general overview), our goal here is to show that massless CTKT models aredominated by a recursively constructed family of mixed-Tate motives. OTIVES OF MELONIC GRAPHS 5
Families of melonic graphs.
The melonic and melon-tadpole graphs thatoccur in the massless CTKT models are all constrained by the condition that allvertices have valence 4, because of the form (1.1) of the action functional. In order tostudy the recursive properties of the Grothendieck classes associated to these graphs,however, it is convenient to consider them as a subfamily of a larger family of graphs,which include melonic graphs with vertices of arbitrary valences.Moreover, in the typical description of melonic graphs, one assumes that the mel-onic insertions are separated by edge propagators (equivalently, one performs an in-sertion by first splitting an edge into three edges by the insertion of two valence-twovertices and then replaced the middle edge by a number of parallel edges). Again, inour setting it is more convenient to consider these graphs as a subfamily of a largerfamily of melonic graphs where an edge can be split into a number of subedges andeach of them replaced by a set of parallel edges. The typical case of graphs withonly valence-four internal vertices and including edge propagators will guide us inthe choice of the examples illustrating the main recursive construction.We also consider graphs with external edges and graphs (vacuum bubbles) withoutany external edges. Instead of following the usual physics convention of regardingexternal edges as half-edges (flags), we consider then as edges with a valence-onevertex. In this setting, when considering non-vacuum graphs for the CTKT case, wewill allow formal valence-one vertices (to mark the external edges) in addition to thevalence-four vertices of the self-coupling interactions.We will not treat separately the melon-tadpole graphs. Indeed our more generalclass of graphs includes the operation of bisecting an edge with an intermediatevertex and a melon-tadpole graph is simply obtained by grafting together at thevertex two melonic graphs with this operation performed on one of their edges. Sincethe Grothendieck classes for graphs joined at a vertex is just a product, these classesare easily derived from the ones in the family we work with.1.4.
Summary of the paper. In § § PAOLO ALUFFI, MATILDE MARCOLLI, AND WALEED QAISAR features of Grothendieck classes of melonic graphs. For example, we prove that theGrothendieck class of a melonic graph can be expressed as a polynomial with posi-tive coefficients in the class S of the moduli space M , , i.e., S = [ P (cid:114) { , , ∞} ].Extensive computer evidence also suggests the following: Conjecture.
The polynomial expressing the Grothendieck class of a melonic graphis log-concave (in the sense of [29] ). It is well known (see [24]) that the log-concavity property often reflects some deeperunderlying geometric structure, in the form of some kind of Hodge–de Rham relations.It seems likely that log-concavity of these Grothendieck classes as functions of S mayindeed be pointing to some richer geometric structure.In § §
3. As an exampleof the type of result we obtain, consider the family consisting of graphs Γ n of theformwith n interlocked circles. Let P n ( u, v ) be the Hodge–Deligne polynomial of thecomplement Z n of the affine graph hypersurface determined by Γ n . That is, P n ( u, v ) = (cid:80) e p,q u p v q , where e p,q = (cid:80) k ( − k h p,q ( H kc ( Z n )) (see e.g., [15]). As a consequence ofProposition 4.1, the following holds: P n ( u, v ) = ( uv − n ( uv ) n +1 · A n ( uv − , where the polynomial A n ( t ) is determined by the equality of formal power series (cid:80) n ≥ A n ( t ) r n = (cid:80) k ≥ a k ( r, t ), with (cid:88) k ≥ a k ( r, t ) s k k ! = e rs cos(( r − rt ) s ) . Alternative expressions for A n ( t ) are given in §
5; in fact, the information carried bythe polynomials A n ( t ) may be encoded in a rational generating function.In § § OTIVES OF MELONIC GRAPHS 7
We describe a procedure for studying the structure of valence-four melonic vacuumbubbles in terms of their tree structure, and we identify certain families of recursiverelations, in the form of “melonic vacuum stars”.In § Melonic graphs
The construction of melonic graphs.
A graph with two vertices and n par-allel edges connecting them is variously referred to in the literature as a melon graph,a banana graph, or a sunset graph. In the spirit of botanical egalitarianism, we willuse the “banana” terminology when referring to these basic building blocks, and call“melonic” the result of iterating the operation of replacing edges of a graph by stringsof bananas. (We call this operation the ‘bananification’ of the edge.)Thus, the basic iterative operation constructing melonic graphs is the following: r a a a a We allow arbitrary sizes a , . . . , a r for the banana components. Edges ought to bedirected in order to determine the order of inclusion of the bananas; in fact this willbe done implicitly in what follows, since it does not affect the invariant (Grothendieckclass) we are computing. A melonic graph is obtained by applying this operation toan initial single edge, then applying it iteratively to any edge of the resulting graphs.We can refer to the initial edge as the graph obtained ‘at stage 0’; the applicationof the iterative process at any stage may be encoded by a tuple(( a , . . . , a r ) , p, k )to represent the replacement of one single edge in the k -th banana constructed atstage p . Example . The constructionmay be represented by the tuples((1 , , , , , ((1 , , , , ((1 , , , ,
3) : • the first operation replaces the single edge at stage 0 with a string consisting ofa 1-banana, a 3 banana, and a 5 banana; this is stage 1; • the second operation replaces one edge in the second banana constructed instage 1 with a string consisting of a 1-banana and a 2-banana; this is stage 2; • and the third operation replaces one edge in the third banana constructed instage 1 with a string consisting of a 1-banana, a 3-banana, and a 1-banana. This isstage 3. PAOLO ALUFFI, MATILDE MARCOLLI, AND WALEED QAISAR
Following this sequence of operations with ((2 , , ,
1) would replace one edge in thefirst banana produced at stage 2 (which actually consists of a single edge) with astring consisting of a 2 banana followed by a 3 banana, producing the graph(As observed below, the same graph admits different constructions.) (cid:121)
Formally, we can make the following definition.
Definition 2.1.
An ‘ n -stage’ (or ‘depth n ’) melonic construction is a list T =( t , . . . , t n ) of tuples t s = ( b s , p s , k s ) such that(i) b s = ( a , . . . a r s ) is a tuple of positive integers, of length | b s | := r s ≥
1. (Thus,the tuple is non-empty.)(ii) p s is an integer, 0 ≤ p s < s ;(iii) k s is an integer, 1 ≤ k s ≤ | b p s | .(iv) p s > s >
1. (By (ii), p = 0.)(v) For all t i = (( a , . . . a r i ) , p i , k i ), i = 1 , . . . , n , and all j = 1 , . . . , r i , at most a j tuples t s = ( b s , p s , k s ) have p s = i, k s = j .The length n of the melonic construction is its ‘depth’. (cid:121) The motivation behind these requirements should be evident from the interpreta-tion discussed above. For example, (v) expresses the constraint that the j -th bananaconstructed at stage i has enough edges to accommodate later replacements. Definition 2.2. A melonic graph is a graph determined by a melonic constructionby the procedure explained above. (cid:121) Every melonic construction determines a melonic graph up to graph isomorphism.Of course different melonic constructions may determine the same melonic graph.We say that two constructions T (cid:48) , T (cid:48)(cid:48) are ‘equivalent’ if the resulting graphs areisomorphic. Example . The construction { ((2) , , , ((1 , , , , , ((1 , , , , , , } deter-mines a melonic graph as follows:The construction { ((2) , , , ((1 , , , , , , , ((1 , , , , } produces an isomor-phic graph. OTIVES OF MELONIC GRAPHS 9 (cid:121)
Also: The graph in Example 2.1 was obtained from the melonic construction { ((1 , , , , , ((1 , , , , ((1 , , , , , ((2 , , , } . The same graph can be obtained by the (shorter) construction { ((1 , , , , , ((2 , , , , , ((1 , , , , } . Similarly, the second construction in Example 2.2 produces the same graph as the(longer) construction { ((2) , , , ((1 , , , , , ((1 , , , , , ((1 , , , , } . In both cases, the shorter construction is obtained by implementing the replacementof the (single) edge in a 1-banana produced at stage s (underlined) by inserting theappropriate tuple (also underlined) directly at stage s .2.2. Reduced melonic constructions.
Constructions such as Example 2.2 suggestthe following definition.
Definition 2.3.
We say that a construction is reduced if it does not prescribe thereplacement of the edge of a 1-banana past stage 0. (cid:121)
Formally, this requirement prescribes that(vi) For all t i = (( a , . . . a r i ) , p i , k i ), i = 1 , . . . , n : If a j = 1, then k s (cid:54) = j for all s such that p s = i .The process illustrated above—replacing 1-bananas by their descendants—maybe performed on every melonic construction, and produces an equivalent reducedconstruction. Therefore: Lemma 2.1.
Every melonic graph admits a reduced construction.
Reduced constructions suffice in order to define melonic graphs, but it is importantto consider non-reduced constructions as well; these may appear in intermediate stepsof the recursive computation we will obtain in § Melonic graphs and bipartite rooted trees.
There is a convenient way to vi-sualize a melonic construction as a labeled bipartite tree. Each tuple (( a , . . . , a r ) , p, k )may be viewed as a rooted tree a a a r with (black) leaves labeled by the integers a i . The (white) root will be attached tothe k -th leaf of the p -th tree; this grafting procedure builds a rooted tree encodingthe same information as a melonic construction. Item (v) in Definition 2.1 amountsto the requirement that the valence of a (black) node labeled a be at most a + 1; thatis, at most a ‘descending’ edges can be adjacent to such a vertex.The tree corresponding to a melonic construction has one white node for each tuplein the construction; thus, the depth of the melonic construction equals the numberof white nodes in the corresponding tree. Example . The rooted trees corresponding to the two melonic constructions inExample 2.2 are
As noted in Example 2.2, these non-isomorphic labeled trees determine isomorphicmelonic graphs. (cid:121)
The ‘reduced’ condition (vi) is the requirement that nodes labeled 1 necessarily beleaves. Every tree can be reduced (cf. Lemma 2.1) by ‘sliding up’ trees grafted atnodes labeled 1, as the case encountered in Example 2.1 illustrates.
21 3 3 1 135 1 3 1 1352 2312
We also note that the melonic graphs determined in Definition 2.2 have arbitraryvertex valences, while in a specific physical theory the valences are constrained bythe terms in the action. The additional generality is needed for the recursion formulawe will obtain in §
3; we will choose families of graphs with fixed valence in most ofthe examples illustrating the recursion in §
4, 5, and 6.
OTIVES OF MELONIC GRAPHS 11
Valence-four melonic graphs.
We will be especially interested in the case inwhich the valence of all internal vertices of the melonic graph is 4. The correspondingmelonic constructions consist of tuples of the type t s = ((1 , , , p s , k s )where k s = 1 ,
2, or 3. The building blocks of these graphs areUp to equivalence, a melonic construction ( t , . . . , t m ) with t s = ((1 , , , p s , k s ) asabove is determined by the tuple (0 , p ± , . . . , p ± n ), where each p s for s > p + s if k s = 2 and p − s if k s = 1 or 3. For example, (0 , + , + , + , + ) indicates thatat each stage the new splitting (1 , ,
1) is performed on one of the 3 parallel edges atthe previous stage. The corresponding melonic graph may be drawn as follows:2.5.
Melonic vacuum bubbles.
We will also consider the ‘vacuum’ flavor of theseconstructions, in which the two external vertices are identified; for example
Definition 2.4. A vacuum melonic graph is a melonic graph without valence-1 ver-tices. (cid:121) Vacuum melonic graph in which every vertex has valence 4 may be obtained byiteratively applying the basic bananification ((1 , , , p s , k s ) starting from a 4-banana.For example, the string of circles depicted above is produced by the construction(((4) , , , ((1 , , , , , ((1 , , , , , ((1 , , , , , ((1 , , , , , while the construction(((4) , , , ((1 , , , , , ((1 , , , , , ((1 , , , , , ((1 , , , , Note that all vacuum valence-4 melonic graphs may also be constructed by startingfrom a 2-banana, performing iteratively the basic (1 , ,
1) bananifications, and thenremoving the two extra valence-2 vertices produced at the beginning. Indeed, the4-banana itself admits such a description: the melonic construction(((2) , , , ((1 , , , , § Grothendieck classes of melonic graphs
The Grothendieck ring of varieties.
For V K the category of varieties over afield K (which we can here assume to be K = Q ), the Grothendieck group of varieties K ( V K ) is the abelian group generated by isomorphism classes [ X ] of varieties X ∈ V K with the inclusion-exclusion relation[ X ] = [ Y ] + [ X (cid:114) Y ]for closed subvarieties Y ⊂ X . This group may be given a ring structure by defining[ X ] · [ Y ] = [ X × Y ] and extending by linearity. Grothendieck classes, sometimesreferred to as virtual motives, behave like a universal Euler characteristic for algebraicvarieties. Grothendieck classes usually provide more computable information aboutthe nature of the motive of a variety. In particular, a Grothendieck class is Tate ifit is contained in the subring generated by the Lefschetz motive L = [ A ] (the classof an affine line), or equivalently in the ring generated by T := L −
1. Since theformulas we will obtain will naturally be polynomials in this class, and we will alsobe interested in expressing them in terms of the class S := L −
2, we highlight theirdefinitions.
Definition 3.1.
We will denote by T the class of the ‘torus’ in the Grothendieck ringof varieties, i.e., T = [ A (cid:114) A ] = L − ∈ K ( V K ). We will also denote by S the classof the complement of three points in P : S = [ P (cid:114) { , , ∞} ] = T − (cid:121) Varieties whose motive is in the category of mixed Tate motives will have a TateGrothendieck class. The converse holds conditionally (see [6] for a discussion of thispoint).3.2.
Kirchhoff–Symanzik polynomials.
We consider the Kirchhoff–Symanzik poly-nomial of a graph G with n edges(3.1) Ψ G ( t ) = (cid:88) T ⊂ G (cid:89) e/ ∈ E ( T ) t e , as a polynomial in variables t = ( t , . . . , t n ) associated to the edges of G , with the sumtaken over all the spanning trees of the graph. This is a homogeneous polynomial of OTIVES OF MELONIC GRAPHS 13 degree (cid:96) = b ( G ), the number of loops of G . Thus, we can consider the associatedprojective graph hypersurface(3.2) X G = { t = ( t : · · · : t n ) ∈ P n − | Ψ G ( t ) = 0 } . Up to renormalization of divergences, the Feynman parameter form of the Feynmanintegral for the graph G , for a massless scalar field theory, is of the form(3.3) U ( G, p ) = Γ( n − D(cid:96)/ π ) (cid:96)D/ (cid:90) ∆ n V G ( t, p ) − n + (cid:96)D/ Ψ G ( t ) D/ dt · · · dt n as a function of the external momenta p , with V G ( t, p ) the second Symanzik polyno-mial (defined in terms of cut sets of G ), D the spacetime dimension, and the integra-tion performed on the n -simplex. In particular (modulo divergences) the Feynmanintegral (3.3) can be regarded as the integration of an algebraic differential formon a locus defined by algebraic equations (that is, a period) on the complement ofthe hypersurface X G , hence the interest in investigating the nature of the motive of P n − (cid:114) X G through the computation of its Grothendieck class. For a general intro-ductory survey of parametric Feynman integrals and their relations to periods andmotives of graph hypersurfaces see [27].In the following we will consider both graphs with external edges and graphs (vac-uum bubbles) with no external edges. From the point of view of the parametricFeynman integral, the contribution of the external edges with their assigned exter-nal momenta is encoded only in the second Symanzik polynomial V G ( t, p ), while thevariables t = ( t e ) run over internal edges. Thus, as long as the exponent satisfies (cid:96)D/ ≥ n , with (cid:96) the number of loops, n the number of (internal) edges, and D thespacetime dimension, the Feynman integral is computed on the complement of thegraph hypersurface defined by Kirchhoff–Symanzik polynomial Ψ G ( t ) that only de-pends on the internal edges of G . The Grothendieck class of the affine complement ofthe hypersurface of a graph G (including external edges) and of the same graph withthe external edges removed are simply related by a product by a power of L (the classof the affine line), hence it is equivalent to compute one or the other. For the purposeof computing Grothendieck classes, considering all graphs (both vacuum bubbles andnon-vacuum graphs) for a massless scalar theory with a self-interaction term of order N , so that the corresponding Feynman graphs have (internal) vertices of valence N , isequivalent to considering all vacuum bubble graphs for a massless scalar theory withself-interaction terms of orders v ≤ N . We will work for convenience with graphswith the external edges included.Up to the issue of renormalization, the Feynman integral (3.3) can then be seenas a period of the graph hypersurface complement. The nature of the motive of thegraph hypersurface complement (detected by its Grothendieck class) then providesinformation on the kind of numbers that can be obtained as periods. The regular-ization and renormalization of the integral (3.3) can also be dealt with geometricallyin terms of blowups or deformations. We will not discuss this in the present paper and we refer the reader to [27] for an overview and to the references therein for moreinformation.3.3. Grothendieck classes of graph hypersurface complements.
In previouswork, especially [3] and [4], we have focused on the essentially equivalent informationgiven by the complement of the affine cone ˆ X G in its ambient affine space, and studiedits class in the Grothendieck group of varieties (the ‘motivic Feynman rule’ of [3]).For short, we will refer to this class as the Grothendieck class of the graph or of thecorresponding melonic construction.
Definition 3.2.
The
Grothendieck class of G (or of any of its melonic constructions)is the class U ( G ) = [ A n (cid:114) ˆ X G ] ∈ K ( V K ) of the complement of ˆ X G in its naturalambient affine space A n , with n the number of edges of G . (cid:121) By construction, U ( G ) is the class of a variety of dimension equal to the numberof edges of G .In this section we will use the melonic constructions introduced in § U ( G ). • This invariant is ‘multiplicative’, in the sense that U (Γ ∪ Γ ) = U (Γ ) · U (Γ )if Γ , Γ are graphs joined at one vertex (or disjoint); • For Γ =a loop, U (Γ) = T (with T as in Definition 3.1); • For Γ =a single edge, U (Γ) = L = T + 1; • If Γ (cid:48) is obtained from Γ by splitting an edge, then U (Γ (cid:48) ) = ( T + 1) · U (Γ); • If Γ is an m -banana, m >
0, then(3.4) U (Γ) = B m := m T m − + T · T m − ( − m T + 1([2] and [4, Corollary 5.6]); • More generally: if e is not a bridge or a looping edge, then for suitable polyno-mials f m , g m , h m in T ,(3.5) U (Γ me ) = f m U (Γ) + g m U (Γ /e ) + h m U (Γ (cid:114) e ) , where: – Γ me stands for the graph obtained from Γ by replacing e with m parallel edgesjoining the same vertices as e ; – Γ (cid:114) e = Γ e is Γ with e deleted; and – Γ /e is Γ with e contracted. OTIVES OF MELONIC GRAPHS 15
This is a weak form of a deletion-contraction relation. Inductively, the coefficients f m , g m , h m are determined by their value for m ≤
2; in fact, we have(3.6) f m = T m − ( − m T + 1 , g m = m T m − − T m − ( − m T + 1 , h m = T m + ( − m TT + 1as obtained in [4, Corollary 5.7]. Formulas for bridges and looping edges are easier,as they follow immediately from the multiplicativity property.3.4. Recursion formulas for the Grothendieck classes.
The properties listedabove, and particularly identity (3.5), lead to recursion formulas for the computationof the Grothendieck class of the melonic graph associated with a melonic construc-tion, or equivalently of the corresponding tree. (We will use the two descriptionsinterchangeably.) We will abuse language and use both U ( T ) and U ( G ) for theGrothendieck class of the melonic graph G resulting from a melonic construction T .The recursion formulas are based on the following observations.Let G be a melonic graph given by a melonic construction (tree) T . • By Lemma 2.1, we may assume that the melonic construction is reduced, i.e.,nodes labeled 1 are leaves of the tree T . • If T has depth 1, i.e., the corresponding melonic construction consists of a singletuple (( a , . . . , a r ) , , U ( T ) = r (cid:89) i =1 B a i = r (cid:89) i =1 (cid:18) a i T a i − + T T a i − ( − a i T + 1 (cid:19) . Indeed, the graph G consists of a string of bananas in this case.If T = ( t , . . . , t n ) has higher depth, consider the last state t n . By construction,the black nodes of t n are all leaves of T . • If t n = (( a ) , p, k ), then an equivalent tree of depth n − t n and increasing the label of the k -th leaf of t p by a − +a k b k a −1 b Indeed, this step of the construction simply replaces 1 edge in the k -th banana of t p by a parallel edges. • If t n = ((1 , . . . , (cid:124) (cid:123)(cid:122) (cid:125) r times ) , p, k ), then let T (cid:48) = ( t , . . . , t n − ) be the construction obtainedby omitting the last stage. Then U ( T ) = ( T + 1) r − U ( T (cid:48) ). Indeed, the effect of t n isto split one edge in the k -th banana of t p a total of r − r • We may therefore assume that t n = (( a , . . . , a r ) , p, k ) with r > a m = max( a , . . . , a r ) >
1. The effect of t n is to replace one edge in the k -th bananaof t p by a string of a , . . . , a m , . . . , a r -bananas; this is the same as replacing that edgeby a string of a , . . . , a m − , , a m +1 , . . . , a r -bananas, and then replacing the resultingsingle edge e by a m parallel edges. ea m e Let G (cid:48) be the graph obtained from G by replacing the a m -banana by the singleedge e . With notation as in (3.5), we have G = G (cid:48) a m e . Claim 3.1.
The edge e is not a bridge (or a looping edge) of G (cid:48) .Proof. This follows from the assumption that T be reduced. Indeed, as a consequencethe k -th banana of t p does not consist of a single edge; hence removing one edge of thisbanana does not disconnect the graph. Since the edge e is one edge in a subdivisionof one edge of the k -th banana of t p , removing it does not disconnect the graph. (Andthe construction never produces looping edges, therefore e is not a looping edge.) (cid:3) It follows that we can use (3.5) to relate U ( G ) to the Grothendieck classes of G (cid:48) and associated graphs:(3.7) U ( G ) = f a m U ( G (cid:48) ) + g a m U ( G (cid:48) /e ) + h a m U ( G (cid:48) (cid:114) e )with f a m , g a m , h a m as in (3.6).Now:— G (cid:48) is a melonic graph: its construction T (cid:48) is obtained from T by replacing t n = (( a , . . . , a m − , a m , a m +1 , . . . , a r ) , p, k )by t (cid:48) n = (( a , . . . , a m − , , a m +1 , . . . , a r ) , p, k ) . Pictorially: a a r a m a a a r a OTIVES OF MELONIC GRAPHS 17 —The contraction G (cid:48) /e is also a melonic graph: its construction T (cid:48)(cid:48) is obtainedfrom T by omitting a m in t n , i.e., replacing t n = (( a , . . . , a m − , a m , a m +1 , . . . , a r ) , p, k )by t (cid:48)(cid:48) n = (( a , . . . , a m − , a m +1 , . . . , a r ) , p, k ) . Since r >
1, the tuple ( a , . . . , a m − , a m +1 , . . . , a r ) is non-empty, as needed (cf. Defi-nition 2.1). Pictorially: r a a a a r a m a m −1 aa a m +1 —The deletion G (cid:48) (cid:114) e is not a melonic graph; it is obtained by replacing one edge ofthe k -th banana of t p by two disconnected strings of bananas attached at the verticesof that edge:Let T (cid:48)(cid:48)(cid:48) be the list obtained from T by omitting t n and decreasing by 1 the order b k ofthe k -th banana in t p . Since T is assumed to be reduced, b k >
1; therefore, T (cid:48)(cid:48)(cid:48) is stilla melonic construction. (Note that, however, T (cid:48)(cid:48)(cid:48) may be non-reduced. This is thereason forcing us to consider non-reduced melonic constructions.) The graph G (cid:48) (cid:114) e is obtained from the melonic graph corresponding to T (cid:48)(cid:48)(cid:48) by attaching two strings ofbananas to two vertices, and it follows that U ( G (cid:48) (cid:114) e ) = (cid:32) m − (cid:89) i =1 B a i (cid:33) (cid:32) r (cid:89) i = m +1 B a i (cid:33) U ( T (cid:48)(cid:48)(cid:48) ) . In conclusion, (3.7) may be rewritten U ( T ) = f a m U ( T (cid:48) ) + g a m U ( T (cid:48)(cid:48) ) + (cid:32) m − (cid:89) i =1 B a i (cid:33) (cid:32) r (cid:89) i = m +1 B a i (cid:33) h a m U ( T (cid:48)(cid:48)(cid:48) ) , or, more explicitly: Proposition 3.2.
With notation as above, U ( T ) = T a m − ( − a m T + 1 U ( T (cid:48) )+ (cid:18) a m T a m − − T a m − ( − a m T + 1 (cid:19) U ( T (cid:48)(cid:48) )+ (cid:32) m − (cid:89) i =1 B a i (cid:33) (cid:32) r (cid:89) i = m +1 B a i (cid:33) T a m + ( − a m TT + 1 U ( T (cid:48)(cid:48)(cid:48) ) . Since T (cid:48) , T (cid:48)(cid:48) , T (cid:48)(cid:48)(cid:48) all correspond to melonic graphs with fewer edges than G , thecorresponding Grothendieck classes are recursively known, and determine U ( G ) = U ( T ). Corollary 3.3.
The graph hypersurface of a melonic graph G determines a mixedTate motive; the Grothendieck class U ( G ) is a polynomial in L of degree equal to thenumber of edges of G .Proof. The recursion implies immediately that U ( G ) is a polynomial in T , thereforein L = T + 1. By construction, U ( G ) is the class in the Grothendieck group of avariety of dimension equal to the number of edges of G , so the statement follows. (cid:3) Positivity and log-concavity.
The class of a melonic graph can of course alsobe written as a polynomial in the class S = T − P (cid:114) { , , ∞} ]. Remarkably,these polynomials are ‘positive’, in the following sense. Corollary 3.4.
Let G be a melonic graph. Then U ( G ) = P ( S ) for a polynomial P ( t ) = a n t n + · · · + a t + a ∈ Z [ t ] with nonnegative integer coefficients.Proof. Given the recursion, it suffices to observe that the classes of banana graphs, B m ( T ) = B m ( S + 1), and the coefficients f m , g m , h m are all positive as polynomialsin S . The key observation is the following. Claim 3.5.
The class T m − ( − m T + 1 = ( S + 1) m − ( − m S + 2 is positive in S ; in fact, T m − ( − m T + 1 = m − (cid:88) j =1 m (cid:88) i =1 (cid:18) m − ij − (cid:19) S j + (cid:26) if m is even if m is odd . This is a straightforward computation, left to the reader. Given Claim 3.5, itfollows immediately that B m = m T m − + T T m − ( − m T + 1 , f m = T m − ( − m T + 1 , h m = T m + ( − m TT + 1are positive in S . As for g m = m T m − − T m − ( − m T + 1 , the required positivity follows from the fact that for all m, i ≥ , j (cid:18) m − j (cid:19) ≥ (cid:18) m − ij − (cid:19) which is clear, as (cid:0) m − j (cid:1) = (cid:0) m − j − (cid:1) + (cid:0) m − j (cid:1) ≥ (cid:0) m − j − (cid:1) ≥ (cid:0) m − ij − (cid:1) for i ≥ (cid:3) OTIVES OF MELONIC GRAPHS 19
Example . The melon-tadpole graph of Figure 2 consists of a 4-banana tadpole,with class B = ( T +1)( T +2 T − T , and of a melonic part which may be constructedby (cid:0) ((4) , , , ((1 , , , , (cid:1) i.e., by the labeled tree
141 3
The recursion obtained above computes the Grothendieck class of this melonic graphto be T ( T + 1) ( T + 3 T − . The conclusion is that the Grothendieck class for the graph in Figure 2 equals T ( T + 1) ( T + 3 T − T + 2 T −
1) = ( S + 1) ( S + 2) ( S + 4 S + 2)( S + 5 S + 2) :indeed, the graph may be obtained by splitting one edge in each of the two components(which has the effect of multiplying each Grothendieck class by T +1), and then joiningthe resulting graphs at the newly created vertices, i.e., multiplying together the tworesulting Grothendieck classes. (cid:121) Positivity as a polynomial in the class T is a torification of the Grothendieck class,which may or may not be induced by a geometric torification of the underlying variety,see [26]. The presence of a torified Grothendieck class has consequences in terms of“geometry over the field with one element”, [8], [26]. One can similarly ask whetherthe positivity of the Grothendieck class as a function of S is induced by an underlyinggeometric structure and whether such a structure carries arithmetic significance. Forexample, the Grothendieck class of the moduli spaces M ,n of genus zero curves withmarked points have the simple expression[ M ,n ] = (cid:18) S n − (cid:19) ( n − S , with [ M , ] = S . However, these classes are not positive in S ,while the classes of the M ,n moduli spaces satisfy positivity (both in S and in T ),see [26].Another feature of the polynomials expressing the classes in terms of S appears tobe the following. Conjecture 3.1.
Let G be a melonic graph, and let U ( G ) = a + a S + · · · + a n S n be itsGrothendieck class. Then the sequence a , a , . . . , a n is log concave, i.e., a i − a i +1 ≤ a i for < i < n . We have verified this conjecture for all melonic graphs with ≤
13 edges and forhundreds of individual examples from the families of melonic graphs considered inthis paper.
Example . As polynomials in S , the Grothendieck classes of all possible melonicgraphs with 7 edges are( S + 1) ( S + 2) ( S + 1) ( S + 2) ( S + 1) ( S + 2) ( S + 2) ( S + 1) ( S + 2) ( S + 3)( S + 1) ( S + 2) ( S + 3)( S + 1) ( S + 2) ( S + 4)( S + 1) ( S + 2) ( S + 5)( S + 1) ( S + 2) (cid:0) S + 4 S + 2 (cid:1) ( S + 1) ( S + 2) (cid:0) S + 4 S + 2 (cid:1) ( S + 1) ( S + 2) (cid:0) S + 5 S + 2 (cid:1) ( S + 1) ( S + 2) (cid:0) S + 5 S + 5 (cid:1) ( S + 1) ( S + 2) (cid:0) S + 6 S + 7 S + 3 (cid:1) ( S + 1) ( S + 2) (cid:0) S + 6 S + 7 S + 3 (cid:1) ( S + 1) ( S + 2) (cid:0) S + 6 S + 9 S + 3 (cid:1) ( S + 1) ( S + 2) (cid:0) S + 8 S + 15 S + 12 S + 3 (cid:1) ( S + 1) ( S + 2) (cid:0) S + 8 S + 19 S + 16 S + 5 (cid:1) ( S + 1) ( S + 2) (cid:0) S + 10 S + 26 S + 31 S + 17 S + 4 (cid:1) One may verify that all these polynomials are log-concave (in the sense that thecoefficients of their expansions are log-concave sequences). The number of distinctGrothendieck classes for melonic graphs with n edges is1 , , , , , , , , , , , , . . . respectively as n = 1 , , , . . . (cid:121) The log-concavity property of the Grothendieck classes implies similar propertiesfor the image of these classes under any motivic measure, meaning a ring homomor-phism µ : K ( V ) → R . Such measures include the topological Euler characteristicand the Hodge–Deligne polynomials (for complex varieties) or the counting of points(for varieties over finite fields). As discussed in [24], the presence of a log-concavestructure is usually a sign of the presence of an underlying richer kind of structure,in the form of Hodge-de Rham relations. These can be seen as a broad combinatorialgeneralization of the setting of the Grothendieck standard conjectures for algebraic OTIVES OF MELONIC GRAPHS 21 cycles. Such combinatorial Hodge-de Rham relations arise, for example, in the con-text of the log-concavity property of characteristic polynomial of matroids, [1]. Thus,the observed log-concavity of the Grothendieck classes of the graph hypersurface com-plements as a function of the S variable suggest the presence of a more interestingunderlying geometric structure in this Hodge-de Rham sense.While the Grothendieck classes are positive in the class S and display this intriguingproperty, we will persist in using T in most of the examples that follow, since thecoefficients of the powers of T in these classes tend to be smaller.4. Explicit computations, I
The recursion obtained in § § § Example . For a simple valence-4 example that can be computed without employ-ing the full recursion from §
3, we can consider the graphwith n circles. A corresponding melonic construction is (0 , − , − , − , . . . , ( n − − ).This construction is non-reduced; a reduced alternative is simply the 1-stage con-struction ((1 , , , , , , . . . , , , . The corresponding Grothendieck class is a product of classes of 3-bananas and( T + 1)-factors, accounting for the external and internal single edges. Explicitly, theclass equals B n · ( T + 1) n +1 = T n ( T + 1) n +1 . for n circles. (cid:121) Example . At the opposite end of the spectrum, and more interestingly, considerthe valence-4 melonic graphs Γ n constructed by (0 , + , + , + , . . . , ( n − + ). Theseare graphs of the form with n circles.4.1. Recursion for the Γ n graphs. The graph Γ n has 4 n + 1 edges, so by Corol-lary 3.3 its Grothendieck class is a polynomial in T of degree 4 n + 1. For n = 1 , . . . , § n = 1 : T ( T + 1) · ( T + 1) n = 2 : T ( T + 1) · ( T + 3 T ) n = 3 : T ( T + 1) · ( T + 5 T + 4 T − n = 4 : T ( T + 1) · ( T + 7 T + 12 T − n = 5 : T ( T + 1) · ( T + 9 T + 24 T + 14 T − T − n = 6 : T ( T + 1) · ( T + 11 T + 40 T + 48 T − T − T ) n = 7 : T ( T + 1) · ( T + 13 T + 60 T + 110 T + 40 T − T − T + 8)Identifying the pattern underlying these expressions is an interesting challenge. • Define polynomials a k ( r, t ) ∈ Z [ r, t ] for k ≥ e rs cos(( r − rt ) s ) = (cid:88) k ≥ a k ( r, t ) s k k ! ; • In turn, define polynomials A n ( t ) ∈ Z [ t ] for n ≥ (cid:88) k ≥ a k ( r, t ) = (cid:88) n ≥ A n ( t ) r n . (Since t only appears in the product rt in (4.1), it is clear that A n ( t ) is indeed apolynomial, of degree at most n . In fact, deg A n = n .) Proposition . With Γ n as above, U (Γ n ) = T n ( T + 1) n +1 · A n ( T ) for n ≥ . Proposition 4.1 may be easily verified for low values of n ; our computer implemen-tation takes a few seconds to verify it for n = 1 , . . . , § A n ( t ) is of course just one choiceamong many. An alternative (and perhaps simpler) formulation will be given in § Corollary . U (Γ n ) = T n ( T + 1) n +1 · (cid:88) ≤ i ≤ j (cid:18) n + i j (cid:19)(cid:18) ji (cid:19) ( − j − i T i = ( S + 1) n ( S + 2) n +1 · (cid:88) ≤ i,j (cid:18) n − ji (cid:19)(cid:18) i + j − j (cid:19) n − i − j S i . The straightforward details are left to the reader. (cid:121)
OTIVES OF MELONIC GRAPHS 23
Example . A similar pattern holds for vacuum graphs analogous to those consid-ered in Example 4.2. Let Γ (cid:48) n denote the graphwith n circles. As observed at the end of §
2, these graphs are also melonic: theirconstruction is ((4) , ,
1) for two circles and((4) , , , ((1 , , , , , ((1 , , , , , . . . , ((1 , , , n − , n ≥ n ≥
2, Corollary 3.3 implies that U (Γ (cid:48) n ) is a polynomial ofdegree 4 n − T . Applying the recursion obtained in § U (Γ (cid:48) n ) n = 2 , . . . , n = 2 : T ( T + 1) · ( T + 2 T − n = 3 : T ( T + 1) · ( T + 4 T + T − n = 4 : T ( T + 1) · ( T + 6 T + 7 T − T − n = 5 : T ( T + 1) · ( T + 8 T + 17 T + 2 T − T ) n = 6 : T ( T + 1) · ( T + 10 T + 31 T + 24 T − T − T + 4) n = 7 : T ( T + 1) · ( T + 12 T + 49 T + 70 T − T − T − T + 8) • Define rational functions a (cid:48) k ( r, t ) ∈ Z [ t ]( r ) for k ≥ (cid:32) ( r − rt ) − r s (cid:33) = (cid:88) k ≥ a (cid:48) k ( r, t ) s k k ! ;that is, let a (cid:48) k ( r, t ) = 0 for k odd and a (cid:48) (cid:96) ( r, t ) = (cid:96) )! r (cid:96) ( t − r ) (cid:96) (1 − r ) (cid:96) . • Define polynomials A (cid:48) n ( t ) ∈ Z [ t ] for n ≥ (cid:88) k ≥ a (cid:48) k ( r, t ) = (cid:88) n ≥ A (cid:48) n ( t ) r n . (Again, A (cid:48) n ( t ) is clearly a polynomial, and deg A (cid:48) n = n .) Proposition . With Γ (cid:48) n as above, U (Γ (cid:48) n ) = T n − ( T + 1) n − · A (cid:48) n ( T ) for n ≥ . Again, Proposition 4.3 may be easily verified by computer, using the recursionformula obtained in §
3, for (hundreds of) low values of n . Proposition 4.3 will alsobe proved in § Corollary . U (Γ (cid:48) n ) = T n − T n − (cid:88) ≤ i ≤ j (cid:18) n + i − j − (cid:19)(cid:18) ji (cid:19) ( − j − i T i . Relations of vacuum and non-vacuum graphs.
A particularly careful readermay notice the following relation from the data shown above:(4.3) A (cid:48) n ( t ) = A n ( t ) − A n − ( t ) . This relation is not a coincidence; it follows from a general formula relating Grothen-dieck classes of melonic vacuum graphs to classes of related non-vacuum graphs. Wewill prove this formula in § U (Γ n ) | U (Γ (cid:48) n +1 ) :for example, A (cid:48) ( t ) = ( t + 5 t + 2)( t + 2 t − · A ( t ) . The relation (4.4) will also be obtained as a corollary of a more general result onmelonic vacuum graphs, Proposition 6.3 in § (cid:121) More examples of computations of Grothendieck classes for melonic vacuum graphswill be given in § § T i ( T + 1) j in the Grothendieck classes for the valence-4 graphs considered in this section. Theexample (0 , + , + , + ),with Grothendieck class T ( T + 1) ( T + 3)( T + 3 T − T + 1) , shows that T n ( T + 1) n +1 is not a common factor of the Grothendieck classes of all n -stage valence-4 melonic constructions.5. Explicit computations, II
Rational generating functions for Γ n graphs. While § § OTIVES OF MELONIC GRAPHS 25
Proposition 5.1.
With notation as in Examples 4.2 and 4.3, and setting A ( t ) = A (cid:48) ( t ) = 1 , (cid:88) n ≥ A n ( t ) r n = 1 − r − (2 + t ) r + 2 r ; (cid:88) n ≥ A (cid:48) n ( t ) r n = (1 − r ) − (2 + t ) r + 2 r . Proof.
We verify that the polynomials A n ( t ), A (cid:48) n ( t ) defined by these expansions agreewith those given in Examples 4.2 and 4.3.Concerning A n ( t ), let τ = ( r − rt ) ; then1 − r − (2 + t ) r + 2 r = 1 − r (1 − r − iτ )(1 − r + iτ ) = 12 (cid:18) − r − iτ + 11 − r + iτ (cid:19) . The terms in the power series expansion of this expression are combinations of powersof ( r − iτ ) and ( r + iτ ), so they may be obtained as the coefficients of s k k ! in12 (cid:0) e ( r + iτ ) s + e ( r − iτ ) s (cid:1) = e rs · e iτs + e − iτs e rs cos( τ s ) . This recovers the description of A n ( t ) given in Proposition 4.1.The argument for A (cid:48) n ( t ) is of course analogous. Again setting τ = ( r − rt ) , wehave (1 − r ) − (2 + t ) r + 2 r = 12 (cid:18) − r − r − iτ + 1 − r − r + iτ (cid:19) = 12 (cid:32) − i τ − r + 11 + i τ − r (cid:33) and the terms in the power series expansion of this expression are the coefficientsof s k k ! in 12 (cid:16) e i τ − r + e − i τ − r (cid:17) = cos (cid:18) τ − r (cid:19) , recovering the description of A (cid:48) n ( t ) in Proposition 4.3. (cid:3) Graphs Γ vn with arbitrary valence. We will discuss some families of valence-4 vacuum graphs in §
6. The non-vacuum graphs to which the first formula applieshave a natural generalization for arbitrary valence: we can let Γ vn be the graphs withmelonic construction (cid:0) ((1 , v − , , , , ((1 , v − , , , , ((1 , v − , , , , . . . , ((1 , v − , , n − , (cid:1) for v ≥
3. For example, the graphs Γ n have the form while the graphs Γ n look likeThe first several classes U (Γ n ) are n = 1 : ( T + 1) · T n = 2 : ( T + 1) · ( T + T ) n = 3 : ( T + 1) · ( T + 2 T ) n = 4 : ( T + 1) · ( T + 3 T + T ) n = 5 : ( T + 1) · ( T + 4 T + 3 T ) n = 6 : ( T + 1) · ( T + 5 T + 6 T + T ) n = 7 : ( T + 1) · ( T + 6 T + 10 T + 4 T )It is natural to guess that for n ≥ U (Γ n ) = ( T + 1) n +1 · C n ( T )with(5.1) C n ( T ) = n (cid:88) i =0 (cid:18) in − i (cid:19) T i . This may be proven by induction on the number of circles: the m = 2 case offormula (3.5) yields the recursion C n +1 = T · ( C n + C n − ) , which determines all C n from C = T , C = T ( T + 1), confirming (5.1). One canpackage this result as a generating function and draw the following conclusion: Proposition 5.2.
For n ≥ , U (Γ n ) = ( T + 1) n +1 · coefficient of r n in the expansion of − T r − T r . A similar, but understandably more complex expression holds for arbitrary va-lence v . Proposition 5.3.
Let v ≥ . OTIVES OF MELONIC GRAPHS 27 • The class U (Γ vn ) is a multiple of T n ( T + 1) n +1 : U (Γ vn ) = T n ( T + 1) n +1 · A vn ( T ) for a polynomial A vn ( t ) of degree ( v − n . • The polynomial A vn ( t ) is the coefficient of r n in the series expansion of therational function α n ( r, t ) = N ( r, t ) /D ( r, t ) , where N ( r, t ) = 1 + t + (( − v − − t v − ) r t = 1 − (cid:32) v − (cid:88) i =0 ( − v − i t i (cid:33) r and D ( r, t ) = 1 + (cid:32) − vt v − − v − (cid:88) i =0 ( − v − i ( i + 2) t i (cid:33) r + (cid:32) ( − v T v − + v − (cid:88) i =0 ( − v − i ( v − − i ) t v − i (cid:33) r . A formal proof of Proposition 5.3 may be constructed along the lines we will provideexplicitly for the case v = 4 in § Example . Consider the case v = 10; the rational function α ( r, t ) is − (1 − t + t − t + t − t + t ) r − (2 − t + 4 t − t + 6 t − t + 8 t + t ) r + (8 t − t + 5 t − t + 3 t − t + t ) r and the coefficient of r in the series expansion of this rational function is a poly-nomial of degree 91: t +103 t +4794 t + · · ·− t + · · · +866304 t − t +4096 . According to Proposition 5.3, the Grothendieck class for the melonic graph con-structed by(((1 , , , , , ((1 , , , , , ((1 , , , , , · · · , ((1 , , , , T ( T + 1) · (cid:0) T + 103 T + · · · − T + · · · − T + 4096 (cid:1) . This may be verified by applying the explicit recursion obtained in § (cid:121) Vacua
In this section we focus on melonic vacuum graphs. We first observe that there isa close relation between Grothendieck classes of vacuum graphs and of related non- vacuum graphs. For this discussion, the graphs are not necessarily assumed to bemelonic; however, the result will explain melonic relations such as the one observedin (4.3).
Vacuum and non-vacuum graphs relations.
Assume Γ v is a graph with adistinguished edge: Γ v and assume this edge is not a bridge in Γ v . Consider two associated graphs: thegraph Γ obtained by cutting the edge, and the graph Γ obtained by inserting a newedge crossing the given edge, with vertices as indicated: ΓΓ Lemma 6.1. U (Γ v ) = U (Γ) − T ( T + 1) U (Γ) T ( T + 1) Proof.
This is an application of the formula for the effect on Grothendieck classes ofadding one parallel edge to a given (non-bridge, non-looping) edge in a graph, i.e., thecase m = 2 of (3.5). Place two valence-2 vertices on the joined edge in Γ v , creatingan edge e in a graph Γ (cid:48) ; by construction, e is neither a bridge nor a looping edge.Then U (Γ (cid:48) ) = ( T + 1) U (Γ v ) , U (Γ (cid:48) /e ) = ( T + 1) U (Γ v ) , while Γ (cid:48) (cid:114) e = Γ. Replacing e by two parallel edges produces Γ without the twoexternal edges. Applying (3.5) then gives U (Γ)( T + 1) = f ( T + 1) U (Γ v ) + g ( T + 1) U (Γ v ) + h U (Γ) , that is (cf. (3.6)) U (Γ)( T + 1) = T ( T + 1) U (Γ v ) + TU (Γ) , with the stated result. (cid:3) In the applications we have in mind, Γ may be a melonic non-vacuum graphsconstructed by ((1 , a , . . . , a r − , , , , t , . . . , t n ;the graph Γ v will then be the (melonic) vacuum graph obtained by joining the twovalence-1 vertices of Γ, and Γ is the non-vacuum graph constructed by((1 , , , , , ((1 , a , . . . , a r − , , , , t (cid:48) , . . . , t (cid:48) n OTIVES OF MELONIC GRAPHS 29 where t (cid:48) i = ( b i , p i + 1 , k i ) if t i = ( b i , p i , k i ), i = 2 , . . . , n . Lemma 6.1 shows thatthe class U (Γ v ) of the vacuum graph is determined by the classes U (Γ), U (Γ) of theassociated non-vacuum graphs.For example, with notation as in §
4, Lemma 6.1 implies that U (Γ (cid:48) n +1 ) = U (Γ n +1 ) − T ( T + 1) U (Γ n ) T ( T + 1) ;with U (Γ n ) = T n ( T + 1) n +1 A n ( T ) and U (Γ (cid:48) n ) = T n − ( T + 1) n − A (cid:48) n ( T ) as in §
4, thisrelation gives T n ( T + 1) n − A (cid:48) n +1 ( T ) = T n +1 ( T + 1) n +3 A n +1 ( T ) − T ( T + 1) T n ( T + 1) n +1 A n ( T ) T ( T + 1) , that is, A (cid:48) n +1 ( T ) = A n +1 ( T ) − A n ( T ) , and this proves (4.3).6.2. Tree structure for valence-four vacua.
Next, we consider specifically mel-onic vacuum graphs in which every vertex has valence 4. These graphs may be drawnas loopless unions of ovals:The information carried by a vacuum melonic graph in valence 4 is equivalent to theinformation of a tree, for example the treefor the graph shown above. Every node of this tree corresponds to one of the ovals,and two nodes are connected by an edge if and only if the corresponding ovals meet.Given a tree, a corresponding melonic construction is obtained in the evident way by associating one arbitrary edge of the tree with a 4-banana and labeling the otheredges with appropriate ((1 , , , ∗ , ∗ ) tuples as prescribed by adjacencies in the tree.For example, the edges of the above tree could be marked as follows (where we alsonumbered the edge of the tree to reflect the stage of the corresponding tuple in themelonic construction; many other choices are possible): ((1,3,1),4,2)((4),0,1)((1,3,1),1,1)((1,3,1),5,1)((1,3,1),6,2)((1,3,1),1,1)((1,3,1),2,1)
123 4 56 7 leading to the melonic construction(6.1) ((4) , , , ((1 , , , , , ((1 , , , , , ((1 , , , , , ((1 , , , , , ((1 , , , , , ((1 , , , , . Alternatively, one could label one node of the tree by a 2-banana and the remainingnodes by ((1 , , , ∗ , ∗ ) tuples; the corresponding construction will produce a vacuummelonic graph with two extra valence-2 vertices. This strategy is used below inExample 6.1.6.3. Recursion relations for vacuum bubbles.
It is natural to ask whether a sim-ple recursion may exist between the Grothendieck classes of vacuum melonic graphs,reflecting the tree-like structure underlying them. The only instance known to us ofsuch a recursion goes as follows. Assume a branch of the tree projects out of the mainbody; let U n denote the Grothendieck class of the vacuum melonic graph obtainedby adding n edges to such a branch. n Claim 6.2.
For n ≥ , (6.2) U n +1 = T ( T + 1) ( T + 2) U n − T ( T + 1) U n − . OTIVES OF MELONIC GRAPHS 31
We will prove this formula in §
7; in fact we will prove that this formula holds evenif the starting graph is not melonic. This will be our main tool in the proofs of thepropositions stated thus far, as well as Proposition 6.3, stated below.
Example . Let Σ sn be the vacuum melonic graph corresponding to the star-shapedtree n s with s rays and n nodes along each ray. For example, Σ is the following melonicvacuum star:Interpreting the central node as a 2-banana (thus adding two valence-2 vertices tothe corresponding circle) leads to the following melonic construction for Σ sn :((2) , , , ((1 , , , . . . , , (cid:124) (cid:123)(cid:122) (cid:125) s ‘3’ ) , , , ((1 , , , , , . . . , ((1 , , , , s ) , ((1 , , , , , . . . , ((1 , , , s, , ((1 , , , s, , . . . , ((1 , , , s, ,. . . , ((1 , , , n − s, , . . . , ((1 , , , n − s, § T + 1) to account for the two additional valence-2 vertices arising inthe construction). On the basis of extensive data, one can formulate the followingstatement. Proposition . Let σ sn ( t ) be the polynomials defined by the expansion − r + (( s − t − ( s − r − (2 + t ) r + 2 r = 1 + (cid:88) n ≥ σ sn ( t ) r n +1 . Then for s, n ≥ U (Σ sn ) = T sn ( T + 1) sn − A n ( T ) s − σ sn ( T ) , where A n ( t ) is the polynomial appearing in Proposition 4.1. For example, according to the above definition, σ ( t ) = t + 22 t + 139 t + 290 t − t − t − t + 88 , and one finds T ( T + 1) A ( T ) σ ( T )= T + 263 T + 34211 T + 2935019 T + · · · + 26065315469197312 T , matching the result of the computation of the Grothendieck class U (Σ )by means of the basic recursion obtained in § (cid:121) The proof of Proposition 6.3 is given in §
6. We record the following consequence,which calls for a more geometric explanation. The relation (6.3) below suggests thatthe complement of the hypersurface ˆ X Σ sn may be realized as a fibration over productsof complements of ˆ X Γ n . This suggests the possible presence of interesting geometricrelations between these families of graph hypersurfaces. Corollary 6.4. If s ≤ n , then U (Γ n ) s − divides U (Σ sn ) . More precisely, (6.3) U (Σ sn ) = T n ( T + 1) n − s U (Γ n ) s − σ sn ( T ) . OTIVES OF MELONIC GRAPHS 33
Proof.
The given equality follows from the formula given in Proposition 6.3 and theexpression for U (Γ n ) obtained in Proposition 4.1. (cid:3) Remark . Corollary 6.4 implies the divisibility relation (4.4) observed in §
4. In-deed, the graph Σ n consists of a string of 2 n + 1 circles: n nn +12 That is, Σ n = Γ (cid:48) n +1 , with notation as in Example 4.3. For this graph, Corollary 6.4states that U (Γ n ) divides U (Γ (cid:48) n +1 ), and this is precisely the assertion in (4.4). (cid:121) Proofs
Proposition 4.1 and Proposition 4.3 will be proved in the equivalent form presentedin Proposition 5.1. For clarity we will focus on the case of valence 4 given in thesepropositions; the same method could be used to prove Proposition 5.3.The statement we will prove will actually be substantially more general than Propo-sitions 4.1 and 4.3: it consists of a recursion ruling the Grothendieck classes of graphsobtained by extending any given graph by a tower of 3-bananas.Let G be a (not necessarily melonic) graph, and let e be an edge of G . Let G n bethe graph obtained by applying a chain of (1 , , e : nG : G n : e Theorem 7.1.
The generating function for the Grothendieck classes U ( G n ) is ratio-nal, with denominator independent of G . More precisely, there exists a polynomial P ( T , ρ ) with integer coefficients such that (cid:88) n ≥ U ( G n ) ρ n = P ( T , ρ )1 − T ( T + 1) ( T + 2) ρ + 2 T ( T + 1) ρ This statement focuses on the fact that the generating function is rational, and givesan explicit form for its denominator, which depends on the bananification processitself rather than on the graph G . The graph G determines the numerator P ( T , ρ );precise formulas will be given below in Theorem 7.2. In practice, Theorem 7.1 and afew case-by-case explicit computations of U ( G n ) for low values of n determine P ( T , ρ ). Proof.
Denote by H n the graph obtained from G n by replacing the last 3-bananificationwith a 2-bananification: H : n Denote U ( G n ) by U n , U ( H n ) by V n . Assume n ≥
2. Consider the graph G (cid:48)(cid:48) obtainedby splitting one of the parallel edges of the top banana in G n − into three edges; let e (cid:48) be the central edge so produced, and note that e (cid:48) is not a bridge or a looping edgeof G (cid:48)(cid:48) . eG : The contraction G (cid:48) := G (cid:48)(cid:48) /e (cid:48) may be obtained from G n − by splitting the same edgeof the top banana into two edges, and the deletion H (cid:48) = G (cid:48)(cid:48) (cid:114) e (cid:48) may be obtainedfrom H n − by attaching two external edges to the vertices of the top (2-)banana: G : H : By (3.5), we have V n = f U ( G (cid:48)(cid:48) ) + g U ( G (cid:48) ) + h U ( H (cid:48) ) U n = f U ( G (cid:48)(cid:48) ) + g U ( G (cid:48) ) + h U ( H (cid:48) ) , where f , f , etc., are as in (3.6). We now note that U ( G (cid:48)(cid:48) ) = ( T + 1) U ( G n − ) = ( T + 1) U n − U ( G (cid:48) ) = ( T + 1) U ( G n − ) = ( T + 1) U n − U ( H (cid:48) ) = ( T + 1) U ( H n − ) = ( T + 1) V n − ;further, f ( T + 1) + g ( T + 1) = T ( T + 1) , f ( T + 1) + g ( T + 1) = T ( T + 1) while h = T , h = ( T − T . The above formulas can then be rewritten(7.1) V n = T ( T + 1) U n − + T ( T + 1) V n − U n = T ( T + 1) U n − + ( T − T ( T + 1) V n − . OTIVES OF MELONIC GRAPHS 35
These imply( T − V n = ( T − T ( T + 1) U n − + ( T − T ( T + 1) V n − = ( T − T ( T + 1) U n − + (cid:0) U n − T ( T + 1) U n − (cid:1) = U n − T ( T + 1) U n − and therefore U n +1 = T ( T + 1) U n + ( T − T ( T + 1) V n = T ( T + 1) U n + T ( T + 1) (cid:0) U n − T ( T + 1) U n − (cid:1) = T ( T + 1) ( T + 2) U n − T ( T + 1) U n − . (This proves Claim 6.2.) Now, for n ≥
2, the coefficient of ρ n +1 in the product (cid:0) − T ( T + 1) ( T + 2) ρ + 2 T ( T + 1) ρ (cid:1) · (cid:88) n ≥ U n ρ n equals U n +1 − T ( T + 1) ( T + 2) U n + 2 T ( T + 1) U n − = 0 . This product is therefore a polynomial P ( T , ρ ), and this proves the statement. (cid:3) The argument shows that(7.2) P ( T , ρ ) = (cid:0) − T ( T + 1) ( T + 2) ρ + 2 T ( T + 1) ρ (cid:1) · (cid:88) n ≥ U n ρ n = U ( G ) + ( U ( G ) − T ( T + 1) ( T + 2) U ( G )) ρ + (cid:0) U ( G ) − T ( T + 1) ( T + 2) U ( G ) + 2 T ( T + 1) U ( G ) (cid:1) ρ . If e is not a bridge, then the argument proves the same recursion for n ≥
1; it followsthat the coefficient of ρ in P ( T , ρ ) is 0 in this case. Maybe a little surprisingly, thesame conclusion holds if e is a bridge (as we will prove below); thus, the polyno-mial P ( T , ρ ) is of degree 1 in ρ . This polynomial is determined by U ( G ) and thedeletion U ( G (cid:114) e ), as we will see below.In fact, Theorem 7.1 and the direct computation of a few values of U ( G n ) sufficeto determine the numerator. Example . The melonic valence-4 vacuum graphs corresponding to the trees n have melonic construction obtained by extending (6.1):((4) , , , ((1 , , , , , ((1 , , , , , ((1 , , , , , ((1 , , , , , ((1 , , , , , ((1 , , , , , ((1 , , , , , ((1 , , , , , . . . , ((1 , , , n + 6 , . Using the recursion obtained in §
3, we can compute the following Grothendieckclasses: n = 0 : T ( T + 1) ( T + 13 T + 56 T + 80 T − T − T + 8) n = 1 : T ( T + 1) ( T + 14 T + 64 T + 94 T − T − T + 12 T + 8) n = 2 : T ( T + 1) ( T + 3)( T + 14 T + 64 T + 96 T − T − T − T + 16)and this is (more than) enough information to determine P ( T , ρ ): if U , U , U arethese three classes, the product (cid:0) − T ( T + 1) ( T + 2) ρ + 2 T ( T + 1) ρ (cid:1) · ( U + U ρ + U ρ )equals T ( T + 1) ( T + 13 T + 56 T + 80 T − T − T + 8) − T ( T + 1) (2 T + 17 T + 39 T + 9 T − T − T + 4) ρ modulo ρ . As expected, the coefficient of ρ vanishes. The polynomial P ( T , ρ ) mustequal this degree 1 polynomial in ρ . (cid:121) In general, P ( T , ρ ) is determined by the Grothendieck classes of G and (if e is nota bridge) G (cid:114) e , if the latter is known. Theorem 7.2.
With notation as above, let r = T ( T + 1) ρ . Then we have (cid:88) n ≥ U ( G n ) ρ n = 1 − r − ( T + 2) r + 2 r · U ( G ) if e is a bridge in G , and (cid:88) n ≥ U ( G n ) ρ n = U ( G ) + (cid:0) ( T − U ( G (cid:114) e ) − U ( G ) (cid:1) r − ( T + 2) r + 2 r if e is not a bridge in G .Proof. The argument proving Theorem 7.1 shows that the coefficient of ρ n in P ( T , ρ )is 0 for n ≥
3, and for n ≥ e is not a bridge, as observed above. If e is a bridge, U = ( T + 1) B (3) U ( G (cid:114) e ) = T ( T + 1) U ( G (cid:114) e ) = T ( T + 1) U , since G is obtained by replacing the central split of e , a bridge, with a 3-banana. Bythe same token, V = ( T + 1) B (2) U ( G (cid:114) e ) = T ( T + 1) U ( G (cid:114) e ) = T ( T + 1) U . OTIVES OF MELONIC GRAPHS 37
By (7.1) we have U = T ( T + 1) U + ( T − T ( T + 1) V = T ( T + 1) ( T + 3) U . On the other hand, T ( T + 1) ( T + 2) U − T ( T + 1) U = T ( T + 1) ( T + 2) U − T ( T + 1) U = T ( T + 1) ( T + 3) U . This verifies that the coefficient of ρ in P ( T , ρ ) (see (7.2)) equals 0 in this case aswell. Therefore, in all cases we have P ( T , ρ ) = U ( G ) + ( U ( G ) − T ( T + 1) ( T + 2) U ( G )) ρ . If e is a bridge, the coefficient of ρ in P ( T , ρ ) is U ( G ) − T ( T + 1) ( T + 2) U ( G ) = − T ( T + 1) U ( G )since U ( G ) = U = T ( T + 1) U = T ( T + 1) U ( G ) as we observed above. Therefore P ( T , ρ ) = U ( G ) − T ( T + 1) U ( G ) ρ = (1 − r ) U ( G )if e is a bridge, and this gives the first formula.If e is not a bridge, splitting it into three and 3-bananifying the central edge gives,arguing as in the proof of Theorem 7.1, U ( G ) = T ( T + 1) U ( G ) + ( T − T ( T + 1) U ( G (cid:114) e )and therefore U ( G ) − T ( T + 1) ( T + 2) U ( G ) = − T ( T + 1) U ( G ) + ( T − T ( T + 1) U ( G (cid:114) e ) . It follows that the degree-1 term in P ( T , ρ ) in this case is( − U ( G ) + ( T − U ( G (cid:114) e )) T ( T + 1) ρ , and this completes the proof of the second formula. (cid:3) The fact that the formulas in Theorem 7.2 depend on r = T ( T + 1) ρ explains whythe specific examples worked out in Propositions 4.1 and 4.3 included powers of T and T + 1 as stated. We recover these results in the next two examples. Example . Define the polynomials A n ( t ) by the power series expansion (cid:88) n ≥ A n ( t ) r n = 1 − r − (2 + t ) r + 2 r (cf. Proposition 5.1). Then the first formula in Theorem 7.2 reads (cid:88) n ≥ U ( G n ) ρ n = (cid:32)(cid:88) n ≥ A n ( t ) r n (cid:33) · U ( G ) = (cid:32)(cid:88) n ≥ A n ( T ) T n ( T + 1) n ρ n (cid:33) · U ( G ) . Equivalently,(7.3) U ( G n ) = T n ( T + 1) n A n ( T ) · U ( G ) . If G consists of a single edge, then with notation as in § G n = Γ n , and U ( G ) = T + 1, therefore (7.3) gives U (Γ n ) = T n ( T + 1) n +1 A n ( T ) , proving Proposition 4.1 (in the form given in Proposition 5.1). (cid:121) Example . Now let G be a 2-banana, and let e be one of its (two) edges. Thegraph G (cid:114) e is a single edge. Therefore U ( G ) = T ( T + 1) , U ( G (cid:114) e ) = T + 1 , and the second formula in Theorem 7.2 states that (cid:88) n (cid:54) =0 U ( G n ) ρ n = T ( T + 1) + (cid:0) ( T − T + 1) − T ( T + 1) (cid:1) r − ( T + 2) r + 2 r = (cid:0) T − r (cid:1) − ( T + 2) r + 2 r · ( T + 1)With notation as in §
4, the graph G n (consisting of a chain of n +1 circles) equals Γ (cid:48) n +1 ,with two extra valence-2 vertices on the first circle. That is, U (Γ (cid:48) n ) = U ( G n − )( T + 1) . Now, since r = T ( T + 1) ρ , U ( G n − ) = T n − ( T + 1) n − · coeff. of r n − in the expansion of (cid:0) T − r (cid:1) ( T + 1)1 − ( T + 2) r + 2 r hence U ( G n − )( T + 1) = T n − ( T + 1) n − · coeff. of r n − in the expansion of (cid:0) T − r (cid:1) − ( T + 2) r + 2 r and therefore U (Γ (cid:48) n ) = T n − ( T + 1) n − · coeff. of r n in the expansion of r (cid:0) T − r (cid:1) − ( T + 2) r + 2 r . This holds for n ≥
1; setting (as in §
4) the constant term of the relevant series to 1amounts to adding 1 to this rational function, and1 + r (cid:0) T − r (cid:1) − ( T + 2) r + 2 r = (1 − r ) − ( T + 2) r + 2 r verifying Proposition 4.3, in the form given in Proposition 5.1. (cid:121) Example . As a final example, we will prove Proposition 6.3, by induction on thenumber s of rays. For s = 1, the statement reproduces Proposition 4.3; so we onlyneed to prove the induction step, and we may assume s > s − n to Σ sn , view Σ sn as the graph obtained by adding a chainof 3-bananas to one of the edges e of the central circle in G = Σ s − n . OTIVES OF MELONIC GRAPHS 39 e Since e is not a bridge, we can apply the second formula given in Theorem 7.2. Wewrite it as follows:1 − r − ( T + 2) r + 2 r U (Σ s − n ) + r ( T − − ( T + 2) r + 2 r U (Σ s − n (cid:114) e ) . The class U (Σ sn ) is the coefficient of ρ n in this expression (i.e., T n ( T + 1) n times thecoefficient of r n ). We will deal with the two summands separately. • By induction, the first summand equals U (Γ n )( T + 1) · T ( s − n ( T +1) s − n − A n ( T ) s − · coeff. of r n +1 in 1 − r + (( s − T − ( s − r − ( T + 2) r + 2 r and U (Γ n ) = T n ( T + 1) n +1 A n ( T ), so this equals T sn ( T + 1) sn − A n ( T ) s − · coeff. of r n +1 in 1 − r + (( s − T − ( s − r − ( T + 2) r + 2 r . • In the second summand, Σ s − n (cid:114) e consists of a join of s − n -circles,therefore its Grothendieck class U (Σ s − n (cid:114) e ) is the ( s − U (Γ n ), up toan appropriate factor of ( T + 1) to account for the fact that Σ s − n (cid:114) e has no valence-2vertices and no external edges. For example, here is a picture contrasting the join of3 graphs Γ (on the left) with Σ (cid:114) e (on the right):It follows that U (Σ s − n (cid:114) e ) = U (Γ n ) s − ( T + 1) s = T ( s − n ( T + 1) s − n − A n ( T ) s − . Therefore, the second summand equals T ( s − n ( T + 1) s − n − A n ( T ) s − · coeff. of ρ n in r ( T − − ( T + 2) r + 2 r . Now, the coefficient of ρ n equals T ( T + 1) times the coefficient of r n , so this may berewritten as T sn ( T + 1) sn − A n ( T ) s − · coeff. of r n in r ( T − − ( T + 2) r + 2 r or equivalently T sn ( T + 1) sn − A n ( T ) s − · coeff. of r n +1 in r ( T − − ( T + 2) r + 2 r . Putting the summands back together, we see that U (Σ sn ) equals T sn ( T +1) sn − A n ( T ) s − times the coefficient of r n +1 in − r + (( s − T − ( s − r − ( T + 2) r + 2 r + r ( T − − ( T + 2) r + 2 r = 1 − r + (( s − T − ( s − r − ( T + 2) r + 2 r and this verifies the induction step, concluding the proof of Proposition 6.3. (cid:121) Acknowkedgment.
The first author acknowledges support from a Simons Foun-dation Collaboration Grant, award number 625561, and thanks the University ofToronto for hospitality. The second author is partially supported by NSF grant DMS-1707882, and by NSERC Discovery Grant RGPIN-2018-04937 and Accelerator Sup-plement grant RGPAS-2018-522593, and by the Perimeter Institute for TheoreticalPhysics. The third author worked on parts of this project as summer undergraduateresearch at the University of Toronto.
References [1] K. Adiprasito, J. Huh, E. Katz,
Hodge theory for combinatorial geometries , Ann. of Math. (2)188 (2018), no. 2, 381–452.[2] P. Aluffi, M. Marcolli,
Feynman motives of banana graphs , Commun. Number Theory Phys. 3(2009), no. 1, 1–57 [arXiv:0807.1690].[3] P. Aluffi, M. Marcolli,
Algebro-geometric Feynman rules , Int. J. Geom. Methods Mod. Phys. 8(2011), no. 1, 203–237 [arXiv:0811.2514].[4] P. Aluffi, M. Marcolli,
Feynman motives and deletion-contraction relations , in “Topology ofalgebraic varieties and singularities”, 21–64, Contemp. Math., 538, Amer. Math. Soc., 2011[arXiv:0907.3225].[5] P. Aluffi, M. Marcolli,
A motivic approach to phase transitions in Potts models , Journal ofGeometry and Physics, Vol.63 (2013) 6–31 [arXiv:1102.3462].
OTIVES OF MELONIC GRAPHS 41 [6] Y. Andr´e,
An introduction to motivic zeta functions of motives , in “Motives, quantum fieldtheory, and pseudodifferential operators”, 3–17, Clay Math. Proc., 12, Amer. Math. Soc., 2010.[arXiv:0812.3920][7] A. Baratin, S. Carrozza, D. Oriti, J. Ryan, M. Smerlak,
Melonic phase transition in group fieldtheory , Lett. Math. Phys., Vol. 104 (2014), N.8, 1003–1017 [arXiv:1307.5026].[8] D. Bejleri, M. Marcolli,
Quantum field theory over F , J. Geom. Phys. 69 (2013), 40–59.[9] D. Benedetti, R. Gurau, S. Harribey, Line of fixed points in a bosonic tensor model ,arXiv:1903.03578v3[10] S. Bloch, P. Vanhove,
The elliptic dilogarithm for the sunset graph , J. Number Theory 148(2015), 328–364. [arXiv:1309.5865][11] V. Bonzom, R. Gurau, V. Rivasseau,
Random tensor models in the large N limit: Uncoloringthe colored tensor models , Phys. Rev. D85 (2012) 084037 [arXiv:1202.3637].[12] V. Bonzom, R. Gurau, A. Riello, V. Rivasseau,
Critical behavior of colored tensor models inthe large N limit , Nucl. Phys. B853 (2011) 174195 [arXiv:1105.3122].[13] V. Bonzom, V. Nador, A. Tanasa, Diagrammatic proof of the large N melonic dominance inthe SYK model , Lett. Math. Phys., Vol. 109 (2019) 2611–2624 [arXiv:1808.10314].[14] S. Carrozza, A. Tanasa, O ( N ) random tensor models , Lett. Math. Phys. 106 (2016), no. 111531–1559 [arXiv:1512.06718].[15] V. I. Danilov, A. G. Khovanski˘ı, Newton polyhedra and an algorithm for calculating Hodge–Deligne numbers , Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986) , 925–945.[16] E. Fusy, A. Tanasa,
Asymptotic expansion of the multi-orientable random tensor model , Theelectronic journal of combinatorics 22(1) (2015), P1.52 [arXiv:1408.5725][17] J.B. Geloun, R. Toriumi,
Parametric representation of rank d tensorial group field the-ory: Abelian models with kinetic term (cid:80) s | p s | + µ , J. Math. Phys., Vol. 56 (2015) 093503[arXiv:1409.0398].[18] D.J. Gross, V. Rosenhaus, A generalization of Sachdev-Ye-Kitaev , Journal of High EnergyPhysics (2017), Article number: 93 [arXiv:1610.01569].[19] R. Gurau,
Topological graph polynomials in colored group field theory , Ann. Henri Poincar´e,Vol. 11 (2010) 565–584. [arXiv:0911.1945][20] R. Gurau,
The /N expansion of colored tensor models , Ann. H. Poincar´e 12 (2011) 829–847[arXiv:1011.2726].[21] R. Gurau, Colored group field theory , Commun. Math. Phys., Vol. 304 (2011) 69–93[arXiv:0907.2582].[22] R. Gurau,
Random Tensors . Oxford University Press, 2016.[23] R. Gurau, J.P. Ryan,
Colored tensor models – a review , SIGMA 8 (2012) 020 [arXiv:1109.4812].[24] J. Huh,
Combinatorial applications of the Hodge-Riemann relations , in “Proceedings of theInternational Congress of Mathematicians–Rio de Janeiro 2018. Vol. IV. Invited lectures”,pp. 3093–3111, World Scientific, 2018.[25] I.R. Klebanov, G. Tarnopolsky,
Uncolored random tensors, melon diagrams, and the SYK mod-els , Phys. Rev. D95 (2017), no. 4 046004 [arXiv:1611.08915].[26] Yu.I. Manin, M. Marcolli,
Moduli operad over F , in “Absolute arithmetic and F -geometry”,pp. 331–361, Eur. Math. Soc., 2016.[27] M. Marcolli, Feynman motives , World Scientific, 2010.[28] M. Marcolli, G. Tabuada,
Feynman quadrics-motive of the massive sunset graph , J. NumberTheory, Vol. 195 (2019), 159–183. [arXiv:1705.10307][29] R. Stanley,
Log-concave and unimodal sequences in algebra, combinatorics, and geometry , in“Graph theory and its applications: East and West (Jinan, 1986)”, pp. 500–535, Ann. NewYork Acad. Sci., vol. 576, 1989.[30] A. Tanasa,
Generalization of the Bollob´as–Riordan polynomial for tensor graphs , J. Math.Phys., Vol. 52 (2011), 073514 [arXiv:1012.1798][31] E. Witten,
An SYK-like model without disorder , arXiv:1610.09758.
Florida State University, Tallahassee, USA
E-mail address : [email protected] California Institute of Technology, Pasadena, USAUniversity of Toronto, Toronto, CanadaPerimeter Institute for Theoretical Physics, Waterloo, Canada
E-mail address : [email protected] University of Toronto, Toronto, Canada
E-mail address ::