Strings from Feynman Graph counting : without large N
aa r X i v : . [ h e p - t h ] M a r QMUL-PH-11-16WITS-CTP-080
Strings from Feynman Graph counting :without large N
Robert de Mello Koch a, and Sanjaye Ramgoolam b, a National Institute for Theoretical Physics ,Department of Physics and Centre for Theoretical PhysicsUniversity of Witwatersrand, Wits, 2050,South Africa b Centre for Research in String Theory, Department of Physics , Queen Mary University of London , Mile End Road, London E1 4NS, UK
ABSTRACT
A well-known connection between n strings winding around a circle and permutationsof n objects plays a fundamental role in the string theory of large N two dimensional YangMills theory and elsewhere in topological and physical string theories. Basic questions inthe enumeration of Feynman graphs can be expressed elegantly in terms of permutationgroups. We show that these permutation techniques for Feynman graph enumeration,along with the Burnside counting lemma, lead to equalities between counting problems ofFeynman graphs in scalar field theories and Quantum Electrodynamics with the countingof amplitudes in a string theory with torus or cylinder target space. This string theoryarises in the large N expansion of two dimensional Yang Mills and is closely related tolattice gauge theory with S n gauge group. We collect and extend results on generatingfunctions for Feynman graph counting, which connect directly with the string picture.We propose that the connection between string combinatorics and permutations has im-plications for QFT-string dualities, beyond the framework of large N gauge theory. [email protected] [email protected] ontents S n . . . . . . . . . . . . . . . . . . . . 102.3 Permutations and Strings beyond 2dYM . . . . . . . . . . . . . . . . . . 122.4 A useful theorem in combinatorics . . . . . . . . . . . . . . . . . . . . . . 12 (Σ , Σ ) φ theory and string theory 18 , Σ ). . . . . . . . . . . . . 215.3 Cycle Index formulae related to double cosets . . . . . . . . . . . . . . . 235.4 Action of Vertex symmetry group on Wick contractions . . . . . . . . . . 255.5 Number of Feynman graphs from strings on a cylinder . . . . . . . . . . . 265.6 The symmetry factor of a Feynman graph from strings on a cylinder . . . 27 φ theory and other interactions . . . . . . . . . . . . . 346.4 Interpretation in terms of 2dYM string . . . . . . . . . . . . . . . . . . . 35
11 Summary and Outlook 53A Semi-direct product structure of Feynman graph symmetries. 56B Functions on the double-coset 58
B.1 QED counting in terms of representation theory . . . . . . . . . . . . . . 60
C Feynman graphs with GAP 61
C.1 Vacuum graphs of φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61C.2 φ with external edges . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61C.3 Symmetry factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62C.4 Action of S v on a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 D Integer sequences 63 Introduction
There is a basic connection between winding states of strings and partitions of numbers. Ifwe have a string wound around a circle, it can have a winding number which is a positiveinteger. For multiple strings, the winding numbers can be added to define a total windingnumber. In the sector of the multi-string Hilbert space with total winding number n ,the partitions of n , denoted p ( n ), correspond to the different types of string states. Thenumber p ( n ) also plays an important role in connection with the symmetric group S n , oforder n !, consisting of the re-arrangements of the integers { , · · · , n } . It is the numberof conjugacy classes in S n .In generic string theories, a string state is characterized by a winding number alongwith additional vibrational and momentum quantum numbers. In simple cases, the wind-ing numbers completely characterize the string state. A particularly striking example ofthis simplification occurs in the string dual of two dimensional Yang Mills theory (2dYM)on a Riemann surface Σ G with genus G . This theory, for U ( N ) gauge group, can be solvedexactly [1]. The coefficients in the large N expansion of amplitudes in 2dYM defined onRiemann surfaces can be expressed in terms of sums over symmetric group elements. Thiscombinatoric data has an interpretation in terms of branched covers of Σ G by a stringworldsheets Σ h , Riemann surfaces of genus h related to the order in the 1 /N expansion[2, 3, 4]. This provides a beautiful realization of the ideas of [5]. These spaces of coversare also called Hurwitz spaces and they are spaces of holomorphic maps from Σ h to Σ G .These spaces are reviewed from the point of view of 2dYM in [6].Of particular interest are manifolds Σ G,B , with genus G and B boundaries. Theobservables of 2dYM are defined as functions of the boundary holonomies taking valuesin the gauge group U ( N ). The gauge-invariant functions of these group elements aremulti-traces, which are also classified for fixed degree n , by partitions. This leads toan interpretation of the boundary partition functions in terms of covering spaces of themanifold with boundary. Generically, the partition function sums over branched covers.The derivative of the map at some points on the worldsheet can vanish, and the images ofsuch points are called branch points . A particularly simple situation occurs when the YangMills theory is defined on a cylinder, and the area is taken to zero. Then the partitionfunction sums only over unbranched covers .The large N expansion of 2dYM can also be expressed in terms of lattice gauge the-ory with S n gauge group which can be formulated following Wilson and letting edgeholonomies take values in S n , with an appropriate plaquette action. This theory is topo-logical and will be called S n TFT (topological field theory). So along with the original U ( N ) description, there is the picture of S n TFT, as well as the Hurwitz space descrip-tion. The Hurwitz space connects most directly to the equations formulated on the stringworldsheet, since the holomorphy condition is the localization locus for the string pathintegral. Explicit worldshseet actions have been proposed [6, 7, 8, 9, 10]. The 2d S n TFT3an be viewed as the string field theory of this Hurwitz space string theory. In this paperwe will find the 2 d S n TFT particularly useful.Other examples where the string-permutation connection is important is Matrix stringtheory[17]. An orbifold superconformal field theory with target space S N ( R ) arises asan IR limit of N = 8 SYM in two dimensions. Conjugacy classes in S N correspond tostring winding numbers which are dual to momenta of gravitons in the Matrix Theoryinterpretation. Yet another example is the connection between Belyi maps (a special classof branched covers with sphere target space) and correlators of the Gaussian Hermitianmatrix model [18, 19].From the above we take the lesson that the connection between permutations andstrings is rather generic. In many cases, it is a tool to implement the ideas of stringsemerging from the large N expansion in quantum field theories. Now a large number ofcounting problems can be formulated in terms of permutations acting on various typesof sets. Some of these counting problems, such as the ones relevant to 2d Yang Millsand to Belyi maps, involve permutations acting on each other, for example by groupmultiplication. It will not surprise anybody that the question of counting Feynman graphsin quantum field theories, without large N , can be formulated in terms of symmetric groupsacting on some sets having to do with vertices and edges.Investigating the counting of Feynman graphs more carefully reveals that, in fact,these counting problems can be formulated in terms of rather intrinsic properties of thesymmetric groups themselves. The nature of the specific interactions determine the formof the vertices, and in turn certain subgroups of symmetric groups, which can be viewed asthe symmetry of all the vertices. Then we draw in an idea from the study of Belyi mapsand their associated ribbon graphs, which is called cleaning . This is the constructionof associating to a given graph, a new graph obtained by introducing a new type ofvertex in the middle of the existing edges, dividing them into half-edges. This allows thedescription of Wick contractions in terms of permutations which are pairings. This wasexploited recently in a physics setting in [19].This line of thinking culminates in an elegant way to compute numbers of Feynmangraphs of real scalar fields, which coincides with the results in a classic in the mathematicalliterature on graph counting [20]. Our derivation does not rely on the notion of graphsuperposition used in [20] and appears more direct, at least when approaching thesegraphs from the perspective of Wick contractions in perturbative quantum field theory.We also show that the formulae counting Feynman graphs, as well as those describingsymmetry factors of individual graphs, can be interpreted as observables in the 2d S n TFT. The number n will depend on the number of vertices and edges in the Feynmangraph calculation. We extend our considerations to QED, deriving new counting resultson Feynman graphs. Again we obtain an expression of Feynman graph counting problemsin terms of observables in the 2d S n TFT. An interesting aside is a surprising connectionbetween the counting of QED Feynman graphs and the counting of ribbon graphs.4hile the string-permutation connections underlie the emergence of strings from large N
2d YM, our results suggest that these same connections could also lead to the emergenceof strings from quantum field theories such as that of a scalar field or QED.We now give an outline of the paper. In section 2, we review the connection betweenstrings and permutations, which plays a prominent role in the string theory of large N U ( N ) gauge group. We describe the perspective of lattice gaugetheory with S n gauge group and explain its topological nature by drawing on existing2dYM literature. We also review the Hermitian matrix model and its connection to Belyimaps, as another realization of the string-permutations connection. Finally, we mentionthe Burnside Lemma from combinatorics, which is used very generally for counting orbitsof group actions. In section 3, we review Feynman rules for φ theory. We recall howthe symmetry factors get multiplied with additional group theory factors associated withglobal or gauge symmetry groups, and space-time integrals. Our main focus will be thecombinatorics of the graphs and their symmetry factors.In section 4, we describe a method of enumerating Feynman graphs and calculatingtheir symmetry factors, which relies on a pair of combinatoric objects. The first of thepair, which we call Σ , captures the form of the vertices. The second piece of data, whichwe call Σ , is associated with the Wick contractions. Given a graph, this data can beconstructed by putting a new vertex in the middle of each edge. We can imagine thesevertices to be coloured differently from the ones already present in the graph. This splitseach existing edge into a pair of half-edges. We associate labels { , , · · · } with each ofthese half-edges. This is explained in the context of real scalar field theory, in the firstinstance, with vacuum graphs in φ theory. This formulation leads to simple one-liners inGAP (mathematical software of choice for group theory) [22] for calculations of symmetryfactors and enumeration. This is explained with examples in Appendix CIn section 5, we use the Burnside Lemma to count Feynman graphs, and obtain thefirst hints of stringy geometry of maps to a torus. The picture of maps to a torus has someintricacies, and it turns out that a deeper understanding of the combinatorics leads to asimpler picture in terms of strings covering a cylinder. This is is achieved by first recallingfrom the maths literature that graph counting formulae are expressed in terms of a certaindouble coset [20]. We find that the formulation of graph counting in terms of the pair(Σ , Σ ) finds a natural meaning in terms of the double coset. This allows us to exhibitthe counting of Feynman graphs in scalar field theory as an observable in 2d S n TFT ona cylinder. Following standard constructions in covering space theory, the S n data is usedto construct unbranched covers of the cylinder. Some of the counting formulae that followfrom the double coset picture can be understood using the idea of Fourier transformationon symmetric groups. Calculations explaining this are in Appendix B. Interestingly verysimilar calculations are relevant to recent results in correlators of BPS operators in N = 4super-Yang-Mills [21].We pursue, following [20], the application of the double coset picture to derive gener-5ting functions for graph counting in section 6. We also explain here how the formulaechange when we generalize our considerations from φ to φ and other interactions. Gen-eralizations away from vacuum Feynman graphs to the case with external edges is alsoexplained. The double coset formulae lead most directly to the set of all Feynman graphs,included disconnected ones. Counting formulae for the connected ones are obtained here.Appendix D contains explicit lists for Feynman graph counting. The first few terms agreewith existing physics literature. A few of the series we consider are listed in the OnlineEncyclopaedia of Integer sequences [23], but the majority are not listed there.In section 7, we show that these approaches work in a simple way for the ribbongraphs which arise in doing the 1 /N expansion. While the ribbon graphs for large N aretraditionally drawn using double lines, they can also be defined in a way closer to ordinarygraphs, with single lines but with the difference that the vertices have a cylic order (seee.g. [24] [25]). This allows cyclic symmetry of the half-edges at the vertices in testingequivalence of differently labelled ribbon graphs, but not arbitrary permutations. Takingthis into account, we express the question of how many Feynman diagrams correspond tothe same ribbon graph (embedded graph) in group theoretic terms, associated with thesymmetry breaking from permutation group to cyclic group.In section 8, we apply these ideas to QED or Yukawa theory, giving the connectionto 2d S n TFT and deriving generating functions. In section 9, we adapt the counting toQED, with the vanishings due to Furry’s theorem taken into account. Again we obtainthe 2d S n TFT connection and the generating functions. Manipulations of the doublecoset description for QED leads to a somewhat surprising connection between QED graphcounting and ribbon graph counting. We explain this by describing a bijection betweenthese graphs. The arrows on electron loops provide the cyclic symmetry which is key toribbon graphs. A consequence is that the number of QED/Yukawa vacuum graphs with2 v vertices is equal to the number of ribbon graphs with v edges.In section 10, we discuss our results with a view to extracting some lessons for gaugestring duality. Ribbon graphs give a vivid physical picture of how strings arise fromquantum field theory, but an equally compelling and arguably simpler physical picture isthat of strings winding on a circle being described by permutations. This latter picturehas been exploited here to give new connections between QFT Feynman diagrams andstring counting. The natural question is whether this connection can be extended toextract from Feynman graphs of QFT, without large N , something more than stringycombinatorics to include space-time dependences of correlators and S-matrices.In section 11, we summarize the paper and discuss some concrete avenues for futurework. Appendix A describes a semi-direct product structure of the Automorphism groups,whose orders give symmetry factors of Feynman graphs, and points out a difference be-tween the notion of Automorphism most commonly used in the mathematics literatureon graphs and the one relevant to symnmetry factors of Feynman graphs.6 Review of strings and permutations
Consider a string wrapping a circle. The winding number is a topological characteristicof the map. In string theory, the string is part of the worldsheet, the circle is viewed aspart of target space. Let the target circle be parametrized by X with X ∼ X + 2 π . Letthe string be parametrized by σ with σ ∼ σ + 2 π . A string with winding number k isdescribed by X = kσ (2.1)For multiple strings, we may have distinct winding numbers. Adding the winding numbersgives the degree of the map from the strings.Given such a configuration of winding strings, we may label the inverse images of afixed point on the circle from 1 to n . Following the inverse images as we move round thetarget circle yields a permutation. The two possible winding configurations at n = 2 areshown in Figure 1. Figure 1: Winding strings and permutationsIn ordinary string theory, the winding number is one of the quantum numbers speci-fying a string state. In some topological settings, the string winding number is the onlyrelevant quantum number. Recall that 2d Yang Mills on a target space with a boundary has the partition function Z = X R (Dim R ) − G − B e − AC ( R ) Y i χ R ( U i ) (2.2) G is the genus of the surface, B is the number of boundaries, U i are the holonomies atthe boundaries and A is the area of the target and C ( R ) is the quadratic Casimir [1].The choice of fixed boundary holonomies is illustrated in Figure 2.7igure 2: The 2dYM partition function with boundary holonomies is known exactlyFor a target cylinder, we have Z ( U , U ) = X R χ R ( U ) χ R ( U ) e − AC ( R ) (2.3)in terms of characters of U ( N ) elements. The power of the dimension is zero because G = 0 , B = 2. The characters have an expansion in multi-traces. These traces can beconstructed from permutations as follows tr ( σU ) ≡ U i i σ (1) U i i σ (2) · · · U i i σ ( n ) (2.4)This is a very useful formula when developing a string interpretation for Wilson loops in2dYM [26].Using these observables for the cylinder, we define Z ( σ , σ ) = Z dU dU tr ( σ U † ) tr ( σ U † ) Z ( U , U ) (2.5)This integral can be done, and in the zero area limit A = 0 we find Z ( σ , σ ) = X R ⊢ n χ R ( σ ) χ R ( σ ) (2.6)The notation R ⊢ n means that R is running over partitions of n (row lengths of Youngdiagrams) which parametrize irreducible representations of S n , and the χ R ( σ ) are char-acters of S n elements. Using the invariance of the character under conjugation, we canreplace χ R ( σ ) = 1 n ! X γ ∈ S n χ R ( γσ γ − ) (2.7)8his gives Z ( σ , σ ) = 1 n ! X γ ∈ S n X R ⊢ n χ R ( σ ) χ R ( γσ γ − ) (2.8)Since P γ γσ γ − is a central element in the group algebra of S n , we can use Schur’sLemma to combine the product of characters into a single character Z ( σ , σ ) = 1 n ! X γ ∈ S n X R ⊢ n d R χ R ( σ γσ γ − ) (2.9)Now use the Fourier expansion on the symmetric group, which allows the delta functionto be written in terms of characters, to obtain Z ( σ , σ ) = X γ δ ( σ γσ γ − ) (2.10)When the gauge group is a product U ( N ) × U ( N ) × U ( N ), we have to specify aholonomy for each gauge group in the product. Denote them by U, V, W . Recalling thatthe character of a direct product is the product of the characters we have the zero areapartition function for a cylinder Z ( U , V , W ; U , V , W ) = X R,S,T χ R ( U ) χ S ( V ) χ T ( W ) χ R ( U ) χ S ( V ) χ T ( W )) (2.11)The boundary observables in this case, in a basis appropriate for a string interpretation,are tr ( σU ) tr ( ρV ) tr ( τ W ). In terms of permutations Z ( σ , ρ , τ ; σ , ρ , τ ) = X γ ∈ S n × S n × S n δ ( σ ◦ ρ ◦ τ γ σ ◦ ρ ◦ τ γ − ) (2.12)We have discussed above the classification of string maps to a circle target space, inmotivating the choice of boundary observables in large N Riemann existence theorem . Consider the equivalence classes of holomorphic maps fromthe worldsheet (Σ h ) to the target (Σ G ) with two maps f and f defined to be equiva-lent if these exists a holomorphic invertible map φ : Σ h → Σ G such that f = f ◦ φ .Given a holomorphic map (branched cover) with L branch points and of degree n , wecan obtain a combinatoric description by picking a generic base point and labeling theinverse images as { , , · · · , n } . By following the inverse images of a closed path startingat the base point and encircling each branch point, we can get a sequence σ , σ , ..., σ L ofpermutations. For G >
0, there are also permutations for the a, b cycles of Σ G , denoted9 i , t i for i = 1 · · · G . Two equivalent holomorphic maps are described by permutations σ , σ , ..., σ L , s , t , · · · , s G , t G and σ ′ , σ ′ , ..., σ ′ L , s ′ , t ′ , · · · , s ′ G , t ′ G which are related by con-jugation σ i = ασ ′ i α − , s i = αs ′ i α − , t i = αt ′ i α − . This correspondence between sequencesof permutations and holomorphic maps is captured in the Riemann existence theorem (seefor example [25]). Relations in the fundamental group of the Riemann surface puncturedat the branch points are reflected in the sequence of permutations.The delta function in (2.10) enforces a relation in the fundamental group of the cylin-der. When this partition function is generalized beyond zero area, the sum (2.10) ismodified to include additional permutations which can be interpreted as a counting ofbranched covers. Even at zero area, but for other Riemann surfaces, there are additionalpermutations associated with branch points. We will come back to this point in thediscussion of (2.16). S n In lattice gauge theory[27] for two dimensions, we discretize (e.g triangulate or give amore general cell decomposition for) the Riemann surface. To each edge we associate agroup element. To each plaquette, we associate a weight which depends on the product ofgroup elements along the boundary of the plaquette. Let us choose the plaquette weightto be Z P ( σ ) = δ ( σ ) = X R ⊢ n d R n ! χ R ( σ ) (2.13)where σ is the product of group elements around the boundary of the plaquette. Thepartition function for the discretized manifold is the product of weights for all the plaque-ttes. The partition function can be shown to be invariant under refinement of the latticeso that the result depends only on the topology of the Riemann surface. This is provedin [28]. The language used there is that of continuous groups, but by replacing integralswith sums, it applies equally well to finite groups. In [28] surfaces with a finite area A are considered. For the topological aspects of the theory that interest us, it is enoughto consider A = 0. For manifolds with boundary one fixes the group elements at theboundaries. As a simple example, consider a lattice theory defined on a disc. The targetmight be discretized with either one plaquette or two plaquettes as shown in Figure 3.Starting from the discretization given in (a) we find Z ( σ , σ ) = X γ Z P ( σ γ − ) Z P ( σγ )= X γ δ ( σ γ − ) δ ( σ γ )= δ ( σ σ ) (2.14)10hich is precisely the partition function obtained using the discretization shown in (b).The generalization to finer discretization works in exactly the same way.Figure 3: Two discretizations of the disc with the same boundary condition.For the cylinder Z ( σ , σ ) = X γ δ ( σ γσ γ − ) (2.15)which is of course (2.10). For a 3-holed sphere, Z ( σ , σ , σ ) = 1( n !) X γ ,γ δ ( σ γ σ γ − γ σ γ − )= 1 n ! X R χ R ( σ ) χ R ( σ ) χ R ( σ ) d R (2.16)This is also derived in [29] from a more axiomatic approach.These formulae from lattice gauge theory of S n arise from the leading the large N limitof the U ( N ) gauge theory at zero area. The cylinder case is special in that this is the fullanswer to all orders in the 1 /N expansion. The answer in (2.16) is obtained after usingthe leading large approximation of Ω, which is a sum over elements in S n weighted bypowers of N . See for example 6.1.2 of [6] for a discussion of Ω. At subleading orders, theΩ factor contains a sum over permutations, which can be interpreted in terms of sourcesof curvature in the S n bundle. From the Hurwitz space string interpretation of 2dYM,there are branch points in the interior.The S n lattice gauge theory perspective for U ( N ) gauge theory at large N is empha-sized in [30]. Computations of 2dYM partition functions for general Riemann surfaceswith boundaries expressed in the symmetric group basis, and the connection with Hurwitzspace counting is given in [4, 6, 26]. The connection to topological field theory with S n gauge group was also observed in [31]. 11o summarize, the large N expansion of 2dYM with U ( N ) gauge group on a Riemannsurface Σ G,B (genus G , with B boundaries) can be expressed in terms of symmetric groups.This combinatoric data arising has an interpretation as 2d gauge theory with S n gaugegroup. This connects directly with the string theory interpretation in terms of maps froma string worldsheet Σ h (genus h related to the order in the 1 /N expansion) using classicresults relating the space of branched covers (Hurwitz space) to symmetric groups.We will find in this paper that counting problems of Feynman graphs can be expressedin terms of certain generalizations of the 2dYM results, which can be expressed elegantlyin terms of the 2d S n gauge theory perspective, and also admit a connection to branchedcovers. The connection between strings and permutations also has interesting implications for thecorrelators of the Gaussian hermitian one-Matrix model [19]. The computation of corre-lators can be mapped to the counting of certain triples of permutations which multiply tothe identity. These count holomorphic maps from world-sheet to sphere target with threebranch points on the target. Holomorphic maps with three branch points are related,by Belyis theorem, to curves and maps defined over algebraic numbers. Thus, the stringtheory dual of the one-matrix model at generic couplings has worldsheets defined over thealgebraic numbers and a sphere target. For related ideas see [32, 33].Finally, yet another connection between strings and permutations that is relevant toour present discussion is provided by Matrix String Theory, defined by the IR limit oftwo-dimensional N = 8 SYM. This limit is strongly coupled and a nontrivial confor-mal field theory describes the IR fixed point. The conformal field theory is the N = 8supersymmetric sigma model on the orbifold target space ( R ) N /S N , formed from theeigenvalues of the Higgs fields X I of the theory [17]. If we go around the space-like S ofthe world-sheet, the eigenvalues can be interchanged. Concretely X I ( σ + 2 π ) = X g ( I ) ( σ )where g takes value in the Weyl group of U ( N ) which is the symmetric group S N . Thesetwisted sectors correspond to configurations of strings with various lengths. Consequently,twisted sectors with given winding number are labeled by the conjugacy classes of theorbifold group S N . Let G be a finite group that acts on a set X . For each g in G let X g denote the set ofelements in X that are fixed by g . Burnside’s lemma asserts the following formula for the12umber of orbitsNumber of orbits of the G -action on X = 1 | G | X g ∈ G | X g | . (2.17)Thus the number of orbits is equal to the average number of points fixed by an elementof G . This is called the Burnside Lemma or the Burnside counting theorem. Usefulreferences for the Burnside Lemma are [41] [42], the former also provides other usefulcombinatoric background. There are many excellent articles and textbooks that deal with the Feynman graph ex-pansion of perturbative quantum field theory. Here we will simply review those aspectsmost relevant for our goals. For a relevant reference that has more details see [43].Perturbative quantum field theory expresses quantities of interest (for example, anamplitude A ) as a power series expansion in the small coupling gA ( g ) = ∞ X k =0 A v g v (3.1)The coefficients A v are obtained by summing Feynman graphs with v vertices D v A v = X D v C D v N D v F D v (3.2) C D v is the symmetry factor of graph D v , N D v is a group theory factor coming fromthe color combinatorics of global or gauge symmetry groups and F D v is the result ofintegrating over the loop momenta in D v . For concreteness assume that g is the strengthof a φ interaction. In this case the factor N D v is 1. If we canonically quantize thetheory, we expand about the free theory using Wick’s Theorem. The Feynman graphsare used to compute the sum over all possible Wick contractions. Different ways of doingthe Wick contractions can lead to the same Feynman graph. The symmetry factors C D v keep track of this. Perturbation theory expands the exponential of the gφ interaction.Consequently, g v come with a 1 /v ! from the Taylor series of the exponential. Accountingfor this factor, the number of Wick contractions leading to Feynman graph D v is v ! C D v .This observation can be used to generate an interesting sum rule for the symmetry factors.The sum of graphs with v vertices and E = 2 n external lines reproduces the complete setof Wick contractions for 4 v + 2 n fields. The total number of Wick contractions is equalto the number of pairs that can be formed from the 4 v + 2 n fields and is also equal to the13um over D v of v ! C D v . Consequently v ! X D v C D v = (4 v + 2 n − C D k starts by giving a set of rules todetermine these automorphisms. For gφ theory the rules are • for a closed propagator (one with both ends attached to the same vertex) there is aswap which exchanges the endpoints. • for p propagators which each start at the same vertex and end at the same vertex(the starting and ending points might not be distinct) there are p ! transformationsthat permute the propagators. • for n vertices that have the same structure there are n ! transformations that permutethe vertices. • for d identical graphs there are d ! transformations that permute the graphs.Denote the automorphism group of a Feynman graph D by Aut( D ). The order | Aut( D ) | is the product of the numbers obtained applying each rule above. A Feynmangraph built using v vertices comes with a coefficient C D v = (4!) v | Aut( D ) | (3.4)Applying the rules to compute symmetry factors can be tricky. The excellent textbook[44]suggests that “When in doubt, you can always determine the symmetry factor by countingequivalent contractions”. As v grows, this quickly becomes hopeless. One of the results ofthis paper is give conceptually simple permutation group algorithms for symmetry factors(Appendix C).One basic quantity of interest is the number of graphs N v = X D v F D v is14nimportant (we are implicitly assuming they are all roughly the same size) so that wemay, for simplicity, work in zero dimensions which amounts to setting F D v = 1.Instead of focusing the discussion on any particular amplitude, it is useful and stan-dard, to study the generating functional of correlation functions Z [ J ] = Z [ Dφ ] e iS + i R d xJφ (3.6)where S = Z d x (cid:18) ∂ µ φ∂ µ φ − m φ − gφ (cid:19) (3.7)The value of Z at J = 0 computes the sum of all vacuum graphs. Derivatives of Z withrespect to J compute the sum of all Feynman graphs with E external legs h | φ ( x ) φ ( x ) · · · φ ( x E ) | i = 1 i E δ n Z [ J ] δJ ( x ) δJ ( x ) · · · δJ ( x E ) (cid:12)(cid:12)(cid:12)(cid:12) J =0 (3.8)Another quantity of interest is the logarithm of Z , W = ln Z . The value of W at J = 0computes the sum of all connected vacuum graphs. Derivatives of W with respect to J , at J = 0, compute the sum of all connected Feynman graphs with E external legs.This suggests another interesting question: how many connected Feynman graphs N conn v are there? The connection between Z and W holds for graphs with the symmetry factor C D v included. The relation between N conn v and N v (which does not include the symmetryfactors) is different. (Σ , Σ ) Consider a vacuum Feynman graph in φ theory. Let v be the number of vertices. At v = 1 we have a single graph, given in figure 4. At v = 2 we have three graphs givenFigure 4: One vertex vacuum graph in φ theoryin figure 5. To get a systematic counting, we describe these graphs in terms of somenumbers. One way to do this is to label the vertices 1 , , · · · , v , and list the edges as [ ij ].15igure 5: Two vertex vacuum graphs in φ theoryThis method is well-known in graph theory and quickly leads to elegant results for graphswhere there are no edges connecting a vertex to itself, and no multiple edges between agiven pair of vertices. In the case at hand, we do have loops and we do have multipleedges between the same vertices. For this reason, the graphs at hand are sometimes calledmulti-graphs in the mathematics literature.For the graphs we consider, it is more convenient to give a combinatoric description,by putting a new vertex in the middle of each edge. Each edge is thus divided into twohalf-edges. Each half-edge is labeled by a number from 1 , , · · · v . Using this labeling,the graphs shown in figure 4 and figure 5 have been labeled in figure 6 and figure 7respectively. We have drawn the original vertices as black dots and the new vertices aswhite dots. Figure 6: Numbering the half-edgesNow the graphs can be described by specifying the list of 4-tuples of numbers at theblack vertices and the pairs of numbers at the white vertices. We thus introduce twoquantities Σ , Σ . For the v = 1 graph, with the labeling chosen in figure 6Σ = < , , , > Σ = (12)(34) (4.1)For the three graphs at v = 2, inspection of figure 7 shows that we have fixed( a ) Σ = < , , , >< , , , > Σ = (12)(34)(56)(78)( b ) Σ = < , , , >< , , , > = (23)(16)(47)(58)( c ) Σ = < , , , >< , , , > Σ = (15)(26)(37)(48) (4.2)Clearly, we could have labeled things differently producing different pairs Σ , Σ . Theserelabelings are elements of S in the case v = 1 and elements of S in the case v = 2.More generally, we have S v . It is useful to consider breaking the half-edges, leaving uswith 4 v half-edges dangling from v v half-edges from 2 v v ! permutations of the vertices themselves. These permutationsform a subgroup of S v called the wreath product S v [ S ]. It is the semi-direct productof ( S ) v ⋊ S v . The physics reader is not required to have any prior knowledge of thesegroups. All the relevant facts we will need from the math literature will be quoted as weneed them.The symmetry group of the Feynman graph, whose order appears in the denominatoras the symmetry factor, is obtained from the action of the symmetric group S v on thepairs (Σ , Σ ). A permutation in S v is a map from integers { , , · · · v } to these integers { , , · · · , v } , the map being one-one and onto. It can be written as an ordered list of17igure 9: Splitting the half-edgesthe images of { , , · · · , v } , e.g a cyclic permutation of { , , , } can be written as2341 (4.3)Alternatively we use cycle notation, (1234) for the example at hand. The image of everyinteger in the bracket is the one to the right, the image of the last is the first.A permutation σ acts on Σ , Σ by taking the integers i in these tuples to σ ( i ). Wedenote this action by σ : (Σ , Σ ) → ( σ (Σ ) , σ (Σ )) (4.4)If (Σ ′ , Σ ′ ) = ( σ (Σ ) , σ (Σ )) for some σ , then both pairs (Σ ′ , Σ ′ ) and (Σ , Σ ) representthe same Feynman graph. We may say that Feynman graphs correspond to orbitsof the S v action on the pairs (Σ , Σ ).As we saw in the examples above we can fix the form of Σ . The subgroup of S v which preserves this is the S v [ S ]. Then different Feynman graphs correspond to orbitsof S v [ S ] acting on the pairings Σ . These are nothing but elements of the conjugacyclass [2 v ] in S v . This leads directly to the computation of numbers of Feynman graphsand symmetry factors using GAP software, which is the mathematical package for grouptheory. See Appendix C. φ theory and string theory In this section we develop a number of key ideas. As we saw in the last section, to generateall possible Feynman graphs, we can fix Σ and allow Σ to vary. Using the action σ : (Σ , Σ ) → ( σ (Σ ) , σ (Σ )) (5.1)of σ ∈ S v , the condition σ (Σ ) = Σ fixes σ to be in the subgroup S v [ S ] of S v . Per-mutations Σ that can not be related by σ ∈ S v [ S ] are distinct Feynman graphs. This18eads to the realization of Feynman graphs as orbits of the vertex symmetry group actingon Wick contractions. A simple application of the Burnside Lemma in section 5.1 thengives an explicit formula (5.4) for the number of Feynman graphs. The formula has animmediate interpretation as a sum over covers of a torus. This suggests an interpretationas string worldsheets mapping to a torus. However, the sum over maps to torus have someconstraints which seem non-trivial to implement geometrically. This motivates a deeperlook at the combinatorics, which reveals a somewhat simpler connection to strings witha cylinder target space. In section 5.2 we explain that Feynman graphs are elements ofa double coset. This description is already present in [20], where it is derived using anoperation called graph superposition. Our derivation does not use this operation and isobtained more directly using the half-edges introduced in Section 4. The pair (Σ , Σ )encountered there finds a natural interpretation in terms of the double coset. This is akey result with a number of implications. It allows us to express the counting of Feynmangraphs as well as their symmetry factors in terms of data in S n TFT. In turn this S n TFT data is used to construct covers of the cylinder. The figure 10 summarizes the keymessage of this section. This will set the stage for Section 6 where generating functionsand counting sequences will be given.Figure 10: Double coset connection
A vacuum Feynman graph in φ theory is constructed by starting with v vertices, eachhaving 4 edges attached, and connecting up the edges, as shown in Figure 11. Withoutany loss of generality label the edges of the first vertex (1 , , , , , , .. v ) to go witheach vertex), but we have made a choice. This is a choice ofΣ = h , , , ih , , , i · · · (5.2)19here the angled brackets are completely symmetric and the different brackets can befreely interchanged. The symmetry of Σ is S v [ S ]. To complete the diagram we need to give the Wick contractions. The Wick contrac-tions are specified by choosing an element Σ in [2 v ]. For example(12)(34) .. (4 v − , v ) (5.3)corresponds to the disconnected graph with a bunch of eights.To construct all possible Feynman graphs we need to allow Σ to run over all possibleWick contractions holding Σ fixed to the choice we made above. This is easily doneby acting on Σ with S v [ S ]. It follows that the distinct Feynman graphs correspond toorbits of the vertex symmetry group S v [ S ] acting in the set of pairings [2 v ], which areelements of a conjugacy class in S v . If we apply the Burnside Lemma, we see that thenumber of Feynman graphs is equal toNumber of Feynman Graphs = 14 v v ! X γ ∈ H X σ ∈ [2 v ] δ ( γσγ − σ − ) (5.4)This is a sum over certain covers of the torus of degree 4 v . The worldsheet is also a torus.One of the monodromies is constrained to lie in the subgroup H = S v [ S ], the other liesin the conjugacy class C = [2 v ]. Constrained sums of this sort where the permutationsare constrained to lie in a given conjugacy class are naturally motivated from holomorphicmaps. Here we are constraining one permutation to lie in a subgroup and one to lie in aconjugacy class. Such a constraint is entirely possible in S n TFT, but the the geometricalinterpretation in terms of Hurwitz space is, at this point, less clear. Subsequent formulaewe derive will show the emergence of a cylinder rather than a torus. The formulae thenbecome more symmetric with respect to exchange of H and H .For purposes of counting applications, the equation (5.4) is easy to implement in GAP,although related techniques involving cycle indices introduced in Section 6 will be moreefficient.Apart from the number of Feynman graphs, it is also interesting to compute theautomorphism group of a given graph. This appears, for example, in the denominatorfor the formula (3.4) for the symmetry factor. With our choice of a fixed Σ , a graph isuniquely specified by Σ . For this reason we will denote the automorphism group of agiven graph by Aut([Σ ] H ). The [Σ ] H denotes an equivalence class under conjugation by H . The automorphism group is given by those elements of H which leave Σ invariant.The order of the automorphism group is | Aut([Σ ] H ) | = X γ ∈ H δ ( γ Σ γ − Σ − ) (5.5) Some authors prefer the notation S ≀ S v . We follow [20] for the reason he gives : that the S v [ S ]notation connects nicely with the substitution formula for cycle indices
20 symmetry γ and a Wick contraction Σ specify an unbranched cover of the torus, ofdegree 4 v with γ constrained to lie in H and Σ constrained to be made of 2-cycles.Recall from section 3 that the number of Wick contractions leading to a particularFeynman graph is v ! C D v = v ! | ( S ) v || Aut([Σ ] H ) | (5.6)Noting that Aut(Σ ) = ( S ) v ⋊ S v we find v ! | ( S ) v || Aut([Σ ] H ) | = | Aut(Σ ) || Aut( D ) | (5.7)This formula has a very natural interpretation: start by making a choice of Σ . To deter-mine a Wick contraction, we need to specify the cycle Σ . Any other Wick contractioncontributing to the same graph can be obtained by swapping vertices or swapping the halfedges at a given vertex. The full set of these transformations is performed by Aut(Σ ).If we have an element of Aut(Σ ) that leaves Σ invariant (this by definition belongs toAut( D )) we do not get a new Wick contraction. Thus, the formula (5.6) is an applicationof the Orbit-stablizer theorem [34]. (Σ , Σ ) . Read [20] derives explicit counting formulae for graphs by developing a formulation interms of double cosets. He arrives at the double cosets by a procedure of graph super-position. We will arrive at the same double coset descriptions using the procedure ofseparating the edges into half-edges, and keeping track of the permutations which containthe information of how the half-edges are glued to make up the graph (See Figure 9).There are two related double coset descriptions relevant to φ graphs. The first one isa double coset of S n × S n . The second is a double coset of S n and can be obtained by agauge fixing of the S n × S n picture. Let us start with this description. The elements ofthe double coset are pairs ( σ , σ ) ∈ ( S n × S n ) (5.8)with the equivalence ( σ , σ ) ∼ ( ασ β , ασ β ) (5.9)where n = 4 v, α ∈ S v , β ∈ H = S v [ S ] and β ∈ H = S v [ S ].To understand why this double coset counts Feynman graphs of φ theory considergraphs with v , Σ description(Σ , Σ ) ∈ ( < v >, [2 v ]) (5.10)21s explained in section 4 the graph splits up into 2 v bi-valent white vertices and v · · · v , so we haveedges { W , W · · · W v } . Label the edges connected to the black vertices { , , · · · , v } ,so we have { B , B , · · · B v } . All possible relabellings of the edges of the white verticesare paramatrized by σ ∈ S v . All possible relabellings of the edges of the 4-valent blackvertices are parametrized by σ ∈ S v . Given any ( σ , σ ) we can construct a graph bygluing W σ ( i ) ↔ B σ ( i ) (5.11)for i = 1 · · · v .Clearly by considering all possible ( σ , σ ) we can get all possible graphs.If we replace the i by α ( i ), we get the same graph, since the labelings do not matter. Sowe learn that ( ασ , ασ ) and ( σσ , σσ ) produce the same graph. Hence we are interestedin equivalence classes ( σ , σ ) ∼ ( ασ , ασ ) (5.12)We also know that the disconnected graph of bi-valent vertices has a symmetry of H = S v [ S ], so that σ and σ β with β ∈ H is the same graph. In the same way, σ ∼ σ β with β ∈ H . This leads us to the conclusion that Feynman graphs areequivalence classes of the equivalence relation( σ , σ ) ∼ ( ασ β , ασ β ) (5.13)completing the demonstration that Feynman graphs are in one-to-one correspondencewith elements of a double coset.Figure 11: Double coset connectionIf we choose α = β − σ − , then any pair is mapped to(1 , β − σ − σ β ) (5.14)22o now only the combination σ − σ appears which is in S n , and we have equivalences byright multiplication with H and left multiplication with H . We learn that S n \ ( S n × S n ) / ( H × H ) = H \ S n /H (5.15)From (5.13) we can consider first doing the coset by H and H , to we have an action of S n on S n /H × S n /H . This is the description we are using when we impose S n relabelingequivalences on (Σ , Σ ) which are nothing but a parametrization of S n /H × S n /H . Sowe have now learnt the group theoretic meaning of the Σ , Σ , which we wrote earlieras a combinatoric description of the graph that followed (at the beginning of section 4)immediately from introducing the white vertices to separate the existing vertices and keeptrack of the Wick contractions.This line of argument makes the generalization to ribbon graphs clear. Not any per-mutation of the legs connecting to a black vertex is allowed: to preserve the genus of theribbon the cyclic order of the labeling must be respected. Consequently, H = S v [ S ]must be replaced by H = S v [ Z ]. We pursue this direction in section 7. In this section we would like to make a connection with the classic results of Read [20] onthe counting of locally restricted graphs. The starting point of [20] it to count equivalenceclasses of pairs ( a , a ) in ( S n × S n ), where the pair ( a , a ) is equivalent to ( b , b ) if( b , b ) = ( xa g , xa g ) (5.16)where x ∈ S n and g ∈ H , g ∈ H . This is equivalent to counting elements of the doublecoset S n \ ( S n × S n ) / ( H × H ) = H \ S n /H (5.17)The number of equivalence classes is counted by N ( Z ( H ) ∗ Z ( H )), where the Z ’s are thecycle indices, and N is a star-product for multivariable polynomials defined as follows:Consider two polynomials P and Q in variables f , f ..f n P = X j ⊢ n P j ,j ..,j n f j f j · · · f j n n Q = X j ⊢ n Q j ,j ..,j n f j f j · · · f j n n (5.18)where j is a partition of n : j + 2 j + · · · + nj n = n . We abbreviate the coefficients as P j , Q j . The function N of a star product is now defined by N ( P ∗ Q ) = X j ⊢ n P j Q j Sym( j ) (5.19)23here Sym( j ) is the symmetry of the conjugacy class corresponding to partition j Sym( j ) = n Y i =1 ( j i ) i i !For our applications we will sometimes want to change the numerator group, in which casethe symmetry of the conjugacy class will change. This generalization of Read’s formulais discussed in section 8.1.Read [20] also proves N ( Z ( H ) ∗ Z ( H )) = X p ⊢ v Z S v [ S ] p Z S v [ S ] p Sym ( p ) = 1 | H || H | n ! X a ,a ∈ S v ν ( a , a ) (5.20)where ν ( a , a ) is the order of the intersection a H a − ∪ a H a − . We can rewrite theabove as N ( Z ( H ) ∗ Z ( H )) = 1 | H || H | n ! X a ,a ∈ S v X u ∈ H X u ∈ H δ ( a u a − a u − a − )= 1 | H || H | X b ∈ S v X u ∈ H X u ∈ H δ ( u bu − b − ) (5.21)which is reminiscent of a cylinder partition function for gauge theory with S v gaugesymmetry, and with holonomies at the boundaries restricted being summed in H and H respectively.Since these formulae will come up, in slight variations, repeatedly, we make somecomments on notation. We will denote the number of points in H \ G/H as N ( H , H )when G is just the symmetric group S n , or N ( H , H ; G ) more generally. The function of N of star products will also sometimes written to make the numerator group explicit. Sothe previous formulae can be expressed as N ( H , H ) = N ( Z ( H ) ∗ Z ( H )) (5.22)or N ( H , H ; G = S n ) = N ( Z ( H ) ∗ Z ( H ); G = S n ) (5.23)Note that we can also write (5.21) as1 | H || H | X R ⊢ S v X u ∈ H X u ∈ H χ R ( u ) χ R ( u ) (5.24)The sums over u and u produce projection operators which project onto the trivialrepresentation so that the only representations R which contribute are the representations24f S v which contain the trivial of S v [ S ] and the trivial of S v [ S ]. Each such R contributesthe product of the multiplicities with which the trivial of H and H appear.Number of Feynman Graphs = X R ⊢ v M R H M R H (5.25)We have used the notation M R H for the multiplicity of the one-dimensional representationof H when the irreducible representation R of S v is decomposed into representations ofthe subgroup H . We now have two different ways to compute the number of Feynman graphs: as thenumber of orbits of the vertex symmetry group acting on Wick contractions, which leadsto 1(4!) v v ! X u ∈ S v [ S ] X σ ∈ [2 v ] δ ( u σu − σ − ) (5.26)or in terms of the star product of the cycle indices N ( Z ( H ) ∗ Z ( H )). We will demonstrate,using some general group theory, the equivalence of (5.26)and (5.21).Noting that H = S v [ S ], (5.26) and (5.21) are equivalent if X σ ∈ [2 v ] δ ( u σu − σ − ) = 1 | H | X b ∈ S v X u ∈ H δ ( u bu − b − ) (5.27)To see that this relation holds, suppose there is a solution to the first delta function. σ isconjugate by some b to σ ≡ (12)(34) .. (4 v − , v ) b − σ b = σ (5.28)We know that all elements commuting with σ are in H ≡ S v [ S ]. From (5.27) we have u σu − = σ (5.29)So u b − σ bu − = b − σ b (5.30)It follows that bu b − = u (5.31)25or some u in H . The b that takes σ to σ can be multiplied by an arbitrary element of H . So if we replace the sum over σ ∈ [2 v ] by a sum over b, u , we will be over countingby | H | . So we conclude that the equality (5.27) is correct.It is instructive to note that the set of elements [2 v ] can be thought as the set ofcosets S v /H , where H = S v [ S ]. To see this, start by noting that S v acts on σ byconjugation to generate all the elements in [2 v ]. For any σ ∈ [2 v ], we have σ = ασ α − (5.32)for some α ∈ S v . If we multiply α on the right by β ∈ S v [ S ], we get the same σ . Wecan write [2 v ] = S v /H (5.33)Thus, we can rewrite (5.26) asNumber of Feynman Graphs = 1(4!) v v ! X u ∈ H X σ ∈ S v /H δ ( u σu − σ − ) (5.34)A similar argument implies that < v > = S v /H (5.35)Here < v > stands for the space of Σ ’s.Using that fact that the expression (5.21) is symmetric under the exchange of H and H , we can also writeNumber of Feynman Graphs = 12 v (2 v )! X u ∈ H X Σ ∈ S v /H δ ( u (Σ ) , Σ ) (5.36)Thus, by Burnside’s Lemma the number of Feynman graphs is also equal to the number oforbits of S v [ S ] acting on the set of vertex labels. Note that (5.36) looks slightly differentfrom (5.34) because Σ is a permutation, but Σ is not. The action of substituting i → u ( i )for some permutation u ∈ S v can be achieved by conjugation for Σ but not for Σ . In this subsection we show that the formula (5.24) for counting Feynman graphs in φ theory is computing an observable in S n TFT. Recall from section 2.1 that the expectationvalue of the observables tr ( σ U † ) , tr ( σ U † ) on the cylinder with boundary holonomies U and U are Z ( σ , σ ) = Z dU dU tr ( σ U † ) tr ( σ U † ) Z ( U , U ) = X γ δ ( σ γσ γ − ) (5.37)26umming this expectation value over permutations in the subgroups H , H Z ( H , H ) ≡ | H || H | X µ ∈ H X µ ∈ H Z ( µ , µ )= 1 | H || H | X µ ∈ H X µ ∈ H X σ ∈ S n δ ( µ σµ σ − ) (5.38)we recover the formula (5.21) for counting Feynman graphs in φ theory.There is an interesting subtlety we should comment on. In the large N expansion of U ( N ) 2dYM, we encounter observables which can be parametrized using permutations.For observables constructed from gauge-invariant polynomials of degree n in the holonomy U , the partition functions are functions only of the conjugacy class of σ in S n . In theabove expression (5.38), the sums over µ i do not run over entire S n conjugacy classes, butare restricted to H i .As explained in section 2.2 the observables in the Ω → N S n TFT. The quantity in (5.38) is a generalization of the observ-ables one gets from the large N expansions of U ( N ) 2dYM, but it is still an observablein the lattice S n TFT. The boundary observables are not invariant under conjugation by S n elements at the boundary. The S n conjugation symmetry is broken to H and H re-spectively. There is a general principle of Schur-Weyl duality which relates the symmetricgroup constructions to unitary groups (see [6, 35, 36, 37] for applications in gauge-stringduality). The wreath products of symmetric groups have also appeared in connectionwith symmetrised traces in the context of constructing eighth-BPS operators [38]. Wetherefore expect that these more general S n TFT observables can also be expressed interms of some construction with gauge theory involving unitary groups. We will leavethis clarification for the future.As mentioned in Section 2, the S n TFT is closely related to covering space theory. Inthe next section we will encounter expressions of the form (5.38) but without the sumover σ . We will show how cutting and gluing constructions of covering spaces use thedata µ , µ , σ . We have already argued that the Feynman graphs of φ theory correspond to elements ofthe double coset, S v \ ( S v × S v ) / ( S v [ S ] × S v [ S ] ) (5.39)27he S v on the left is the diagonal subgroup of the product group, i.e pairs of the form( σ, σ ). The symmetry factor of a Feynman graph corresponding to an orbit with repre-sentative ( σ , σ ) is the size of the stabilizer group. This can be computed by calculating X γ ∈ S v X µ ∈ S v [ S ] X µ ∈ S v [ S ] δ ( γσ µ σ − ) δ ( γσ µ σ − ) (5.40)Using one of the delta functions to perform the sum over γ we obtain X µ ∈ S v [ S ] X µ ∈ S v [ S ] δ ( σ µ − σ − σ µ σ − ) (5.41)By defining σ = σ − σ , we can write the formula for the symmetry factor asSym( σ ) = X µ ∈ S v [ S ] X µ ∈ S v [ S ] δ ( µ − σµ σ − ) (5.42)This expression relates more directly to the equivalent description of the double coset(5.39) as S v [ S ] \ S v / × S v [ S ] (5.43)Comparing to (2.10) we see that this is an observable in S n TFT, with H , H observableson the boundaries. This is illustrated in Fig. 12.Consider a cover of the cylinder. Choose a point on one boundary circle and label theinverse images of that point { , , · · · , n } . Following the inverse images of these pointsalong the boundary circle leads to a permutation µ which is constrained to be in H .Likewise there is a permutation µ of { ′ , ′ , · · · , n ′ } at the other boundary. Following apath on the cylinder which joins the two points will produce a permutation σ . We arefixing σ to lie in a fixed element of H \ S n /H .From the point of view of S n lattice TFT, the line joining the two chosen points onthe boundary circle, associated with a fixed permutation σ is a Wilson line. Given theclose relation, between S n lattice TFT and Hurwitz spaces, we can map the Wilson lineobservable to a construction in covering spaces of the cylinder.First notice that the cylinder can be obtained by gluing two ends of a square [ AA ′ B ′ B ]in Figure 13. This is a topological quotient which identifies the edge AB with A ′ B ′ . Toconstruct a covering space associated with the data µ , σ, µ subject to µ − σµ σ − = 1, weconsider cutting the square further into the rectangles [ AA ′ C ′ C ] and [ DD ′ B ′ B ] (Figure14). The cylinder is recovered by the gluings (Fig 14) AC = A ′ C ′ DB = D ′ B ′ S n lattice TFTobservable A’B B’ ABA AB = A’B’
Figure 13: Gluing a square to get cylinder CC ′ = DD ′ (5.44)Now take n copies of these pairs of rectangles, with labels [ A i A ′ i C ′ i C i ] and [ D i D ′ i B ′ i B i ],with i ranging from 1 to n , as in Figure 15. For the application to (5.38) we have n = 4 v .The union of rectangles { [ A A ′ C ′ C ] , · · · , [ A n A ′ n C ′ n C n ] } is quotiented by identifyingedges A i C i to the edges A ′ i C ′ i by the permutation µ . A ′ i C ′ i = A µ ( i ) C µ ( i ) (5.45)The rectangles [ D i D ′ i B ′ i B i ] are quotiented by identifications D ′ i B ′ i = D µ ( i ) B µ ( i ) (5.46)Finally we identify D i D ′ i = C σ ( i ) C ′ σ ( i ) (5.47)29 ’A A’BCD C’D’ Figure 14: Gluing a pair of rectangles to get cylinderThe covering map is specified by mapping each of the n rectangles, to the original rect-angles without labels in the obvious way (see Figure 15)The condition µ − σµ σ − = 1 ensures that if we consider the inverse image of theclosed contractible path shown in blue in Figure 16, the inverse image is also contractible.This type of construction is a standard part of covering space theory in connectionwith the Riemann existence theorem (see for example [39]).The generalizations of (2.8) visible in (5.38) and (5.42) involve restricting the sumsover µ , µ to specific subgroups H , H and leaving the σ unsummed. From the point ofview of S n TFT these constraints are easily implemented, since the degrees of freedom oneach edge form the whole group, where elements and subgroups can be chosen. Howeverthe boundary observables are not invariant under conjugation by S n and have no dual inthe large N S n elements and monodromies of covers, it is not surprising that a definite covering spaceconstruction for specific µ , µ , σ exists. We have provided such an explicit cutting-and-gluing construction here. There are clear analogies between the construction given andthe implementation of twisted boundary conditions in D-branes on orbifolds [40]. Thepresence of a Wilson line defect located at a line joining the 2 boundaries of the cylinderin the S n TFT picture suggests that there should be a D-brane interpretation. We leavethis as a future direction of research. Developing the connection of the lattice S n TFT tothe axiomatic approach to TFT and branes in [46] maybe a useful approach.In the standard discussions of the Riemann existence theorem for (branched) covers,one has a two-way relation. On the geometrical side, there are maps f with the equiva-lence f = f ◦ φ for holomorphic automorphisms φ (see more details in [6, 25]). On thecombinatoric side there are permutations with conjugation equivalence. From permuta-tions we can construct covers, and vice versa. The equivalence classes map to each other.Here we have permutations, but with some restrictions having to do with choices H , H .We can still construct some covers with this data. We have not fully articulated the cor-rect equivalences on the geometrical side for a 1-1 correspondence. The construction we30 B C D D’ C’ B’ A’ A’ B’ C’ A B C D A’ n B’ n C’ n C n D n D’ D’ n A n B n D AB A’C’D’ B’ C Figure 15: Gluing copies of rectangle pairs to get covering of cylinder - identificationsdetermined by µ , µ , σ gave should provide useful hints for the precise definitions on the geometrical side, whichwill provide the equivalence. We leave this as a problem for the future. In section 5.3 a formula expressing the number of Feynman graphs in terms of the starproducts of two cycle indices was given. This result is useful because formulas for thecycle index of the wreath product of two symmetric groups are known. Using these resultswe will write down rather explicit formulas for the number of vacuum graphs in φ theoryin section 6.1. 31 ’A A’BCD C’D’ Figure 16: Closed contractible path
Let d i denote the total number of vacuum Feynman graphs in φ theory with i vertices.To obtain an analytic formula for d i we will need the cycle indices of two wreath products, S n [ S ] and S n [ S ]. The known generating functions for these wreath products are Z S ∞ [ S ] [ t, ~x ] ≡ X n t n Z S n [ S ] ( ~x ) = e P ∞ i =1 ti i ( x i +8 x i x i +6 x i x i +3 x i +6 x i ) (6.1) Z S ∞ [ S ] [ t, ~x ] ≡ X n t n Z S n [ S ] ( ~x ) = e P ∞ i =1 ti i ( x i + x i ) (6.2)To compute N ( Z ( H ) ∗ Z ( H )) take the product of the coefficients of Q i x p i i in the twocycle indices, and then further take the product with Q i i p i p i !. This can be accomplishedif we make the substitution x i → √ iy i , to obtain˜ Z S ∞ [ S ] [ t, y i ] = Z S ∞ [ S ] [ t, x i = √ iy i ]˜ Z S ∞ [ S ] [ t, y i ] = Z S ∞ [ S ] [ t, x i = √ iy i ] (6.3)and then replace the ( y i ¯ y i ) p i with p i !. This leads to the following formula for the numberof φ vacuum graphs with v vertices d v = I t − v dt t I t − v dt t ∞ Y i =1 Z dy i d ¯ y i π e − P k y k ¯ y k ˜ Z S ∞ [ S ] [ t , y i ] ˜ Z S ∞ [ S ] [ t , ¯ y i ] (6.4)The integrals over y i and ¯ y i ensure that only terms with equal powers of y i and ¯ y i con-tribute, and they implement the substitution ( y i ¯ y i ) p i → p i !. The contours of integrationover t and t are both counter clockwise and they both encircle the origin.We could contemplate many ways of refining this result. For example, can we deter-mine how many of these vacuum graphs are connected? Before turning to this question,32t is instructive to ask how we can identify if a graph is connected or disconnected fromits description in terms of the pair Σ , Σ . A Feynman graph (Σ , Σ ) with v vertices isdisconnected if for some subgroup G = S v − p × S p with p > σ (Σ ) ∈ G for all σ ∈ Aut(Σ ). Let c i denote the number of connected Feynman graphs with i vertices.The total number of vacuum graphs with i vertices is the coefficient of g i in the partitionfunction Z ≡ ∞ X i =1 d i g i = ∞ Y i =1 (cid:20) − g i (cid:21) c i (6.5)This formula, given for example in [20], can be used to determine the c i once the d i havebeen computed. It is now straight forward to determine the number of vacuum diagrams.See Appendix D for numerical results. The next natural generalization is to consider Feynman graphs with E external legs.Summing Feynman graphs with E external legs produces an E -point correlation function.For the generic case when all E -points corresponds to different spacetime events, graphsobtained by permuting labels of the external points give distinct contributions to thecorrelation function. For this reason the automorphism group of the graph does notinclude any elements that permute the labels of external legs.For a graph with v vertices and E external legs, we obtain a total of 4 v + E halfedges. Σ is now an element of the conjugacy class [2 v + E ] of S v + E , while Σ contains v E numbers specifying how the half edges connecting to external points arelabeled. Since we do not have automorphisms that permute the half edges connectingto external points, H = Aut(Σ ) = S v [ S ] × ( S ) E . We can also write the subgroup as S v [ S ], with the understanding that it is acts by keeping fixed the last E integers from theset { , , · · · , v + E } that S v + E acts on. Consequently, Feynman graphs in φ theorywith v vertices and E external points are in one-to-one correspondence with elements ofthe double coset ( S v [ S ] × S E ) \ S v + E /S (4 v + E )2 [ S ] (6.6)To compute the number of graphs the only new cycle index we need is Z S v [ S ] × S E ( x , · · · , x n ) = x E Z S v [ S ] ( x , · · · , x n ) (6.7)It is now straight forward to obtain the number of Feynman graphs in φ theory with E external lines N ( H , H ; G ) = N ( Z ( H ) ∗ Z ( H ); S v + E ) = X p ⊢ (4 v + E ) Z S n [ S ] × S E p Z S v + E [ S ] p Sym( p ) (6.8)33he notation p ⊢ (4 v + E ) indicates that p runs over all partitions of 4 v + E .We can again refine this counting by asking how many of these graphs are connected.For concreteness, focus on the case E = 2. Let d ,i denote the number of Feynman graphswith two external lines and i vertices and let c ,i denote the number of these graphs thatare connected. We can again write down a partition function which gives the total numberof graphs with two external lines and i vertices as the coefficient of g i . In this formula thenumber of vacuum graphs c i and d i defined in the last subsection participate. We find ∞ Y i =1 (cid:20) − g i (cid:21) c i ∞ X k =0 c ,k g k ! = ∞ X k =1 d ,i g i ! (6.9)This formula can be used to determine the c ,i once the d ,i have been computed using(6.8). See Appendix D for numerical results. φ theory and other interactions Although our discussion has focused on φ theory it should be clear that our methods aregeneral. To illustrate this we will now consider a φ interaction. A graph with v verticeshas 3 v half edges. Σ is now an element of the conjugacy class [2 v ] of S v , while Σ contains v H = S v [ S ] and H = S v [ S ]. Vacuum Feynman graphs in φ theory with v vertices are in one-to-one correspondence with elements of the double coset S v [ S ] \ S v /S v [ S ] (6.10)To compute the number of graphs the only new cycle index we need can be read from thegenerating function Z S ∞ [ S ] [ t ; x , x , · · · ] = ∞ X n =0 t n Z S n [ S ] ( x , x , · · · , x n )= e P ∞ i =1 ti i ( x i +3 x i x i +2 x i ) (6.11)It is now straight forward to obtain the number of vacuum Feynman graphs in φ N ( Z ( H ) ∗ Z ( H )) = X p ⊢ v Z S v [ S ] p Z S v [ S ] p Sym( p ) (6.12)Feynman graphs with E external legs are in one-to-one correspondence with elementsof the double coset ( S v [ S ] × S E ) \ S v + E /S (3 v + E )2 [ S ] (6.13)34sing the generating function Z S n [ S ] × S E ( x , · · · , x n ) = x E Z S n [ S ] ( x , · · · , x n ) (6.14)it is straight forward to compute the number of Feynman graphs with v vertices and E edges N ( H , H ; S v + E )) = N ( Z ( H ) ∗ Z ( H ); S v + E ) = X p ⊢ (3 v + E ) Z S v [ S ] × S E p Z S v + E [ S ] p Sym( p )(6.15)The notation p ⊢ (3 v + E ) indicates that p runs over partitions of (3 v + E ). See AppendixD for numerical results. Cubic graphs have recently played a role in studies of N = 8SUGRA [49] as well as in the classification of N = 2 4-dimensional gauge theories [48].The φ theory in 6 dimensions is known to be asymptotically free [50]. Hence this case isof special interest.Finally, consider a theory with both cubic and quartic vertices. To count the vacuumgraphs having v cubic and v quartic vertices, we would repeat the above discussion with S v [ S ] × S v [ S ] replacing the groups S v [ S ] (for φ ) or S v [ S ] (for φ ). We have already argued that the number of vacuum graphs in φ theory is computing anobservable in the Lattice TFT with S n gauge group. The counting of Feynman graphsin this section also has a formulation in Lattice TFT: One simply uses (5.38) with theappropriate H , H described above.The formula (5.42) for the symmetry factor as an amplitude is also directly applicablehere. The identification of Feynman graphs as elements of the double coset S n \ ( S n × S n ) / ( H × H ) is a nice unifying picture. The groups H and H are the symmetries of the interaction(black) and bivalent (white) vertices respectively. This allows a simple generalization fromordinary graphs to ribbon graphs: symmetries must by definition preserve the genus ofthe ribbon graph. The cyclic order of the labels at a black vertex must be respected if thegenus is to be preserved. Thus, for example, the replacement H = S v [ S ] → H = S v [ Z ]takes us from counting Feynman graphs in φ theory to counting ribbon graphs in Tr φ theory. 35igure 17: Cyclic versus symmetric for Feynman vertex versus ribbon vertexIt is now clear that ribbon graphs can also identified with elements of a double coset.For example, the vacuum graphs with v vertices of a matrix model with Tr φ interactionare elements of the coset S v \ ( S v × S v ) / ( S v [ Z ] × S v [ S ]) (7.1)where we have identified H = S v [ Z ] and H = S v [ S ]. Using the insights of the previous subsection and repeating the argument of section 5,we can generate all possible ribbon graphs by allowing S v [ Z ] to act on the set of allpossible Wick contractions, that is, the conjugacy class [2 v ] of S v . Each orbit of S v [ Z ]is a distinct ribbon graph. Thus, the number of ribbon graphs can be obtained, usingBurnside’s Lemma, asNumber of ribbon graphs = 14 v v ! X γ ∈ S v [ Z ] X σ ∈ [2 v ] δ ( σγσ − γ − )= 1 | H | X γ ∈ H X σ ∈ S v /H δ ( σγσ − γ − ) (7.2)Using the results of section 5.4, we can also writeNumber of ribbon graphs = 12 v (2 v )! X γ ∈ S v [ S ] X σ ∈ [4 v ] δ ( σγσ − γ − )= 1 | H | X γ ∈ H X σ ∈ S v /H δ ( σγσ − γ − ) (7.3)36here is also a way of writing this that is manifestly symmetric between exchange of H , H Number of ribbon graphs = 1 | H || H | n ! X R ⊢ n X γ ∈ H X γ ∈ H χ R ( γ ) χ R ( γ )= X R ⊢ n M R H M R H (7.4)Recall the notation M R H for the multiplicity of the one-dimensional representation of H when the irrep R of S v is decomposed into representations of the subgroup H . Ribbon graphs come equipped with a natural notion of a genus. For this reason notall of the contractions leading to a given Feynman graph will produce the same ribbongraph. As an example, consider vacuum graphs in φ theory with v = 1 vertex. All threecontractions give the same Feynman graph, but two of them give a genus zero ribbongraph and one gives a genus one ribbon graph : see Figure 18.The line of thinking developed above allows an elegant answer to the question of howmany ribbon graphs are there for a given Feynman graph. In this case Σ can be chosenas a permutation made of 4-cyclesΣ = (1234)(5678) ... (4 v − , v − , v − , v ) (7.5)The symmetry group of this permutation S v [ Z ]. This is the set of permutations in S v which commutes with Σ . The Wick contractions are described by Σ which are pairings,i.e permutations in the conjugacy class [2 v ] of S v . Ribbon graphs are orbits of S v [ Z ]action (by conjugation) on the permutations in [2 v ]. We can first decompose the setof all possible the pairings in [2 v ] into orbits of the group S v [ S ]. Each such orbit isa Feynman graph. Then decompose each orbit into orbits of the subgroup S v [ Z ] of S v [ S ]. The elements within one orbit of S v [ S ] will fall generically into multiple orbits of S v [ Z ], corresponding to multiple ribbon graphs. We can view the genus of the worldsheetdescribed by the ribbon graph as an invariant of the orbits of S v [ Z ] acting on 2-cycles. Iftwo 2-cycles are in the same orbit, they must be associated with the same genus. Considerthe permutation obtained by multiplying Σ with any Wick contraction which belongs toa fixed orbit of S v [ Z ]. Denote this permutation by σ and the number of cycles in thispermutation by c . The genus g of each S v [ Z ] orbit is related to the number of cycles inthe permutation c and number of vertices v by2 − g = c − v (7.6)This is obtained by applying the Riemann Hurwitz formula to maps the case of mapswith three branch points determined by Σ , Σ , σ [19].37t is straightforward to perform, in software such as GAP, the decomposition intoorbits of S v [ S ], and then refine the decomposition according to S v [ Z ].Figure 18: At v = 1 the three possible Wick contractions give the same Feynman graph.Two of them give a genus zero ribbon graph and one gives a genus one ribbon graph.A number of interesting directions can be contemplated. Different ribbon graphswith the same underlying Feynman graph will have the same space-time integrals butpossibly different genus. Our methods provide information about how the space-timedependence allows a certain range of genera. When we are dealing with vacuum graphs,each Feynman graph contributes a number. However, our techniques can be applied tographs that have external legs in which case we have explicit space-time dependences.Presumably a Feynman graph with a certain “complexity” - reflected in its space-timedependence, will allow a certain range of genera. The more complex it becomes, the moregenera it will allow. This could be studied quantitatively with the current set-up. In this section we would like to count the number of ribbon graphs using cycle indices.For 4-valent ribbon graphs, the total number of graphs for v vertices, is obtained by usingthe formula N ( Z ( S v [ Z ]) ∗ Z ( S v [ S ]) ) (7.7)Thus, we need the cycle index of S v [ Z ]. The cycle index of any cyclic group is Z Z n ( ~x ) = 1 n X d ⊢ n ϕ ( d ) x n/dd (7.8)where ϕ ( d ) is the Euler totient function. For Z we find Z Z ( ~x ) = 14 ( x + x + 2 x ) (7.9)38sing known results for the cycle index of wreath products we now find Z S ∞ [ Z ] [ t, ~x ] ≡ X n t n Z S n [ Z ] ( ~x ) = e P ∞ i =1 ti i ( x i + x i +2 x i ) (7.10)It is now straight forward to obtain explicit answers for the number of ribbon graphs.The methods at hand also apply to ribbon graphs with arbitrary numbers and typesof traces. For a ribbon graph with v single traces, v double traces, v triple traces etc.we would use the group S v × S v [ Z ] × S v [ Z ] · · · to count graphs. We have focused on real fields corresponding to particles which are neutral. In this sectionwe explain how our methods can be extended to the case of complex fields correspondingto charged particles. We will refer to the charged particle as an electron with a view toapplications to QED. Since are just counting Feynman graphs, we do not track minussigns coming from the anti-commuting nature of fermion fields. As far as the number ofdiagrams or their symmetry factors goes, there is no difference between QED or Yukawatheory. Of course, in QED certain diagrams vanish automatically due to Furry’s Theorem.In the next section we will consider the problem of counting the QED diagrams that remainafter Furry’s Theorem is applied.The fact that we are dealing with charged particles implies that each edge will nowhave a preferred direction, determined by the flow of charge. The model we have in mindhas a cubic vertex in which the charged particle interacts with a neutral particle. Thisstructure of the vertex matches both the Yukawa interaction and the coupling of chargedfermions to a gauge field. We will refer to the neutral particle as the photon. Thus, eachvertex has 3 edges: an incoming electron, an outgoing electron and a photon. Considervacuum Feynman graphs made of 2 v vertices. We again clean the graph, by introducingnew vertices in the middle of each edge. Label the vertices so that 1 , ..., v are attachedto the incoming electrons, 2 v + 1 , · · · , v go with the outgoing electrons and 4 v + 1 · · · v go with the photons.So we can describe the vertices byΣ = v Y i =1 < i, v + i, v + i > (8.1)The Wick contractions are pairings of the formΣ = v Y i =1 ( i, v + τ ( i )) · σ (8.2)39igure 19: QED verticesThe permutation σ is in [2 v ] inside the S v which acts on the last 2 v among the { , · · · , v } .The permutation τ is a general permutation of 2 v objects. It tells us which of the first2 v (incoming electrons) go with which of the 2 v + 1 · · · v (outgoing electrons). The sym-metry γ of Σ is just S v of exchanging the 2 v brackets. Because all legs are distinct thereis no symmetry exchanging the legs at a vertex. Given the form of Σ we see that this isthe diagonal S v of the S v × S v × S v acting on the electron, anti-electron and photonlabels. We will call this Diag ( S v ).The automorphisms of the Feynman graph are those elements γ ∈ Aut(Σ ) = Diag ( S v )which also leave fixed the Σ pairing determined by ( σ , τ ). The order of this group againdetermines the symmetry factor of the Feynman graph.Distinct Feynman graphs are orbits of the permutationsDiag ( S v ) → S v × S v × S v (8.3)acting on Σ ( σ , τ ), where the σ , τ are described above. We know that the stabilizer of σ ∈ [2 v ] in S v is conjugate to S v [ S ]. This leads to S v /S v [ S ] = [2 v ] (8.4)The permutation τ mixes the first 2 v incoming electrons with the next 2 v outgoingelectrons. The stabilizer of such a permutation is the diagonal S v which simultaneouslymoves the first 2 v with the next 2 v , so that τ is running over the coset( S v × S v ) / Diag ( S v ) (8.5)We use the subscript 2 because it is the diagonal of the two S v . This is enough toconclude that the Feynman graphs are in one-to-one correspondence with the points ofthe double cosetDiag ( S v ) \ ( S v × S v × S v ) / (Diag ( S v ) × S v [ S ]) (8.6)The Diag is the diagonal of the first two S v in S v × S v × S v .40 .1 Double cosets of product symmetric groups To count the number of Feynman graphs, we need to count the number of points in thedouble coset H \ ( S v × S v × S v ) /H (8.7)where H = Diag ( S v ) and H = Diag ( S v ) × S v [ S ]. It is instructive to approach thisproblem by first considering cosets H \ G/H (8.8)where G is a general product of symmetric groups S n × S n · · · × S n k , and H , H aregeneral subgroups. Let us denote by N ( H , H ; G ) (8.9)the number of points in this double coset. The case N ( H , H ; G = S n ) = N ( H ∗ H ) isthe one we already discussed in sections 5.2 and 5.3. In the case at hand, we need N ( H , H ; S v × S v × S v ) (8.10)As a consequence of the fact that the numerator group has changed, when we computethe star product, it will turn out that we should multiply the coefficients of Z ( H ) , Z ( H ),and now weight them by the symmetry factors appropriate for S n × S n × · · · S n k . Thisis a generalization of Read’s original formula [20] so it is worth discussing in some detail.When the numerator group G is a product of symmetric groups, a subgroup such as H will have some number of elements in each conjugacy class of G , which is specified bythree partitions. { p (1) , p (2) · · · p ( k ) } (8.11)For the case at hand k = 3. The symmetry of such a conjugacy class isSym( { p ( i ) } ) = Y i Sym( p ( i ) ) = Y i =1 2 v Y j =1 j p ( i ) j p ( i ) j ! (8.12)We will restrict the discussion to k = 3, but the generalization to any k is immediate.For a subgroup H of G it is natural to define Z H → G = X p (1) ,p (2) ,p (3) Z H → Gp (1) ,p (2) ,p (3) Y i,j,k x p (1) i i y p (2) j j z p (3) k k (8.13)41here the coefficients Z H → Gp (1) ,p (2) ,p (3) keep track of the number of permutations in the sub-group with the cycle structure specified by the 3 partitions. For two subgroups H , H ,we can use the two cycle indices (with respect to G ) to define a generalization of the starproduct in (5.19) as follows N ( H , H ; G ) = X p Z H → Gp (1) ,p (2) ,p (3) Z H → Gp (1) ,p (2) ,p (3) Sym( { p ( i ) } ) (8.14)We can understand this formula by adapting the reasoning in [20]. For any ρ ∈ G , theproduct h ρ h gives | H | × | H | elements, that all belong to the same equivalence class inthe coset (8.9), as h runs over H and h runs over H . Some elements will be repeated.When this happens h ρh = ˜ h ρ ˜ h (8.15)which implies that ˜ h − h = ρ ˜ h h − ρ − (8.16)Clearly ˜ h − h ∈ H ∩ ρH ρ − . Given ˜ h , ˜ h and an element of H ∩ ρH ρ − , h and h are uniquely determined. Consequently the number of times that ρ is repeated, ν ( ρ ), is | H ∩ ρH ρ − | . Every element in G which appears in the equivalence class of ρ comeswith this same multiplicity in H ρH . Hence, the number of distinct elements in theequivalence class of ρ n ( ρ ) = | H || H | ν ( ρ ) (8.17)Consider X ρ ∈ G ν ( ρ ) (8.18)For an equivalence class containing ρ , we get a contribution n ( ρ ) ν ( ρ ) = | H || H | . This isindependent of the equivalence class. If there are N d equivalence classes, we get X ρ ∈ G ν ( ρ ) = N d | H || H | (8.19)Rearranging we find N d = 1 | H || H | X ρ ∈ G ν ( ρ ) (8.20)42o compute the sum over ρ of ν ( ρ ) we can choose u ∈ H and u ∈ H and count howmany ρ ’s obey u = ρu ρ − (8.21)Since u and u are conjugate, they have the same cycle structure. Thus, to compute thesum over ρ of ν ( ρ ) we need to fix a cycle structure, multiply the number of elements in H with this cycle structure by the number of elements in H with this cycle structure,then multiply by the number of elements of G that leave this cycle structure invariant,and then finally sum over cycle structure. This is precisely what (8.14) is computing. In this section we will apply the formulas obtained in the previous section to obtainexplicit results for the number of Feynman graphs. We need two cycle indices. For H = Diag ( S v ) we have Z H → G = X p ⊢ v p ) v Y j =1 ( x j y j z j ) p j (8.22)For H = Diag ( S v ) × S v [ S ], we have Z H → G = X q ⊢ v X r ⊢ v q ) Z S v [ S ] r v Y j =1 ( x j y j ) q j z r j j (8.23)To calculate the N ( H , H ; G ), we need to multiply like terms in the (8.22) and (8.23),weighted by an appropriate symmetry factor. Picking up like terms forces q = p = r sothat N ( H , H ; G ) = X p ⊢ v p ) Z S v [ S ] p Sym( p ) · (Sym( p )) = X p ⊢ v Z S v [ S ] p Sym( p ) (8.24)For the wreath product Z S ∞ [ S ] [ t ; x , x , · · · ] ≡ ∞ X n =0 Z S n [ S ] ( x , · · · , x n ) t n = e P ∞ i =1 ti i ( x i + x i ) (8.25)43o get the desired counting from this, we need to replace Q i x p i i wth i p i p i !. Equivalently,do a replacement x i → iy i and expand in y i replacing y p i i with p i !. In terms of˜ Z [ t ; y i ] = Z S ∞ [ t ; x i = iy i ]= e P ∞ i =1 ti i ( iy i +2 y i ) (8.26)the counting of vacuum graphs with 2 v vertices can be written as N ( H ( v ) , H ( v ); G ( v )) = I t − v dtt (cid:0)Y i Z ∞ dy i e − y i (cid:1) ˜ Z [ t ; y , y · · · ] (8.27)See Appendix D for explicit numerical results obtained using these formulas. The number of Feynman graphs is counted by the formula1 | H || H || G | X u ∈ H X u ∈ H X γ ∈ G δ ( u γu γ − ) (8.28)We have already established that G = S v × S v × S v H = Diag ( S v ) H = Diag ( S v ) × S v [ S ] (8.29)This is a cylinder partition function in S ∞ × S ∞ × S ∞ gauge theory which is Schur-Weyl dual to the large N limit of U ( N ) × U ( N ) × U ( N ) gauge theory. As before thecorrespondence holds in the zero area limit. Above we have counted the total number of Feynman diagrams in QED or Yukawa theory.In the case of QED, Furry’s Theorem proves a fermion loop punctuated with an oddnumber of photons vanishes [44]. In this the number of QED Feynman graphs, notvanishing by the Furry constraint, are counted. Enumeration of some Feynman graphswith their symmetry factors for this case is given in [43].44igure 20: Even cycle lengths and Furry’s theorem
Following the discussion of the last section, each QED Feynman graph is specified byΣ = v Y i =1 < i ; 2 v + i ; 4 v + i > Σ ( σ , τ ) = v Y i =1 ( i, v + τ ( i )) · σ (9.1)Recall that τ specifies how incoming and outgoing electrons are connected. Consequently,the number of cycles appearing in τ is equal to the number of fermion loops in the graphand the length of each cycles is equal to the number of photons decorating the loop. Itis clear that the constraint that each fermion loop has only an even number of photonsis easily imposed in the (Σ , Σ ) description by requiring that τ consists of permutationswhich only have cycles of even length, as shown in Fig 20.The orbits of the group G = S v × S v × S v acting on the pairs (Σ , Σ ) correspondto the inequivalent Feynman graphs. The set of γ ∈ G which preserve the pair (Σ , Σ )generates the automorphism group of the graph and the order of this automorphism groupis the symmetry factor. Preserving Σ forces γ to be in the diagonal S v of G :For 1 ≤ i ≤ vi → γ ( i )2 v + i → v + γ ( i )4 v + i → v + γ ( i ) (9.2)We can write this as γ ◦ γ ◦ γ , where the first γ acts on the first 2 v according to the second45ine of (9.2), while leaving the subset { v + 1 , · · · , v } fixed ; the second γ moves only thesubset { v + 1 , · · · , v } according to the 3rd line; and the third γ moves only the subset { v + 1 , · · · , v } according to the 4th line.The symmetry factor of a Feynman graph specified by the standard Σ and a generalΣ is | Aut( (cid:2) Σ (cid:3) H ) | = X γ ∈ S v δ ( γ ◦ γ ◦ γ Σ γ − ◦ γ − ◦ γ − Σ − ) (9.3)Σ is not a permutation in G , since the τ mixes the first 2 v with the second 2 v . Themultiplication can be viewed in S v or S v × S v A simple application of the Burnside Lemma implies that the number of Feynmangraphs remaining after Furry’s Theorem is applied is1(2 v )! X γ ∈ S v X σ ∈ [2 v ] ∈ S v X τ ∈ S v :[ τ ] even δ ( γ ◦ γ ◦ γ Σ ( σ , τ ) γ − ◦ γ − ◦ γ − Σ − ( σ , τ ) )(9.4)Another approach to the same counting problem would be to implement the Furryconstraint in the double coset language. Recall the double coset relevant for QED isDiag ( S v ) \ ( S v × S v × S v ) / Diag ( S v ) × S v [ S ] (9.5)Using (5.21) with the G, H , H identified according to (9.5) the size F of this doublecoset is counted by F = 1(2 v )!(2 v )!2 v v ! X σ ∈ S v X τ ∈ S v X ρ ∈ S v [ S ] X b ,b ,b ∈ S v δ S v × S v × S v ( σ ◦ σ ◦ σ b ◦ b ◦ b τ ◦ τ ◦ ρ b − ◦ b − ◦ b − ) (9.6) b acts on the incoming electrons, b on the outgoing electrons and b on the photons. σ permutes vertices in the graph, τ permutes electron lines and ρ permutes photon lines.This can be simplified to F = 1(2 v )!(2 v )!2 v v ! X σ ∈ S v X τ ∈ S v X ρ ∈ S v [ S ] X b ,b ,b ∈ S v δ S v × S v ( σ ◦ σ b ◦ b τ ◦ τ b − ◦ b − ) δ S v ( σb ρb − ) (9.7)Solving the second delta function for σ , we can plug back into the first delta function anddo a redefinition of the sums P b ,b to absorb the b , so as to get F = 1(2 v )!2 v v ! X τ ∈ S v X ρ ∈ S v [ S ] X b ,b ∈ S v S v × S v ( ρ ◦ ρ b ◦ b τ ◦ τ b − ◦ b − )= 1(2 v )!2 v v ! X ρ ∈ S v X τ ∈ S v [ S ] X b ,b ∈ S v δ S v ( ρb τ b − ) δ S v ( ρb τ b − )= 1(2 v )!2 v v ! X ρ ∈ S v [ S ] X b ,b ∈ S v δ S v ( b − ρ − b b − ρb )= 12 v v ! X ρ ∈ S v [ S ] X τ ∈ S v δ S v ( τ ρτ − ρ − ) (9.8)In the last line we have recognized that, given the interpretation of b , b it is clear that τ = b − b is the permutation τ in (9.1). The Furry constraint is now easily implemented.Thus, the number of Feynman graphs remaining after the Furry constraint is implementedis given by Number of Feynman graphs for QED with Furry constraint= 12 v v ! X τ ∈ S v : even X ρ ∈ S v [ S ] δ S v ( τ ρτ − ρ − ) (9.9)The permutation τ is constrained to have even cycles only and we are summing overelements of S v [ S ]. For each element of S v [ S ], the weight is the number of permutationswith even cycles only, which commute with the given permutation in S v [ S ].From the first line of (9.8) we see that an equivalent double coset description of thecounting is S v \ ( S v × S v ) / Diag ( S v [ S ]) (9.10)This gives a slightly simpler connection to observables for 2d Yang-Mills with cylindertarget space. Namely we have a connection to U ( ∞ ) × U ( ∞ ) rather than U ( ∞ ) × U ( ∞ ) × U ( ∞ ) as described earlier. Comparing the expressions (9.8) and (9.9) with (7.4) for ribbon graph counting, it becomesclear that the counting of QED Feynman graphs can be matched with counting ribbongraphs. The restriction to vertices of valency 4 in (7.4) is being relaxed to allow arbitraryvalencies in (9.8) and arbitrary even valencies in (9.9). We will now explain the bijectionbetween ribbon graphs and QED/Yukawa graphs which explains the equality of countingformulae.The general QED/Yukawa vacuum graph has loops with vertices where the photonjoins the loop. These loops determine 2 v cycles which form a permutation in S v . Tobuild a bijection to ribbon graphs think of the photon labels in Figure 19 as attached47o the vertices. The photon contractions determine an element of [2 v ]. At the centreof each electron loop draw a point and radiate edges (spokes) to intersect the edges ofthe loop, one spoke between each pair of electron-photon vertices. Use the arrow on theelectron propagator to move each end of the photon propagator along the electron looptowards the intersection of the spoke. Erase the electron loop, leaving the spokes andthe photon propagators connecting them. This gives a graph with vertices of arbitraryvalency, but with the vertices being equipped with a cyclic order, which is nothing but aribbon graph, completing the construction of the bijection. This process is illustrated inFigure 21. The bijection demonstrates that the number of QED vacuum graphs with 2 v vertices is the same as the number of ribbon graphs with v edges (the photon propagatorsbecome edges). Figure 21: QED graphs to ribbon graphsThis means that the total valency of the vertices in the ribbon graph is 2 v . Eachvertex can have any integer valency compatible with this constraint. Thus, following thediscussion (7.3) and (7.7), the number of QED/Yukawa vacuum graphs with 2 v verticesis Number of Feynman graphs for QED = X p ⊢ v N ( Z ( H ) ∗ Z ( H p )) (9.11)where p is a partition of 2 v , the group H is S v [ S ] and the group H p is S p [ Z ] × S p [ Z ] × · · · S p v [ Z v ] (9.12)The function N of two cycle index polynomials is as defined in (5.19). For QED with theFurry constraint implemented, the valencies are constrained to be even so thatNumber of Feynman graphs for QED with Furry constraint48 X p ⊢ vp even N ( Z ( H ) ∗ Z ( H p )) (9.13)Given the connection between ribbon graphs and Belyi pairs [19] we see that QEDcounts the number of clean Belyi pairs (bi-partite embedded graphs with bivalent whitevertices) with degree v and black vertices of any valency. QED with the Furry constraintcounts the number of such Belyi pairs with black vertices of even order only.
10 Discussion
We have seen that the counting of Feynman diagrams can be interpreted in terms of stringamplitudes with a cylinder target space. The nature of the interaction vertices determinesa permutation symmetry group H and the Wick contractions determine a permutationsymmetry group H . The counting could also be written in terms of commuting pairs ofpermutations, where one permutation lives in one of these subgroups, and another onelives in a coset involving the other group. This latter realization can be interpreted interms of covering maps of a torus target space, such as those that arise in the stringtheory of 2d Yang Mills for torus target space [3, 4], except that there is a constraint onthe two permutations associated with the cover, by following the inverse image of pathsalong the a and b cycle.In 2d Yang Mills, one encounters a sum over commuting permutations s , s which isequivalent to counting the set of all covers of the torus by a torus. This sum is invariantunder the SL (2 , Z ) of the space-time torus, which acts on these permutations by S : (cid:18) s s (cid:19) → (cid:18) s s − (cid:19) T : (cid:18) s s (cid:19) → (cid:18) s s s (cid:19) (10.1)and obeys the SL (2 , Z ) relations : S (cid:18) s s (cid:19) → (cid:18) s − s − (cid:19) ( ST ) (cid:18) s s (cid:19) → (cid:18) s s (cid:19) (10.2)In the case at hand, the SL (2 , Z ) of space-time is broken in that s is summed over H while s is summed over S v /H = [2 v ]. 49hile making two choices of observable at each end of a cylinder seems like a naturalconstruction for string theory with a cylinder, the constraints on the covering maps ofthe torus look rather intricate, and are not of any sort that has been discussed in theliterature on topological strings. Nevertheless, there might well be a string constructionwhich sums over maps with constraints of this sort.In any case, the Feynman diagram combinatorics suggests two emergent dimensionsof cylinder or torus, with one formulation possibly more appealing than the other. In theconstruction of correlators of general trace operators in the Hermitian Matrix model, thereis another 2-dimensional target space emerging, which is the sphere with three punctures[19].In the application of Feynman graphs to QFT, the Feynman graph counting is onlypart of the answer. When there are external edges, each edge is associated with a space-time position (or momentum), each of which takes values in R . The computation ofGreens functions h φ ( x ) φ ( x ) · · · φ ( x E ) i is a sum over Feynman diagrams, which live in DC ( v, E ) = ( S v [ S ] × S E ) \ S v + E /S v + E/ [ S ] (10.3)This has an action of S E . Projecting down the S E orbits gives another double coset.There is a fibration ( S v [ S ] × S E ) \ S v + E /S v + E/ [ S ] ← S E ↓ ( S v [ S ] × S E ) \ S v + E /S v + E/ [ S ] (10.4)We may write G ( x , x , · · · , x E ) = X Σ G ( x , x , · · · x E ; Σ) (10.5)where Σ lives in the double coset (10.3). The Bose symmetry of G ( x , x , · · · , x E ) arisesas an S E invariance of the summands G ( x σ (1) , x σ (2) , · · · x σ ( E ) ; σ (Σ)) (10.6)This shows that in some sense the space-time coordinates ( R ) E and the double coset DC ( v, E ) associated with covers of a cylinder may usefully be viewed as being on anequal footing when we work out the physics. It is tempting to infer that this is indicativeof an underlying R × Cylinder or R × Torus in four dimensional QFTs. We have shownthat the extra two dimensions emerge from the Feynman graph combinatorics, but itremains to be seen if there is a dynamical six dimensional picture along these lines, andhow it might relate to other appearances of six dimensions in 4D QFTs such as [51].A very interesting problem is to interpret the full correlators or S-matrices, in termsof the cylinder (or torus) emerging from combinatorics, with the R which is manifestly50here to begin with. This would be an example of a holographic dual of a quantum fieldtheory, analogous to AdS/CFT, where from the point of view of the dual gauge theory,an emergent S along with an emergent radial direction, resulting in M × S , with theoriginal R of the gauge theory realized as the boundary of the M .If this is indeed possible with φ theory, it would be an example of gauge-string dualitywith an interesting difference from all the examples known so far. It would not rely onthe large N expansion, and ribbon graphs thickening into string worldsheets. Rather itwould be a realization through the fundamental link between permutations and strings,of the physical expectation of holography [52, 53]. This makes it very important to knowwhether this works or not. For some general recent discussions of the scope of holographysee [54]. Another direction for relating Feynman graphs to string amplitudes, by directlyidentifying the integrand on the moduli space of complex structures of string worlsheets,has been advocated in [55]. The ideas emerging from the permutations-strings connectionmay well have interesting overlaps with this programme, which we will leave to futureinvestigation.If there are indeed new full-fledged gauge-string dualities for theories such as φ in fourdimensions, or φ in six dimensions or indeed QED or QED-like theories, there are someobvious questions to be addressed. What is the map between physical parameters? Inthe φ theory, Feynman graphs with v vertices are weighted with g v . We have associatedthem with cylinder worldsheets covering a target space cylinder, with maps of degree 4 v .If the cylinder has area A , the Nambu-Goto action will weight such strings with e − nA .This suggests that the QFT coupling constant g should be identified with e − A .Another obvious question : What is the string coupling of this dual string theory?Since all the details of the field theory interaction translate into the group H at a bound-ary of the cylinder amplitude, interaction in the field theory do not translate directlyinto interactions of the string. A related question is whether there is some modificationof the Feynman graph counting problem on the field theory side which leads to highergenus string worldsheets covering the cylinder. This requires further exploration of thesort of string amplitudes that come from various QFT Feynman diagrams. The addedmotivation for these investigations is that the stringy picture leads directly to powerfulcounting results, such as those in Appendix D. We have argued that there is a link between Feynman graph counting and string ampli-tudes by recognizing the combinatorics of worldsheet maps to a cylinder in the Feynmangraph combinatorics. The string theory in question involves the one that appears in thelarge N expansion of 2dYM with U ( N ) gauge group. This string theory also has a space-time description in terms of topological lattice theory with S n (for all n , hence probablybetter described as S ∞ ) as gauge group. The observables needed to describe Feynman51raph combinatorics are slightly more general than the ones that appear in U ( N ) 2dYM.They involve boundary observables which are not invariant under the entire S n in thelattice TFT description, but only certain specified subgroups H , H .Several worldsheet approaches to the string theory of 2dYM have been proposed[6, 7, 8, 9, 10]. It will be very interesting to develop, in these approaches, the boundaryobservables which count the Feynman graphs and their symmetries. The geometrical de-scription in terms of gluing world-sheets in section 5.6 should be useful. An interestinggoal would be to find new worldsheet methods for calculating Feynman graph combina-torics, which may be efficient for obtaining asymptotic results. We have developed a method to think about the combinatorics of Feynman graphs, in-volving pairs (Σ , Σ ) of combinatoric data. This was discussed from the point of viewof double cosets, which provide a link to generating functions. From the point of view ofconstructing the Feynman graphs, perhaps the most immediate lesson extracted from thestudy of the pairs (Σ , Σ ) is thatA Feynman graph is an orbit of the vertex symmetry group acting on theWick contractions.It is a simple exercise to obtain both the vertex symmetry group and the set of Wickcontractions given a quantum field theory. We have listed this data in the table below forthe theories that we have considered here. The data is relevant to vacuum graphs - thereare simple generalizations including external edges.Group acts on Feynman Graph problem S v [ S ] permutations in [2 v ] of S v φ theory S v [ S ] permutations in [2 v ] of S v/ φ theory S v [ S ] × S v [ S ] permutations in [2 (3 v v ] of S v +4 v φ + φ theory S v [ S ] All permutations in S v Yukawa/QED S v [ S ] All even-cycle permutations in S v Furry QED S v [ Z ] permutations in [2 v ] of S v Large N expansion of Matrix φ The double coset picture allows us to recognize that one can also consider the symmetryof the Wick contractions (e.g S v [ S ] in case of φ ) acting on the cosets associated withthe vertices ( S v /S v [ S ] in this case). 52 We showed that the counting of vacuum graphs in QED/Yukawa and QED with Furryconstraint implemented can be mapped to counting of ribbon graphs. In the case of QEDwith Furry constraint, there is an even-only restriction on the vertices. We know, fromthe theory of Dessins d’Enfants [12], that there is an action of the absolute Galois group
Gal ( ¯ Q / Q ) on ribbon graphs. (For recent applications of Dessins d’Enfants to HermitianMatrix Model correlators, supersymmetric gauge theories and extensive references to theMathematical literature see [13, 19, 14, 15].) This means that there is an action of theGalois group on the QED vacuum diagrams. The list of orders of the vertices is a Galoisinvariant [16], so the Furry theorem restriction continues to allow the Galois action toclose.A related fact is that sums over permutations restricted by conjugacy classes are knownto be sums over Galois group orbits. X σ ∈ T X σ ∈ T X σ ∈ T δ ( σ σ σ ) (10.7)Given the link we have observed between counting ribbon graphs and counting QEDgraphs, we have Galois actions on the QED/Yukawa and Furry QED vacuum graphs. Wedo not know if such is the case for the vacuum graphs of scalar field theories. The answerdepends on whether sums of the following form X σ ∈ H X σ ∈ H X σ ∈ S n δ ( σ σ − σ σ ) (10.8)for H = S v [ S ] , H = S v [ S ] are related to Galois theory. We know if H = S v [ Z ],because the connection between QED and ribbon graphs we described, these sums countDessins. For H = S v [ S ] we do not know an argument to relate to Galois actions.
11 Summary and Outlook
We have shown that the counting of Feynman graphs and their symmetry factors in scalarfield theory, e.g φ or φ theories, as well as QED, can be mapped to string amplitudes.The string amplitudes are of the type that appear in the string dual of large N YangMills theory, for which there are several proposed worldsheet constructions. The large N S n TFT. The counting problems related toFeynman graphs have been expressed as observables in this S n TFT. We used this S n TFTdata to construct covers of the cylinder. The covers are inrepreted as string world-sheets,which are also of cylinder topology. The form of the interactions in the QFT determines53he observables at the two ends of the spacetime cylinder, which constrain the windingsof the string worldsheets to belong to certain subgroups of the permutation group.The formulation in terms of string amplitudes is directly related to some classic formu-lae on counting of graphs in papers by Read[20]. We have found it useful to think aboutthese formulae in terms of an operation of introducing an additional vertex in the middleof each edge of the Feynman graph. This separates each edge into a pair of half-edges.We label the half-edges with numbers 1 , , · · · , n , and associate a quantity Σ describingthe vertices of the Feynman graph. The newly-added vertices are described by a quantityΣ . This operation is called “cleaning” in the context of ribbon graphs and related Belyimaps. In that case two permutations σ , σ play the role of Σ , Σ . In the case at hand,Σ is generically not a permutation, but can be viewed as labeling a coset of permutationgroups.There are groups H , H which are symmetry groups of Σ and Σ respectively. Thecounting of Feynman graphs is equivalent to the counting of elements in the double cosetof the form H \ ( Permutation group ) /H (11.1)For the case of scalar field theories the permutation group is something of the form S n .In QED, it is a product group.The double coset connects directly to a string amplitude with cylinder target space,where H , H are associated with the boundaries. The similarities in the current approachbetween Feynman graphs and ribbon graphs, allows us to formulate in group theoreticterms, and derive nice formulae for questions such as the total number of types of ribbongraphs (summed over genus). Further it allows us to express as a group theory problemthe counting of the number of ribbon graphs which correspond to the same Feynmangraph. We may interpret this by saying that large N ribbon graphs (say of matrix φ theory) arise from ordinary Feynman graphs (of φ theory) by a symmetry breaking of S v [ S ] to S v [ Z ].The formulae we obtain for QED Feynman graph counting turn out, upon simplifi-cation, to be related to the counting of ribbon graphs. We give an explanation of thisrelation by mapping QED graphs to ribbon graphs. The key point is the cyclic nature ofribbon graph vertices which map to orientations of fermion loops, in the correspondencewe describe between ribbon graphs and QED graphs.We outline some avenues for extensions of this work. It will be desirable to understandin terms of double cosets and string amplitudes and to derive generating functions forthe refinements of graph counting that occur in QFT. We have made some steps inthe direction of connected graphs, and a more systematic understanding for the case ofmultiple external legs will be useful. In QFT, it is a familiar fact that the generatingfunctional of one-particle irreducible graphs is related to the generateing functional ofconnected graphs by a Legendre transform. Clarifying the implications for double cosets54nd strings will be desirable. Extending to other quantum field theories is an obviousdirection. This would give new counting results for the relevant Feynman graphs, andcould also help address some conceptual issues on the meaning of the string amplitudeinterpretation, with a view to exploring the possibility that the stringy combinatoricswe have uncovered is merely the tip of an iceberg, the bulk of which is a full-fledgedQFT-string duality, which does not involve large N .With regard to precise information on the asymptotic growth of amplitudes in per-turbation theory, the counting sequences of Feynman graphs and their asymptotics are ofinterest. For the case of vacuum graphs in φ theory, there is an asymptotic result [57] e / (4 v )!(4!) v (2 v )!2 v v ! (11.2)For general φ r , with v vertices, we need rv to be even for non-vanishing vacuuum diagrams,i.e rv = 2 m for some m , and the result [57] is e − ( r − / (2 m )!2 m m ! r ! v v ! (11.3)We have given a string interpretation of these sequences. Strings at large quantum num-bers often become classical. Can the above asymptotic result, e.g in φ , be explained byan appropriate semi-classical string ? For QED/Yukawa Feynman graphs or for QED,with the Furry constraint, we hope that the counting sequences and analytic formulae wehave described, along with techniques such as those of [57], will allow the determinationof the asymptotics. Acknowledgements
We thank C. S. Chu, A. Hanany, V. Jejjala, T.R Govindarajan, N. Mekaryaa, K. Pa-padodimas,J.Pasukonis, R.Russo, W. Spence, B. Stefanski, R. Szabo, G. Travaglini, P.Van Hove for useful discussions. SR is supported by an STFC grant ST/G000565/1.RdMK is supported by the South African Research Chairs Initiative of the Departmentof Science and Technology and National Research Foundation. This project was initiatedat the Mauritius workshop on quantum fields and cosmological inflation, June 2011, spon-sored by NITHEP (National Institute for theoretical Physics, South Africa). We thankNITHEP, the Physics Department at the University of Mauritius, and the enthusiasticaudience of students and faculty which contributed to a stimulating environment. Wethank the organizers of the Corfu summer institute 2011 for an opportunity to presentsome of the results of this paper. 55
Semi-direct product structure of Feynman graphsymmetries.
In this section we will explain that the group of automorphisms of a Feynman graph canbe realized as the semi-direct product of two subgroups defined shortly. Towards this end,it is useful to recall the definition of the semi-direct product. Given two groups G and G , and a group homomorphism ψ : G → Aut( G ), the semi-direct product of G and G with respect to ψ is denoted G ⋊ ψ G . As a set G ⋊ ψ G is the Cartesian product G × G . Multiplication is defined using ψ as( g , h ) ∗ ( g , h ) = ( g ψ h ( g ) , h h ) (A.1)for all g , g ∈ G and h , h ∈ G . The identity element e is ( e G , e G ) and( g, h ) − = ( ψ h − ( g − ) , h − ) (A.2)The set of group elements ( g , e G ) for a normal subgroup of G ⋊ φ G isomorphic to G ,while the set of elements ( e G , h ) form a subgroup isomorphic to G .To make our discussion concrete, again consider gφ theory. Recall from Section 4that any graph can be specified, after introducing a new type of vertex (say white whenthe original vertices are colored black) between existing edges and labelling the resultinghalf-edges, by data Σ associated with the vertices and Σ with the edges. For a graphwith v vertices, Σ is a collection of v = v Y r =1 Σ ( r )0 (A.3)Σ is a product of two cycles. The symmetric group S v acts by permuting the half edges.We can define the Automorphism group of the graph using the data Σ and Σ . It isthe group of permutations γ ∈ S v which have the property γ (Σ ) = Σ γ (Σ ) γ − = Σ (A.4)By γ (Σ ) we mean the operation which acts on, on each factor of Σ , as follows < i j k l > → < γ ( i ) γ ( j ) γ ( k ) γ ( l ) > (A.5)The action on Σ can also be written as above, or equivalently in terms of conjugation in S v . In testing the first equality, we treat each angled-bracket as completely symmetric.Two types of actions can be automorphisms : vertices (i.e. black dots) can be swapped,and propagators (i.e. white dots) can be swapped. Construct the group G E which acts56nly on the propagators and the group G V that acts on the vertices. The elements of G V act as a non-trivial permutation on the r index running over the vertices. The elementsof G E act trivially on the r index.For any given Feynman graph we can argue that Aut( D ) = G E ⋊ ψ G V . Towards thisend we will prove that G V acts as an automorphism of G E . For each element ν ∈ G V there is a permutation γ ∈ S v which permutes the vertices ν (Σ ( r )0 ) = Σ ( γ ( r ))0 The elements of G E leave the vertices fixed ǫ (Σ ( r )0 ) = Σ ( r )0 ǫ ∈ G E We have ν − ǫν (Σ ( r )0 ) = ν − ǫ ((Σ ( γ ( r ))0 ) = ν − ((Σ ( γ ( r ))0 ) = Σ ( r )0 Thus, ν − ǫν ∈ G E . Defining ψ ν ( ǫ ) = ν − ǫν , it follows that ψ ν ψ ν ( ǫ ) = ψ ν ν ( ǫ ) whichshows that we indeed have a homomorphism ψ : G V → Aut( G E ).To understand why Aut( D ) = G E ⋊ ψ G V , consider the product law for the semi-directproduct ( ǫ , ν ) ∗ ( ǫ , ν ) = ( ǫ ψ ν ( ǫ ) , ν ν ) = ( ǫ ν ǫ ν − , ν ν )Acting on the graph D means acting on Σ and Σ as defined above. The left handside applies ǫ ν ǫ ν to D . The right hand side applies ǫ ψ ν ( ǫ ) ν ν = ǫ ν ǫ ν − ν ν = ǫ ν ǫ ν completing the demonstration.Figure 22: For this Feynman graph v = 3.An example is in order. For the graph in Figure 22, we haveΣ = < >< >< > Σ = (1 10)(4 11)(3 8)(2 5)(6 9)(7 12) (A.6)57rom the figure we read off the generators (1 4)(10 11), (2 3)(5 8), (9 12)(6 7) for G E .Since these generators commute, G E is a group of order 8. The group G V will permutethe vertices A , B and C of the Feynman graph. There are 6 possible permutations of thevertices. To obtain the generators of G V consider (for example) the permutation whichswaps B and C , shown in figure 23. The relevant permutation is σ BC =(1 2)(3 4)(5 10)(69)(7 12)(8 11). This is indeed an endomorphism of G E since σ BC (1 4)(10 11) σ − BC = (2 3)(5 8) σ BC (2 3)(5 8) σ − BC = (1 4)(10 11) σ BC (9 12)(6 7) σ − BC = (6 7)(9 12)The same is true of the other elements of G V . G V has order 6 and G E has order 8 so thatAut( D ) is order 48. This Feynman graph thus comes with a coefficient (4!) /
48 = 288.Figure 23: The two Feynman graphs shown are related by swapping vertices B and C .Note that there is a notion of Graph automorphism used in graph theory [58]. Thisonly includes G V , since the standard labelling of graphs is to label the vertices and listthe edges as pairs of numbers associated with the vertices they are incident on. Thedistinction is discussed in [59]. B Functions on the double-coset
In this Appendix we will explain how to build a complete set of functions on S n \ ( S n × S n ) / ( H × H ). This Appendix makes use of techniques similar to what was used in [21].A basis of functions on S n is given by the matrix elements of irreducible representa-tions. Start with an element in the group algebra of ( S n × S n ) labelled by representationsand states in the representations.˜ O R ,R i j ; i j = X σ ,σ ∈ S n D R i j ( σ ) D R i j ( σ ) σ ⊗ σ (B.1)58e can make it invariant under left action of S n and under right action of H × H by taking O R ,R i j ; i j = 1 n ! | H || H | X α ∈ S n X β ∈ H X β ∈ H ( α ⊗ α ) ˜ O R ,R i ,j ; i j β ⊗ β (B.2)Some basic manipulations lead to O R ,R i j ; i j = 1 n ! | H || H | X σ ,σ ∈ S n X α ∈ S n X β ∈ H X β ∈ H D R i j ( ασ β ) D R i j ( ασ β ) σ ⊗ σ = 1 n ! | H || H | X σ ,σ ∈ S n X α ∈ S n X β ∈ H X β ∈ H σ ⊗ σ D R i k ( α ) D R k l ( σ ) D R l j ( β ) D R i k ( α ) D R k l ( σ ) D R l j ( β )= X σ ,σ X µ ,µ σ ⊗ σ D R k l ( σ ) D R k l ( σ ) δ R ,R δ i ,i δ k ,k B R ,µ l ( H ) B R ,µ j ( H ) B R ,µ l ( H ) B R ,µ j ( H ) (B.3)The label µ runs over the multiplicity with which the trivial irrep. of H appears in thedecomposition of the S n irrep. R with respect to the subgroup. Thus O R ,R i j ; i j = δ R R X σ ,σ δ i i σ ⊗ σ D R l l ( σ − σ ) δ R ,R B R ,µ l ( H ) B R ,µ j ( H ) B R ,µ l ( H ) B R ,µ j ( H )= δ R R X σ,σ δ i i ( σ σ − ⊗ σ ) δ R ,R D R l l ( σ ) B R ,µ l ( H ) B R ,µ j ( H ) B R ,µ l ( H ) B R ,µ j ( H ) (B.4)The sum over σ is trivial, so it suffices to consider O Rj ,j = X σ ∈ S n σ D Rl l ( σ ) B R,µ l ( H ) B R,µ j ( H ) B R,µ l ( H ) B R,µ j ( H ) (B.5)From this O R ,R i j ; i j is reconstructed by using δ R ,R δ i i .Now define an element in the group algebra of S n labeled by multiplicities ( ν , ν )of the identity irrep of ( H , H ) appearing in a decomposition of irrep R of S n . Usingorthogonality of the branching coefficients we have O Rν ,ν ≡ X j ,j B R,ν j ( H ) B R,ν j ( H ) O Rj ,j = X σ σD Rl l ( σ ) B R,ν l ( H ) B R,ν l ( H ) (B.6)59o we see that a complete set of functions on the coset S n \ ( S n × S n ) / ( H × H ) (B.7)can be labelled by the representations R of S n which contain the trivial of H and of H .We have a complete set of elements in the group algebra of ( S n × S n ), which areinvariant under the left action of S n and the right action of H × H .This can be viewed as a derivation of the formula for Feynman graph counting interms of multiplicities given in formulas (7.4) and (5.25). B.1 QED counting in terms of representation theory
To count the number of QED vacuum graphs with 2 v vertices we need to evaluate F QED (2 v ) = 12 v v ! X σ ∈ S v [ S ] X γ ∈ S v δ ( σγσγ − σ )= 1(2 v )! 12 v v ! X R ⊢ v X σ ∈ S v [ S ] X γ ∈ S v d R χ R ( σγσ − γ − )= 1(2 v )! 12 v v ! X R ⊢ v X σ ∈ S v [ S ] X γ ∈ S v d R D Rij ( σ ) D Rjk ( γ ) D Rkl ( σ − ) D Rli ( γ − )= 12 v v ! X R ⊢ v X σ ∈ S v [ S ] χ R ( σ ) χ R ( σ ) (B.8)We have used the orthogonality of matrix elements X γ ∈ S v D Rjk ( γ ) D Rli ( γ − ) = (2 v )! d R δ kl δ ji (B.9)The product of characters can be be expanded using the Clebsch-Gordan (inner-product)multiplicities. C ( R, R,
Λ) is the number of times the representation Λ of S v appears inthe tensor product R × R , when this is decomposed in terms of the diagonal S v . Thus F QED (2 v ) = 12 v v ! X R ⊢ v X σ ∈ S v [ S ] C ( R, R, Λ) χ Λ ( σ )= X R ⊢ v C ( R, R, Λ) M Λ Sv [ S (B.10)The multiplicity M Λ Sv [ S is the number of times the identity representation of S v [ S ] ap-pears when the representation Λ of S v is decomposed into the irreducible representationsof the subgroup S v [ S ].Similar manipulations in the case of Furry QED leads to expressions involving, notClebsch-Gordan multiplicities, but Clebsch-Gordan coefficients, along with the matrixelements for P τeven τ ⊗ τ in R ⊗ R . 60 Feynman graphs with GAP
We have explained different points of view on the enumeration of Feynman graphs, perhapsthe most intuitive and useful is that Feynman graph is an orbit of the vertex symmetrygroup acting on the Wick contractions. For the case of φ theory, we have the wreathproduct S v [ S ] acting on the conjugacy class of permutations in S v consisting of 2-cycles,which we denoted [2 v ]. Calculations with this formulation are easy to implement directlyin the GAP software for group theoretic computations [22]. C.1 Vacuum graphs of φ We illustrate with the sequence 1 , , , , ... of vacuum Feynman graphs in φ theory.The count of 3 for v = 2 vertices can be obtained from GAP using the commands.gap > C := ConjugacyClass( SymmetricGroup(8), ( 1,2) (3,4) (5,6) ( 7,8) ) ;;gap > G := WreathProduct ( SymmetricGroup(4), SymmetricGroup(2) ) ;;gap > Length(OrbitsDomain ( G , C ) ) ;[ 3 This directly implements the formulation of vacuum Feynman graphs as orbits of S v [ S ] on the conjugacy class [2 v ] described in section 4. The numbers 7 ,
20 can alsobe recovered without much trouble with this method. For the purposes of just gettingthe number of vacuum graphs, this method is an overkill. Better methods with cycleindices are described in Section 5. We found the use of GAP to be very useful as a tool tocheck in formulating the correct group theoretical formulations of various Feynman graphcounting problems. The command OrbitsDomain ( G , C ) actually gives a nested listof lists of pairings. The list runs over inequivalent Feynman graphs. For each Feynmangraph, there is a list of different Wick contractions which lead to the given graph. Hencethese orbits encode not just the number of Feynman graphs, but the Feynman graphsthemselves.
C.2 φ with external edges It is a small generalization to consider graphs with external edges. For φ theory graphswith E external edges we act with the wreath product S v [ S ] on the conjugacy class[2 v + E ] of S v +2 E . The relevant sequence for E = 2 is 1 , , , , , , ... . The count of7 for v = 2 and E = 2 can be obtained from GAP as followsgap > C := ConjugacyClass( SymmetricGroup(10), ( 1,2) (3,4) (5,6) (7,8) (9,10) ) ;;gap > G := WreathProduct ( SymmetricGroup(4), SymmetricGroup(2) ) ;;gap > Length(OrbitsDomain ( G , C ) ) ;[ 7 61omments made above are again applicable: there are better methods to count thesegraphs which use cycle indices.
C.3 Symmetry factor
The symmetry factor of a given Feynman graph can be constructed using GAP. Thefollowing commands compute the symmetry factor of the diagram in Figure 22.gap > H1 := WreathProduct( SymmetricGroup(4) , SymmetricGroup(3) ) ;gap > Sig1 := (1,10) ( 4,11) (3,8) (2,5) (6,9) ( 7,12) ;gap > for g in H1 doif OnPoints( Sig1 , g ) := Sig1 thenCountsym := Countsym +1 ; fi ; od ; Countsym ;[ 48Simple modifications of these commands can construct the group, study its subgroupsetc. , but these will have no immediate interest for us. C.4 Action of S v on a set An alternative way to code the graphs is to label the vertices { , · · · , v } . Consider listsof 2 v unordered pairs. { ( i , j i ) , ( i , j ) ... ( i v , j v ) } (C.1)Put the constraint that v X k =1 δ ( i, i k ) + δ ( i, j k ) = 4 (C.2)for all i from 1 , ..v . This imposes the condition that all vertices are 4-valent.There is an action of S v on this set. We are interested in the orbits. This can beprogrammed in GAP and gives an alternative method to get the sequence of vacuumFeynman graphs in φ theory.The method of using S v is more efficient in generating the whole set of Feynmangraphs with GAP, than the S v [ S ] method. But it does not automatically have the edgesymmetries, unlike the S v [ S ] approach. It only gives the automorphism G V discussed inAppendix A. 62 Integer sequences
In this Appendix we collect the numerical results we have obtained by counting Feynmangraphs. • The number of vacuum graphs in φ theory, with v vertices, starting with v = 1, isgiven by the sequence1 , , , , , , , , , , , , , , , .. (D.1)This sequence is listed in “The On-Line Encyclopedia of Integer Sequences” [23],where it is described as the sequence of 4-regular multi-graphs (loops allowed). Thissequence is derived using (5.21) or (6.4). • The number of connected vacuum graphs in φ theory with v vertices, starting with v = 1, is given by the sequence1 , , , , , , , , , , , , , ... (D.2)This sequence was generated using (6.5). • The number of graphs in φ theory with E = 2 external legs, with v vertices, startingwith v = 0, is given by the sequence1 , , , , , , , , , , , , , ... (D.3)This sequence is generated by the formula (6.8) with E = 2. • The number of connected graphs in φ theory with E = 2 external legs, with v vertices, starting with v = 0, is given by the sequence1 , , , , , , , , , , , , , ... (D.4)This sequence is generated by the formula (6.9). These graphs have been tabulatedin [56]. Comparing the Table II of [56], we find a match between the numberof diagrams at first and second order in perturbation theory. At third order inperturbation theory we have 10 graphs compared to the 8 graphs listed in [56]. Thetwo graphs not listed in [56] are obtained from graph • The number of graphs in φ theory with E = 4 external legs, with v vertices, startingwith v = 0, is given by the sequence3 , , , , , , , , , , , , ... (D.5)This sequence is generated by the formula (6.8) with E = 4. • The number of vacuum graphs for φ theory, with 2 v vertices, starting from v = 12 , , , , , , , , ... (D.6)This sequence was generated using (6.12). • The number of connected vacuum graphs in φ theory, with 2 v vertices, startingfrom v = 1 2 , , , , , , , , ... (D.7)This sequence was generated using (6.5). • The number of graphs in φ theory with E = 2 external legs, with 2 v verticesstarting from v = 1, is5 , , , , , , , , ... (D.8)This sequence was generated using (6.15) with E = 2. • The number of QED/Yukawa vacuum graphs with 2 v vertices, which equals thetotal number of ribbon graphs with 2 v edges is given by2 , , , , , , , , , ... (D.9)This sequence was generated using (8.27). • The number of connected QED/Yukawa vacuum graphs with 2 v vertices, startingfrom v = 1, is given by2 , , , , , , , , , ... (D.10)This sequence was generated using (6.5).64 Vacuum graphs in QED after implementing the constraint due to Furry’s theorem.1 , , , , , , , , ... (D.11)This sequence was generated using (9.13). • Connected Vacuum graphs in QED after implementing the constraint due to Furry’stheorem. 1 , , , , , , , , , ... (D.12)This sequence was generated using (6.5). The 1 , , References [1] A. A. Migdal, “Recursion Equations in Gauge Theories,” Sov. Phys. JETP , 413(1975).[2] D. J. Gross, “Two-dimensional QCD as a string theory,” Nucl. Phys. B400 (1993)161-180. [hep-th/9212149].[3] D. J. Gross, W. Taylor, “Two-dimensional QCD is a string theory,” Nucl. Phys.
B400 (1993) 181-210. [hep-th/9301068].[4] D. J. Gross, W. Taylor, “Twists and Wilson loops in the string theory of two-dimensional QCD,” Nucl. Phys.
B403 (1993) 395-452. [hep-th/9303046].[5] G. ’t Hooft, “A Planar Diagram Theory for Strong Interactions,” Nucl. Phys. B (1974) 461.[6] S. Cordes, G. W. Moore, S. Ramgoolam, “Large N 2-D Yang-Mills theory and topo-logical string theory,” Commun. Math. Phys. (1997) 543-619 [hep-th/9402107] ;“Lectures on 2-d Yang-Mills theory, equivariant cohomology and topological field the-ories,” Nucl. Phys. Proc. Suppl. (1995) 184-244. [arXiv:hep-th/9411210 [hep-th]].[7] P. Horava, “Topological rigid string theory and two-dimensional QCD,” Nucl. Phys. B463 , 238-286 (1996). [hep-th/9507060].[8] M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, “Holomorphic anomalies in topo-logical field theories,” Nucl. Phys. B (1993) 279 [arXiv:hep-th/9302103].[9] M. Aganagic, H. Ooguri, N. Saulina, C. Vafa, “Black holes, q-deformed 2d Yang-Mills, and non-perturbative topological strings,” Nucl. Phys.
B715 , 304-348 (2005).[hep-th/0411280]. 6510] N. Caporaso, M. Cirafici, L. Griguolo, S. Pasquetti, D. Seminara, R. J. Szabo,“Topological Strings, Two-Dimensional Yang-Mills Theory and Chern-Simons Theoryon Torus Bundles,” Adv. Theor. Math. Phys. , 981-1058 (2008). [hep-th/0609129].[11] L. Schneps, “The Grothendieck theory of Dessins d’Enfants,” London MathematicalSociety, Lecture Notes Series 200.[12] A. Grothendieck, “Esquisse d’un Programme,” in [11][13] S. K. Ashok, F. Cachazo and E. Dell’Aquila, “Children’s drawings from Seiberg-Witten curves,” arXiv:hep-th/0611082.[14] V. Jejjala, S. Ramgoolam, D. Rodriguez-Gomez, “Toric CFTs, Permutation Triplesand Belyi Pairs,” JHEP (2011) 065. [arXiv:1012.2351 [hep-th]].[15] A. Hanany, Y. -H. He, V. Jejjala, J. Pasukonis, S. Ramgoolam, D. Rodriguez-Gomez,“The Beta Ansatz: A Tale of Two Complex Structures,” JHEP (2011) 056.[arXiv:1104.5490 [hep-th]].[16] G. Jones and M. Streit “Galois groups,monodromy groups and cartographic groups,”in “Geometric Galois actions 2.The inverse Galois problem, moduli spaces and Mappingclass groups,” edited by L. Schneps and P. Lochak, Lon. Math. Soc. Lecture Note Series243.[17] R. Dijkgraaf, E. P. Verlinde, H. L. Verlinde, “Matrix string theory,” Nucl. Phys. B500 , 43-61 (1997). [hep-th/9703030].[18] M. Bauer and C. Itzykson, “ Triangulations ” in [11][19] R. d. M. Koch, S. Ramgoolam, “From Matrix Models and quantum fields to Hurwitzspace and the absolute Galois group,” [arXiv:1002.1634 [hep-th]].[20] R.C. Read, “The enumeration of locally restricted graphs,” Journal London Math.Soc. 34 (1959), 417-436.[21] T. W. Brown, P. J. Heslop, S. Ramgoolam, “Diagonal multi-matrix correlators andBPS operators in N=4 SYM,” JHEP , 030 (2008). [arXiv:0711.0176 [hep-th]],T. W. Brown, P. J. Heslop, S. Ramgoolam, “Diagonal free field matrix correlators,global symmetries and giant gravitons,” JHEP
A11 (1996) 3885-3933. [hep-th/9412110].[27] K. G. Wilson, “Confinement of Quarks,” Phys. Rev.
D10 , 2445-2459 (1974).[28] E. Witten, “On quantum gauge theories in two-dimensions” Commun. Math. Phys. :153-209,1991.[29] M. Fukuma, S. Hosono, H. Kawai, “Lattice topological field theory in two-dimensions,” Commun. Math. Phys. (1994) 157-176. [hep-th/9212154].[30] A. D’Adda, P. Provero, “Two-dimensional gauge theories of the symmetric groupS(n) and branched N coverings of Riemann surfaces in the large N limit,” Nucl. Phys.Proc. Suppl. (2002) 79-83. [hep-th/0201181].[31] R. Dijkgraaf, “Mirror symmetry and elliptic curves,”published in The Moduli Spaceof Curves, Procedings of the Texel Island Meeting, April 1994 (Birkhauser).[32] T. W. Brown, “Complex matrix model duality,” Phys. Rev.
D83 , 085002 (2011).[arXiv:1009.0674 [hep-th]].[33] R. Gopakumar, “What is the Simplest Gauge-String Duality?,” [arXiv:1104.2386[hep-th]].[34] http://en.wikipedia.org/wiki/Orbit-stabilizer theorem (2008) 255 [arXiv:0804.2764 [hep-th]].[36] Y. Kimura and S. Ramgoolam, “Branes, anti-branes and brauer algebras in gauge-gravity duality,” JHEP (2007) 078 [arXiv:0709.2158 [hep-th]].[37] R. d. M. Koch, M. Dessein, D. Giataganas, C. Mathwin, “Giant Graviton Oscilla-tors,” [arXiv:1108.2761 [hep-th]].[38] J. Pasukonis and S. Ramgoolam, “From counting to construction of BPS states inN=4 SYM,” JHEP (2011) 078 [arXiv:1010.1683 [hep-th]].6739] C. L. Ezell, “Branch point structure of covering maps onto nonorientable surfaces,”Transactions of the American Mathematical Society Volume 243, September 1978[40] M. R. Douglas and G. W. Moore, “D-branes, quivers, and ALE instantons,”arXiv:hep-th/9603167.[41] Peter J. Cameron, “Topics, Techniques, Algorithms.” Cambridge University Press,1994 (reprinted 1996).[42] http://en.wikipedia.org/wiki/Burnside’s lemma[43] P. Cvitanovic, B. E. Lautrup, R. B. Pearson, “The Number And Weights Of FeynmanDiagrams,” Phys. Rev.
D18 , 1939 (1978).[44] M.E. Peskin and D.V. Schroeder, “Introduction to quantum field theory,” BoulderColorado:Westview Press, 1995.[45] P. Cvitanovic, “Asymptotic Estimates and Gauge Invariance,” Nucl. Phys.
B127 ,176 (1977).[46] G. W. Moore and G. Segal, “D-branes and K-theory in 2D topological field theory,”arXiv:hep-th/0609042.[47] A.V. Alexeevski and S. M. Natanzon, “Hurwitz numbers for regular coveringsof surfaces by seamed surfaces and Cardy Frobenius algebras of finite groups,”arXiv:0709.3601.[48] A. Hanany, N. Mekareeya, “Tri-vertices and SU(2)’s,” JHEP (2011) 069.[arXiv:1012.2119 [hep-th]].[49] Z. Bern, J. J. M. Carrasco, H. Johansson, “Perturbative Quantum Gravity as aDouble Copy of Gauge Theory,” Phys. Rev. Lett. (2010) 061602. [arXiv:1004.0476[hep-th]].[50] M. Srednicki,“Quantum field theory,” Cambridge, UK: Univ. Pr. (2007).[51] E. Witten, “Perturbative gauge theory as a string theory in twistor space,” Commun.Math. Phys. (2004) 189 [arXiv:hep-th/0312171].[52] G t Hooft, “Dimensional Reduction in Quantum Gravity,” Utrecht Preprint THU-93/26, gr-qc/9310006[53] L. Susskind, “The World as a hologram,” J. Math. Phys. , 6377-6396 (1995).[hep-th/9409089]. 6854] S. El-Showk, K. Papadodimas, “Emergent Spacetime and Holographic CFTs,”[arXiv:1101.4163 [hep-th]].[55] R. Gopakumar, “From free fields to AdS,” Phys. Rev. D70 , 025009 (2004).[hep-th/0308184],R. Gopakumar, “From free fields to AdS. 2.,” Phys. Rev.
D70 , 025010 (2004).[hep-th/0402063],R. Gopakumar, “From free fields to AdS: III,” Phys. Rev.
D72 , 066008 (2005).[hep-th/0504229].[56] H. Kleinert, A. Pelster, B. M. Kastening, M. Bachmann, “Recursive graphical con-struction of Feynman diagrams and their multiplicities in phi**4 theory and in phi**2A theory,” Phys. Rev.