A generalization of the Virasoro algebra to arbitrary dimensions
aa r X i v : . [ h e p - t h ] M a y Pacs numbers: 02.10.Ox, 04.60.Gw, 05.40-a
A generalization of the Virasoro algebra to arbitrary dimensions
Razvan Gurau ∗ Perimeter Institute for Theoretical Physics,31 Caroline St. N, ON N2L 2Y5, Waterloo, Canada
Colored tensor models generalize matrix models in higher dimensions. They admit a1/N expansion dominated by spherical topologies and exhibit a critical behavior stronglyreminiscent of matrix models. In this paper we generalize the colored tensor models tocolored models with generic interaction, derive the Schwinger Dyson equations in the large N limit and analyze the associated algebra of constraints satisfied at leading order by thepartition function. We show that the constraints form a Lie algebra (indexed by trees)yielding a generalization of the Virasoro algebra in arbitrary dimensions. Keywords: Random tensor models, 1/N expansion, critical behavior
I. INTRODUCTION
Random matrix models [1–3] generalize in higher dimensions to random tensor models [4–8]and group field theories [9–12] (see [13–18] for some further developments). The Feynman graphsof GFT in D dimensions are built from vertices dual to D simplices and propagators encodingthe gluing of simplices along their boundary. Parallel to ribbon graphs of matrix models (dual todiscretized surfaces), GFT graphs are dual to discretized D dimensional topological spaces. Tensormodels are notoriously hard to control analytically and one usually resorts to numerical simulations[19–21]. Progress has recently been made in the analytic control of tensor models with the adventof the 1 /N expansion [22–24] of colored [25–27] tensor models. This expansion synthesizes severalevaluations of graph amplitudes [28–36] and provides a straightforward generalization of the familiargenus expansion of matrix models [37, 38] in arbitrary dimension. The coloring of the fields allowsone to address previously inaccessible questions like the implementation of the diffeomorphismsymmetry [36, 39, 40] or the identification of embedded matrix models [41] in tensor models. Thesymmetries of generic tensor models have recently been studied using n-ary algebras [42, 43].The critical behavior of matrix models is most conveniently addressed using the loop equations[44–46] in conjunction with the 1 /N expansion. The loop equations translate in a set of constraints(obeying the Virasoro algebra) satisfied by the partition function and provide the link betweenmatrix models and continuum conformal field theories. Generic matrix models exhibit multi criticalpoints [47] which are at the core of their applications to string theory [48, 49], two dimensionalgravity [3], critical phenomena [1, 50, 51], black hole physics [52], etc.Recently the investigation of the critical behavior of the simplest colored tensor models hasbeen performed [53] mapping the dominant family of graphs (generalizing the planar [37] graphsof matrix models) on certain species of colored trees.However, up to now, the colored tensor models considered posses only one interaction term (onecoupling constant). It is well known that multi critical points for matrix models appear only whenone adds multiple interaction terms. A first question we will solve in this paper is to write a coloredtensor model with generic interactions. In order to access the critical behavior of such a modelone must derive a generalization of the loop equation in higher dimensions. We will derive the ∗ [email protected] closed set of Schwinger-Dyson Equations (SDE) obeyed by a generic colored tensor models in thelarge N limit which we translate in constraints on the partition function. We will prove that theconstraints form a Lie algebra yielding a higher dimensional generalization of the Virasoro algebra.This paper is organized as follows. In section II we recall the derivation of the loop equationsand the link with the Virasoro algebra in matrix models. In section III we introduce an algebraindexed by colored trees. In section IV we derive the SDEs at leading order in the 1 /N expansionof colored tensor models and translate them into constraints on the partition function which weidentify with the generators of our algebra. Section V draws the conclusions of this work. II. MATRIX MODELS AND THE VIRASORO ALGEBRA
This section is a quick digest of [44, 46] and presents the classical derivation of the loop equationsand their link with the Virasoro algebra in matrix models. In the spirit of our subsequent treatmentof colored tensor models, we will start from a colored matrix model [54–56] of three independentnon hermitian matrices M , M and M , defined by the partition function Z = Z [ dM ][ dM ][ dM ] e − N Tr[ V ( M ,M ,M )] ,V ( M , M , M ) = M M † + M M † + M M † − λM M M − ¯ λM † M † M † , (1)with [ dM ] = Q a,b dM ab d ¯ M ab . As the integral is Gaussian, one can explicitly integrate over twocolors to obtain the partition function as an integral over one matrix Z = Z [ dM ] e − N Tr[ V ( M M † )] V ( M M † ) = M M † + ln( I − λ ¯ λ M M † ) = M M † + X j ( λ ¯ λ ) j j ( M M † ) j . (2)To pass from a model with one coupling constant to a generic matrix model, one attributes toevery operator in the effective action for the last color an independent coupling constant replacing ( λ ¯ λ ) j j by t j Z = Z [ dM ] e − N Tr[ V ( MM † )] , V ( M M † ) = X j =1 t j ( M M † ) j . (3)The Schwinger-Dyson equations (SDE) of a generic matrix model write0 = Z [ dM ] δδM ab (cid:16) [( M M † ) n M ] ab e − N Tr[ V ( MM † )] (cid:17) = D n X k =0 [( M M † ) k ] aa [( M † M ) n − k ] bb E − N D X j =1 j t j [( M M † ) n M ] ab [ M † ( M M † ) j − ] ba E , (4)which, summing over a and b , becomes D n X k =0 Tr[(
M M † ) k ]Tr[( M M † ) k ] n − k E − N X j j t j D Tr[(
M M † ) n + j ] E = 0 . (5)Every insertion of an operator Tr[( M M † ) j ] in the correlation function can be re expressed as aderivative of V ( M M † ) with respect to t j . Consequently the SDEs become L n Z = 0 , for n ≥ , L n = N δ ,n − N ∂∂t n + 1 N n − X k =1 ∂ ∂t k ∂t n − k + ∞ X j =1 j t j ∂∂t n + j , (6)where the derivatives w.r.t. t j , with j ≤ L n ’s respect the commutationrelations of (the positive operators of) the Virasoro algebra[ L m , L n ] = ( m − n ) L m + n for m, n ≥ . (7)Note that as we only deal with L m , m ≥ L ′ m = L m − N P n − k =1 ∂ ∂t k ∂t n − k also respect [ L ′ m , L ′ n ] = ( m − n ) L ′ m + n . If one specializesthe operators we define below for D = 2 (and takes into account the cyclicity of the trace), oneobtains the operators L ′ m .This classical result is our guide towards deriving SDEs and loop equations for colored tensormodels in arbitrary dimensions. III. A LIE ALGEBRA INDEXED BY COLORED, ROOTED, D-ARY TREES
As the higher dimensional generalization of the Virasoro algebra we obtain is rather non trivialwe will first present it in full detail and only later identify it with the algebra of constraints satisfiedby the partition function. The operators in our algebra are indexed by colored rooted D -ary trees.Trees and D -ary trees are well studied in the mathematical literature [57]. The colored rooted D -ary trees index the leading order in the 1 /N expansion of colored tensor models [53]. A. Colored rooted D-ary trees: Definitions A colored rooted D -ary tree T with |T | vertices is a tree with the following properties • It has a root vertex, denoted ( ), of coordination D . • It has |T | − D + 1 (i.e. each of them has D descendants). • It has ( D − |T | + 1 leaves of coordination 1 (i.e. with no descendants). • All lines have a color index, 0 , , . . . , D −
1, such that the D direct descendants (leaves orvertices) of a vertex (or of the root) are connected by lines with different colors.We will ignore in the following the leaves of the tree, as they can automatically be added oncethe vertices and lines of the colored tree are known. A crucial fact in the sequel is that a coloredrooted D -ary tree admits a canonical labeling of its vertices. Namely, every vertex can be labeledby the list of colors V = ( i , . . . , i n ) of the lines in the unique path connecting V to the root ( ).The first color, i , is the color of the line in the path ending on the root (and i n is the color of theline ending on V ). For instance the vertex (0) is the descendant connected to the root ( ) by theline of color 0, and the vertex (01) is the descendant of the vertex (0) connected to it by a line ofcolor 1 (see figure 1 for an example of a canonically labeled 3-ary tree with |T | = 7 vertices).In the sequel “tree” will always mean a colored rooted D -ary tree. A tree is completely identifiedby its canonically labeled vertices, hence it is a set T = { ( ) , . . . } . We denote i, . . . , i | {z } n a list of n (22) (222)
10 2 21 210 1 20 1 00 1 22 0 210 (2)(0) (01) (20) ( )
FIG. 1. A colored rooted D -ary tree. identical labels i . We will use the shorthand notation V for the list of labels identifying a vertex.Let a tree T , and one of its vertices V = l, . . . , k, i, . . . , i | {z } n , (8)with k = i . The colors l, . . . , k might be absent. The successor of color j of V , denoted s j T [ V ],is the vertex s j T [( l, . . . , k, i, . . . i | {z } n )] = ( l, . . . , k, i, . . . i | {z } n , j ) if it existsif not ( l, . . . , k, i, . . . i | {z } n ) if j = i ( l, . . . , k ) if j = i . (9)The colored successor functions are cyclical, namely if a vertex does not have a descendant in thetree of color j , then its “successor of color j ” is the first vertex one encounters, when going form V to the root, whose label does not end by j . For the example of figure 1 we have. s [( )] = (0) s [( )] = ( ) s [( )] = (2) s [(0)] = ( ) s [(0)] = (01) s [(0)] = (0) s [(01)] = (01) s [(01)] = (0) s [(01)] = (01) s [(2)] = (20) s [(2)] = (2) s [(2)] = (22) s [(20)] = (2) s [(20)] = (20) s [(20)] = (20) s [(22)] = (22) s [(22)] = (22) s [(22)] = (222) s [(222)] = (222) s [(222)] = (222) s [(222)] = ( ) . (10)We call V the maximal vertex of color i in a tree T if V = i, . . . i | {z } n such that s i T [( i, . . . i | {z } n )] = ( ) . (11)In figure 1 the maximal vertex of color 2 is (222), the maximal vertex of color 0 is (0) and themaximal vertex of color 1 is the root ( ) itself.We define the branch of color i of T , denoted T i , the tree T i = n ( X ) | ( i, X ) ∈ T o . (12)The branch T i can be empty. The root of the branch T i , ( ) ∈ T i corresponds to the vertex ( i ) ∈ T .The rest of the vertices of T (that is the root ( ) and all vertices of the form ( k, U ) ∈ T , k = i )also form canonically labeled tree ˜ T i , the complement in T of the branch T i . In figure 1, thebranch of color 2 is the tree T = { ( ) , (2) , (22) , (0) } , as all the vertices (2), (22), (222) and (20)belong to T . Its complement is ˜ T = { ( ) , (0) , (01) } .Two colored rooted D -ary trees T and T can be joined (or glued) at a vertex V ∈ T . For allcolors i , denote the maximal vertices of color i of T ( i, . . . , i | {z } n i ) , s i T [( i, . . . , i | {z } n i )] = ( ) . (13)The glued tree T ⋆ V T , is the tree canonically labeled T ⋆ V T = ( X ) for all ( X ) ∈ T , ( X ) = ( V, . . . )( V, Y ) for all ( Y ) ∈ T ( V, i, . . . , i | {z } n i +1 , Z ) for all ( V, i, Z ) ∈ T . (14)This operation can be seen as cutting all the branches starting at V in T , gluing the tree T at V , and then gluing back the branches at the maximal vertices of the appropriate color in T . Thevertices of T \ ( V ) and T \ ( ) map one to one onto the vertices of ( T ⋆ V T ) \ ( V ), and both( V ) ∈ T , ( ) ∈ T map to ( V ) ∈ T ⋆ V T , thus |T ⋆ V T | = |T | + |T | −
1. An example is given infigure 2, where the leaves are not drawn. (0) (01) ( ) 0 1 20 1 (2) (20) (22) (2)(0) (01) (20) ( ) (22) (200) (21) (0) (1) (2) (11)(00) (211) (2000) (222) (222) (221) (2222) (2221) FIG. 2. Gluing of two trees at a vertex T ⋆ (2) T . For any tree T , with maximal vertex of color i , ( i, . . . , i | {z } n i ), the maximal vertex of color i in thebranch T i will have one less label ( i, . . . , i | {z } n i − ). One can glue the tree { ( ) , ( i ) } on ( i, . . . , i | {z } n i − ) ∈ T i . All thevertices of T i are unchanged by this gluing, its only effect being to introduce a new vertex, ( i, . . . , i | {z } n i )which becomes the new maximal vertex of color i . Subsequently, one can glue the complement ofthe branch i , ˜ T i on this new maximal vertex T ′ = (cid:16) T i ⋆ ( i, . . . i | {z } ni − ) { ( ) , ( i ) } (cid:17) ⋆ ( i, . . . i | {z } ni ) ˜ T i . (15)The two trees T and T ′ have the same number of vertices |T ′ | = | ˜ T i | + |T i | = |T | , and the verticesof the initial tree T map one to one on the vertices of the final T ′ . As none of the vertices of ˜ T i starts by a label i , it follows that the maximal vertex of color i in T ′ is ( i, . . . , i | {z } n i ). Thus( i, V ) ∈ T ↔ ( V ) ∈ T ′ , ( V ) = ( i, . . . i | {z } n i , U ) , ( W ) ∈ T , ( W ) = ( i, V ) ↔ ( i, . . . i | {z } n i , W ) ∈ T ′ . (16)Most importantly, it is straightforward to check that the mapping is consistent with the successorfunctions V, W ∈ T ↔ V ′ , W ′ ∈ T ′ such that s i T [ V ] = W ⇔ s i T ′ [ V ′ ] = W ′ . (17)We will say that the two trees T and T ′ are equivalent , T ∼ T ′ , and extended by transitivity ∼ to an equivalence relation between rooted trees. An example is presented in figure 3. (2220)(22201)
21 2100 1 2 0 10 10( ) (0) (2) (22) (222)
20 1 20 110 2 21 210 1 20 1 00 1 22 0 210 (2)(0) (01) (20) ( ) (222) (22) FIG. 3. Two equivalent trees
T ∼ T ′ , with T ′ = (cid:16) T ⋆ (22) { ( ) , (2) } (cid:17) ⋆ (222) ˜ T . The equivalence class of a tree [ T ] has |T | members all obtained by choosing a vertex V = i, j, k . . . , l in T performing the elementary operation ∼ on the colors i followed by j followed by k and so on up to l . B. Colored rooted D-ary trees: Properties
In this section we prove a number of lemmas concerning the gluing of trees, ⋆ . All this propertiescan be readily understood in terms of the graphical representation of the trees. In the sequel wewill deal with three trees T , T and T . We denote ( i, . . . i | {z } n i ) the maximal vertex of color i in T and( i, . . . i | {z } q i ) the maximal vertex of color i in T . Lemma 1. If ( V ) = ( k, U ) ∈ T then ( T ⋆ V T ) k = T k ⋆ U T , and, for i = k , ( T ⋆ V T ) i = T i . Proof:
The composite tree T ⋆ V T is T ⋆ V T = ( i, X ) for all ( i, X ) ∈ T , ( i, X ) = ( k, U, . . . )( k, U, Y ) for all ( Y ) ∈ T ( k, U, i, . . . , i | {z } n i +1 , Z ) for all ( k, U, i, Z ) ∈ T . (18)It follows that i = k : ( T ⋆ V T ) i = n ( X ) for all ( i, X ) ∈ T o = T i , (19)and the branch of color k of T ⋆ V T is( T ⋆ V T ) k = ( X ) for all ( k, X ) ∈ T , ( k, X ) = ( k, U, . . . )( U, Y ) for all ( Y ) ∈ T ( U, i, . . . , i | {z } n i +1 , Z ) for all ( k, U, i, Z ) ∈ T , (20)As T k = { ( X ) | ( k, X ) ∈ T } it follows that ( U ) ∈ T k and T k ⋆ U T = ( X ) for all ( X ) ∈ T k , ( X ) = ( U, . . . ) ⇔⇔ ( k, X ) ∈ T , ( k, X ) = ( k, U, . . . )( U, Y ) for all ( Y ) ∈ T ( U, i, . . . , i | {z } n i +1 , Z ) for all ( U, i, Z ) ∈ T k ⇔ ( k, U, i, Z ) ∈ T . (21) Lemma 2.
For any three trees, ( T ⋆ V T ) ⋆ V T = T ⋆ V ( T ⋆ ( ) T ) . Proof:
Note that the label of any vertex in T starts by the empty label ( ), hence the joiningat the root writes T ⋆ ( ) T = ( Y ) for all ( Y ) ∈ T ( i, . . . , i | {z } q i +1 , Y ) for all ( i, Y ) ∈ T . (22)The maximal vertices of color i in T ⋆ ( ) T are ( i, . . . , i | {z } q i + n i ). Both operations lead to a tree withvertices ( X ) for all ( X ) ∈ T , ( X ) = ( V, . . . )( V, Y ) for all ( Y ) ∈ T ( V, i, . . . i | {z } q i +1 , Y ) for all ( i, Y ) ∈ T ( V, i, . . . i | {z } q i + n i +1 , Z ) for all ( V, i, Z ) ∈ T . (23) Lemma 3.
For any two distinct vertices V = W ∈ T , we denote W ′ the image of W in the tree T ⋆ V T and V ′ the image of V in T ⋆ W T . Then ( T ⋆ V T ) ⋆ W ′ T = ( T ⋆ W T ) ⋆ V ′ T . Proof:
We distinguish two cases. Either V and W are not ancestor to each other in T (both( W ) = ( V, . . . ) and ( V ) = ( W, . . . )). Hence W ′ = W and V ′ = V . In this case, both operationslead to ( X ) for all ( X ) ∈ T , ( X ) = ( V, . . . ) and ( X ) = ( W, . . . )( V, Y ) for all Y ∈ T ( V, i, . . . , i | {z } n i +1 , Z ) for all ( V, i, Z ) ∈ T ( W, Y ) for all Y ∈ T ( W, i, . . . , i | {z } q i +1 , Z ) for all ( W, i, Z ) ∈ T . (24)Or one of them (say V ) is ancestor to the other (hence ( W ) = ( V, j, A ) ). In this case V ′ = V , but W ′ = V, j, . . . j | {z } n j +1 , A . Both operations lead to the tree ( X ) for all X ∈ T , X = ( V, . . . )( V, Y ) for all Y ∈ T ( V, i, . . . i | {z } n i +1 , Z ) for all ( V, i, Z ) ∈ T with ( V, i, Z ) = ( V, j, A, Z ′ ) = ( W, Z ′ )( V, j, . . . j | {z } n j +1 , A, Y ) for all Y ∈ T ( V, j, . . . j | {z } n j +1 , A, i, . . . i | {z } q i +1 , Z ) for all ( V, j, A, i, Z ) = (
W, i, Z ) ∈ T . (25) Lemma 4.
Let V ∈ T and W ∈ T then ( T ⋆ V T ) ⋆ W ′ T = T ⋆ V ( T ⋆ W T ) , where again W ′ denotes the image of W in the tree T ⋆ V T . Proof:
The image of W , W ′ = V, W . Both joinings lead to the tree ( X ) for all X ∈ T X = ( V, . . . )( V, Y ) for all Y ∈ T Y = ( W, . . . )( V, W, Y ) for all Y ∈ T ( V, W, i, . . . i | {z } q i +1 , Z ) for all ( W, i, Z ) ∈ T ( V, i, . . . i | {z } n i +1 , Z ) for all ( V, i, Z ) ∈ T if ( W ) = ( i, . . . , i | {z } w i )( V, i, . . . i | {z } q i + n i +1 , Z ) for all ( V, i, Z ) ∈ T if ( W ) = ( i, . . . , i | {z } w i ) . (26) Lemma 5.
We have ( T ⋆ ( ) T ) i ∼ ( T ⋆ ( ) T ) i . Proof:
We have T ⋆ ( ) T = ( k, Y ) for all ( k, Y ) ∈ T ( i, . . . i | {z } n i +1 , Y ) for all ( i, Y ) ∈ T , (27)hence ( T ⋆ ( ) T ) i = ( Y ) for all ( i, Y ) ∈ T ( i, . . . i | {z } n i , Y ) for all ( i, Y ) ∈ T = ( Y ) for all Y ∈ T i ( i, . . . i | {z } n i , Y ) for all Y ∈ T i . (28)Note that the vertices ( Y ) ∈ T i start necessarily by at most n i − i . The maximal vertex ofcolor i in ( T ⋆ ( ) T ) i is ( i, . . . , i | {z } n i + q i − ). The tree ( T ⋆ ( ) T ) i is equivalent with T ′ obtained by mappingthe vertices( i, V ) ∈ ( T ⋆ ( ) T ) i → ( V ) ∈ T ′ ( W ) ∈ T , ( W ) = ( i, V ) → ( i, . . . , i | {z } n i + q i − , W ) . (29)Iterating n i times we see that all the vertices belonging initially to T i will, at one step, acquire n i + q i − i and lose a label i at all the other n i − T willonly lose a label i at all n i steps. Thus ( T ⋆ ( ) T ) i is equivalent to the tree ( i, . . . , i | {z } q i , Y ) for all Y ∈ T i ( Y ) for all Y ∈ T i , (30)which we recognize by eq. (28) to be ( T ⋆ ( ) T ) i . C. The Lie algebra indexed by colored rooted D-ary trees
We now define a Lie algebra of operators indexed by the trees. We associate to every tree avariable t T , and we denote | R | the coordination of the root of T . Consider the operators L T defined as L T = ( − ) | R | N D − D | R | ∂ | R | Q i, T i = ∅ ∂t T i + X T t T X V ∈T ∂∂t T ⋆ V T , L { ( ) } = N D + X T t T X V ∈T ∂∂t T , (31)where N is some parameter (destined to become the large N parameter of the tensor model). Wewill not consider the most general domain of this operators. Namely, in stead of defining them forarbitrary functions f ( t T ) we restrict their domain to class functions f ( t [ T ] ), with t [ T ] = P T ′ ∼T t T ′ . Theorem 1.
When restricted to class functions, the operators L T form a Lie algebra with com-mutator h L T , L T i f ( t [ T ] )= X V ∈T L T ⋆ V T f ( t [ T ] ) − X V ∈T L T ⋆ V T f ( t [ T ] ) . (32) Proof:
We start by evaluating L T L T f . We have L T L T f = h ( − ) | R | N D −| R | D ∂ | R | Q ∂t T j + X T ′ t T ′ X V ′ ∈T ′ ∂∂t T ′ ⋆ V ′ T i (33) h ( − ) | R | N D −| R | D ∂ | R | f Q ∂t T i + X T t T X V ∈T ∂f∂t T ⋆ V T i = ( − ) | R | + | R | N D − D ( | R | + | R | ) ∂ | R | + | R | f Q ∂t T j Q ∂t T i +( − ) | R | N D −| R | D (cid:16) X k X V ∈T k ∂ | R | f Q j = k ∂t T j ∂t T k ⋆ V T + X T t T X V ∈T ∂ | R | +1 f Q j ∂t T j ∂t T ⋆ V T (cid:17) +( − ) | R | N D −| R | D X T ′ t T ′ X V ′ ∈T ′ ∂ | R | +1 f Q ∂t T i ∂t T ′ ⋆ V ′ T + X T ′ t T ′ X V ′ ∈T ′ (cid:16) X V ∈T ′ ⋆ V ′ T ∂f∂t ( T ′ ⋆ V ′ T ) ⋆ V T + X T t T X V ∈T ∂ f∂t T ′ ⋆ V ′ T ∂t T ⋆ V T (cid:17) , h L T , L T i f =( − ) | R | N D −| R | D X k X V ∈T k ∂ | R | f Q j = k ∂t T j ∂t T k ⋆ V T (34) − ( − ) | R | N D −| R | D X k X V ∈T k ∂ | R | f Q j = k ∂t T j ∂t T k ⋆ V T + X T t T X V ′ ∈T X V ∈T ⋆ V ′ T ∂f∂t ( T ⋆ V ′ T ) ⋆ V T − X T t T X V ′ ∈T X V ∈T ⋆ V ′ T ∂f∂t ( T ⋆ V ′ T ) ⋆ V T . We reorganize the terms. For the terms in the first line we use lemma 1 and identify thederivatives with respect to ( T ⋆ V T ) k (resp. ( T ⋆ V T ) k ) . For the terms in the second line, thevertex V can either be the image of a vertex in T or of a vertex in T (resp. T ). Separating theterm with V the image of V ′ ∈ T we get h L T , L T i f =( − ) | R | N D −| R | D X V ∈T \ ( ) ∂ | R | f Q j ∂t ( T ⋆ V T ) j (35) − ( − ) | R | N D −| R | D X V ∈T \ ( ) ∂ | R | f Q j ∂t ( T ⋆ V T ) j + X T t T X V,V ′∈T V = V ′ (cid:16) ∂f∂t ( T ⋆ V ′ T ) ⋆ V T − ∂f∂t ( T ⋆ V ′ T ) ⋆ V T (cid:17) + X T t T X V ′ ∈T (cid:16) ∂f∂t ( T ⋆ V ′ T ) ⋆ V ′ T − ∂f∂t ( T ⋆ V ′ T ) ⋆ V ′ T (cid:17) + X T t T X V ′ ∈T X V ∈T \ ( ) ∂f∂t ( T ⋆ V ′ T ) ⋆ V T − X T t T X V ′ ∈T X V ∈T \ ( ) ∂f∂t ( T ⋆ V ′ T ) ⋆ V T , where, by a slight abuse of notations, we denote V and V ′ also the images of V and V ′ under ⋆ operations. By lemma 3, the terms in the third line cancel (after exchanging V and V ′ in thesecond term). Using lemma 4 the terms in the last line recombine with the ones in the first twolines. Finally the terms in the fourth line rewrite using lemma 2. We thus obtain h L T , L T i f = X V ∈T \ ( ) L T ⋆ V T f − X V ∈T \ ( ) L T ⋆ V T f + X T t T X V ∈T (cid:16) ∂f∂t T ⋆ V ( T ⋆ ( ) T ) − ∂f∂t T ⋆ V ( T ⋆ ( ) T ) (cid:17) . (36)Note that in both trees T ⋆ ( ) T and T ⋆ ( ) T , the root has a nonempty branch of color i ifat least one of T i or T i is non empty. The two roots have then equal coordination denoted | R | .Adding and subtracting, the commutator becomes h L T , L T i f = X V ∈T L T ⋆ V T f − X V ∈T L T ⋆ V T f − N D − D | R | (cid:16) ∂ | R | f Q t ( T ⋆ ( ) T ) i − ∂ | R | f Q t ( T ⋆ ( ) T ) i (cid:17) , (37)and the last term cancels due to lemma 5 and taking into account that f is a class function f ([ T ]),thus ∂ t T f = ∂ t T ′ f if T ∼ T ′ .1 IV. SCHWINGER DYSON EQUATIONS IN THE LARGE N LIMIT OF COLOREDTENSOR MODELS
In this section we first recall the independent identically distributed (i.i.d.) colored tensor modeland its 1 /N expansion. We then generalize it to a colored model with a generic potential, derive,in the large N limit the SDEs of the model and translate them into a set of equations (involvingthe operators L T ) for the partition function Z . We will closely parallel the derivation of the loopequations in section II. A. The i.i.d. colored tensor models with one coupling
We denoted ~n i , for i = 0 , . . . , D , the D -tuple of integers ~n i = ( n ii − , . . . , n i , n iD , . . . , n ii +1 ),with n ik = 1 , . . . , N . This N is the size of the tensors and the large N limit defined in [22–24]represents the limit of infinite size tensors. We set n ij = n ji . Let ¯ ψ i~n i , ψ i~n i , with i = 0 , . . . , D , be D +1 couples of complex conjugated tensors with D indices. The independent identically distributed(i.i.d.) colored tensor model in dimension D [24–27] is defined by the partition function e − N D F N ( λ, ¯ λ ) = Z N ( λ, ¯ λ ) = Z d ¯ ψ dψ e − S ( ψ, ¯ ψ ) ,S ( ψ, ¯ ψ ) = D X i =0 X n ¯ ψ i~n i ψ i~n i + λN D ( D − / X n D Y i =0 ψ i~n i + ¯ λN D ( D − / X n D Y i =0 ¯ ψ i~n i . (38) P n denotes the sum over all indices n ij from 1 to N . The tensor indices n ij need not be simpleintegers (they can for instance index the Fourier modes of an arbitrary compact Lie group, or evenof a finite group of large order [58]). Rescaling ψ i~n i = N − D/ P i~n i leads to S ( ¯ P , P ) = N D/ (cid:16) D X i =0 X ~n ¯ P i~n i P i~n i + λ X ~n D Y i =0 P i~n i + ¯ λ X ~n D Y i =0 ¯ P i~n i (cid:17) . (39)The partition function of equation (38) is evaluated by colored stranded Feynman graphs [25–27].The tensors have no symmetry properties under permutations of their indices (i.e. all ψ i~n i , ¯ ψ i~n i areindependent). The colors i of the fields ψ i , ¯ ψ i induce important restrictions on the combinatoricsof stranded graphs. We have two types of vertices, say one of positive (involving ψ ) and one ofnegative (involving ¯ ψ ). The lines always join a ψ i to a ¯ ψ i and possess a color index. Any Feynmangraphs G of this model is a simplicial pseudo manifold [26] and the colored tensor models providea statistical theory of random triangulations in dimensions D , generalizing random matrix models.The tensor indices n jk are preserved along the strands. The amplitude of a graph with 2 p verticesand F faces (closed strands) is [24] A ( G ) = ( λ ¯ λ ) p N − p D ( D − + F . (40)The n -bubbles of the graph are the maximally connected subgraphs made of lines with n fixedcolors. For instance, the D -bubbles are the maximally connected subgraphs containing all but oneof the colors. They are associated to the 0 simplices (vertices) of the pseudo-manifold. We label B b i ( ρ ) the D -bubbles with colors { , . . . , D } \ { i } (and ρ labels the various bubbles with identicalcolors). We denote B [ D ] the total number of D bubbles, which respects [24] p + D − B [ D ] ≥ , (41)2where p is half the number of vertices of the graph.A second class of graphs crucial for the 1 /N expansion of the colored tensor model are the jackets [22–24, 31]. Definition 1.
Let τ be a cycle on { , . . . , D } . A colored jacket J of G is the ribbon graph madeby faces with colors ( τ q (0) , τ q +1 (0)) , for q = 0 , . . . , D , modulo the orientation of the cycle. A jacket J of G contains all the vertices and all the lines of G (hence J and G have the sameconnectivity), but only a subset of faces. The jackets (further studied in [35, 41]) are ribbon graphs,completely classified by their genus g J . For a colored graph G we define its degree [23, 24] Definition 2.
The degree ω ( G ) of a graph is the sum of genera of its jackets, ω ( G ) = P J g J . The number of faces of a graph evaluates as a function of its degree [23, 24] ω ( G ) = ( D − (cid:16) p + D − B [ D ] (cid:17) + X i ; ρ ω ( B b i ( ρ ) ) . D − ω ( G ) = D ( D − p + D − F . (42)The 1 /N expansion of the colored tensor model is encoded in the remark that ω ( G ), which isa positive number, has exactly the combination of p and F appearing in the amplitude of a graph(40), thus A ( G ) = ( λ ¯ λ ) p N D − D − ω ( G ) . (43)The free energy F N ( λ, ¯ λ ) of the model admits then an expansion in the degree F N ( λ, ¯ λ ) = F ∞ ( λ, ¯ λ ) + O ( N − ) , (44)where F ∞ ( λ, ¯ λ ) is the sum over all graphs of degree 0. The degree plays in dimensions D ≥ Lemma 6.
If the degree vanishes (i.e. all jackets of G are planar) then G is dual to a D -sphere. We conclude this section with the following lemma.
Lemma 7.
Let G be a graph (with colors , . . . D ) and B b D ( ρ ) its D -bubbles with colors , . . . , D − .Then ω ( G ) ≥ D X ρ ω ( B b D ( ρ ) ) . (45) Proof:
Consider a jacket J of G . By eliminating the color D in its associated cycle we obtaina cycle over 0 , . . . , D − J b D ( ρ ) for each of its bubbles. As graphs, J b D ( ρ ) areone to one with disjoint subgraphs of J (obtained by deleting the lines of color D and joining thestrands ( π − ( D ) , D ) and ( D, π ( D )) in mixed faces corresponding to ( π − ( D ) , π ( D )) in J b D ( ρ ) [23]),consequently g J ≥ X ( ρ ) g J b D ( ρ ) . (46)Every jacket J b D ( ρ ) is obtained as subgraph of exactly D distinct jackets J (corresponding to insertingthe color D anywhere in the cycle associated to J b D ( ρ ) ). Summing over all jackets of G we obtain X J g J ≥ D X ρ X J b D ( ρ ) g J b D ( ρ ) . (47)3 B. Leading order graphs
The leading order graphs of the colored tensor model have been analyzed in detail in [53]. Wepresent below reader’s digest of these results. We are interested in understanding in more depththe structure of leading order vacuum graphs in D dimensions. Leading order vacuum graphcan be obtained from leading order two point graphs by reconnecting the two external lines (andconversely, cutting any line in a leading order vacuum graph leads to a leading order two pointgraph). We detail below the two point graphs.A D -bubble with two vertices B b i ( ρ ) has D ( D − internal faces, hence, by equation (42), the degree(and the topology) of a graph G and of the graph G / B b i ( ρ ) obtained by replacing B b i ( ρ ) with a line ofcolor i (see figure 4) are identical . ... i−1i+1i i i FIG. 4. Eliminating a D -bubble with two vertices. It can be shown [53] that, for D ≥
3, a leading order two point graph must possess a D -bubblewith exactly two vertices. Eliminating this bubble, we obtain a leading order graph having two lessvertices. The new graph must in turn possess a bubble with two vertices, which we eliminate, andso on. It follows that the leading order 2-point graphs must reduce after a sequence of eliminationsof D -bubbles to the graph with a single D -bubble and only two vertices of figure 4. It is moreuseful to take the reversed point of view and start with the graph of figure 4 and insert D -bubbleswith two vertices on its lines. This insertion procedure preserves colorability, degree and topology.The leading order 2-point connected graphs (with external legs of color say D ) are in one to onecorrespondence with colored rooted ( D + 1)-ary trees. Order ( λ ¯ λ ) : The lowest order graph consists in exactly one D -bubble with two vertices (andexternal lines say of color D ). We represent this graph by the tree with only the root vertex ( )decorated with ( D + 1) leaves. The leaves have colors 0 , . . . D . On G , we consider “active” all linesof colors j = D and the line of color D touching the vertex λ . They correspond to the leaves of thevertex. See figure 5, where the vertex λ is dotted and the inactive line is represented as dashed. ... ( ) FIG. 5. First order.
Order ( λ ¯ λ ) : At second order we have D + 1 graphs contributing. They come from insertinga D -bubble with two vertices on any of the D + 1 active lines of the first order graph. All theinterior lines of the new D -bubble are active, and so is the exterior line touching its vertex λ . Saywe insert the new bubble on the active line of color j . This graph corresponds to the tree { ( ) , ( j ) } ,see figure 6 for the case j = 0. Order ( λ ¯ λ ) p +1 : We obtain the graphs at order p + 1 by inserting a D -bubble with two verticeson any of the active lines of a graph at order p . The interior lines (and the exterior line touching the This elimination is a 1-Dipole contraction for one of the two lines of color i touching B b i ( ρ ) [24]. D 0D−1 ......
D 0 D D10 D10 ( )(0)
FIG. 6. Second order. vertex λ ) of the new bubble are active. We represent this by connecting a vertex of coordination D + 2, with D + 1 active leaves, on one of the active leaves of a tree at order p . The new tree lineinherits the color of the active line on which we inserted the D -bubble. At order ( λ ¯ λ ) p we obtaincontributions from all rooted colored ( D + 1)-ary trees with p vertices.Our tree is a colored version of Gallavotti-Nicolo [59] tree. The vertices of the tree representcertain subgraphs of G . We call them melons M and we identify them as the 1-particle irreducible(1PI) amputated 2-point sub-graphs of G . The intuitive picture is that a melon is itself made ofmelons within melons. The D -bubbles with only two vertices are obviously the smallest melons.The largest melon is the graph itself.A rooted tree is canonically associated to a partial order. The partial ordering correspondingto the tree we have introduced is M ≥ M if either M ⊃ M or ∃ N ( ρ ) , M ∪ (cid:0) ∪ ρ N ( ρ ) (cid:1) ∪ M is a 2-pointamputated connected sub-graph of G with external points ¯ λ ∈ M and λ ∈ M , (48)and ≥ is transitive.The line connecting M towards the root on the tree (i.e. going to a greater melon) inherits thecolor of the exterior half-lines of M . An example in D = 3 is given in figure 7 where the dottedvertices of G are λ , the inactive line of G is dashed and the active leaves are implicit. We identifythe melons by their external point λ . Since the active external line of a melon is always chosento be the one touching the vertex λ , the root melon in an arbitrary graph is the one containingthe external point ¯ λ , e.g. M in figure 7. Note that M ⊃ M , M , M , M , hence it is theirancestor. Also M ∪ M ∪ M forms a two point function with external point ¯ λ ∈ M and as M ⊂ M , the melon M is the ancestor of M , M , M , M , M , M , M .The vacuum leading order graphs (also called melonic) are obtained by reconnecting the twoexterior half lines of a melonic two point graphs with a line. Their amplitude is N D . If a graph isa melonic graph with D + 1 colors, all its D bubbles are melonic graphs with D colors. This is easyto see, as the reduction of a D bubble with two vertices represents the reduction of a D − D bubbles which contain the two vertices. When reducing the graph toits root melon, one by one all its D -bubbles reduce to D -bubbles with two vertices.Moreover, the D -ary trees of the D -bubbles are trivially obtained from the tree of the graphby deleting all lines (and leaves) of color D . We will be needing below the following obvious fact:given a melonic graph and one of its D bubbles (say B b D (1) ), all the lines of color D connecting onit either separate it from a different D -bubble (and are tree lines in the associated colored rootedtree) or they connect the two external points of a 1PI amputated two point subgraph with D − B b D (1) (i.e. they connect the two external points of a melon in B b D (1) ), in which case they areleaves of the associated tree.5 M1M2 M3M4 M5 M6M7 M8 M9 M10M11 M12 M13M1 M2M3M8M10M9 M13M12M11M7 M5M6 M4
00 0 3 33 1101 32 3012
FIG. 7. A melon graph and its associated colored GN rooted tree.
C. From one to an infinity of coupling constants
Inspired by section II, we generalize the colored tensor model with one coupling to a model withan infinity of couplings and derive the SDEs of the general model. First we integrate all colors saveone, and second we “free” the couplings of the operators in the effective action for the last color.When integrating all colors save one the partition function becomes Z = Z dψ D d ¯ ψ D e − S D ( ψ D , ¯ ψ D ) S D ( ψ D , ¯ ψ D ) = X ¯ ψ D~n D ψ D~n D + X B b D ( λ ¯ λ ) p Sym( B b D ) Tr B b D [ ¯ ψ D , ψ D ] N − D ( D − p + F B b D (49)where the sum over B b D runs over all connected vacuum graphs with colors 0 , . . . D − D -bubbles with colors 0 , . . . D −
1) and p vertices. The operators Tr B b D [ ¯ ψ D , ψ D ] inthe effective action for the last color are tensor network operators. Every vertex of B b D is decoratedby a tensor ψ D~n D or ¯ ψ D~ ¯ n D , and the tensor indices are contracted as dictated by the graph B b D . Wedenote v , ¯ v the positive (resp. negative) vertices of B b D , and l iv ¯ v the lines (of color i ) connectingthe positive vertex v with the negative vertex ¯ v . The operators writeTr B b D [ ¯ ψ D , ψ D ] = X n (cid:16) Y v, ¯ v ∈B b D ¯ ψ D~ ¯ n ¯ vD ψ D~n vD (cid:17)(cid:16) D − Y i =0 Y l iv ¯ v ∈B b D δ n vDi ¯ n ¯ vDi (cid:17) , (50)where all indices n are summed. Note that, as all vertices in the bubble belong to an unique line ofa given color, all the indices of the tensors are paired. The scaling with N of an operator computesin terms of its degree N − D ( D − p + ( D − D − p + D − − D − ω ( B b D ) = N − ( D − p + D − − D − ω ( B b D ) , (51)thus the effective action for the last color writes (dropping the index D ) S D ( ψ, ¯ ψ ) = X ¯ ψ ~n ψ ~n + N D − X B ( λ ¯ λ ) p Sym( B ) N − ( D − p − D − ω ( B ) Tr B [ ¯ ψ, ψ ] , (52)6Attributing to each operator its coupling constant and rescaling the field to T = ψN − D − , weobtain the partition function of colored tensor model with generic potential Z = e − N D F ( g B ) = Z d ¯ T dT e − N D − S ( ¯ T ,T ) ,S ( ¯ T , T ) = X ¯ T ~n T ~n + X B t B N − D − ω ( B ) Tr B [ ¯ T , T ] . (53)It is worth noting that, although in the end we deal with an unique tensor T , the colors arecrucial to the definition of the tensor network operators in the effective action. The initial vertex ofthe tensor model described a D simplex. The tensor network operators describe (colored) polytopesin D dimensions obtained by gluing simplices along all save one of their faces around a point (dualto the bubble B ). This is in strict parallel with matrix models, where higher degree interactionsrepresent polygons obtained by gluing triangles around a vertex.When evaluating amplitudes of graphs obtained by integrating the last tensor T , the tensornetwork operators act as effective vertices (for instance each comes with its own coupling constant).It is however more convenient to represent the Feynman graph of the path integral (53) still asgraphs with D + 1 colors. The effective vertices are the subgraphs with colors 0 , . . . , D −
1, andencode the connectivity of the tensor network operators.The partition function of eq. (53) provides a natural set of observables of the model: the multibubble correlations defined as D Tr B (1) [ ¯ T , T ] Tr B (2) [ ¯ T , T ] . . . Tr B ( ρ ) [ ¯ T , T ] E = ρ Y i =1 (cid:16) − N − h D − − D − ω ( B ( i ) ) i ∂∂t B ( i ) (cid:17) Z . (54)When introducing an infinity of coupling constants, we did not change the scaling with N of theoperators. The graphs G contributing to the connected multi bubble correlations are connectedvacuum graphs with D + 1 colors and with ρ marked subgraph corresponding to the insertionsTr B ( ν ) [ ¯ T , T ]. Taking into account the scaling of the insertions, the global scaling of such graphs is D Tr B (1) [ ¯ T , T ] Tr B (2) [ ¯ T , T ] . . . Tr B ( ρ ) [ ¯ T , T ] E c ≤ N D − D − ω ( G ) N − ρ ( D − P ρ D − ω ( B ( ρ ) ) ≤ N D − ρ ( D − − D ! ω ( G ) , (55)where we use lemma 7.In the large N limit, the connected correlations receiving contributions from graphs of degree 0(melonic graphs) dominate the multi bubble correlations. All their bubbles are necessarily melonic,in particular the insertions Tr B ( i ) [ ¯ T , T ]. As we have seen in section IV B, the melonic D -bubbles(i.e. melonic graphs with D colors) are one to one with colored rooted D -ary trees T . The tensornetwork operators, eq. (50), of melonic bubbles can be written directly in terms of T . Whenbuilding a D -bubble starting from T , each time we insert a melon corresponding to a vertex V ∈ T we bring a T and a ¯ T tensor for the two external points of the melon. We denote the indices of T by ~n V and the ones of ¯ T by ~ ¯ n V , and we getTr B [ ¯ T , T ] ≡ Tr T [ ¯ T , T ] = Y V ∈T (cid:16) T ~n V ¯ T ~ ¯ n V D − Y i =0 δ n iV ¯ n isi T [ V ] (cid:17) , (56)where s i T is exactly the colored successor function defined in section III. As this operator dependsexclusively of the successor functions, it is an invariant for an equivalence class of trees T ∼T ′ ⇒ Tr T [ ¯ T , T ] = Tr T ′ [ ¯ T , T ], hence the action and the partition function depend only on the7class variables t [ T ] = P T ′ ∼T t T . Taking into account that the melonic bubbles have degree 0 (andredefining the coupling of the tree T = { ( ) } ), the action writes S ( ¯ T , T ) = X T t T Tr T [ ¯ T , T ] + S r ( ¯ T , T ) , (57)where S r correspond to non melonic bubbles. D. Schwinger Dyson equations
Consider a melonic bubble corresponding to the tree T with root ( ) . We denote δ T n, ¯ n = Y V ∈T D − Y i =0 δ n iV ¯ n isi T V ] . (58)The SDEs are deduced starting from the trivial equality X ~p,n Z δδT ~p h T ~n ( )1 δ ~ ¯ n ( )1 ~p (cid:16) Y V ∈T \ ( ) T ~n V ¯ T ~ ¯ n V (cid:17) δ T n, ¯ n e − N D − S i = 0 , (59)which computes to X n Z n δ ~n ( )1 ~ ¯ n ( )1 (cid:16) Y V ∈T \ ( ) T ~n V ¯ T ~ ¯ n V (cid:17) δ T n, ¯ n + X V =( ) T ~n ( )1 ¯ T ~ ¯ n V δ ~ ¯ n ( )1 ~n V (cid:16) Y V ∈T \{ ( ) ,V } T ~n V ¯ T ~ ¯ n V (cid:17) δ T n, ¯ n − N D − T ~n ( )1 δ ~ ¯ n ( )1 ~p (cid:16) Y V ∈T \ ( ) T ~n V ¯ T ~ ¯ n V (cid:17) δ T n, ¯ n × h X T t T X V ∈T ¯ T ~ ¯ n V δ ~n V ~p (cid:16) Y V ′ ∈T \ V T ~n V ′ ¯ T ~ ¯ n V ′ (cid:17) δ T n, ¯ n + δS r δT ~p io e − N D − S . (60)The second line in eq. (60) represents graphs in which a line of color D on a melonic bubbleconnects the ¯ T on the root melon ( ) to a ¯ T on a distinct melon V . Hence it can not be a melon(see the end of section IV B). The last term represents a melonic bubble connected trough a line toa non melonic bubble (coming from δS r δT ). Thus it can not be a melon either. Taking into accountthat we have one line explicit in both graphs, (hence a factor N − ( D − ), and that the scaling ofTr T [ T, ¯ T ] is N D − , in both cases the correlations scale at most like1 Z D . . . E ≤ N D − D − . (61)The first term in eq. (60) factors over the branches T i of T . We denote ( ) ,i the root of thebranch T i . Recall that, for a non empty branch T i , the vertex s i T [( ) ] = ( i ) ∈ T maps on the root( ) ,i ∈ T i . For each branch we evaluate X n iR , ¯ n i ( )1 δ n i ( )1 ¯ n i ( )1 δ n i ( )1 ¯ n isi T δ n i [ si T ] − ¯ n i ( )1 = N if s i T [( ) ] = ( ) δ n i [ si T ] − ¯ n isi T = δ n i [ si T i ] − ,i ] ¯ n i ( )1 ,i if not , (62)8thus, denoting | R | the coordination of the root ( ) ∈ T , we get X ~n R ,~ ¯ n R δ ~n R ~ ¯ n R δ T n, ¯ n = N D −| R | D − Y i =0 T i = ∅ δ T i n, ¯ n . (63)The third term in eq. (60) computes X ~ ¯ n ( )1 ~n V δ ~ ¯ n ( )1 ~n V δ T n, ¯ n δ T n, ¯ n = X ~ ¯ n ( )1 ~n V δ ~ ¯ n ( )1 ~n V D − Y i =0 δ n i [ si T − ¯ n i ( )1 D − Y i =0 δ n iV ¯ n isi T [ V ] δ T \ ( ) n, ¯ n δ T \
Vn, ¯ n = D − Y i =0 δ n i [ si T − ¯ n isi T [ V ] δ T \ ( ) n, ¯ n δ T \
Vn, ¯ n = δ T ⋆ V T n, ¯ n , (64)hence the SDEs write, for every rooted tree T , with | R | non empty branches starting from theroot N D −| R | D D − Y i =0 T i = ∅ Tr T i [ ¯ T , T ] E − N D − X T t T X V ∈T D Tr T ⋆ V T [ ¯ T , T ] E = D . . . E ( − ) | R | N D − D | R | (cid:16) ∂ | R | Q i, T i = ∅ ∂t T i (cid:17) Z + X T t T X V ∈T ∂∂t T ⋆ V T Z = D . . . E , (65)where D . . . E denotes the non melonic terms of eq. (61). Taking into account the definition of L T in eq. (31), we obtain L T Z = D . . . E ⇒ lim N →∞ (cid:16) N − D Z L T Z (cid:17) = 0 , ∀T . (66)Recall that Z = e − N D F ( t B ) depends only on class variables t [ T ] . At leading order in 1 /N onlymelonic graph contribute to the free energy F ( t B ), hence lim N →∞ F ( t B ) = F ∞ ([ t [ T ] ]). The SDEsat leading order imply D − Y i =0 T i = ∅ (cid:16) ∂F ∞ ( t [ T ] ) ∂t T i (cid:17) − X T t T X V ∈T ∂F ∞ ( t [ T ] ) ∂t T ⋆ V T = 0 , ∀T . (67)The most useful way to employ the SDEs is the following. Consider a class function ˜ Z = e − N D ˜ F satisfying the constraints at all orders in N , L T ˜ Z = 0. Its free energy in the large N limit,˜ F ∞ ( t [ T ] ) = lim N →∞ ˜ F ( t [ T ] ) respects eq. (67), hence ˜ F ∞ ( t [ T ] ) = F ∞ ( t [ T ] ), that is the N → ∞ limitof ˜ Z and Z coincide.Note that an SDE at all orders can be derived for the trivial insertion X ~p Z δδT ~p h T ~p e − N D − S i = 0 → (cid:16) N D + X B |B| t |B| ∂∂t B (cid:17) Z = 0 , (68)where |B| denotes the number of vertices of the bubble B . The above operator, which at leadingorder is L ( ) , should be identified with the generator of dilations [60].9 V. CONCLUSIONS
We have generalized the colored tensor models to colored tensor models with generic interac-tions, derived the Schwinger Dyson equations at leading order and established that (at leadingorder) the partition function satisfies a set of constraints forming a Lie algebra. Much remains tobe done in order to fully characterize the critical behavior of the colored tensor models. We presentbelow a non exhaustive list of topics one needs to address.First, although the algebra of melonic bubbles observable closes at leading order in 1 /N , it doesnot closes at all orders (in contrast with matrix models, for which the algebra of loop observablescloses at all orders). Neither does the algebra of tensor networks corresponding to all bubbles.Indeed, if one attempts to derive the full SDEs, one generates terms associated to the addition oflines of color D on the D -bubbles with colors 0 , . . . D −
1. In order to obtain a full set of observables,one must also include tensor network operators for the corresponding graphs. The full SDEs canbe derived, but their algebra is somewhat more involved than the one at leading order.A second line of inquiry is to study the algebra of constraints L T . As colored rooted D -ary treescan be indexed in many alternative ways, in is yet unclear whether this algebra is an entirely newone or some relabeling of an already known algebra. One should study in the future its (unitary)representations, central extension, etc.. Although we do not yet know what is the continuumsymmetry this algebra encodes, as the generator of dilations is one of its generators, we expectsthe continuum theory to be scale invariant.Third, the equation (67) completely defines the free energy at leading order. One can easilywrite a solution of this equation as a perturbation series in the coupling constants. The perturbativesolution is ill adapted to the study of F ∞ ( t [ T ] ). The differential equation (67) constitutes a muchbetter starting point for the study of the leading order multi critical behavior of generic coloredtensor models in arbitrary dimensions. ACKNOWLEDGEMENTS
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