The expansion of colored tensor models
Abstract
In this paper we perform the expansion of the colored three dimensional Boulatov tensor model. As in matrix models, we obtain a systematic topological expansion, with increasingly complicated topologies suppressed by higher and higher powers of . We compute the first orders of the expansion and prove that only graphs corresponding to three spheres contribute to the leading order in the large limit.
I Introduction
Random tensor models and Group Field Theories (GFT) laurentgft ; quantugeom2 generalize random matrix models mm ; mmgravity ; ambj3dqg ; sasa1 to higher dimensions. The Feynman graphs of GFT are built from vertices dual to simplices, and propagators encoding the gluing of simplices along boundary simplices. Parallel to ribbon graphs of matrix models (dual to discretized surfaces), GFT graphs are dual to discretized dimensional topological spaces. For the simplest GFT models GFT the Feynman amplitude of a graph, reproduces the partition function of discretized BF theory FreidelLouapre ; gftnoncom ^{1}^{1}1More involved GFT models newmo2 ; newmo3 ; newmo4 ; newmo5 have been proposed in an attempt to implement the Plebanski constraints and reproduce the gravity partition function..
In contrast with random matrix models, the usual GFT models suffer from two major problems. First the Feynman graphs of GFTs are dual not only to manifolds and pseudo manifolds but also to more singular spaces sing . Second, and even more problematic, no equivalent of the expansion or of the notion of planarity Brezin:1977sv crucial in matrix models had yet been found in GFTs. This is one of the most important challenges GFTs and tensor models face today Alexandrov:2010un . The recently introduced “colored GFTs” color ; PolyColor (CGFT) solve the first problem sing and generate only pseudo manifolds. In this paper we prove that they also solve the second problem, namely CGFTs admit a topological expansion. We present in this paper the systematic expansion at all orders of CGFT and explicitly compute the first terms. We prove that at leading order in only graphs dual to the three sphere contribute. To establish this result we will rely on one hand on results and methods introduced in FreiGurOriti ; sefu1 ; sefu2 ; sefu3 ; param concerning amplitudes of CGFT graphs and on the other on results in combinatorial topology and manifold crystallization theory Lins ; FG . Almost none of the concepts and techniques we use can be applied to non colored GFT models.
Ii The Colored Boulatov Model
Let be some compact multiplicative Lie group, and denote its elements, its unit, and the integral with respect to the Haar measure. Let , be four couples of complex scalar (or Grassmann) fields over three copies of , . We denote the delta function over with some cutoff such that is finite, but diverges (polynomially) when goes to infinity. For (denoting the character of in the representation ) respectively we can chose
(1) 
The partition function of the colored Boulatov model color over is the path integral
(2) 
with normalized Gaussian measure of covariance
(3) 
and interaction (denoting )
(5)  
where . We call “black” the vertex involving the ’s and “white” the one involving the ’s.
The half lines of the CGFT vertex (represented in figure 1) have a color . The group elements in eq. (5) are associated to the “strands” (represented as solid lines) common to the half lines and . The vertex is dual to a tetrahedron and its half lines represent the triangles bounding the tetrahedron. The strand , common to the half lines and , represents the edge of the tetrahedron common to the triangles and .
The CGFT lines (figure 1) always connect two vertices of opposite orientation (i.e. a black and a white vertex). They have three parallel strands associated to the three arguments of the fields. A line represents the gluing of two tetrahedra (of opposite orientations) along triangles of the same color.
The strand structure of the vertex and propagator is fixed. One can represent a CGFT graph either as a stranded graph (using the vertex and propagator in figure 1) or as a “colored graph” with (colored) solid lines, and two classes of oriented vertices. Some examples of CGFT graphs are given in figure 2. We denote them from left to right , , , , and .
The lines of a vacuum CGFT graph are oriented (say from the black to the white vertex). The closed strands of form “faces” and are labeled by couples of colors. A vacuum CGFT graph must have the same number of black and white vertices. In this paper we will only deal with connected graphs. We denote , , the sets of vertices, lines and faces of . Also, we denote the set of lines of color and the set of faces of colors of . The Feynman amplitude of is
(6) 
where the notation (which we sometimes omit) signifies that the line belongs to the face and (resp. ) if the orientations of and coincide (resp. are opposite). The functions are invariant under cyclic permutations and conjugation of their arguments hence the amplitude of a graph does not depend on the orientation of the faces or on their starting point.
The first ingredient in our expansion is the scaling of the coupling in eq. (5). In sefu2 it is proved that obeys
(7) 
and that the bound is optimal (that is there exist graphs at any order saturating it). In order to obtain a sensible large limit, the scaling of the couplings and must be chosen such that the maximally divergent graphs have uniform degree of divergence at all orders.
Iii Ribbon Graphs
To any CGFT graph one associates two classes of ribbon graphs: its bubbles color and its jackets sefu3 . We denote in the sequel , and .
Bubbles. The bubbles color of a CGFT graph are the maximally connected subgraphs with three colors. They are dual to the vertices of the gluing of tetrahedra^{2}^{2}2 Recently an alternative definition for bubbles has been proposed in Bonzom:2010ar . Although interesting in itself, this definition is somewhat idiosyncratic, and it seems preferable to use the more standard notion of bubbles dual to vertices of the gluing of tetrahedra.. The bubbles admit two representations, either as colored graphs or as ribbon graphs color ; PolyColor . The ribbon graph of a bubble with colors is obtained by deleting all the strands containing the color . The bubbles of the graph (figure 2) are represented in figure 3.
We denote the set of all the bubbles of and the set of bubbles of colors .
For a bubble , we denote , and the sets of its vertices, lines and faces. The graph has four valent vertices (), while its bubbles have three valent vertices (). We have
(8)  
(9) 
with the genus of the bubble . A graph is dual to an orientable pseudo manifold. If all its bubbles are planar then it is dual to an orientable manifold sing .
Jackets. A second class of ribbon graph associated to are its jackets sefu3 . A jacket of is the ribbon graph obtained from by deleting all the faces with colors and . A CGFT graph has three jackets. The three jackets of are represented in figure 4, where the labels are associated to the faces.
The jackets of have four valent ribbon vertices. The reader might be worried that, while the vertices of the jacket with faces deleted (the one originally identified in sefu3 ) are simple ribbon vertices, the ones of the other two jackets (with the faces and deleted) appear twisted in figure 4. This is just an illusion: permuting the half lines and and respectively and on every jacket vertex eliminates all the twists. The sets of vertices, lines and faces of a jacket are , and .
Face routing. In non identically distributed matrix models GW ; GW1 ; GW2 the amplitude of a Feynman graph is computed via a “routing” algorithm, a digested version of which we present below.
To every ribbon graph (with sets of vertices, lines and faces denoted , and ) one associates a dual graph . The construction is standard (see for instance param ; GW2 and references therein). The vertices of , correspond to the faces of , its lines to the lines of and its faces to the vertices of . The lines of admit (many) partitions in three disjoint sets: a tree in , (), a tree in the its dual , (), and a set , ( ) of “genus” lines (param ).
We orient the faces of such that the two strands of every line have opposite orientations. We set a face of as “root” (denoted ). Consider a faces sharing some line with the root (that is the two strands of belong one to and the other to ). The group element appears exactly once in the argument of and
(10) 
where we set as the last line of and as the first line of . By our choice of orientations and eq. (10) becomes
(11) 
This trivial multiplication has two consequences. First the face is canonically associated to the line . Second, the face becomes a root face in the graph , obtained from by deleting and connecting and into a new face (see figure 5). Iterating for all faces except the root we get
(12) 
If is planar, is the exterior face of the tree in . The group elements corresponding to lines of touching leafs (vertices of coordination one in ) appear consecutively , and drop from the root face. Iterating for all line in we get
(13) 
for any base group . Remark that only the argument of the root changes under routing.
An Example: GFT. The dimensional GFT (with ) is a non identically distributed matrix model. The couplings do not need to be rescaled in this case. The free energy admits a familiar “genus expansion”.
By face routing one can integrate all group elements with and by a tree change of variables FreiGurOriti one eliminates all group elements with . One is left with an integral over the genus lines corresponding to a “super rosette graph” param with only one vertex and one face. The super rosette is obtained from by deleting the lines in and contracting the lines in . The particular super rosette to which a graph is reduced depends on the routing trees and , but all super rosettes associated to a graph have the same genus . One can define as the equivalence class of all super rosettes of genus . For a super rosette, each genus line appears twice in the argument of the last function. We expand in characters and integrate the genus lines (by the “third Filk move” in the dual super rosette param ). Each genus line brings a factor , hence the amplitude of a super rosette is , for all super rosettes of genus ^{3}^{3}3To correctly identify the scaling with one must use sliced functions, . . The amplitude of equals the one of the super rosette class to which it belongs. The genus expansion of the free energy writes
(14) 
with a combinatorial factor counting the graphs which reduce to the super rosette class i.e. all graphs of genus . Of course in dimensions, as the super rosette amplitudes can be computed explicitly one completely forgets about them, indexes the expansion of the free energy by the genus and concludes that higher and higher genus graphs are suppressed by larger and larger powers of the cut off.
Iv Dipoles
The second ingredient we need to establish our results are the Dipole moves Lins ; FG encoding homeomorphisms of pseudo manifolds (we will make a precise statement later). We will identify the various bubbles, faces and lines below by their colors (in superscript) and their vertices (in subscript).
1Dipoles. Consider a line of color with end vertices and (denoted ) in a graph . Call ( and ) the end vertex of the line of color ( and ) touching , and ( and ) the end vertex of the line of color ( and ) touching (see figure 6). The vertices and belong each to some 3bubble of colors , and . The two bubbles might coincide or might be different. If they are different and at least one of them is planar then the line is called an 1Dipole.
A 1Dipole can be contracted, that is the line together with the vertices and can be deleted from the graph and the remaining lines reconnected respecting the coloring (see figure 6). In the dual gluing a 1Dipole of color represents two tetrahedra sharing the triangle (of color 3) such that the vertices opposite to the triangle (duals to and ) are different. The contraction translates in squashing the two tetrahedra, merging the two vertices, and coherently identifying the remaining triangles , and (see figure 6).
In this picture it is clear why one of or is required to be planar. If both points opposite to the triangle were isolate singularities, the squashing of tetrahedra would decrease the number of singular points and would not be a homeomorphism. It is however a homeomorphism as long as one of the points is regular^{4}^{4}4See FG , especially the remark on page 93 in the proof of the main theorem..
The vertices and belong to the same faces , and (, , ), but distinct faces , and (, , , and , ). They also belong to the same bubbles , and , (, , ) but different bubbles ( and ). We track the effect of the 1Dipole contraction on the graph . Taking the planar bubble, the contraction

deletes the vertices and and the line .

glues on to form a new line (and similarly for colors and ).

transforms the face into a face (and similarly for and ) .

glues the face on the face to form a new face (and similarly for and ).

transforms the bubble into a bubble (and similarly for and )

glues on to form a new bubble .
The bubbles , and transform trivially under contraction. Call , , and (, , and ) the vertices, lines, faces and genus of one of these bubbles before (after) contraction. We have
(15) 
The bubble (with , and ) is glued on (with , and ) to form the new bubble (with , , and ) and
(16) 
Thus if . If is dual to a conical singularity () then the new bubble is dual to an identical singularity and the two dual pseudo manifolds are homeomorphic FG . Were we to allow a contraction when both we would merge two conical singularities into a unique (more degenerate) conical singularity.
Amplitude. Suppose that all lines enter and exit . We denote the group element associated to , etc. and use the shorthand notation for etc. The contribution of all faces containing and/or to the amplitude of is
(17)  
(18)  
(19)  
(20) 
where denotes the product of the remaining group elements along the face and similarly for the rest. We first change variables to , (and similarly for and ). The integral over is trivial. Forgetting the primes we obtain
(21)  
(22)  
(23)  
(24) 
We change again variables to (and similarly for and ) to obtain
(25)  
(26)  
(27)  
(28) 
We integrate using and (hence ) and eq. (17) becomes
(29)  
(30)  
(31)  
Remark that, ignoring , the integrand of eq. (29) corresponds to the graph with the 1Dipole contracted. But reproduces the external face of a ribbon graph obtained by cutting the vertex in the bubble . The latter is a planar ribbon graph hence by eq. (13) can be replaced by . Recalling that the number of vertices decreases by we obtain that the amplitudes of and (the graph with the 1Dipole contracted), are proportional
(32) 
2Dipoles. A 2Dipole of colors (see figure 7) is a couple of lines connecting the same two vertices and , and such that the faces and are different. The 2Dipole forms a face . Like the 1Dipoles, the 2Dipoles can be contracted (by deleting the lines and forming the 2Dipole and reconnecting the rest of the lines respecting the colors). This is represented in figure 7.
After contraction the two faces and are glued into a unique face . A 2Dipole is dual to two tetrahedra sharing two triangles (of colors 2 and 3 for figure 7) such that the edge opposite to the two triangles in each tetrahedron (dual to the faces and ) are different. The contraction translates in squashing the two tetrahedra and coherently identifying the remaining boundary triangles. This move always represents a homeomorphism FG . Denoting the graph obtained from after contracting the 2Dipole, a short computation along the lines of the one for 1Dipoles yields
(33) 
The Dipole contraction moves can be inverted into Dipole creation moves. The fundamental result we will use in the sequel FG is that two pseudo manifolds dual to colored graphs and are homeomorphic if and are related by a finite sequence of 1 and 2Dipole creation and contraction moves. We call two such graphs “equivalent”, .
V Bubble routing and Core Graphs
In the literature one finds two classes of results (bounds and evaluations) for amplitudes of GFT graphs. They are expressed either in terms of the number of vertices (sefu1 ; sefu2 ) or in terms of the number of bubbles FreiGurOriti ; sefu3 . In order to build the expansion in CGFT we need to strike the right balance between the vertices and the bubbles of a graph. This is achieved by a bubble routing algorithm.
Bubble routing. We start by choosing a set of roots of for all colors . For the color , if all the bubbles are planar we chose one of them as root and denote it . If there exist non planar bubbles we set a non planar bubble as principal root , and the other non planar bubbles as “branch roots” . We denote the set of roots of by . We repeat this for all colors (and denote the set of all roots of ).
We associate to the bubbles of a “ connectivity graph”. Its vertices represent the various bubbles . Its lines are the lines of color in . They either start and end on the same bubble (in which case they are “tadpole” lines in the connectivity graph), or not. A particularly simple way to picture the connectivity graph is to draw with lines , and much shorter than the lines . We chose a tree in the connectivity graph, (and call the rest of the lines “loop lines”). For a branch root , the line incident on it and belonging to the path in connecting to the principal root is represented as dashed. All the other lines in are represented as solid lines. An example is given in figure 8.
All the solid lines in are 1Dipoles and we contract them. We end up with a connectivity graph with vertices corresponding to the roots . The remaining lines of color cannot be 1Dipoles (they are either tadpole lines or they separate two non planar roots). The number of 1Dipoles of color contracted is . Neither the number nor the topology of the bubbles of the other colors, , and is changed under these contractions.
Having exhausted a complete set of 1Dipoles of color , we repeat the procedure for the 1Dipoles of color . The routing tree is chosen in the graph obtained after contracting the 1Dipoles of color and depends on , . The contraction of 1Dipole of color changes the 012 connectivity graph but it cannot create new 1Dipoles of color : the topology of the bubbles is unaffected by reducing 1Dipoles of color , hence the lines of color will still either be tadpole lines or separate two non planar roots . After a full set of 1Dipole contractions indexed by four distinct routing trees we obtain a Core Graph^{5}^{5}5If is dual to a manifold and one further reduces a full set of 2Dipoles one recovers a “gem” graph of Lins ..
Definition 1 (Core Graph).
A colored graph with vertices is called a Core Graph at order if, for all colors , it either has a unique (planar or non planar) bubble or all its bubbles are non planar.
The amplitude of the graph and of the Core Graph obtained after routing are related by
(34) 
The Core Graph one obtains by routing is not independent of the routing trees . The same graph leads to several equivalent Core Graphs, all at the same order , . One can prove that all equivalent Core Graphs at the same order have the same amplitude. Only the creation/contraction of dipoles of color can change the number of bubbles of colors , and the latter only create/annihilate planar bubbles. It follows that the numbers of bubbles of colors of and are equal and consequently the total numbers of 1Dipole creations and contractions are equal. As and have the same number of vertices, the total numbers of 2Dipole creations and contractions are also equal and .
We denote the set of equivalence classes of Core Graphs at order under the equivalence relation . The amplitude is a well defined function of the equivalence class . Under an arbitrary routing any graph will fall in a unique equivalence class . The free energy of the colored Boulatov model admits a topological expansion in Core Graphs classes
(35) 
where is a combinatorial factor counting all the graphs routing to a Core Graph class at order . The scaling with is entirely captured by the Core Graph amplitude . A Core Graphs class is dual to a specific pseudo manifold. Note however that the same pseudo manifold is represented by an infinity of classes at higher and higher orders in .
Core Graphs are in three dimensions the appropriate generalization of the super rosettes of two dimensional GFT. The only ingredient missing at this point is some estimate of their amplitude.
Theorem 1 (The Core Graph bound).
The amplitude of a Core Graph at order , , with set of bubble respects
(36) 
Proof: We denote the set of lines and faces of by and . The amplitude of the Core Graph is
(37) 
Denote the jacket of with the faces faces and deleted. The idea is to use the jacket graph to integrate explicitly as many group elements as possible. Indeed, routing the faces of the jacket graph will associate a line to all (save one) of its faces. When integrating all (save one) of the functions of the faces of the jacket graph will contribute , as . The effect of this integrations over the rest of the functions is exceedingly complicated to track. However we will not need to do it, as we will just use a naive bound