Renormalization in combinatorially non-local field theories: the Hopf algebra of 2-graphs
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Renormalization in combinatorially non-local fieldtheories: the Hopf algebra of 2-graphs
Johannes Th¨urigen
Mathematisches Institut der Westf¨alischen Wilhelms-Universit¨at M¨unsterEinsteinstr. 62, 48149 M¨unster, Germany, EUInstitut f¨ur Physik/Institut f¨ur Mathematik der Humboldt-Universit¨at zu BerlinUnter den Linden 6, 10099 Berlin, Germany, EU
E-mail: [email protected]
Abstract:
It is well known that the mathematical structure underlying renormalizationin perturbative quantum field theory is based on a Hopf algebra of Feynman diagrams. Aprecondition for this is locality of the field theory. Consequently, one might suspect thatnon-local field theories such as matrix or tensor field theories cannot benefit from a sim-ilar algebraic understanding. Here I show that, on the contrary, the renormalization andperturbative diagramatics of a broad class of such field theories is based in the same wayon a Hopf algebra. These theories are characterized by interaction vertices with graphsas external structure leading to Feynman diagrams which can be summed up under theconcept of “2-graphs”. From the renormalization perspective, such graph-like interactionsare as much local as point-like interactions. They differ in combinatorial details as I ex-emplify with the central identity for the perturbative series of combinatorial correlationfunctions. This sets the stage for a systematic study of perturbative renormalization aswell as non-perturbative aspects, e.g. Dyson-Schwinger equations, for a number of combi-natorially non-local field theories with possible applications to quantum gravity, statisticalmodels and more. ontents
Locality in QFT allows to perform perturbative renormalization order by order. Sub-tracting divergences is described mathematically by a Hopf algebra of Feynman diagrams[1–4]. For a given diagram, the coproduct separates divergent subdiagrams. Then therenormalization operation subtracting the divergences of this subdiagram is based on theHopf algebra’s antipode. Combinatorially, one needs a closed set of diagrams in which itis possible to “separate” divergent diagrams and counter them by single vertices with thesame external structure. It is well known that this works in many cases for local, i.e. point-like, interactions [1–10]. But there are also examples of renormalizable field theories withcertain “non-local” interactions for which a Connes-Kreimer type Hopf algebra be found[11–14]. More specifically, these are combinatorially non-local field theories such as non-commutative quantum field theory and matrix field theory [15–19], and its generalizationfield theories of higher rank tensor fields [20–22], in particular also group field theory [23–26]. It is thus a natural question whether this is just a coincidence in specific examples ora more general feature, and whether the (Hopf algebra) mechanisms are different or thesame.Standard quantum field theories on a D -dimensional space are local in the sense thatthey have point-like interactions, for example φ ( xxx ) n for a scalar field φ on coordinates x , ..., x D . This relates to energy-momentum conservation at interaction vertices via Fouriertransformation S ia [ φ ] = Z d xxx λ n φ ( xxx ) n = λ n Z d xxx n Y i =1 Z d ppp i ˜ φ ( ppp i )e i ppp i · xxx = λ n Z n Y i =1 d ppp i δ (cid:18) n X i =1 ppp i (cid:19) n Y i =1 ˜ φ ( ppp i )(1.1)where δ is the Dirac distribution. It constrains interactions to have incoming and outgoingenergy-momentum being equal. In particular, correlation functions and effective verticeshave the same external constraint structure, as well as each Feynman diagram in their– 1 – = ∼ = Figure 1 . Comparison of the combinatorial structure of an order- n interaction vertex in acombinatorially local theory (left) and non-local theory (right). In both cases, there are n fields(red ellipses, here n = 4) which depend on a number of arguments (green lines). In the local case,Eq. (1.1), due to Lorentz (or some other “rotational”) symmetry these arguments form a vectorwhich is constrained by a single delta distribution (blue circle) such that all of this structure isequivalently captured by a vertex (black) with n half edges (red lines). On the other hand, inthe non-local case, Eq. (1.2)) the arguments are convoluted pairwaise (blue dots); to capture thisstructure it is necessary to add a second set of edges (green lines) leading to the notion of a 2-graph. perturbative expansion. For perturbatively renormalizable quantum field theories, this ex-ternal structure of interaction vertices allows to subtract the divergent part of an amplitudeidentifying it in subgraphs. Remarkably, the structure of this renormalization procedure,independent of a specific renormalization scheme, is captured by the Hopf algebra of Feyn-man diagrams [1–4]. Thereby, the external structure of vertices induced by locality of theinteractions plays a crucial role.The aim of this work is to show that also a broad class of “non-local” field theories ischaracterized by the same Hopf-algebraic structure of perturbative renormalization. Thus,it is not locality in the sense of point-like interactions but an appropriate class of externalstructures which is crucial for perturbative renormalization. The field theories under con-sideration are characterized by an external constraint structure which pairs single entriesof “momenta”. For example, let φ again be a scalar field, that is a function of r arguments ppp = ( p , ..., p r ) where each argument p a is in a d -dimensional manifold. Then, a genericinteraction of order n , with n · r even, has the form S ia [ φ ] = λ n Z n Y i =1 d ppp i Y ( ia,jb ) δ ( p ai + p bj ) n Y i =1 ˜ φ ( ppp i ) (1.2)where the product over pairs ( ia, jb ) means that for each argument p ai there is a convolu-tion δ ( p ai + p bj ) with exactly one other argument p bj . As a consequence the diagrammaticrepresentation of interactions is not just a vertex in a graph but has to capture this pairingof arguments (see Fig. 1). In effect, each interaction has the combinatorial structure of agraph itself, i.e. the necessary class of external structures is graphs.Consequently, the Feynman diagrams of a combinatorially non-local field theory arecertain gluings of vertex graphs, or equivalently, they are graphs with the additional struc-ture of “strands” at each edge. Following a strand through a cycle (loop) of the graphgives a face. Thus, such “strand graphs” are actually two-dimensional objects. Indeed,they are two-dimensional combinatorial complexes in a specific sense (Prop. 2.12). Thisis well known in the case of fields with r = 2 arguments, that is matrices; these generate– 2 –ombinatorial maps, also called ribbon or fat graphs, which are dual to n -angulations ofsurfaces [27]. With more arguments r >
2, one can use additional structure to extendthis duality to n -angulations of r -dimensional (pseudo) manifolds [28–30]. Here I want toconsider such diagrams in full generality allowing vertices of arbitrary order n with fields φ i , i = 1 , , ..., n with an arbitrary number of arguments r i (similar to [31]). To emphasizethat such Feynman diagrams are still just a generalization of standard Feynman graphsadding a second layer I will call them and any field theory with such combinatoricsa combinatorially non-local field theory (cNLFT).In this work I show that the Hopf-algebraic structure underlying renormalization incNLFT is very general and independent of any specific theory, along a similar logic asfor local field theory [8]. Renormalizablity of various cNLFTs is known as for exampleGrosse/Wulkenhaar’s non-commutative field theory [16, 17] related to Kontsevich’s matrixmodel [15, 32], tensor-field models [20, 22] and group field theories [23]. In any such casethere is a set of superficially divergent 2-graphs and the Hopf algebra encodes the procedureof identifying (coproduct) and subtracting (antipode) these divergences in a given Feynman2-graph via subgraph contraction. I will generalize the contraction operation from graphsto 2-graphs. This gives rise to a general Hopf algebra of 2-graphs (Thm. 5.1). Under mildconditions on the set of divergent 2-graphs, one can then derive any renormalization Hopfalgebra of a specific theory (such as the known ones for the non-commutative field theory[11, 12] and some tensor-field models [13, 14]) as the quotient of the general Hopf algebrawith some Hopf ideal [8].With the appropriate concept of 2-graphs as well as contraction and insertion opera-tions at hand, the general algebraic structure of renormalization in cNLFT turns out to beexactly the same as for local field theory, but the algebras differ in combinatorial details.In particular, 2-graphs have more structure, and thus less symmetry, than usual graphs.I show this for the central identity (Thm. 4.6), i.e. the action of the coproduct on an infi-nite series over 2-graphs which captures the combinatorics of the perturbative expansion ofGreen’s functions; this formula differs from the one of local field theory in that there occursthe symmetry factor of the boundary graph of each interaction vertex, not just the facto-rial n !. These concrete formula set the stage for explicit (BPHZ) renormalization and theinvestigation of Dyson-Schwinger equations in cNLFT which we will report on elsewhere.The structure of the paper is the following: In Sec. 2 I introduce the concept of 2-graphs, followed by the definition of 2-graph contraction in Sec. 3 with a discussion onrelevant subtleties concerning the connectedness of the 2-graph’s boundary and topology.In Sec. 4 I define the co-algebra and prove the central identity. Finally, in Sec. 5 I introducethe general Hopf algebra and explain how to obtain the Connes-Kreimer Hopf algebra fora specific cNLFT by dividing out a Hopf ideal induced by the theory’s set of divergent2-graphs. I close with two examples, the Hopf algebra for the Grosse-Wulkenhaar modeland for tensorial field theories. – 3 – Combinatorial basis: 2-graphs
For perturbative field theory it is convenient to define graphs in terms of half-edges associ-ated to vertices which are then pairwise combined into edges [8–10]. This captures nicelythe appearance of self-loops, multi-edges and external legs in Feynman diagrams.
Definition 2.1 (graph). A , or simply graph , is a tuple g = ( V , H , ν, ι ) with(1) a set of vertices V ,(2) a set of half-edges H ,(3) an adjacency map ν : H → V associating half-edges to vertices,(4a) an involution on H , that is ι : H → H such that ι ◦ ι = id. The resulting pairs ofhalf-edges are called edges .The involution may have fixed points, half-edges paired to themselves. These are under-stood as external edges (or legs ).Alternatively, one can define a graph in terms of an explicit set of edges: Definition 2.2 (graph with edge set).
A 1-graph is a tuple ( V , H , ν, E ) with the aboveproperties (1) – (3) and(4b) a set of disjoint, two-element subsets of H , (that is E ⊂ H such that e ∩ e = ∅ forall e , e ∈ E ).The two definitions in terms of either property (4a) or (4b) are equivalent [8] since aninvolution (4a) partitions H into sets of either one or two elements where the latter are inone-to-one correspondence to edge sets (4b).The simplest way to extend from such notion of graphs to the general class of Feynmandiagrams of cNLFT is to introduce another pairing of half-edges, only this time betweenthose at a vertex, to capture the strands. However, it can happen that also the strandingat a vertex between different edges is multiple, and there can also be self-loops at a singleedge. Thus it is not sufficient to describe the stranding by an involution but it is necessaryto explicitly introduce a second layer of (dimension-two) half-edges for the strands: Definition 2.3 (2-graph). A is a tuple G = ( V , H , ν, ι ; S , µ, σ , σ ) with the aboveproperties (1) – (3), (4a) and(5) a set of strand sections S (6) an adjacency map µ : S → H associating them to half-edges,(7a) a fixed-point free involution σ : S → S pairing strand sections at a given vertex, thatis for every s ∈ S : ν ◦ µ ◦ σ ( s ) = ν ◦ µ ( s ),(8a) an involution σ : S → S describing the pairing of strands along edges, that is • ι ◦ µ ( s ) = µ ◦ σ ( s ) for all strand sections s ∈ S • every s ∈ S is a fixed point of σ iff µ ( s ) is a fixed point of ι .– 4 –ote that the fullfilment of properties (4a) for ι and (8a) for σ depend on each other:There can only be an edge between two half-edges when both have the same number ofadjacent strand sections. And if there is an edge, than the adjacent strands have to bepaired along that edge. In that sense ι and σ together define stranded edges . One couldcollapse the two defining properties into one (in particular ι is eventually redundant), butsince the involutions act on different objects (half-edges and strands) and both are neededfor some purpose, it is clearer to separate them. Definition 2.4 (corollae, vertex graph).
It is common to refer to the preimage ν − ( v )of a graph’s vertex v as corolla . Its cardinality gives the degree of the vertex d v = | ν − ( v ) | .In a 2-graph G , also half edges have a corolla, that is µ − ( h ) for a half-edge h ∈ H G , as wellas a degree d h = | µ − ( h ) | . Thus, the full of a vertex is ( ν ◦ µ ) − ( v ) for v ∈ V G .For the entire structure at the vertex v ∈ V it is necessary to include the pairing ofthe strand sections in the 2-corolla. I call this the vertex graph g v = ( V v , H v , ν v , ι v ) := (cid:0) ν − ( v ) , ( ν ◦ µ ) − ( v ) , µ | H v , σ | H v (cid:1) (2.1)in which half-edges become vertices and strands become edges. Note that vertex graphs arenot necessarily connected but can have several components. Such a disconnected vertex isalso called a multi-trace vertex. Remark 2.5 (vertex-graph representation).
Since the strand sections S are paired atindividual vertices v ∈ V (property (7)), a 2-graph partitions into its vertex graphs. Thatis, there is a bijective map from 2-graphs to a class of graphs equipped with an additionaledge structure β vg : ( V , H , ν, ι ; S , µ, σ , σ ) (cid:0) { g v } v ∈V , ι, σ (cid:1) . (2.2)The 2-graph can thus be understood as a gluing of vertex graphs along their corollae. I willcall this the vertex-graph representation of a 2-graph. This is a common way physicallyrelevant classes of 2-graphs are defined in the literature. In particular they arise in invarianttensor models, group field theory and spin-foam models where the vertex graphs have beengiven various colourful names, e.g. “bubbles” connected along “0-edges” [33], “atoms”bonded along “patches” [31], or “squid graphs” glued along their “squids” [34].It has to be emphasized that it is essential for the bijectivity of the map that its imageis defined in terms of the set { g v } v ∈V of vertex graphs and not just using the disjoint union F v ∈V g v . A single vertex graph g = g v can have several connected components, g = F j g j .Thus, in the disjoint union F v ∈V g v the information which connected components belongto one vertex in the 2-graph is lost. The map to a vertex-graph representation using thedisjoint union, π vg : ( V , H , ν, ι ; S , µ, σ , σ ) (cid:0) G v ∈V g v , ι, σ (cid:1) . (2.3)is therefore a projection. For the algebraic structures defined below based on contractionand insertion operations it will be crucial that β vg is bijective and conserves the vertex-belonging information. The underlying physical reason is that it is the coupling constantsassociated to vertices which have to be renormalized.– 5 – = ∼ =1 2 34 56 7 1 2 34 765 1 2 34 765 Figure 2 . A combinatorial map ( H , σ, ι ) = ( { , , , , , , } , (1)(234)(576) , (12)(35)(46)) drawnon the plane with counter-clockwise orientation of the vertices (left) and the corresponding 2-graph ( V , H , ν, ι ; S , µ, σ , σ ) in the stranded representation (ribbon graph, middle) and vertex-graph representation (right; dashed lines represent the pairing σ into stranded edges). For eachhalf-edge j ∈ H there are strand sections s ji , s jk ∈ S according to the cycle ( ...ijk... ) of thepermutation σ , for example here s and s adjacent to h , and σ pairs each s ij with s ji . Asexample of the special case n <
3, the univalent vertex v defined by the cycle (1) in σ has twostrand sections s and s ′ at the half-edge h which are paired with each other by σ . The framework of 2-graphs is very general and covers many examples of field theory.Even standard local field theory is covered if one considers trivial 2-graphs with S = ∅ . Theparadigmatic example in mind, though, are theories of tensor fields, in particular matrixfield theories [18, 19] (related to noncommutative field theory) with combinatorial mapsas diagrams and theories with tensorial interactions of arbitrary rank whose diagrams arecoloured graphs [22, 23]: Example 2.6 (combinatorial maps).
Combinatorial maps (also called “ribbon graphs”or “fat graphs” in physics) are a special example of 2-graphs with all edges of degree d h = 2:A (finite) combinatorial map is a triple ( H , σ, ι ) with the above properties (1) and (4a) and a permutation σ : H → H whose cycles are called vertices.In this case it is not necessary to define the strand sections explicitly; like the vertices,they are already encoded in the cyclic structure around vertices given by σ . Thus a σ -cycle v = ( h , h , ..., h n ), n ≥
3, defines the full vertex graph g v = ( { h , h , ..., h n } , { s n , s , s , s , ...s n n − , s n } , ν v , ι v ) (2.4)where ν v encodes the adjacency of each strand s ij to the half-edge h i and ι v pairs s ij with s ji iff σ ( h i ) = h j . Also univalent and bivalent vertices (one-cycles and two-cycles)are captured in this way, one only has to distinguish s and s ′ respectively s , s and s ′ , s ′ . See Fig. 2 for an example.In the same way, the edge involution ι : H → H defines already a pairing σ of theedges’ strands in terms of the orientation given by σ . Then, the explicit 2-graph is given bythe vertex graphs and the bijection β vg , Eq. (2.2). In particular, if all vertices have degree n ≥
3, faces are simply the cycles of σ ◦ ι [35]. Example 2.7 (coloured graphs).
Another standard example are the Feynman diagramsof rank- r tensor theories with invariant interactions [30, 33]. These are ( r + 1)-regular, In contrast to combinatorics literature (e.g. [35]), I include boundaries as fixed-points of ι (externaledges) and not in terms of marked edges. In this way, diagrams which are disconnected upon removing adisconnected boundary are in fact disconnected maps. – 6 – = c c c c Figure 3 . Example of a (4+1)-coloured graph (left) and its corresponding 4-coloured 2-graph (inits vertex-graph representation, right). In the coloured graph, dashed lines represent colour-0 edges,internal and external. The internal ones become the edges in the 2-graph whose strand structureis here completely determined by the colouring of the c , c ∈ { , , , } strands. edge-coloured graphs, or ( r + 1) -coloured graphs for short, that is graphs for which eachvertex is adjacent to exactly one edge decorated with “colour” c = 0 , , ..., r each. Theinterpretation as a Feynman diagram, and thus the bijection to 2-graphs is the following(see also Fig. 3):For a ( r + 1)-coloured graph, the connected components of the r -coloured graph ob-tained by deleting all 0-edges define a set of vertex graphs; that is, the coloured edges arebijective to coloured strand sections in the 2-graph with pairing σ at each vertex accordingto this graph structure. Clearly, all half edges h are r -valent, d h = r . Then the 0-edgesdefine then stranded edges whereby a unique pairing σ of the strand sections follows fromthe condition that only pairings of strand sections of the same colour are allowed. Thismeans that in the result, the r -coloured 2-graph , the involution σ is redundant.This bijection maps only to coloured 2-graphs with connected vertices, that is allvertices have connected vertex graphs. In fact, the bijection is exactly the map π vg discussedin Remark 2.5 which is bijective only upon restriction to 2-graphs with connected vertices.To extend to coloured 2-graphs with disconnected vertices the bijection β vg is needed.For this the coloured graphs lack the additional information which subsets of connectedcomponents form 2-graph vertices after 0-edge deletion.For some purposes it is useful to define the pairings σ and σ explicitly in terms ofsubsets of 2 S like the edges in Def. 2.2: Definition 2.8 (2-graph with edge set). A is a tuple ( V , H , ν, E ; S , µ, S v , S e )with the above properties (1) – (4b) – (6) and(7b) a complete partition of S into disjoint subsets of two elements adjacent to the samevertex, that is S v ⊂ S such that • a ∩ b = ∅ for all a, b ∈ S v , and | a | = 2 for all a ∈ S v , • for every vertex v ∈ V and every s ∈ ( ν ◦ µ ) − ( v ) there is an s ∈ ( ν ◦ µ ) − ( v )such that { s , s } ∈ S v .(8b) a set of disjoint, two-element subsets of S compatible with the edges E , that is S e ⊂ S such that • a ∩ b = ∅ for all a, b ∈ S e and | a | = 2 for all a ∈ S e ,– 7 – for every e = { h , h } ∈ E and every s ∈ µ − ( h ) there is an s ∈ µ − ( h ) suchthat { s , s } ∈ S e . • An h ∈ H is not contained in any edge e ∈ E iff all s ∈ µ − ( h ) are not containedin any strand edge a ∈ S e . Proposition 2.9.
The two 2-graph definitions, Def. 2.3 and Def. 2.8, are equivalent.Proof.
The fixed-point free involution σ partitions S into disjoint 2-element sets S v ⊂ S .Property (7a) says that such pairs are adjacent to the same vertex v ∈ V . Thus, for every s ∈ S adjacent to v ∈ V there is also the unique s = σ ( s ) = s adjacent to v whichgives (7b), and (7a) follows from (7b) in the same way.The same argument applies to the adjacency of pairs { s, σ ( s ) } ∈ S e to an edge { h, ι ( h ) } ∈ E . The equivalence of fixed points of ι and σ is equivalent to the equivalenceof half-edges h ∈ H and their adjacent strands s ∈ µ − ( h ) not occurring in the power sets E and S e respectively. Together this shows that (8a) and (8b) are equivalent. Definition 2.10 (external edges and strand sections).
Half-edges which are not partof any edge are called external , H ext := H \ S e ∈E e . Equivalently, their associated strandsections are called external strands , S ext := S \ S s ∈S e . For some purpose it is meaningfulto define external edges as one-element sets E ext := {{ h }| h ∈ H ext } compatible with theactual (internal) edges. Definition 2.11 (faces).
Let f = ( s , s , ..., s n ) be an 2 n -tuple of distinct strand sections s i ∈ S , i = 1 , , ..., n , which are each mapped to the following one by σ and σ alternating: s σ s σ s σ ... σ s n . (2.5)Then f is an external face iff s and s n are fixed points of σ , i.e. they are external.If σ ( s n ) = s there is an equivalence relation of cyclic permutations s i s i +2 . Theequivalence class [( s , s , ..., s n )] is called an internal face .In the following, F ext denotes the set of external faces, F int of internal faces and F = F ext ∪ F int . As faces are complete chains of strand sections, they are sometimessimply called strands. This is why, to avoid confusion, I refer to the elements s ∈ S asstrand sections .The reason to call the chains of strand sections faces , and the diagrams themselves is that they are indeed 2-dimensional complexes (in the sense of Reidemeister[36]; the reason to use the old combinatorial definition is simply that 2-graphs are alsopurely combinatorial objects [37]). Proposition 2.12.
Recall that according to [36] a complex ( C , dim, ≤ ) is a set C of cells c with adimension map dim : C → N and a partial ordering ≤ that obeys the property (CP): If c > c ′′ and dim ( c ) − dim ( c ′′ ) >
1, then there is a cell c ′ such that c > c ′ > c ′′ .– 8 –or a 2-graph G , define the complex C G := V G ∪ ( E G ∪ E ext G ) ∪ F G (2.6)and assign the dimension 0 , , ν and µ induce a partial ordering: For v ∈ V , e ∈ E and f ∈ F : v < e iff h ∈ e suchthat ν ( h ) = v ; e < f iff there is an h ∈ e and s ∈ f (independent of a chosen representativeof f ) such that µ ( s ) = h ; and v < f iff there is an e ∈ E such that v < e < f . Then (CP)holds by definition.A complex is pure if each cell of non-zero dimension bounds a 0-cell [36]. By definitionof the bounding relation via adjacency maps ν and µ this holds.A complex is n -dimensional iff n is the maximal dimension of cells and for each cellthere is a bounding n -cell [36]. For C G the maximal dimension is n = 2 by definition.However, as ν and µ are not necessarily surjective, there might be vertices or half-edgesnot bounded by a face. Thus C G is 2-dimensional only if both are surjective, that is allvertices and half-edges are not 0-valent. Remark 2.13 ( D -dimensional complexes from edge-coloured graphs). With someadditional structure, 2-graphs can also be bijective to complexes of higher dimension, andeven to discrete (pseudo) manifolds. In fact, bipartite ( r + 1)-coloured graphs (Ex. 2.7) aredual to r -dimensional abstract simplicial pseudo manifolds. This works since the connectedcomponents upon deleting edges of p colours define ( p − The central operation for the coproduct on 2-graphs, and thus for the Hopf algebra, iscontraction of a subgraph. In effect, the components of the subgraph are substituted bysingle vertices with the subgraph’s external structure, usually called the residue . For a 2-graph the residue is (bijective to) a boundary 1-graph. Since such a boundary 1-graph canbe disconnected even for a connected 2-graph it is crucial to keep track of which boundary(1-graph) connected component belongs to which bulk (2-graph) connected component.
Definition 3.1 (subgraph).
For a 2-graph G , a subgraph H is a 2-graph which is onlydifferent from G in having E H ⊂ E G and S eH ⊂ S eG . Then one writes H ⊂ G .Note that E H and S eH must still be compatible due to the 2-graph property (8b). Theyform indeed stranded edges. Definition 3.2 (contraction).
For 2-graphs H ⊂ G the contraction of H in G is definedby shrinking all stranded edges which belong to H . That is, the contracted graph G/H consists of • V G/H = K H the set of connected components of H , that is each connected componentin H is shrunken to a single vertex, • H G/H = H ext H , S G/H = S ext H , only half-edges and strand sections which are external in H remain in G/H , – 9 – ν G/H = π H ◦ ν H (cid:12)(cid:12) H G/H where π H : V H → K H is the projection of vertices on theirconnected component, • µ G/H = µ H (cid:12)(cid:12) S G/H , simply the restriction to the remaining strand sections, • E G/H = E G \ E H , S eG/H = S eG \ S eH , stranded edges of H are deleted and • S vG/H = (cid:8) { s , s n }| ( s ...s n ) ∈ F ext (cid:9) , external faces are shrunken to the strands at thenew contracted vertices.Note that internal faces are deleted completely since S G/H = S ext H .Take as an example the coloured 2-graph of Fig. 3, with explicit labelling of half-edges G =
12 34 56 78 c c . (3.1)It has two edges, and thus 2 = 4 different subgraphs H = , H = , H = , H = G = (3.2)One has to distinguish the cases c = c = c and c = c . In the first case, one has G/H =
12 34 56 78 c c , G/H = c , G/H =
12 3 6 78 c , G/G =
12 78 c (3.3)while for c = c the contractions are G/H =
12 34 56 78 c c , G/H =
12 5 874 c c , G/H =
12 6 873 c c , G/G =
12 78 . (3.4)The last contraction G/G is an example of a multi-trace vertex, i.e. a vertex with discon-nected vertex graph.In field theory, the structure of graphs is relevant up to relabelling their components.In this sense, H and H in Eq. (3.2) are equivalent and so are G/H and G/H . Definition 3.3 (iso/automorphism). An isomorphism j between 2-graphs G and G is a triple of bijections j = ( j V , j H , j S ) where j V : V G → V G , j H : H G → H G and j S : S G → S G such that • ν G = j V ◦ ν G ◦ j − H and µ G = j H ◦ µ G ◦ j − S , • ι G = j H ◦ ι G ◦ j − H , • σ G = j S ◦ σ G ◦ j − S and σ G = j S ◦ σ G ◦ j − S .A 2-graph automorphism is an isomorphism from a 2-graph G to itself.– 10 – efinition 3.4 (unlabelled 2-graphs). Two 2-graphs G and G are equivalent uponrelabelling, G ∼ = G , iff there is a 2-graph isomorphism G G . Such an equivalenceclass is an unlabelled 2-graph , Γ = [ G ] ∼ = = [ G ] ∼ = . Let G be the set of all unlabelled2-graphs and G the set of unlabelled 1-graphs.Thereby I use the convention to denote unlabelled objects by Greek letters whilelabelled ones by Roman letters. Capital letters refer to 2-graphs and small letters to1-graphs. Definition 3.5 (residue).
In analogy to usual Feynman graphs, one refers to the classof 2-graphs without any stranded edge as residues R ∗ ⊂ G , and denotes the set of thosewith a single vertex as R ⊂ R ∗ . Shrinking all stranded edges of a 2-graph results in aresidue. This is commonly defined as the residue map res : G → R ∗ , Γ Γ / Γ . (3.5)Furthermore, for every 2-graph Γ ∈ G there is a trivial subgraph Θ ⊂ Γ without strandededges, called the skeleton (not to be confused with the p -skeleton of a cell complex):skl : G → R ∗ , Γ Θ . (3.6)This can be used to define a subset of 2-graphs with a given type of vertices S ∗ ⊂ R ∗ as G ( S ∗ ) := { Γ ∈ G | skl(Γ) ∈ S ∗ ⊂ R ∗ } . (3.7) Remark 3.6 (boundaries).
The residue is very similar to the usual notion of a boundaryfor 2-dimensional complexes, with subtle but crucial distinctions. Indeed, the union ofvertex graphs Eq. (2.1) of the residue of a 2-graph Γ ∈ G is its boundary. Thus it ismeaningful to define the boundary map ∂ : G → G , Γ ∂ Γ := π vg (res(Γ)) (3.8)where by slight abuse of notation the image of π vg is simply taken as the set of 1-graphs G since there are no stranded edges in the image of res. It is straightforward to check thatthis definition of boundary is equivalent to the notion of boundary for a complex accordingto Prop. 2.12, in particular in the example of D -coloured graphs (Ex. 2.7) to the notion ofboundary of D -dimensional simplicial pseudo manifolds (Rem. 2.13).However, for renormalization in field theory it is necessary to distinguish the boundariesof the different connected components of the 2-graph Γ = F i ∈ I Γ i . To appropriately takethis into account another boundary map is needed, e ∂ : G → P ( G ) , Γ = G i ∈ I Γ i e ∂ Γ := { ∂ Γ i } i ∈ I , (3.9)where here P ( G ) denotes the power set of all multisets of G since any γ ∈ G may appearmultiple times in e ∂ Γ. In fact, this boundary map contains exactly the same information asthe residue. That is, the two are bijective by changing to the vertex-graph representation,– 11 – ∂ Γ = β vg (res(Γ i )) since each connected component Γ i maps to one vertex under the residuemap. Thus, the difference between e ∂ and ∂ mirrors the one between β vg and π vg discussedin Rem. 2.5.Similarly one can distinguish two types of skeletons in the vertex-graph picture, ς : G → G , Γ ς Γ := π vg (skl(Γ)) = G v ∈V Γ γ v and (3.10) e ς : G → P ( G ) , Γ e ς Γ := β vg (skl(Γ)) = { γ v } v ∈V Γ . (3.11)This allows for a definition of 2-graphs with specific vertex types in terms of their vertexgraphs V ⊂ G as G ( V ) := { Γ ∈ G | e ς Γ ∈ P ( V ) } (3.12)which is equivalent to the definition Eq. (3.7) in the sense that for S ∗ ⊂ R ∗ it holds that G ( V ) = G ( S ∗ ) iff P ( V ) = ∂ S ∗ .The important point here is that it makes a difference to compare the external graphstructure on the level of vertex graphs or the single graph given by their disjoint unionTake for example the unlabelled 2-graph Γ = of Fig. 3 or Eq. (3.1). If thedistinguished colours c = c are not the same, the residue res(Γ) = is a multi-tracevertex. Thus for several such components, e.g. Γ ⊔ Γ, there is a difference between ∂ (Γ ⊔ Γ) = ⊔ ⊔ ⊔ and e ∂ (Γ ⊔ Γ) = (cid:26) ⊔ , ⊔ (cid:27) (3.13)as the boundary ∂ misses the information which boundary component belongs to which2-graph component. Remark 3.7 (topology change).
When one can associate manifolds with 2-graphs, con-traction may lead to change of topology. The boundary of a connected component Θ i of asubgraph Θ ⊂ Γ might be disconnected, that is a 1-graph with several connected compo-nents ∂ Θ i = F j γ j , as exemplified in Eq. (3.13) Contraction of such Θ i leads to multi-tracevertex in Γ / Θ. From the field-theory perspective alone, it is not obvious how to understandthe topology of such 2-graphs in cases where they relate to pseudo manifolds (e.g. surfaces(Ex. 2.6) or higher dimensional (Ex. 2.7)):(1.) One could focus simply on the strands (propagation of degrees of freedom) and ignorethe fact that the disconnected parts are adjacent to the same vertex; this would leadto topology change since the contracted component would be separated.(2.) Topology change upon contraction could be avoided by simply assigning to the multi-trace vertex with vertex graph ∂ Θ i = F j γ j the topology of Θ i . But this might notbe unique as there could be various 2-graphs Θ i related to different topologies.(3.) In matrix models such multi-trace vertices are interpreted as singular points. Thissuggests to understand multi-trace vertices as points at which the complex is onlyone-dimensionally connected. In particular, one finds that there is a region in the phase diagram of multi-trace matrix models [38, 39]with the critical behaviour of the continuum random tree [40]. This regime is dominated by “cactus”geometries, that is surfaces connected along single points thus leading to a tree-like fractal structure. – 12 –n fact, in some examples of surfaces, the last possibility is in agreement with the Eulercharacteristic. Consider for example the following contraction of combinatorial mapsΓ / Θ ∼ = , = . (3.14)While Γ has torus topology, Θ is a cylinder. The contraction of Θ in Γ leads to a double-trace vertex whose strands are disconnected. Considering only the strands but not thevertex (1.), Γ / Θ is a sphere different from the original torus (2.). On the other hand, takingthe vertex as a singular point (3.) gives a pinched torus and in fact a naive calculation ofEuler characteristic gives χ (Γ / Θ) = V − E + F = 3 − D -dimensional pseudomanifolds of D -coloured graphs (Ex. 2.7). Algebraic structure can be defined on 2-graphs in the same way as for 1-graphs [8]. Withthe proper definition of a 2-graph (Def. 2.3) and of contraction of subgraphs (Def. 3.2) aproduct, a coproduct as well as an antipode can be defined in a standard way promotingthe set of 2-graphs G to an algebra, coalgebra and Hopf algebra respectively. The crucialdifference to 1-graphs lies in the combinatorial details as I will exemplify with the centralidentity for the action of the coproduct on power series over 2-graphs in Theorem 4.6. Definition 4.1 (2-graph algebra).
Let G := h G i be the Q -algebra generated by all 2-graphs Γ ∈ G with multiplication in terms of the disjoint union defined on the generatorsas m : G ⊗ G → G , Γ ⊗ Γ Γ ⊔ Γ . (4.1)This is clearly associative and commutative and the empty 2-graph is the neutral element.With a unit, that is a linear map u : Q → G , q q this is a unital commutative algebra.The set G ( V ) of 2-graphs with vertex graphs of restricted types V ⊂ G , Eq. (3.12),generates a subalgebra G V := h G ( V ) i since by definition m ( G V ⊗ G V ) ⊂ G V .As common, there is a coproduct ∆ on the algebra of 2-graphs defined on their genera-tors as a sum over all subgraphs tensored with their contraction. For example, the coloured2-graph of Eq. (3.1) with distinguished colours c = c has∆ (cid:18) (cid:19) = ⊗ + ⊗ + ⊗ (4.2)In this way one obtains a coalgebra and furthermore a bialgebra:– 13 – roposition 4.2 (2-graph coalgebra) . Equipping G with the coproduct , a linear mapdefined on its generators as ∆ : G → G ⊗ G , Γ X Θ ⊂ Γ Θ ⊗ Γ / Θ , (4.3) and the counit, a projector defined as ǫ : G → Q , Γ ( if Γ ∈ R ∗ else , (4.4) G is an associative counital coalgebra. Since m and ∆ are compatible in the sense that ∆ isan algebra homomorphism and m is a coalgebra homomorphism, G is furthermore a (unitaland counital ) bialgebra.Proof. For an associative coalgebra one has to show that(∆ ⊗ id) ◦ ∆ = (id ⊗ ∆) ◦ ∆ and ( ǫ ⊗ id) ◦ ∆ = (id ⊗ ǫ ) ◦ ∆ = id . (4.5)The proof is the same as for the 1-graph coalgebra (e.g. [8] Prop. 5.2.1) and for thebialgebra property [8], Prop. 5.2.2.In field theory, one is often interested in the subalgebra G V = h G ( V ) i of 2-graphs withvertex graphs of specific type V ∈ G . But this is not a subcoalgebra since contractionsof 2-graphs in G V might lead to 2-graphs with other vertices, as in the example Eq. (3.3)and Eq. (3.4). Thus, in general G V ∈ G is only a right co-ideal, i.e. ∆( G V ) ⊂ G V ⊗ G .To upgrade it to a subcoalgebra it is thus necessary to extend the set of 2-graphs by allpossible contractions: Definition 4.3 (Contraction closure and subcoalgebras).
Given a set P ⊂ G , a set K ⊂ G is called P - contraction closed iff for all Θ ⊂ Γ ∈ K with Θ ∈ P also Γ / Θ ∈ K .The P - contraction closure of K ⊂ G is the extension of K by all such contractions, P K := (cid:8) Γ = Γ ′ / Θ (cid:12)(cid:12) Θ ⊂ Γ ′ ∈ K , Θ ∈ P (cid:9) . (4.6)For P = G one calls the G -contraction closure of K simply the contraction closure andwrites K . Proposition 4.4 (2-graph subbialgebra) . For a set of 2-graphs G ( V ) of given vertextypes V ⊂ G , the algebra h G ( V ) i is a subbialgebra of G .Proof. Let Γ ∈ G ( V ) be a 2-graph in the contraction closure of 2-graphs of vertextypes V ⊂ G . By definition this means that there are 2-graphs ˜Θ ⊂ ˜Γ ∈ G ( V ) such thatΓ = ˜Γ / ˜Θ. In particular, there can be connected components ˜Γ i and ˜Θ j = ˜Γ i such that˜Γ i / ˜Γ i = res(˜Γ i ) is a connected component of Γ. Thus the vertices of Γ can be of any typeof residues (respectively boundaries) of G ( V ), that is skl(Γ) ∈ res( G ( V )). Since thereare no restrictions on edges, this completely characterizes 2-graphs in the closure G ( V ).A subgraph Θ ⊂ Γ has by definition skl(Θ) = skl(Γ). Thus, it is also in G ( V ). Thecontraction Γ / Θ is in the closure by definition and thus ∆Γ ∈ G ( V ) ⊗ G ( V ), which provesthat G ( V ) is a subcoalgebra. As G ( V ) is also a subalgebra, it is a subbialgebra.– 14 – xample 4.5 (Bialgebra of maps and coloured 2-graphs). Both the example ofcombinatorial maps (Ex. 2.6) and of r -coloured graphs (Ex. 2.7) are classes of 2-graphswhich are characterized by a fixed half-edge degree, d h = 2 respectively d h = r , as wellas a specific edge stranding σ induced by orientation respectively colour structure. Thefirst property, d h = r , defines a set of vertex graphs V r ∈ G . In this case, since theresidue (boundary) of any 2-graph with such vertex type is again of this type, i.e. has fixed d h = r , the set G ( V r ) is already contraction closed. The second property of specific edgestranding is also preserved under contractions. Thus the set of combinatorial maps andthe set of r -coloured 2-graphs each generate a subbialgebra in the bialgebra of all 2-graphs G . In field theory such sets of diagrams are called “theory space” exactly because of thisclosure property.The quantity of interest in field theory is the perturbative expansion of Green’s func-tions which is a formal power series labelled by Feynman diagrams. The correspondingcombinatorial object therefore is the formal series X := X Γ ∈ G Γ | Aut Γ | , or X γ = X Γ ∈ G γ Γ | Aut Γ | , (4.7)that is its restriction to a sum over G γ , connected ∂ Γ = γ ∈ G . The crucial operatorion for the renormalization of Green’s func-tions is the coproduct on the series X γ which is induced by the general case ∆ X [5, 6, 9, 41]: Theorem 4.6 (central identity) . The coproduct of the weighted series over all 2-graphs is ∆ X = X Γ ∈ G (cid:16) Y v ∈ V Γ | Aut γ v | X γ v (cid:17) ⊗ Γ | Aut Γ | . (4.8)The same formula is also true for more restricted formal power series where the sumruns over a subset K ⊂ G such as X γ , the sum over 2-graphs with specific boundary γ ,or the restriction to bridgeless (one-particle irreducible) 2-graphs. The necessary propertyfor such a subset K is to be contraction closed ([8] Corollary 5.3.1).To prove the theorem one needs the inverse operation to contraction which is insertion.However, one has to be careful with the definition since for 2-graphs not any kind of inser-tion is dual to a contraction. To insert a labelled 2-graph G into another G ′ appropriately,one replaces all vertices of G ′ with connected components of G . For this it is necessary thatthe boundaries of the components of G agree with (are isomorphic to) the vertex graphsof G ′ , that is e ∂G ∼ = e ς G ′ , and to specify how edges and strands of G ′ connect to e ∂G : Definition 4.7 (insertion).
Let G and G ′ be 2-graphs and i = ( i H , i S ) : ∂G → ςG ′ a1-graph isomorphism which is component sensitive , that isfor every g ∈ e ∂G there is a g ′ ∈ e ς G ′ such that i ( g ) = g ′ . Then the insertion of G into G ′ along i is G ′ ◦ i G := (cid:0) V G , H G , ν G , E G ∪ i − H ( E G ′ ); µ G , S G , S vG , S eG ∪ i − S ( S eG ′ ) (cid:1) . (4.9)– 15 –ne denotes the set of possible insertions of G into G ′ , i.e. the component-sensitive 1-graphisomorphisms i , as I ( G, G ′ ). Remark 4.8.
The number of possible insertions is the number of component-sensitive 1-graph automorphisms. As each such isomorphism is already fixed by the specific elementin the orbit which it is mapping to, the only choice left is an additional automorphism.Thus one has |I ( G, G ′ ) | = | Aut e ∂G | = | Aut e ς G ′ | = Y g ∈ G V gG ′ ! Y v ∈V G ′ | Aut g v | (4.10)where V gG ′ is the number of vertices in G ′ with vertex graph g .Dropping the restriction to component sensitivity would allow to insert a 2-graph G ′ into G with ∂G ∼ = ςG ′ but e ∂G ≇ e ς G ′ . For example, a 2-graph with two boundarycomponents such as from Eq. (3.1) with distinguished colours c = c couldthen be inserted not only into a vertex of its residue type, the multi-trace vertexbut also in two copies of the vertex . While this is definitely an interesting operationfrom a topological point of view (basically the operation of adding a handle), it cannot beinverted in terms of a contraction. This is because it involves two vertices but contractionyields by definition only one vertex per connected component.With this crucial concept of component sensitivity the proof of the central identity isbasically the same as for usual graphs (see e.g. [8]): Proof.
One can expand the coproduct of a 2-graph Γ which is a sum over contractions ofsubgraphs equivalently as a sum over pairs of 2-graphs (Θ , ˜Γ) whose insertion of one intothe other yields Γ,∆Γ ≡ X Θ ′ ⊂ Γ Θ ′ ⊗ Γ / Θ ′ = X Θ , ˜Γ ∈ G (cid:12)(cid:12)(cid:12) { Θ ′ ⊂ Γ | Θ ′ ∼ = Θ and Γ / Θ ′ ∼ = ˜Γ } (cid:12)(cid:12)(cid:12) Θ ⊗ ˜Γ . (4.11)How many such pairs (Θ , ˜Γ) are there? On the level of their representatives, that isfor labelled 2-graphs H, G, ˜ G , the set of triples ( H ′ , j , j ) of subgraphs H ′ ⊂ G withisomorphisms j : H → H ′ and j : G/H ′ → ˜ G is isomorphic to the set of pairs ( i, j ) of aninsertion place i ∈ I ( H, ˜ G ) and an isomorphism j : ˜ G ◦ i H → G . Therefore the cardinalityof the two sets agrees, |{ H ′ ⊂ G | H ′ ∼ = H and G/H ′ ∼ = ˜ G }|·| Aut H |·| Aut ˜ G | = |{ i ∈ I ( H, ˜ G ) | ˜ G ◦ i H ∼ = G }|·| Aut G | . (4.12) The argument works along the lines of Lemma 5.3.1 of [8] with a straightforward generalization from1-graphs to 2-graphs: From ( H ′ , j , j ) one can construct an insertion place i ′ ∈ I ( H ′ , G/H ′ ) which isisomorphic to i ∈ I ( H, ˜ G ) via j and j . Furthermore this gives an isomorphism j : ˜ G ◦ i H → G since G/H ′ ◦ i ′ H ′ = G . The other way, from ( i, j ) one has H ⊂ ˜ G ◦ i H such that there must be a unique H ′ ⊂ j ( ˜ G ◦ i H ) = G with an induced isomorphism j such that j ( H ) = H ′ . Contraction on both sidesinduces further the isomorphism j . – 16 –sing this identity again on the level of unlabelled 2-graphs one finds∆ X = X Γ ∈ G | Aut Γ | X Θ , ˜Γ ∈ G |{ i ∈ I (Θ , ˜Γ) | ˜Γ ◦ i Θ ∼ = Γ }| | Aut Γ || Aut Θ | |
Aut ˜Γ | Θ ⊗ ˜Γ (4.13)One can change the sums and use for given Θ , ˜Γ ∈ G the simple fact X Γ ∈ G |{ i ∈ I (Θ , ˜Γ) | ˜Γ ◦ i Θ ∼ = Γ }| = |I (Θ , ˜Γ) | (4.14)to find∆ X = X Θ , ˜Γ ∈ G |I (Θ , ˜Γ) | Θ | Aut Θ | ⊗ ˜Γ | Aut ˜Γ | = X Θ , Γ ∈ G |I (Θ , Γ) | Θ | Aut Θ | ⊗ Γ | Aut Γ | (4.15)(where the second step is simply a relabelling).The symmetry-weighted sum over all 2-graphs Θ with a suitable structure of connectedcomponents to be inserted into a 2-graph Γ is Q v ∈ V Γ X γ v / Q γ ∈ G V γ Γ ! in which the denom-inator takes care of implicit automorphisms between isomorphic 2-graphs with the sameboundary. Inserting the number of insertions Eq. (4.10) one has finally∆ X = X Γ ∈ G Y γ ∈ G V γ Γ ! Y v ∈V Γ | Aut γ v | Q v ∈ V Γ X γ v Q γ ∈ G V γ Γ ! ⊗ Γ | Aut Γ | (4.16)which concludes the proof. In this section I show that for every renormalizable combinatorially non-local field theorythere is a Connes-Kreimer type Hopf algebra of divergent Feynman diagrams [2–4]. Besidesthe specific external structure of such diagrams being 2-graphs with a 1-graph boundary,one can use the same logic to construct these algebras as for local field theory [8]: There isa general Hopf algebra of all 2-graphs which can be restricted to the specific set of diagramsof a given theory and further to the subset of divergent diagrams. In the end I will illustratehow this works in the example of matrix field theory and for field theories with tensorialinteractions.By the same arguments as for 1-graphs, the bialgebra of 2-graphs G has a uniquecoinverse element and is thus a Hopf algebra. From the renormalization perspective, theimportant insight in this is that there is group structure on the set of algebra automorphismson G which extends also to algebra homomorphisms φ : G → A mapping to any otherunital commutative algebra A . The multiplication of such homomorphisms is given by the convolution product φ ∗ ψ := m A ◦ ( φ ⊗ ψ ) ◦ ∆ (5.1)which due to (co)associativity of m A and ∆ is also associative and has the neutral element u A ◦ ǫ G . In particular, A can be the algebra of amplitudes of a field theory and φ the eval-uation of the amplitude labelled by a given Feynman diagram Γ ∈ G . For renormalization,the object of interest is then the group inverse of this evaluation map of diagrams.– 17 – heorem 5.1 (Hopf algebra of 2-graphs/group of algebra homomorphisms) . The bialgebraof 2-graphs G is a Hopf algebra, i.e. there exists an antipode S , that is a unique inverseto the identity id : Γ Γ with respect to the convolution product, S ∗ id = id ∗ S = u ◦ ǫ . (5.2) Furthermore, the set Φ GA of algebra homomorpisms from G to a unital commutative alge-bra A is a group with inverse S φ = φ ◦ S for every φ ∈ Φ GA , S φ ∗ φ = φ ∗ S φ = u A ◦ ǫ G . (5.3) The subbialgebra h G ( V ) i generated by 2-graphs with specific vertex graphs V ⊂ G (Prop.4.4) is a Hopf subalgebra of G .Proof. The 1-graph structure contained in the 2-graphs is already sufficient and there aretwo standard ways to prove the unique existence of the antipode [42] which both use thefact that the bialgebra G is a graded bialgebra G = L E ≥ G E with respect to the numberof edges (which follows directly from Def. 3.2). The common first option is to reduce to theaugmentation ideal Ker ǫ = { } ⊕ L E ≥ G E which is a connected ( G is one-dimensional)graded bialgebra, and thus automatically a Hopf algebra (Sec. III.3 in [42]). Here we usethe second construction following [8] where one keeps all the residues r ∈ R ∗ which behaveas group-like elements w.r.t. the coproduct [42] , ∆ r = r ⊗ r , and augment G by formalinverses r − for all = r ∈ R ∗ defining r − r = r r − = . Then the antipode on r ∈ R ∗ is S R ∗ ( r ) := r − from which one can construct the antipode on G as S := X k ≥ ( u ◦ ǫ − S R ∗ ) ∗ k ∗ S R ∗ (5.4)as proven for example in [8].For the inverse of a homomorphism φ : G → A one directly calculates S φ ∗ φ = m A ◦ (( φ ◦ S ) ⊗ φ ) ◦ ∆ = φ ◦ m A ◦ ( S ⊗ id) ◦ ∆ = φ ◦ u G ◦ ǫ G = u A ◦ ǫ G . (5.5)Finally, by definition of the antipode m ◦ (id ⊗ S ) ◦ ∆( h G ( V ) i ) = u ◦ ǫ ( h G ( V ) i ). It followsthat h G ( V ) i S ( G ) = S ( h G ( V ) i ) G and thus S ( h G ( V ) i ) ⊂ h G ( V ) i .Connes-Kreimer-type Hopf algebras for specific renormalizable field theories followfrom the general Hopf algebra of 2-graphs by a further restriction. With Thm. 5.1 onehas already a Hopf subalgebra for a specific class of Feynman diagrams as for example(Ex. 4.5) combinatorial maps in matrix theories or r -coloured 2-graphs in tensor-invarianttheories. One can always further restrict to one-particle irreducible (bridgeless) 2-graphsas a quotient Hopf algebra dividing out the Hopf ideal generated by one-particle reducible2-graphs ([8] Ex. 5.5.1). For renormalization one restricts then further to the subset ofsuperficially divergent diagrams which will turn out to be contraction-closed by definition,and thus will also be a Hopf algebra.Concerning the structure of renormalization one can consider quantum field theoryfrom a purely combinatorial perspective [9, 10, 43]. In this spirit, a local quantum field– 18 –heory is specified by its field content, the interactions and their weights as well as thespacetime dimension D [8, 9]. For combinatorially non-local field theory let us restricthere to a single field type (of which there could still be various, for example with differentnumber d h of arguments [31]). Then a combinatorially non-local scalar field theory issimply given by its interactions, their weights and a dimension : Definition 5.2 (comb. field theory).
A combinatorial cNLFT T = ( G e , G v , ω, d ) con-sists of a dimension d ∈ N , boundary graph sets G e ⊂ G v ⊂ G and a weight map ω : G e ∪ G v → Z (5.6)where G v , or equivalently R v ⊂ R ⊂ G under the bijection β vg , is the set of interactionsand G e (or R e ) is the set of propagators which are graphs with two vertices (or respectively2-valent 2-graph vertices) . Its Feynman diagrams are all 2-graphs with G v -type vertices, G T := G ( G v ) = { Γ ∈ G | e ς Γ ∈ P ( G v ) } ⊂ G , (5.7)and they generate a Hopf algebra G T := h G T i by closure with respect to contractionsaccording to Thm. 5.1. Definition 5.3 (renormalizability and the Hopf algebra of Feynman 2-graphs).
Let T = ( G e , G v , ω, d ) be a combinatorial cNLFT. For a Γ ∈ G T , the superficial degree ofdivergence is ω sd (Γ) = X v ∈V Γ ω ( γ v ) − X e ∈E Γ ω ( γ e ) + d · F (5.8)where F = |F int | is the number of internal faces (Def. 2.11), γ v is the vertex graph of thevertex v , Eq. (2.1), and γ e the “vertex graph” of the edge e , defined as the 1-graph withtwo vertices and as many edges between them as there are strands at the edge e in the2-graph Γ. The set of superficially divergent diagrams in T are bridgeless 2-graphs whosenontrivial components are superficially divergent, P s.d. T := n Γ = G i ∈ I Γ i ∈ G T bridgeless (cid:12)(cid:12)(cid:12) ∀ i ∈ I : if Γ i R then ω sd (Γ i ) ≥ o . (5.9)The cNLFT T is renormalizable iff ω sd depends only on the boundary of 2-graphs and forevery connected Γ ∈ G T : ω sd (Γ) = ω ( ∂ Γ) . (5.10)Then H f2g T = h P s.d. T i is the Connes-Kreimer Hopf algebra of divergent Feynman 2-graphsof the theory T . It is a Hopf subalgebra of G T and G according to Prop. 4.4 and Thm. 5.1as it is contraction closed due to renormalizablity. This dimension d is not the dimension of spacetime but simply the dimension of a single argument ofthe field. It can have various meaning depending on the specific theory. For example, a matrix field theoryderived from D -dimensional non-commutative field theory has d = D/ d is the dimension of the group G which the field arguments are valued in and G is related to the Lorentzgroup if the theory has an interpretation as a quantum theory of gravity [23–26]. In field theory it is usually not necessary to have the two-valent interactions included ( R e ⊂ R v ); With-out restriction to the underlying theory, these are added here for the simple formula of contraction accordingto Def. 3.2 which necessarily leads to such a vertex when contracting a propagator-type subgraph [8]. – 19 –his is the combinatorial structure of renormalization. To perform actual renormaliza-tion of a cNLFT, a renormalization scheme has to be specified by choosing an appropriatelinear operator R on the algebra of amplitudes A on which diagrams are mapped by thehomomorphism φ : H f2g T → A . Based on the existence of the antipode S this allows torecursively define the counterterm map S φR : H f2g → A , S φR ( ) = , S φR (Γ) = − Rφ (Γ) − R (cid:20) X Θ ⊂ Γ S φR (Θ) φ (Γ / Θ) (cid:21) (5.11)which is in the group Φ H f2g A if R is a Rota-Baxter operator [42]. Then the renormalizedFeynman amplitudes are evaluated by S φR ∗ φ . We will exemplify how this works for acNLFT elsewhere and close this work with two examples of renormalizable cNLFT fromthe purely combinatorial perspective. Example 5.4 (Hopf algebra of matrix field theory/Grosse-Wulkenhaar model).
The Feynman diagrams of matrix field theories [17–19] are combinatorial maps which are2-graphs with bivalent edges, d h = 2 (Ex. 2.6). Thus, the interaction vertex graphs can bepolygons with n vertices and disjoint unions thereof (multi-trace vertices).So called φ nD matrix field theory is specified by G e = { } , a set of interactions G v (without multi-traces) of maximal order n and a dimension d = D/ D of a corresponding non-commutative field theoryfor even D [17]. Let the weights be ω | G e = 1 and ω | G v \ G e = 0.According to Eq. (5.8) the superficial degree of divergence is ω sd (Γ) = d · F Γ − E Γ where E Γ = |E Γ | is the number of (internal) edges and F Γ = |F intΓ | is the number of internal faces.The Euler formula for such a 2-graph Γ (dual to a triangulated surface) is2 − g Γ − K ∂ Γ = V Γ − ( E Γ + V ∂ Γ ) + F Γ + E ∂ Γ = V Γ − E Γ + F Γ (5.12)with genus g Γ and K ∂ Γ the number of connected components of the boundary. The numberof boundary vertices/external edges V ∂ Γ = |V ∂ Γ | = |E extΓ | and boundary edges/externalfaces E ∂ Γ = |E ∂ Γ | = |F extΓ | agree since the boundary graphs are polygons. Together withthe relation 2 E Γ + V ∂ Γ = X γ ∈ G v V γ · V γ Γ = n X k =1 kV ( k )Γ (5.13)between edges and the number V γ Γ of interactions in Γ with V γ boundary vertices, or V ( k )Γ interactions of order V γ = k , one obtains ω sd (Γ) ≡ dF Γ − E Γ = − d ( V Γ −
1) + d − n X k =1 kV ( k )Γ − V ∂ Γ ! − d (2 g Γ + K ∂ Γ − . (5.14)For example the Grosse-Wulkenhaar model, a φ D matrix field theory (only V (4)Γ = 0)related to D = 2 d dimensional non-commutative field theory, has [44]2 ω sdΓ = D − D − V ∂ Γ + ( D − V Γ − D (2 g Γ + K ∂ Γ − . (5.15)– 20 –or D = 4 the divergence degree is independent of V Γ and only planar maps Γ, that isgenus g Γ = 0, with single boundary component K ∂ Γ = 1 and a maximal number of V ∂ Γ ≤ φ matrix theory is renormalizable. In particular, theonly Green’s functions which need renormalization are the effective propagator (2-pointfunction) and the regular ( K ∂ Γ = 1) 4-point function. This defines the set of superficiallydivergent diagrams P s.d. gw which generates the Connes Kreimer Hopf algebra of divergent2-graphs for the D = 4 Grosse-Wulkenhaar model.A first attempt to construct the Connes-Kreimer Hopf algebra of quartic non-com-mutative field theory in D = 4 dimensions has been obtained already in [11] and furtherdetailed in [12]. However, this earlier work is lacking the understanding that the externalstructure of 2-graphs are 1-graphs (here d h = 2 regular graphs, i.e. polygons) and thatit is in particular crucial to respect their number of connected components K ∂ Γ . Thesystematic approach here clarifies this issue and applies to renormalizable non-commutativefield theory and matrix field theory of arbitrary interactions and corresponding to anydimension D . Example 5.5 (Hopf algebra of tensorial field theories).
The Feynman diagrams oftensorial field theories of rank r ≥ r -valent edges, d h = r , whichcan be represented as bipartite ( r + 1)-coloured graphs (up to multi-trace vertices, Ex. 2.7).The interaction vertex graphs can be any r -edge-coloured graphs.A φ nd,r tensorial field theory is specified by G e = { ... } , i.e. the propagator vertexgraph with r edges, a set of bipartite r -coloured graphs G v with maximal number of vertices n and a dimension d . The weights for the propagator are ω | G e = 2 ζ , ζ > ω sd (Γ) = dF Γ − ζE Γ + P v ω ( γ v )into a form which is meaningful for the question of renormalizability is more involved thanfor matrices since the 2-graphs are now dual to simplicial pseudo manifolds of dimension r .Generalizing the genus g , the relevant quantity for power counting turns out to be Gurau’sdegree ω g of a coloured graph [45, 46]. The Gurau degree ω g ∈ N is not a topologicalinvariant (except for r = 2 where ω g = g ) but induced from the genus of Heegaard splittingsurfaces of the pseudo manifold [47] (for r = 3, and a generalized notion of splitting surfacesin r ≥ r + 1)-coloured graph is an r -coloured graph, andthus pseudo manifold of dimension r −
1, also the Gurau degree of the boundary plays arole. Applying the Euler formula Eq. (5.12) repeatedly for the various splitting surfacesand using the edge-vertex relation Eq. (5.13) one finds [22, 48] ω sdΓ = − d r ( V Γ − d r − ζ X k kV ( k )Γ − V ∂ Γ ! − d (cid:18) ω g Γ − ω g ∂ Γ ( r − K ∂ Γ − (cid:19) + X v ∈ G v ω ( γ v )(5.16) In earlier work on non-commutative field theory the focus was on the concept of the number of “brokenfaces” B lacking the understanding that this is actually the number of boundary components K ∂ Γ . The issuebecomes apparent with the definition of insertions in [12] which does not respect the boundary structure(see for example Fig. 9 therein where a diagram with K ∂ Γ = 2 is inserted in a vertex v with K γ v = 1). – 21 –here d r := d ( r −
1) is thus the counterpart to the dimension of local field theory from therenormalization perspective, i.e. φ nd,r tensorial field theory behaves similar to φ nD = d r localfield theory. Note that the above example of matrices, r = 2, is covered by this formula .A vanishing Gurau degree gives the maximal contribution to the divergence degree.One can prove that always ω g Γ ≥ ω g ∂ Γ [20]. Thus, − ( ω g Γ − ω g ∂ Γ ) is maximal for ω g Γ = ω g ∂ Γ = 0.The question which φ nd,r tensorial theories are renormalizable is then as usual a question ofhow the maximal order of interactions n is balanced by the dimension d r , with some morefreedom given by the propagator weight 2 ζ . The dependence on the number of 2-graphvertices vanishes, ω sdΓ = d r − d r − ζ V ∂ Γ − d (cid:18) ω g Γ − ω g ∂ Γ ( r − K ∂ Γ − (cid:19) , (5.17)when setting the vertex weights for each vertex v of degree d v to ω ( γ v ) = d r − d v d r − ζ ) . (5.18)Thus, there can be super-renormalizable theories for d r = 2 ζ and otherwise the maximaldegree for renormalizable interaction vertices is n = ⌊ d r d r − ζ ⌋ [22]. The theory is justrenormalizable if the number in the floor bracket is integer.The first example, and a particularly interesting one, is the BenGeloun-Rivasseaumodel [20], i.e. φ , tensorial field theory with ζ = 1. Thus, it has divergence degree ω sdΓ = 6 − V ∂ Γ −
13 ( ω g Γ − ω g ∂ Γ ) − ( K ∂ Γ − . (5.19)This means that the 4-point Green’s function with two boundary components needs renor-malization. It is thus necessary to include the multi-trace vertex in the set G v ofinteractions and all the subtleties discussed in Sec. 3 apply. Furthermore, for V ∂ Γ = 2 thereare divergent diagrams with ω g Γ = 6. Thus, there is some dependence on the bulk topologyin this case. Still, the set P s.d. bgr of divergent 2-graphs is contraction closed since one canprove that the Gurau degree is contraction/insertion invariant ([13], Lemma 4.1). Thenone obtains the Connes-Kreimer Hopf algebra of divergent 2-graphs of the BenGeloun-Rivasseau model as H f2g bgr = h P s.d. bgr i . In the same way this works also for a number ofrenormalizable tensorial theories with other rank r and dimension d [22]. Acknowledgments
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