PHYSICAL REVIEW D 104051 (2011) Diffeomorphisms in...

Diffeomorphisms in group field theories

Aristide Baratin,1,* Florian Girelli,2,† and Daniele Oriti3,‡

1Triangle de la Physique, CPHT Ecole Polytechnique, IPhT Saclay, LPT Orsayand Laboratoire de Physique Theorique, CNRS UMR 8627, Universite Paris XI, F-91405 Orsay Cedex, France

2School of Physics, University of Sydney, Sydney, New South Wales 2006, Australia3Max Planck Institute for Gravitational Physics, Albert Einstein Institute, Am Muhlenberg 1, 14467 Golm, Germany

(Received 12 February 2011; published 27 May 2011)

We study the issue of diffeomorphism symmetry in group field theories (GFT), using the non-

commutative metric representation introduced by A. Baratin and D. Oriti [Phys. Rev. Lett. 105,

221302 (2010).]. In the colored Boulatov model for 3d gravity, we identify a field (quantum) symmetry

which ties together the vertex translation invariance of discrete gravity, the flatness constraint of canonical

quantum gravity, and the topological (coarse-graining) identities for the 6j symbols. We also show how,

for the GFT graphs dual to manifolds, the invariance of the Feynman amplitudes encodes the discrete

residual action of diffeomorphisms in simplicial gravity path integrals. We extend the results to GFT

models for higher-dimensional BF theories and discuss various insights that they provide on the GFT

formalism itself.

DOI: 10.1103/PhysRevD.83.104051 PACS numbers: 04.60.Pp, 02.40.Gh, 04.60.Gw, 04.60.Nc

I. INTRODUCTION

Diffeomorphism symmetry is a crucial aspect of thedynamics of spacetime geometry as described by generalrelativity and its higher derivative extensions. It is tied tothe notion of background independence [1], as the intro-duction of a nondynamical background breaks the fulldiffeomorphism invariance. It also imposes strong con-straints on the allowed dynamics. In fact, for example,the only diffeomorphism invariant action (in 4d) for atensor metric field that involves, at most, its first derivativesis the Einstein-Hilbert action (with cosmological constant);and, in a canonical formalism based on intrinsic metric andconjugate extrinsic curvature, only canonical general rela-tivity is compatible with the algebra of (the canonicalcounterpart of) diffeomorphisms [2].

This fact acquires even more relevance from the point ofview of ongoing efforts to build a quantum theory ofgravity. In background independent approaches [3] aimingat explaining the very origin of spacetime geometry, start-ing from ‘‘pregeometric,’’ discrete, or purely algebraicstructures, the correct implementation of diffeomorphisminvariance is a key guiding principle for the very definitionof the microscopic dynamics. A major open problem inthese approaches, such as in simplicial gravity [4], spinfoam models [5], and group field theories (GFT) [6], is toshow how the dynamics reduce to general relativity in asemiclassical and continuum approximation. A good con-trol over the (pregeometric analogue of) diffeomorphisminvariance is then essential: provided such an approxima-tion does not break this symmetry, general relativity shouldemerge as the dynamics of the metric field defined in terms

of the fundamental degrees of freedom of the theory, atleast at leading order. If the invariance is only approximate,still the requirement that it becomes exact in the continuumlimit is an important guiding principle for the definition ofappropriate coarse-graining and renormalization proce-dures, or to identify the diffeomorphism invariant sectorwhich should be dominant in the limit [7].With the smooth manifold of general relativity replaced

by discrete structures, the issue becomes that of identifyingsuitable transformations of the pregeometric data,1 leavingthe quantum amplitudes invariant, and encoding the (re-sidual) action of the diffeomorphism group. This is knownin the context of Regge calculus [9], where an action ofdiffeomorphisms at the vertices of the Regge triangulationhas been shown to exist around flat solutions. This isunderstood geometrically as the invariance of the Reggeaction upon translations of the vertices, in a local flatembedding of the triangulation in Rd. The invariance isexact in 3d, where the geometry is constrained to be flat; itis only approximate in the 4d case and in the presence of acosmological constant (see [10] and references therein). Inboth cases, the (approximate) invariance can be related todiscrete Bianchi identities. The action of diffeomorphismsin spin foam models has also been studied in the context of3d gravity [11]. In this work, it is shown that the discreteresidual of the local Poincare invariance, classicallyequivalent to diffeomorphism invariance, is responsiblefor (part of) the divergences of the Ponzano-Regge model.A related aspect of diffeomorphisms in spin foam modelsis the algebraic expression of diffeomorphism invariance interms of algebraic identities satisfied by n� j symbols, at

*[email protected]†[email protected]‡[email protected]

1In dynamical triangulations [4,8], all such data are fixed toconstant values, and the only analogue of diffeos is the auto-morphism group of the simplicial complex itself.

PHYSICAL REVIEW D 83, 104051 (2011)

1550-7998=2011=83(10)=104051(22) 104051-1 � 2011 American Physical Society

http://dx.doi.org/10.1103/PhysRevLett.105.221302


http://dx.doi.org/10.1103/PhysRevD.83.104051

the root of the topological invariance of some models, andrecognized to be an algebraic translation of the canonicalgravity constraints [12,13].

GFTs [6] are a higher-dimensional generalization ofmatrix models [14] and provide a second quantization ofboth spin network dynamics and simplicial gravity. TheirFeynman diagrams are dual to simplicial complexes; theamplitudes are given equivalently as spin foam models orsimplicial gravity path integrals [15]. Conversely, any spinfoam model can be interpreted as a Feynman amplitude ofa group field theory [16]. Hence, in the GFT perturbativeexpansion, one obtains a sum over (pre)geometric dataweighted by appropriate amplitudes, augmented by asum over simplicial complexes of arbitrary topology. Inthis paper, we ask ourselves whether the various notions ofdiffeomorphism invariance studied in the literature on dis-crete gravity can be traced back to a symmetry of the groupfield theory.

This task had proven impossible to fulfill up to now. Themain reason was the absence, at the GFT level, of explicitmetric variables, on which (discrete) diffeomorphismswould act. Now, recently, a metric formulation of GFT,completely equivalent to the usual formulations in termsof group variables or group representations, has been de-veloped [15] and used to prove an exact duality betweenspin foam models and simplicial path integrals. Here, weuse this formulation to study the action of discrete diffeo-morphisms in GFT. By doing so, we relate in a clear wayvarious aspects of diffeomorphism invariance in spin foammodels, canonical loop quantum gravity, and simplicialgravity. More precisely, we show that there is a set of fieldtransformations leaving the GFT action invariant, whosegeometrical meaning in the various GFT representationsties together the symmetry of the Regge action and thesimplicial Bianchi identities, the canonical constraints ofloop quantum gravity (adapted to a simplicial complex),and algebraic identities satisfied by n� j symbols.

A key feature of this metric formulation, which recastsGFTs as noncommutative field theories on Lie algebras, isto reveal and to make explicit the noncommutativity of thegeometry in GFTand spin foammodels [17–19]. The actionof discrete diffeomorphisms described in this paper natu-rally incorporates this noncommutativity, as it is generatedby a Hopf algebra [20]. Diffeomorphism invariance in GFTthus takes the formof a deformed (quantum) symmetry. Thedefinition of deformed symmetries in GFT, also consideredin [21], requires to embed the field theory into the largerframework of braided quantum field theories [22].

We work in the colored version of the GFT formalism[23,24], analogous to multimatrix models [14]. The color-ing can be used [25] to define a full homology for the GFTcolored diagrams2 and to unambiguously associate to it a

triangulated pseudomanifold, that is, complexes with point-like topological singularities [27]. The color formalismeliminates more pathological diagrams that are insteadgenerated by standard GFTs [23]. Strikingly, the coloringturns out to be also crucial for recasting the perturbativeexpansion of the (colored) Boulatov model, with a cutoff inrepresentation space, in terms of a topological expansion,and to show that the sum is dominated by manifolds oftrivial topology in the large cutoff limit [28]. This is theGFTanalogue of the ‘‘large-N’’ expansion of matrix models.These are very strong motivations for introducing coloringin GFTmodels. In this paper, we give another one: it is onlyin the colored framework that the action of discrete diffeo-morphisms can be encoded into field transformations.We focus on the topological models—namely, the (col-

ored) Boulatov and Ooguri models for 3d gravity and4d BF theory. The analysis can, however, be extended to4d gravity models obtained by imposing constraints ontopological ones [29].The paper is organized as follows. In Sec. II, we review

the GFT framework in dimension three, in its three knownformulations: the ‘‘group’’ formulation in terms of fieldson a group manifold, the ‘‘spin’’ formulation in terms oftensors in group representations, and the recent ‘‘metric’’formulation in terms of fields on Lie algebras. We illustratehow the duality of GFT representations translates into anexact duality between spin foam models, lattice gaugetheory, and simplicial path integrals.In Sec. III, we introduce a set of field transformations

which, we show, leaves invariant the action of the coloredBoulatov for 3d gravity. These transformations are gener-ated by a Hopf algebra [20], more precisely by the trans-lational part of a deformation of the Poincare group. Thedefinition of deformed (quantum) symmetries on GFTrequires to embed the field theory into the larger frame-work of braided quantum field theories [22]. We exploit theinvariance of the GFT vertex function to give the geomet-rical meaning of the symmetry in the three GFT represen-tations. We find that:(1) in the ‘‘metric’’ representation, the symmetry re-

flects the invariance under translations of each ofthe vertices of the Euclidean tetrahedron patternedby the GFT interaction.

(2) in the ‘‘group’’ representation, the symmetry ex-presses the flatness of the boundary connectionthat the field variables represent.

(3) in the ‘‘spin’’ representation, the symmetry encodesthe topological identities and recursion relations ofthe 6j symbols.

In Sec. IV, we look at the invariance of the GFT ampli-tudes and explain how the GFT symmetry relates to theaction of diffeomorphisms in simplicial path integrals. Theanalysis naturally distinguishes between manifold graphsand pseudomanifold ones. In the case of manifold graphs,we show, both geometrically and algebraically, how to

2For alternative definitions of homology of GFT diagrams,see [26].

ARISTIDE BARATIN, FLORIAN GIRELLI, AND DANIELE ORITI PHYSICAL REVIEW D 83, 104051 (2011)

104051-2

derive discrete Bianchi identities from the invariance of thevertex and propagator functions.

Finally, in Sec. V, we extend the results to the GFTmodel for 4d BF theory and discuss the case of constrainedmodels for gravity. We conclude in Sec. VI with a discus-sion of various issues raised by our analysis and newinsights that it provides on the GFT formalism.

II. COLORED GFTs AND METRICREPRESENTATION

d-dimensional GFTs [6], in their colored version [23],are field theories described in terms of dþ 1 complexfields f’‘g‘¼1��dþ1 defined over d copies of a group G,with a certain gauge invariance. The index ‘ is referred toas the color of the fields. Here, we consider the 3d case andthe Euclidean rotation group G ¼ SOð3Þ, so that each field’‘ is a function on SOð3Þ�3. The gauge invariance condi-tion reads:

8h 2 SOð3Þ; ’‘ðhg1; hg2; hg3Þ ¼ ’‘ðg1; g2; g3Þ:(1)

The dynamics is governed by the action S½’� ¼Skin½’� þ Sint½’�, where the kinetic term couples fieldswith the same colors:

Skin½’� ¼Z½dgi�3

X4‘¼1

’‘ðg1; g2; g3Þ’‘ðg1; g2:g3Þ; (2)

where ½dg�n is the product Haar measure on the groupSOð3Þ�n, and ’‘ are the complex conjugated fields. Theinteraction is homogeneous of degree four and given by

Sint½’� ¼ �Z½dgi�6’1ðg1; g2; g3Þ’2ðg3; g4; g5Þ

� ’3ðg5; g2; g6Þ’4ðg6; g4; g1Þþ �

Z½dgi�6’4ðg1; g4; g6Þ’3ðg6; g2; g5Þ

� ’2ðg5; g4; g3Þ’1ðg3; g2; g1Þ: (3)

The six integration variables in each integral follow thepattern of the edges of a tetrahedron. A field represents atriangle, the three field arguments being associated to itsedges (see Fig. 1). The four triangles of the tetrahedron are

marked by distinct colors.3 When the fields with differentcolors are all identified ’‘ :¼ ’ to a single real field,colored GFTs reduce to standard GFTs.The Feynman expansion of a GFT generates stranded

diagrams, with three strands per propagator, equipped witha canonical orientation of all lines and higher-dimensionalfaces. The propagator and vertex for ’ are drawn in Fig. 2;the vertex for ’ is obtained by reversing the order of alllabels. While the interaction vertex patterns a tetrahedronwith colored triangles, the propagator glues together tetra-hedra along triangles of the same color.Graph amplitudes are built out of propagators and vertex

functions:

P‘ðg; g0Þ ¼Z

dhY3i¼1

�ðg�1i hg0iÞ;

Vðg; g0Þ ¼Z Y4

‘¼1

dh‘Y6i¼1

�ððg‘i Þ�1h‘h�1‘0 g

‘0i Þ;

(4)

which identifies the variables along connected strands,modulo left shift by the gauge variables h arising fromthe invariance (1). The vertex function has an interpretationin terms of lattice gauge theory, where the three groupvariables g‘i and the group variables h‘ are viewed asholonomies along the links of the complex topologicallydual to a tetrahedron, shown in Fig. 3. The g‘i are ‘‘bound-ary’’ holonomies along the links dual to a triangle ‘. The h‘are ‘‘bulk’’ holonomies along the links connecting thetriangles to the center of the tetrahedron. The vertex func-tion simply states that the two-dimensional faces of thecomplex dual [in red (light gray) in Fig. 3] are flat. Thisimplies that the encoding of geometric information inthe model fits a piecewise-flat context, as in simplicialquantum gravity approaches.

FIG. 1 (color online). Geometric interpretation of the GFTfield.

FIG. 2. 3d GFT propagator and vertex.

3The coloring of each field, and thus of each triangle, by asingle label ‘ can be equivalently converted in a coloring of eachvertex of the tetrahedron by a label in the same range. In thissetting, each field triangle is labeled by the three colors of itsthree vertices. This shows that colored GFTs are a field theorygeneralization of double-indexed 3d tensor models [30].

DIFFEOMORPHISMS IN GROUP FIELD THEORIES PHYSICAL REVIEW D 83, 104051 (2011)

104051-3

In gluing two tetrahedra, the propagator function iden-tifies the boundary variables of the shared triangle, up to agroup variable h interpreted as a further parallel transportthrough the triangle.

After integration over all boundary variables g, theamplitude of a closed GFT diagram G takes the form

AG ¼Z Y

l

dhlYf

�

�Y!l2@f

hl

�; (5)

where the products are over the lines l and the faces (loopsof strands) f of the diagram. l and f dually label thetriangles and the edges of the triangulation defined bythe diagram. The ordered products of line variables alongthe boundary @f of the faces are computed by choosing anorientation and a reference vertex for each face f. Thegroup variables are taken to be hl or h

�1l , depending on

whether the orientations of l and f agree or not. In terms oflattice gauge theory, the set of variables ðhlÞl2G gives a

discrete connection on (the complex dual to) the triangu-lation, giving parallel transports from one tetrahedron toanother. The delta functions in (5) impose this connectionto be flat. The model is already seen then as describing adiscrete version of topological 3d BF theory, discretized onthe simplicial complex dual to the GFT diagram.

The ‘‘spin’’ representation of the GFT is obtained usingthe Peter-Weyl expansion of the fields over half-integerspins labeling the representations of SOð3Þ�3. Because ofgauge invariance, the coefficients are proportional to the

SOð3Þ Clebsh-Gordan coefficients Cj1;j2;j3m1;m2;m3

; the interac-tion vertex is expressed in terms of 6j symbols. Theamplitude of a Feynman diagram gives the Ponzano-Regge spin foam model [31]:

AG ¼ Xfjeg

Ye

djeY�

8<: j�1 j�2 j�3

j�4 j�5 j�6

9=;; (6)

where the spins je label the edges of the triangulationassociated to the diagram, dj ¼ 2jþ 1 is the dimension

of the representation j, and the amplitude is a product oftetrahedral 6j symbols. Thus, group and spin representa-tions of the GFT realize explicitly the duality between the

connection (5) and spin foam (6) formulations of thePonzano-Regge model [5,32].A third representation of GFTs, in terms of continuous

noncommutative ‘‘metric’’ variables x 2 suð2Þ � R3, hasbeen recently developed [15] and shown to realize a furtherduality between spin foam models and simplicial pathintegrals. Since the geometrical meaning of the symmetriesstudied in the next sections is best understood in such ametric representation, let us briefly recall here its construc-tion. The representation is obtained using the group Fouriertransform [17,20] of the fields

’ ‘ðx1;x2;x3Þ :¼Z½dgi�3’‘ðg1;g2;g3Þeg1ðx1Þeg2ðx2Þeg3ðx3Þ;

(7)

expressed in terms of plane-wave functions eg : suð2Þ �R3 ! Uð1Þ. The definition of the plane wave depends on achoice of coordinate systems on the group manifold. In thefollowing, we identify functions of SOð3Þ � SUð2Þ=Z2

with functions of SUð2Þ invariant under g ! �g. Using

the parametrization g ¼ e� ~n� ~� and x ¼ ~x � ~� of group andLie algebra elements in terms of the (anti-Hermitian) suð2Þgenerators ~� ¼ ð�1; �2; �3Þ, a convenient representation ofthe plane waves is

egðxÞ :¼ ei Trxg; (8)

where the trace is given by Tr�i�j ¼ ��ij. Note that, since

egðxÞ ¼ eg�1ðxÞ ¼ egð�xÞ, the Fourier transform of the

complex conjugate field relates to the complex conjugateof the Fourier transform as

�’‘ðx1; x2; x3Þ ¼ �’‘ð�x1;�x2;�x3Þ: (9)

The image of the Fourier transform inherits by duality anontrivial (noncommutative) pointwise product from theconvolution product on the group. It is defined on planewaves as

ðeg ? eg0 ÞðxÞ :¼ egg0 ðxÞ; (10)

and extends componentwise to the product of three planewaves and by linearity to the whole image of the Fouriertransform.The first feature of this representation is that the gauge

invariance condition (1) expresses itself as a ‘‘closureconstraint’’ for the triple of variables xi of the dual field.To see this, we consider the projector P onto gauge invari-ant fields:

P x’‘ ¼Z½dh�’‘ðhg1; hg2; hg3Þ; (11)

and note that

dPx’‘¼ C?’‘; Cðx1;x2;x3Þ :¼�0ðx1þx2þx3Þ;(12)

where �0 is the element x ¼ 0 of the family of functions

FIG. 3 (color online). The ‘‘boundary’’ holonomies g‘i are inred (light gray), while the ‘‘bulk’’ holonomies h‘ are in blue(dark gray).


104051-4

�xðyÞ :¼Z½dh�eh�1ðxÞehðyÞ: (13)

These functions play the role of Dirac distributions in thenoncommutative setting, asZ

½d3y�ð�x ? fÞðyÞ ¼ fðxÞ; (14)

where d3y is the standard Lebesgue measure on R3. Wemay thus interpret the variables xi of the dual gaugeinvariant field as the closed edge vectors of a triangle inR3 and further confirm the interpretation of the GFT fieldsas (noncommutative) triangles.

The GFTaction can be written in terms of dual fields andmetric variables by exploiting the duality between groupconvolution and ? product. Given two functions f, h on

SOð3Þ and bf, h their Fourier transform (7), this duality canbe read in the property

Z½dg�fðgÞhðgÞ ¼

Z½d3x�ðf ? h�ÞðxÞ; (15)

where h�ðxÞ :¼ hð�xÞ, and d3x is the Lebesgue measureon R3. Hence, the combinatorial structure of the GFTaction in the metric representation is the same as in groupone, while group convolution is replaced by ? product.Using the short notation ’123

‘:¼ ’‘ðx1; x2; x3Þ, we can

write the action as

S½’� ¼Z½d3xi�3

X4‘¼1

’123‘ ? �’123

‘

þ �Z½d3xi�6’123

1 ? ’3452 ? ’526

3 ? ’6414

þ �Z½d3xi�6 �’146

4 ? �’3453 ? �’526

2 ? �’6411 ; (16)

where it is understood that ? products relate repeated upperindices as ’i ? ’i :¼ ð’ ? ’�ÞðxiÞ, with ’�ðxÞ ¼ ’ð�xÞ.

Feynman amplitudes are built out of propagators andvertex functions:

P‘ðx; x0Þ ¼Z½dh�Y3

i¼1

ð��xi ? ehÞðx0iÞ;

Vðx; x0Þ ¼Z Y4

‘¼1

½dh‘�Y6i¼1

ð��x‘i? eh‘h�1

‘0Þðx‘0i Þ;

(17)

where the �x are given by (13). These have a naturalinterpretation in terms of simplicial geometry, where thex variables on connected strands encode the metric of thesame edge in different frames, related with each other bythe holonomies h. In building up the diagram, propagatorand vertex strands are joined to one another using the ?product.

Under the integration over the holonomy variables, theamplitude of a closed diagram G factorizes into a product

of face amplitudes Af½h�, taking the form of a cyclic ?

product:

Af½h� ¼Z YNf

j¼0

½d3xj� ~wNþ1j¼0 ð�xj ? ehjjþ1

Þðxjþ1Þ; (18)

where the product is over the Nf vertices of G (dual to

tetrahedra) in the loop of strands that bound f. The ordered? product is computed by choosing an orientation and areference vertex for the face f; by convention, we setxNþ1 :¼ x0. The holonomy hjjþ1 parallel transports the

reference frame of j to that of jþ 1. In terms of simplicialgeometry, it encodes the identification, up to parallel trans-port, of the metric variables associated to the edge dual to fin the different frames j.After integration, within all face amplitudes, over all

metric variables xj except for that x0 of the reference

frame, the amplitude of the GFT diagram G takes theform of a simplicial path integral:

AG ¼Z Y

l

½dhl�Ye

½d3xe�eiP

e TrxeHe ; (19)

where the products are over the lines of G and the edges ofthe dual triangulation, and He :¼ Q!

l2@fehl is the holon-

omy along the boundary of the face fe of G dual to e,calculated from a given reference tetrahedron frame. Theexponential term is the (exponential of the) discrete actionof first-order 3d gravity (which is the same as 3d BFtheory), in Euclidean signature. This gives the definiteconfirmation of the interpretation of the xe variables asdiscrete triad variables associated to the edges of thetriangulation dual to the GFT Feynman diagram (edgevectors).Thus, the metric representation of GFT realizes explic-

itly the duality between spin foam models (5) or (6) andsimplicial path integrals (19), generalizing it to arbitrarytransition amplitudes (corresponding to open GFT dia-grams) with appropriate boundary terms arising naturallyin the simplicial action in (19), for fixed triad variables atthe boundary, and boundary observables. This result isgeneral: it extends to BF theories in higher dimensionsand to gravity models obtained as constrained BF theories(see [15,33]).The metric representation has, of course, the advantage

of making the (noncommutative, simplicial) geometry ofGFT and spin foam models more transparent. This will beuseful for the understanding of the symmetries studied inthe next section.

III. GFT (DISCRETE) DIFFEOMORPHISMSYMMETRY

In this section, we introduce a set of field transforma-tions which, we show, leave the GFT action invariant. Wegive the geometrical meaning of such transformations in


104051-5

the different representations and show that they correspondto diffeomorphisms in discrete quantum geometry models.We also derive yet another GFT representation in terms ofthe generators of the symmetry, which are Lie algebra‘‘position’’ variables associated to the vertices of the sim-plex patterned by the GFT field.

The noncommutativity of the metric (triad) space plays acrucial role in the definition and meaning of the symmetrytransformation. This is, in fact, a Hopf algebra (quantum)symmetry, characterized by a nontrivial action on a tensorproduct of fields, due to a nontrivial coproduct. The rele-vant quantum group here, i.e., in this specific GFT modelfor Euclidean 3d gravity with the local gauge group beingSOð3Þ, is a deformation of the Euclidean group ISOð3Þ, theso-called Drinfeld double DSOð3Þ.4

A. Action of DSOð3Þ on fields on SOð3ÞThe Drinfeld double is defined as DSOð3Þ ¼

CðSOð3ÞÞ 2CSOð3Þ, where the group algebra CSOð3Þ actsby the adjoint action on the algebra of functions CðSOð3ÞÞ.It is a deformation of the three-dimensional Euclideangroup ISOð3Þ—more precisely, of the Hopf algebraCðR3Þ 2CSOð3Þ—where the ‘‘rotations’’ belong to thegroup algebra CSOð3Þ and the ‘‘translations’’ are complexfunctions in CðSOð3ÞÞ. A general element can be written asa linear combination of elements f ��, where f 2CðSOð3ÞÞ and � 2 SOð3Þ. The space of functions on thegroup CðSOð3ÞÞ gives a representation of DSOð3Þ, inwhich rotations act by adjoint action on the variable andtranslations act by multiplication:

�ðgÞ��ð��1xgÞ :¼�ð��1g�Þ; �ðgÞ�fðgÞ�ðgÞ;�2CðSOð3ÞÞ: (20)

Choosing as a translation element a generating plane wavelabeled by " 2 suð2Þ � R3, the field � gets multiplied bya phase f"ðgÞ ¼ egð"Þ. Upon the group Fourier transform

introduced in the previous section,

�ðxÞ ¼Z½dg��ðgÞegðxÞ; (21)

this corresponds to the dual action �ðxÞ � �ðxþ "Þ. Wewill also use the dual action of DSOð3Þ, where rotationsact by inverse adjoint action and plane waves labeled by "act, by translation, by �".

Up to now, the transformations are the exact analogue ofthe usual Poincare transformations on functions on flatspace, here replaced by the algebra suð2Þ, while momen-tum space is replaced by the group manifold SOð3Þ. Thedeformation manifests itself as a nontrivial action on atensor product of fields, due to the nontrivial coproducton the translation algebra CðSOð3ÞÞ. Thus,

�1ðg1Þ�2ðg2Þ � 4fðg1 � g2Þ�1ðg1Þ�2ðg2Þ; (22)

where the coproduct 4 is given by

4 fðg1 � g2Þ ¼ fð1Þðg1Þfð2Þðg2Þ ¼ fðg1g2Þ;8f 2 CðSOð3ÞÞ: (23)

Using the fact that eg1g2ð"Þ ¼ ðeg1 ? eg2Þð"Þ, one can checkthat the dual action of the plane wave egð"Þ on a tensor

product is obtained by translating each variable by " andby taking the ? product of the resulting fields with respectto ":

�1ðx1Þ�2ðx2Þ � �1ðx1 þ "Þ ?" �2ðx2 þ "Þ: (24)

This structure is what replaces the usual translation groupR3, and the deformation is consistent with the noncommu-tativity of the algebra of functions on suð2Þ � R3 inducedby the ? product.

B. GFT as a braided quantum field theory

In order to allow the Hopf algebra to act on the poly-nomials of fields defined by the GFT action, the idea is toembed the theory into the algebraic framework of braidedquantum field theories defined by Oeckl [22]. In short, thisconsists of lifting all polynomials of fields to tensor prod-ucts, in order to keep track of the ordering of the fields andfield variables. Commuting fields or field variables requiresto specify braiding maps B12 : X1 � X2 ! X2 � X1 be-tween any two copies of the space of fields. The theory isdefined perturbatively as a braided Feynman diagram ex-pansion, using a braided Wick theorem [22]. In a trivialembedding of GFTs, where all fields commute, into thebraided framework, the braiding maps are chosen to be thetrivial flip maps:

B12: CðSOð3ÞÞ � CðSOð3ÞÞ ! CðSOð3ÞÞ � CðSOð3ÞÞ��ðg1Þ ��ðg2Þ � �ðg2Þ ��ðg1Þ:

We emphasize that such trivial embedding does notmodifythe theory.5 It, however, allows us to define Hopf algebratransformations on the GFT fields.The possibility of using a nontrivial braiding between

fields or field arguments is not employed in usual GFTs, sowe will stick to the usual formalism in what follows. Thechoice of trivial braiding is often made also in the non-commutative geometry literature, even in the presence ofquantum group symmetries. However, since, in general, thetrivial braiding map does not intertwine the action of thequantum group symmetry, this choice leads to a breakingof the symmetry at the level of the n-point functions.In order to make the full theory symmetric, it is mostnatural to use the braiding of the (braided) category of

4The role played by the Drinfeld double in spin foam and GFTmodels has been emphasized already, e.g., in [21,34,35].

5In fact, in this setting, the braided Feynmanology is redun-dant, and the braided amplitudes coincide with the unbraidedones.


104051-6

representations of the quantum group [20]. Thus, in scalarfield theory fully invariant under the Drinfeld doubleDSOð3Þ, this braiding is

B12: CðSOð3ÞÞ � CðSOð3ÞÞ ! CðSOð3ÞÞ � CðSOð3ÞÞ��ðg1Þ ��ðg2Þ � �ðg2Þ ��ðg2g1g�1

2 Þ: (25)

Moreover, the use of a trivial braiding in the presence ofquantum group symmetries has been argued to be theorigin of the (in)famous IR UV mixing [36–38]. Thus,the choice of braiding does affect the physics of the modeland modifies its Feynman amplitudes.

The view we take in this paper opens the way to ageneralization of GFTs which would include a nontrivialbraiding intertwining the quantum symmetry describedbelow. We discuss this possibility in the concluding sec-tion. The suggestion of extending GFTs to fully braidedfield theories has been put forward also in the recent work[21], again following the identification of a quantum groupsymmetry in the GFT context.

C. Symmetries of the GFT action

We have recalled above howDSOð3Þ acts on a functionof a single variable. Here, we define a set of transforma-tions of the GFT fields ’‘ under rotations and translationswhich leave the GFT action invariant. As we illustrate inthe different GFT representations in the next subsection,the translational part of this action, interpreted as ‘‘vertextranslations’’ in the simplex patterned by the field, willencode the action of discrete diffeomorphisms in GFT.

Let us first point out that the requirement of gaugecovariance restricts the number of independent transforma-tions that a field transformation T can undergo. Suchtransformation is, indeed, well-defined on a gauge invari-ant field only if it commutes with the projector (11):

P xðTx’‘Þ ¼ TxðPx’‘Þ:Thus, for instance, the only gauge covariant action of therotations in CSOð3Þ�3,

’‘ðg1;g2;g3Þ�’‘ð��11;‘xg1;�

�12;‘xg2�

�13;‘xg3Þ; (26)

is the diagonal one, �i;‘ :¼ �‘. In the metric representa-

tion, gauge covariance simply means that the transforma-tion preserves the closure �ðx‘1 þ x‘2 þ x‘3Þ of the triangle ‘.In the case of rotations, one can easily go one step furtherand check that the only field transformation that preservesthe kinetic and interaction polynomials is generated by asingle rotation �‘ :¼ �. In the metric representation, thisis the only action of the rotations that respects the gluing ofedge vectors of the tetrahedron patterned by theinteraction.

Let us stress that this symmetry corresponds precisely tothe invariance under local changes of frame in each tetra-hedron and in each triangle that one expects in 3d simpli-cial gravity (see Sec. IV). We thus find such an invariance

implemented as the well-known local gauge invariance inboth the simplicial path integral and pure gauge theoryformulation of the GFT Feynman amplitudes, as well as intheir spin foam representation.We now turn to the more interesting case of translations.

We will define transformations generated by foursuð2Þ-translation parameters "v, where v labels the fourvertices of the interaction tetrahedron, diagrammaticallyrepresented by its dual diagram in Fig. 2. Each vertex ofthis tetrahedron is represented by a certain subgraph, whichwe call the ‘‘vertex graph’’ [25,39]: the vertex graph for thevertex v‘ opposite to the triangle of color ‘ is obtained byremoving all the lines which contain strands of color ‘. Thevertex graph of v3 is pictured in Fig. 4: its three linespattern the three edges 1, 3, 4 sharing v3.The vertices opposite to the triangles ‘ ¼ 2, 3, 4 are

represented by identical (after anticlockwise rotation by�=4, �=2, and 3�=4) diagrams, where 1, 3, 4 are replaced,respectively, by 3, 5, 1; by 5, 6, 4; and by 6, 1, 2 (in thisorder).To define the action of a translation of the vertex v3, we

equip the lines of the vertex graph with an orientation, asdrawn in Fig. 4. Using this convention, each line has an‘‘incoming’’ and an ‘‘outcoming’’ external strand. A trans-lation of v3 generated by "3 2 suð2Þ acts nontrivially onlyon the strands of the vertex graph. In the metric represen-tation, it shifts the corresponding variables x‘i by �"3,whether the strand i comes in or out of ‘:

x‘i � x‘i þ "3 if i is outgoing;

x‘i � x‘i � "3 if i is incoming;(27)

in a way that preserves the closure �ðx‘1 þ x‘2 þ x‘3Þ of eachtriangle ‘. More precisely, the translationT �3 of the vertex

v3 acts on the dual fields as

T "3x’1ðx1;x2;x3Þ :¼w"3’1ðx1�"3;x2;x3þ"3Þ;T "3x’2ðx3;x4;x5Þ :¼w"3’2ðx3�"3;x4þ"3;x5Þ;T "3x’4ðx6;x4;x1Þ :¼w"3’4ðx6;x4�"3;x1þ"3Þ;T "3x’3ðx5;x2;x6Þ :¼ ’3ðx5;x2;x6Þ:

(28)

FIG. 4. Vertex graph for the vertex v3.


104051-7

The same field transformation can be expressed in amore explicit way (without star product) in the grouprepresentation, by group Fourier transform, as follows:

T "3x’1ðg1; g2; g3Þ :¼ eg�11

g3ð"3Þ’1ðg1; g2; g3Þ;

T "3x’2ðg3; g4; g5Þ :¼ eg�13

g4ð"3Þ’2ðg3; g4; g5Þ;

T "3x’4ðg6; g4; g1Þ :¼ eg�14

g1ð"3Þ’4ðg6; g4; g1Þ;

T "3x’3ðg5; g2; g6Þ :¼ ’3ðg5; g2; g6Þ:

(29)

The transformation is immediately extended to the com-plex conjugated fields �’‘ by requiring consistency withcomplex conjugacy, using the property of the plane wavesthat egð"Þ ¼ eg�1ð"Þ.

We see at first glance in (28) the geometric meaning ofthis transformation as a vertex translation, by the way itaffects the arguments of the dual fields interpreted as edgevectors. When translating a vertex of the triangle, onetranslates the two edges sharing this vertex; each edge istranslated in an opposite way due to the orientation of theedges. If the vertex is shared by many edges, all these edgesare translated accordingly, while taking into account theirorientation. The gauge covariance ofT is manifest in bothrepresentations: in the metric representation, this is be-cause the shift of the edge variables preserves the closureof each triangle; in the group representation, this is becausethe arguments g�1

j gk of the plane waves are gauge

invariant.Let us show that these field transformations leave the

GFT action invariant. In fact, they leave invariant the fieldpolynomials in this action, even before integration over thefield variables. We check this in the group representation.Following the definition (22) of the translation algebra on atensor product of fields, the action of the transformationT "3 on the interaction polynomials

’1ðg1; g2; g3Þ’2ðg3; g4; g5Þ’3ðg5; g2; g6Þ’4ðg6; g4; g1Þresults in an overall multiplication by the phase

ðeg�11

g3? eg�1

3g4? eg�1

4g1Þð"3Þ ¼ eg�1

1g3g

�13

g4g�14

g1ð"3Þ ¼ 1:

(30)

The interaction term is, therefore, invariant. Clearly, theordering of fields and field arguments is crucial. Thekinetic term is also invariant, since T �3 acts on the field

polynomials ’‘ðg1; g2; g3Þ �’‘ðg1; g2; g3Þ, for instance,when ‘ ¼ 1, by multiplication by the plane wave:

ðeg�11

g3? eg�1

3g1Þð"3Þ ¼ ðeg�1

1g3g

�13

g1Þð"3Þ ¼ 1:

Thus, we have shown that the transformation generated bytranslation of the vertex v3 is a symmetry of the GFTaction. We can show similarly the invariance of the actionunder translations of the three other vertices v1, v2, and v4

of the tetrahedron.

Before discussing further the meaning of the symmetryin the next section, let us point out that the four symmetrygenerators are not all independent—in other words, thesymmetry is reducible. In fact, there is a global translationof the four vertices of the tetrahedron under which thefields transform trivially. Geometrically, this correspondsto the rather trivial fact that the geometry of a Euclideantriangle is invariant under a global translation of its verti-ces. Such a global translation is defined by choosing anorder for the vertices of each triangle. For example, choos-ing the order v3, v4, v2 for the vertices of the triangle‘ ¼ 1, a global translation acts on ’1 as

’1ðg1; g2; g3Þ � eg�11

g3ð"Þ ? eg�1

2g3ð�"Þ ? eg�1

1g2ð�"Þ

� ’1ðg1; g2; g3Þ ¼ ’1ðg1; g2; g3Þ:

D. Invariance of the vertex and diffeomorphisms

We now want to show how the field symmetry (29), and,more specifically, the invariance of the vertex function, tietogether various notions of (discrete residual of) diffeo-morphisms studied in the literature. To do so, we probe themeaning of such invariance in the different GFT represen-tations. This picture will be completed in the next section,when we will discuss the invariance of the GFT Feynmanamplitudes and n-point functions.(i) Metric representation. The vertex function is given

by the formula (17)

Vðx‘i ; x‘0i Þ ¼Z Y4

‘¼1

½dh‘�Y6i¼1

ð��x‘i? eh‘h�1

‘0Þðx‘0i Þ:

(31)

Fixing the ordering of the variables to the one de-fined by the interaction polynomials, this functioncan be lifted to the group Fourier dual of a tensorproduct in CðSOð3ÞÞ�12, invariant under the (non-commutative) translation (27). As we have alreadyemphasized, this transformation is geometrically in-terpreted as a translation of a vertex of the tetrahe-dron patterned by the interaction. More precisely, the

function Vðx‘i ; x‘0i Þ imposes the variables x‘i , inter-preted as edge vectors expressed in different frames,to match the metric of a Euclidean tetrahedron. Thesymmetry expresses the invariance of the matchingcondition under a translation of each of the verticesin an embedding of this tetrahedron in R3. This isalso how the action of discrete residual of diffeo-morphisms is encoded in 3d Regge calculus.

(ii) Group representation. The vertex function is givenby the formula (4):

Vðg‘i ; g‘0i Þ ¼Z Y4

‘¼1

dh‘Y6i¼1


‘0i Þ:

(32)


104051-8

The invariance of this function under translations(27) of the vertex v3 means that, for all �3 2 suð2Þ,

e Gv3ð"3ÞVðg‘i ; g‘0i Þ ¼ Vðg‘i ; g‘0i Þ; (33)

where the argument of the plane wave is

Gv3¼ ðg11Þ�1g13ðg23Þ�1g24ðg44Þ�1g41: (34)

We thus see that translation invariance reflects, inthe group representation, a conservation ruleGv3

¼ 1. Now, recall that the group field variables

g‘i encode boundary holonomies, along paths con-necting the center of each triangle ‘ to its edges. Inthe interaction term, they define a discrete connec-tion living on the graph dual to the boundary trian-gulation of the tetrahedron, which has the topologyof a two-sphere. As illustrated on the right of Fig. 5,Gv3

is the holonomy along a loop circling the vertex

v3 of the tetrahedron.Thus, the symmetry under the translation of eachvertex says the boundary connection is flat.

Imposing flatness F ¼ 0 of the boundary connec-tion is precisely the role of the Hamiltonian andvector constraints, i.e., the canonical counterpartof diffeomorphisms, in (first-order) 3d gravity[40]. Here, we see that the GFT symmetry resultsin such a constraint on the tetrahedral wave functionconstructed from the GFT field.

(iii) Spin representation. In the spin representation, ob-tained by Plancherel decomposition into SOð3Þrepresentations, the vertex function takes the formof SOð3Þ 6j symbols. This is a standard calculation,starting from the tensorial expression of V in theSOð3Þ representations of spin fj‘i g:

Vj‘im‘

i n‘i

:¼Z Y

i

dg‘i dg‘0i Vðg‘i ; g‘0i Þ

Yi;‘

Dj‘im‘

i n‘i

ðg‘i Þ;

(35)

where DjmnðgÞ are the Wigner representation matri-

ces. After a change of variables g‘i ! h�1‘ g‘i , the

integration over the group elements h‘ present in Vgives projectors onto the invariant tensors:

Zdh‘D

j‘1ðh‘ÞDj‘

2ðh‘ÞDj‘3ðh‘Þ ¼ ji‘ihi‘j; (36)

where the intertwiners i‘ are the normalizedWigner 3j symbols. The 6j symbol, resultingfrom a contraction of four 3j symbols that patternsa tetrahedron, shows up from the contraction of Vji

with the product of intertwinersQ

‘i‘ and theorthogonality of the Wigner matrices.

There is a connection, also pointed out in [12,13,41],between the flatness constraint described in (ii) and thetopological identities (Biedenhard-Elliot) satisfied bythe 6j symbol, which insures the formal invariance of thePonzano-Regge spin foam model under refinement ofthe triangulation. To see how the symmetry relates tosuch identities, let us make, within the integral (35), the(trivial) substitution:

Vðg‘i ; g‘0i Þ !Z

dkeGv3ðk"3k�1ÞVðg‘i ; g‘0i Þ: (38)

G3 is the vertex holonomy given by (34); the factor in frontof V is the evaluation of a central function whosePlancherel decomposition is

Zdkegðk"3k�1Þ ¼ X

j

�jðgÞ�jð"3Þ; (39)

where�j is the SOð3Þ character in the spin j representation,and �j ¼ R

dg�jðgÞeg is its group Fourier transform.6 Wealso decompose into characters the three delta functions inthe expression of Vðg‘i ; g‘0i Þ [see Eq. (32)] associated to theedges i ¼ 1, 3, 4 sharing the vertex v3, with the Plancherelformula �ðgÞ ¼ P

kdk�kðgÞ. We thus obtain an expression

in terms of the spins j‘i and a sum over four additional spinski and j. Elementary recoupling theory then shows that

FIG. 5 (color online). Vertex translation and trivial vertexholonomy.

6Explicitly, �jð"Þ ¼ Jdj ðj"jÞ=j"j, where Jdj is the Bessel func-tion of the first kind associated to the integer dj :¼ 2jþ 1,peaked on the value j"j ¼ dj.


104051-9

Comparing (36) and (40), we obtain the following identities:

8";

8<: j1 j2 j3j4 j5 j6

9=; ¼ X

ki;j

dk1dk3dk4dj�jð"Þ

8<: j1 j2 j3

j k1 k3gf j1 j5 j4

j k4 k1

9=;8<: j3 j6 j4

j k3 k4gf k1 k3 j2k4 j5 j6

9=;: (41)

In turn, these identities imply recursion relations for thesame 6j symbols (see, e.g., [13,42]), interpreted as discreteversions of the Wheeler-DeWitt equation [12]. More gen-erally, we expect that our type of analysis, based on GFTsymmetries, can give a systematic way, also for gravitymodels in higher dimension, to derive algebraic identitiesof the spin foam quantum amplitude from the study of theGFT symmetries.

This gives a clear interpretation of the symmetry in thevarious representations of the GFT, which matches whatwe expect from the action of diffeomorphisms in discreteapproaches. Thus, in the metric representation, the sym-metry encodes the invariance under (noncommutative)translation of the four vertices of the tetrahedron. In thegroup picture, they encode the flatness of the discreteboundary connection, which is the Wheeler-DeWitt

constraint in connection variables and thus the canonicaldiffeomorphism constraints. In the spin picture, they en-code recursion relations for the fundamental spin foamamplitudes (6j symbols) and their behavior under coarse-graining.

E. GFT with vertex variables

We have seen that the invariance of the vertex functionreflects some conservation rules for the holonomies Gv

along loops surrounding the vertices of the tetrahedronpatterned by the interaction. These conservation rules canbe made manifest by integrating out the gauge groupelement h‘ in the vertex function. Using three of the sixdelta functions in (4) to integrate three of the four integra-tion variables h‘, we obtain

Vðg‘i ; g‘0i Þ ¼Z Y4

‘¼1

dh‘Y6i¼1


‘0i Þ

¼Z½dh4��ððg24Þ�1g25ðg35Þ�1g36ðg46Þ�1g44Þ�ððg11Þ�1g12ðg32Þ�1g36ðg46Þ�1g41Þ�ððg11Þ�1g13ðg23Þ�1g24ðg44Þ�1g41Þ:

Thanks to the normalization of the Haar measure, thissimply gives

Vðg; g0Þ ¼ �ðGv1Þ�ðGv2

Þ�ðGv3Þ: (42)

Note that the fourth constraintGv4¼ 1 is a consequence of

the other three, due to the dependence relationG�1

v4ðg3g�1

30 ÞðGv2Gv3

Gv1Þðg30g�1

3 Þ ¼ 1 between the fourvertex holonomies. This is the counterpart of the reduc-ibility of the translation symmetry studied in the previoussection. The dependence relation can be easily understoodas a discrete Bianchi identity for the boundary connectionon the boundary surface of the tetrahedron.

This form of the vertex function suggests yet anotherrepresentation of GFT in terms of vertex variables vi 2suð2Þ instead of edge vectors xi. These vertex variables,which are the generators of the translation symmetry, areintroduced by plane-wave expansion �ðGvi

Þ ¼Rd3vieGvi

ðviÞ of the delta functions on the group.

Writing each of these plane waves as a cubic term, for, e.g.,

e Gv1¼ eðg2

4Þ�1g2

5? eðg3

5Þ�1g3

6? eðg4

6Þ�1g4

4; (43)

suggests to recast the GFT interaction in terms of newfields defined by the Fourier transform


104051-10

c 1ðv2; v3; v4Þ :¼Z

dg1dg2dg3eg�11

g2ðv2Þeg�1

1g3ðv3Þeg�1

2g3ðv4Þ’1ðg1; g2; g3Þ;

c 2ðv1; v3; v4Þ :¼Z

dg3dg4dg5eg�14

g5ðv1Þeg�1

3g4ðv3Þeg�1


c 3ðv1; v2; v4Þ :¼Z

dg5dg2dg6eg�15

g6ðv1Þeg�1

2g6ðv2Þeg�1


c 4ðv1; v2; v3Þ :¼Z

dg6dg4dg1eg�16

g4ðv1Þeg�1

6g1ðv2Þeg�1

4g1ðv3Þ’4ðg6; g4; g1Þ:

(44)

In terms of these new fields, well-defined on gauge invari-ant fields Px’‘, the combinatorics of the GFT interactionpatterns now the combinatorics of the four verticesv1 � � �v4 in the four triangles in a tetrahedron, with aconsequent change in the diagrammatic representation.Using the short notation c 123 :¼ c ðv1; v2; v3Þ, we, infact, obtain

Sint½c � ¼ �Z½d3vi�3 c 234

1 ? c 1342 ? c 124

3 ? c 1234

þ �Z½d3vi�3 �c 321

4 ? �c4213 ? �c

4312 ? �c

4321 :

(45)

Here, the ? product relates repeated indices as c i ? c i ?c i ¼ ðc ? c ? c ÞðviÞ. In each integral, the integration isover three variables v1, v2, v3 only: the value of v4 is puregauge, fixed to an arbitrary value by global translation. Asimilar analysis can be performed for the kinetic term. Interms of the fields c ‘, it is given by

Skin½c � ¼ X‘

Z½d3vi�2 c 123

‘ ? �c123‘ ; (46)

where the integration is over two variables v1, v2 only, thevalue of v3 being arbitrarily fixed using translationinvariance.

We do not analyze further this reformulation of themodel in terms of vertex variables, in this paper. We,however, believe that it will be relevant in many respects.First, the properties of the GFT following from our diffeo-morphism transformations should be simpler to analyze inthis formulation, since it is, in fact, on the vertex variablesthat these transformations act naturally and in the simplestway. Second, the formulation of the GFT amplitudes interms of vertex variables should simplify the analysis oftheir divergences, which are known to be located on three-bubbles of GFT diagrams—namely, at the vertices of thetriangulation, in addition to a global dependence on theoverall topology of the diagrams [11,26,32,39,43].

IV. FROM GFT TO SIMPLICIAL GRAVITYSYMMETRIES

We have seen in Sec. II that the amplitude of a FeynmanGFT diagram, in the metric representation, gives the sim-plicial path integral form of the Ponzano-Regge model. In

this section, we investigate how the GFT symmetry de-scribed above relates to the discrete residual action ofdiffeomorphisms in this model [11].

A. Diffeomorphisms in simplicial path integrals

The amplitude of a closed Feynman GFT diagram G, inthe metric representation, takes the form:

Z� ¼Z Y

l

dhlYe

d3xeeiS�ðxe;hlÞ; (47)

where � is the simplicial complex dual to G, and Sðxe; hlÞis the discrete 3d gravity action

eiS�ðxe;hlÞ :¼ eiP

e TrxeHe ¼ Ye

eHeðxeÞ: (48)

The variables of this action are a discrete metric fxege2� onthe edges of the triangulation and a discrete connectionfhlgl2G on the lines of G. The group element He ¼Q!

l2@fehl is the holonomy along the boundary of the

face fe of G dual to e, computed from a given referencevertex in @fe. In the case of open diagrams and in thepresence of boundary data fðxÞ, the integrand is obtainedby taking the ? product f ? eiS� , with respect to theboundary variables fxege2@�.The action S�ðxe; hlÞ is a discrete version of the contin-

uum action for first-order 3d gravity:

SðB; AÞ ¼Z

TrB ^ F; (49)

where B is the triad frame field and F is the curvature of theconnection A. We recall in the Appendix the local Poincaresymmetry of the continuum theory—namely, the SOð3Þgauge invariance and translation symmetry:

B!BþdA�A!A

�� B!½B;X�A!AþdXþ½A;X�; F!Fþ½F;X�;

(50)

with both X and � scalars with value in suð2Þ. The trans-lation symmetry, typical of BF-type theories, is due to theBianchi identity dAF ¼ 0. As we show in the Appendix,the action of diffeomorphisms in 3d gravity is classicallyequivalent to (a combination of gauge transformations and)translation of the frame field.


104051-11

The action S�ðxe; hlÞ enjoys a discrete version of thesesymmetries [11]. It can, moreover, be shown that, when-ever � triangulates a three-manifold, the discrete residualof translation invariance, and hence of diffeomorphisminvariance, in the discrete path integral (47), is partially7

responsible for the large-spin divergences in the Ponzano-Regge model [11,32].

The discretization of the gauge transformations followsthe usual lattice gauge theory techniques. The generators�v are labeled by vertices of the GFT graph G—equiv-alently by tetrahedra in the triangulation �. The holonomyhl on the oriented lines of G are transformed as hl !�vs

hl��1vt, where vs, vt denote the source and target

vertices of the line l. This means, in particular, that theholonomy He around the boundary of a face fe is trans-formed as He ! �eHe�

�1e , where �e is the generator

associated to the reference vertex in @fe from which theholonomy is computed. The metric variable xe transformsas xe ! �exe�

�1e . Such a gauge transformation, under

which the action is clearly invariant, corresponds to arotation of the reference frame of the e.

The discrete residual of translation invariance is due to adiscrete analogue of the Bianchi identity satisfied by thecurvature elementsHe. In terms of the GFT diagram and itsdual simplicial complex �, this can be understood asfollows. Given a vertex v 2 �, the set of GFT faces fedual to the edges e � v meeting at v defines a cellulardecomposition of a surface Lv, called the link of the vertexv. In GFT language, the link of a vertex is the boundary ofa three-dimensional ‘‘bubble’’ of the diagram. Wheneverthe simplicial complex � defines a triangulated manifold(as opposed to a pseudomanifold), the link of every vertexhas the topology of a two-sphere. Then, for any ordering ofthe edges e � v meeting at v, the curvature elements He

satisfy a closure relation of the type

Y!e�v

ðkevÞ�1Hekev ¼ 1 (51)

for some group-valued functions kev :¼ kevðhlÞ of the vari-ables hl on the links l of Lv. The group elements kev areinterpreted as the parallel transport along paths between afixed vertex in Lv to the reference vertex of each face fefrom which the holonomyHe computed. We have assumedhere that the orientation of the faces fe agrees with a fixedorientation of the sphere Lv; if the orientation of fe isreversed, H�1

e should appear in place of He. Note that nosuch closure identity holds when the Lv has a higher genustopology.

The idea of the works [11,32] was to use the identities(51) to prove a (commutative) discrete analogue of trans-lation symmetry for the discrete action S� ¼ P

eTrxeHe.To do so, the identity (51) is first written in terms of the

projections Pe :¼ TrHe ~� of the curvature elements ontothe Lie algebra:

Xe�v

ðkevÞ�1ðUvePe þ ½�v

e ; Pe�Þkev ¼ 0; (52)

where the scalar Uve and Lie algebra elements �v

e arecertain (complicated) functions of the Pe’s obtained fromthe Campbell-Hausdorff formula [11]. This leads to theinvariance of S� under the following transformation, gen-erated by �v 2 suð2Þ:

xe � xe þUve"

ev � ½�v

e ; "ev�; with

"evð�vÞ ¼ kve�vðkve Þ�1: (53)

Note that, if �v is interpreted as a translation vector in thereference vertex frame of Lv, "

ev encodes the same trans-

lation vector parallel-transported in the reference frame offe. The transformation (53) is a discrete analogue of thetranslation symmetry (50).In the next section, we show that the discrete Bianchi

identity (51) can be related to a vertex translation symme-try in a direct way—that is, without any projection to theLie algebra—provided one takes into account the noncom-mutativity of the translation algebra studied in Sec. III A.This will clarify the relationship between the GFT sym-metry and the discrete Bianchi identities leading to diffeo-morphism invariance in the simplicial path integrals.

B. Simplicial diffeomorphisms as quantumgroup symmetries

To see how the discrete Bianchi identities are tied to theinvariance under noncommutative vertex translation de-fined in Sec. III C, let us fix the value xe of the metric inthe exponential of the action (48) for all edges e which donot touch the vertex v. This defines a function of theremaining nv variables in suð2Þ, labeled by the nv edgese � v sharing v. Choosing an ordering of these edges, as in(51), one can lift this function to an element of the tensorproduct CðSOð3ÞÞ�e�v of nv copies of CðSOð3ÞÞ.Let us now act with the noncommutative translation

xe!xeþ"evð�vÞ; "evð�vÞ¼kve�vðkve Þ�1; (54)

shifting the metric of the edges sharing v by the variables"ev, defined as in (53). The group elements kve paralleltransport the frame of a fixed vertex in Lv and that of thereference vertex of the face fe from which the holonomyHe is computed. Upon such a translation, the function (48)gets transformed into a ? product of functions of �v:Y

e

ei TrxeHe � w!e�v

Ye

ei Trðxeþ"evÞHeð�vÞ: (55)7For a finer analysis of the divergences of the Ponzano-Reggemodel, see [26].


104051-12

Using the rule (10) for the ? product of plane waves, we seethat such a noncommutative translation acts on the actionterm by multiplication by the plane wave:

ei Tr½�vðQ!

e�vðkevÞ�1Hek

evÞ� ¼ 1; (56)

which is trivial due to the Bianchi identity (51).As we have seen in Sec. III C, both the GFT propagator

and vertex functions, which the Feynman integrand (48) isbuilt upon, are invariant under vertex translation xe !xe þ �v, for e � v. The generator �v is interpreted as atranslation vector in a given frame. This is the frameassociated to the reference point of the loop circling v,along which the conserved holonomy is computed. In (30),for example, this is the frame associated to the edge 1 of thetetrahedron patterns by the interaction.8

The transformation (55) has the same geometrical mean-ing: it corresponds to a vertex translation expressed in agiven frame. This frame is the reference vertex frame of thelink Lv. Indeed, recall from the calculation of the Feymanamplitudes in Sec. II that the variable xe present in theaction term is the edge metric in the reference frame of theGFT faces fe dual to the e. Using the parallel transports k

ve ,

one could instead use variables xve , labeling to the sameedge metric but expressed in the reference vertex frame ofLv. These are defined by xe ¼ kvex

ve ðkve Þ�1. Now, in this

frame, a vertex translation acts as xve ! xve þ �v. Thisamounts to act on the original variables xe by the‘‘twisted’’ translation xe ! xe þ "ev, where "evð�vÞ ¼kve�vðkve Þ�1.

Thus, the equality (56), and hence the discrete Bianchiidentity, express the invariance of the exponential of theaction under the (quantum) GFT symmetry defined in theprevious section. Note that, interestingly, the analysis ofthe invariance under simplicial diffeomorphisms distin-guishes the closed GFT diagrams G which define a mani-fold from those defining only a pseudomanifold. In fact, inthe case of nonmanifold graphs, the triangulation hasvertices v for which the link Lv defines a surface withnontrivial topology. For such vertices, there is no analogueof the discrete Bianchi identity (51): the invariance of theexponential of the action (48) under vertex translation is,therefore, broken.

The goal of the next subsection is to illustrate how theserather geometric considerations can be understood in apurely algebraic way. We will show on a simple examplehow the use of braiding techniques could give a systematicway to derive Bianchi identities from the GFT symmetry.

C. Noncommutative translations, invariance of theGFT amplitudes, and Bianchi identities

As spelled out in Sec. II, the integrand of a GFTFeynman amplitude in the metric representation is calcu-lated by sticking together propagator and vertex functionsalong each loop fe of the diagram, using the ? product.This gives a product loop amplitude [see (18)]Y

fe

~wNþ1j¼0 ð�xje

? ehjjþ1Þðxjþ1

e Þ: (57)

The exponential of the discrete BF action (48) is thenobtained by integrating, within each loop, over all metric

variables xje, save one xe :¼ x0e. It was shown in the pre-vious section that the GFT propagator and vertex defineinvariant functions under the noncommutative translation(27). The question we are asking is to which extent thetranslation invariance of the propagator and vertex func-tions induces the invariance (55) and (56) of the actionterm. We will only sketch an answer here with a simpleexample, leaving the full proof to future work.Since the transformation is quantum symmetry, it is

crucial, to answer the above question, to keep track ofthe ordering of the variables in the calculation of theFeynman integrand. It is precisely to keep track of thisordering that the braided quantum field theory formalism[22] uses a perturbative expansion into braided Feynmandiagrams.Note that, to study the behavior of (57) under the trans-

lation of a vertex v, it is enough to restrict the product tothe set of loops fe such that e � v. This amounts toconsidering the contribution of a subdiagram called a‘‘three-bubble’’ [23], which represents the vertex v. Athree-bubble, obtained by erasing all lines having strandsof a given color, is a trivalent ribbon graph dual to the linkLv of a vertex of the dual triangulation.Figure 6 shows the simplest GFT diagram of order two,

dual to a triangulation of the sphere S3 with two three-simplices; Fig. 7 shows the three-bubble obtained by eras-ing all the lines having strands of color 4. The three-bubblecan be drawn as a braided diagram, on the left of Fig. 8: allvertices are put beside each other, all legs up, in the lowerpart of the diagram, in a way that preserves the cyclic order

FIG. 6. Two-vertex Feynman diagram encoding the discretiza-tion of the sphere S3.

8As made clear using the covariance of the plane wave uponconjugacy, the same translation expressed in a different frame,say, that of edge 3, is generated by �0v :¼ k�vk

�1, where k :¼g�13 g1 parallel-transports one frame to another.


104051-13

of the legs on the plane; then, the propagator strands, in theupper part of the diagram, connect the legs with each other.A convenient way to represent these vertices is as a productof three ‘‘cups’’ (see Fig. 8):

The Feynman rules to compute the contribution of thethree-bubble to the amplitude are easily read from (17). Ifone reabsorb the minus sign and group variables of thepropagator into the vertex, we get a contribution of each‘‘cup’’

given by

whereas the propagator strands are just noncommutativedelta functions �xðx0Þ.Upon noncommutative translation x ! xþ �, the

‘‘cup’’ function of two variables xiij; xjij gets transformed as

This can be easily seen by group expansion of the non-commutative delta functions. One can then convince one-self that the translation invariance of the vertex function isthen due to the following identities:

Hence, we see that, by construction, the lower part of thebraided bubble diagram defines a translation invariantfunction of the metric variables.

Now, the contribution of the three-bubble appears in thefinal integrand (57) as a product of loops, as drawn on theright of Fig. 8.

Going from the left to the right diagrams in Fig. 8 by‘‘separating the loops’’ induces reordering of the strands—hence, a reordering of the metric variables. In order toprobe the behavior of (57) under translation, the idea isto associate to a certain braiding map to the separation ofthe loops, induced by the universal R-matrix of DSOð3Þgiven in (25).9 As a direct calculation shows, swapping two

cups (the right one above the left one) with the DSOð3Þbraiding gives

where

denotes the action of h by conjugacy on the two variablesof the cup:

FIG. 8. Three-bubble drawn (i) as a braided diagram and (ii) asa product of loops.

FIG. 7. Three-bubble dual to the vertex 4, obtained from Fig. 6by erasing the strands related to color 4.

9Let us stress that this braiding is merely a technical aid tokeep track of the action of our quantum group symmetry onfunctions of several Lie algebra variables and does not corre-spond here to any nontrivial braiding in the algebra of GFTfields, which we have chosen to be trivial, as in the standard GFTframework.


104051-14

By construction, swapping the cups in this way intertwinesthe translation T �.

Let us now use this braiding to ‘‘separate the loops.’’ Theloop 12�2 �1 is separated as follows:

The next step is to form the loop 23�3 �2 :

Hence, using the DSOð3Þ braiding to separate the loopsfinally gives a twisted product of loops:

Since the braiding map intertwines the translation symme-try, this twisted product defines, by construction, a trans-lation invariant function of the metric variables. The idea isthen to deduce from this an invariance of the nontwistedproduct of loops under a twisted translation. To make thisexplicit at the level of the action term, let us integrate (64)over all the variables, save one in each loop: x112; x

223; x

331.

We are left with

eiTr½x112H12�eiTr½ðh�1�2h�1x

223h�1�1h�2ÞH23�eiTr½ðh

�1�3h�1x

331h�1�1h�3ÞH31�; (65)

where Hij ¼ h�1i hjh

�1�jh�i denotes the holonomy along the

loop ij �j �i . The invariance of this expression underthe translation xij ! xij þ � expresses the invariance ofthe product

ei Tr½x112H112�ei Tr½x223H2

23�ei Tr½x112H331�

under the translation xij ! xij þ "ijð�Þ, where"12ð�Þ :¼ �;

"23ð�Þ :¼ h�1�2h�1�h

�1�1h�1�2;

"31ð�Þ :¼ h�1�3h�1�h

�1�1h�3:

(66)

In any case, the invariance leads to the identity (56), whichreads here

eH12h�1�1h�2H23h

�1�2h�1h

�1�1h�3H31h

�1�3h�1ð�Þ ¼ 1: (67)

This equality holding for all �, it gives a Bianchi identity ofthe type of (51):

H12h�1�1h�2H23h

�1�2h�1h

�1�1h�3H31h

�1�3h�1 ¼ 1: (68)

We thus derived a Bianchi identity from the translationinvariance of the vertex and propagator functions. In thisanalysis, the ‘‘twist’’ elements kve in the Bianchi identity,geometrically interpreted as parallel transport from a fixedpoint to the reference point of each loop, show up incommuting the variables using the DSOð3Þ braiding.

More generally, we expect an analogous algorithm forany planar three-bubble; namely, when the link Lv of thevertex has the topology of a two-sphere. Starting from thethree-bubble drawn as a braided diagram, the bottom part(a product of ‘‘cups’’) gives, by construction, a translation

invariant function of the metric variables xje. The algorithmwill then define a sequence of topological moves corre-sponding to the separation of the loops and inducing a

reordering of the variables xje, and associate to it a certainDSOð3Þ braiding map. This braiding map encodes thebehavior of the amplitude (57) under noncommutativetranslation. The example shown above is particularlysimple, as the braided three-bubble does not involve anycrossing of the propagator strands; in general, an additionalrule should be added in the definition of the braiding map,which would take into account such crossings.Just as in the above example, the action of such a

braiding map on the function defined by the product of

cups will induce a twisting of the variables xje by certain

group-valued functions kjeðhÞ of the holonomies. A condi-tion to obtain Bianchi identities, and hence an invariance

of the action term eiS�ðxe;hlÞ, is that these functions donot depend on the variables j within a loop e: namely,

kje :¼ ke. We conjecture that this condition can be reachedprecisely when the three-bubble is planar—namely, whenall the crossings of the braided diagram are removable bysome topological move. In the presence of nontrivial cross-ings, on the other hand, the braiding map will give aninvariance of the amplitude (57), which will not translateinto any Bianchi identity or an invariance of the action term

eiS�ðxe;hlÞ [obtained from (57) by integration over all the

variables xje, save one per loop]. This reflects a breaking ofthe discrete diffeomorphism symmetry whenever the(closed) GFT graph has nonspherical three-bubbles—namely, for pseudomanifold graphs.Whether this conjecture can be proven remains to be

seen: we leave this for future work. It will also be important


104051-15

to understand how this analysis is affected by the use of anontrivial braiding in the algebra of GFT fields, intertwin-ing the quantum symmetry.

D. Open diagrams and n-point functions

The geometrical and algebraic analysis of the previoustwo sections can be extended to open GFT graphs, withfixed boundary metric or connection data. An open GFTgraph is dual to a simplicial complex with boundaries. Wehave seen that the invariance of the Feynman integrand(exponential of the action) under noncommutative trans-lation of a vertex v of this simplicial complex is due to adiscrete Bianchi identity on the link Lv of the vertex. Weshowed both geometrically and algebraically that, when vis in the bulk, the invariance holds only when Lv has atrivial topology, or, equivalently, when the three-bubbleassociated to v is planar.

The same condition applies when the vertex lies at theboundary. In this case, the link Lv defines an open surface,whose boundary is a loop circling the vertex v: this is thelink @Lv of v in the boundary triangulation. Now, in the‘‘group’’ representation, the boundary data encodes aboundary connection. One can then easily convince oneselfthat a discrete ‘‘Bianchi identity’’ on the link Lv simplysays that the holonomy of this boundary connection, along@Lv, is trivial. Such a discrete Bianchi identity, and hencethe invariance of the Feynman integrand under noncom-mutative translation of the vertex, hold when the link Lv

has a trivial (disk) topology.We had already noticed, at the level of the GFT vertex,

that our symmetry implies (in the group representation)flatness of the boundary connection. In fact, dealing with aflat boundary connection means that the holonomies alongall (3d) contractible loops are trivial. Now, the loop @Lv

circling the boundary vertex v is contractible preciselywhen the link Lv has a trivial topology; the invarianceunder translation then holds and expresses precisely thatthe holonomy is trivial. Thus, the behavior of the Feynmanintegrand under noncommutative translation of the bound-ary vertices indeed encodes the flatness of the boundaryconnection—namely, what we expect as a result of diffeo-morphism invariance.

More generally, for the GFT graphs dual to manifolds,the behavior of the Feynman amplitudes under our quan-tum GFT symmetry is consistent with what we know about

discrete diffeomorphisms at the quantum level from ca-nonical (discrete) 3d gravity, as well as its covariant pathintegral formulation. Since not much is known about theaction of diffeomorphisms in simplicial gravity on pseu-domanifold, we conclude that we are not missing anyexpected feature of discrete diffeomophism invariance, inour trivially braided GFT formalism, as far as it can be seenat the present stage of development.Given the interpretation of our GFT symmetry as the

counterpart of diffeomorphism invariance, it is natural toask whether the GFT n-point functions respect the sym-metry. We know this is not the case: sticking to the usualGFT formalism, we have used a trivial braiding in thealgebra of fields, which does not commute with the actionof our symmetry transformations. As it is well-known, thisleads generically to a breakdown of the symmetry at thequantum level. In the context and spirit of the braidedquantum field theory formalism, it would be more naturalto use a nontrivial braiding intertwining the symmetry andhence fully implement the covariance of the n-point func-tions. However, the consequences of such a nontrivialbraiding—although currently under investigation—are dif-ficult to forecast, at this stage. In fact, it should clear fromthe above analysis of the amplitudes that the properties ofGFT n-point functions in this trivially braided GFT contextdo not seem to indicate inconsistencies, a specific physicalreason why a nontrivial braiding would be necessary, orany problem with the implementation of diffeomorphisminvariance. On the contrary, none of the expected featuresof diffeomorphisms seems to be missing in this formalism.

V. DIFFEOMORPHISMS IN TOPOLOGICALMODELS IN HIGHER DIMENSIONS

The analysis of the previous sections can be extended tohigher dimensions, for models describing BF theory, in arather straightforward manner. Here, we consider theOoguri GFT [41] for 4d BF theory, generalized to includecolors. The variables are complex scalar fields ’‘, with‘ ¼ 1; � � � ; 5, defined on G�4 ¼ SOð4Þ�4, which satisfythe gauge invariance condition

8h 2 SOð4Þ;’‘ðhg1; hg2; hg3; hg4Þ ¼ ’‘ðg1; g2; g3; g4Þ 8‘:

(69)

The action of the model is S½’� ¼ Skin½’� þ Sint½’�, with

Skin½’� ¼Z½dg�4 X4

‘¼1

’‘ðg1; g2; g3; g4Þ �’‘ðg1; g2; g3; g4Þ;

Sint½’� ¼ �Z½dg�10’1ðg1; g2; g3; g4Þ’2ðg4; g5; g6; g7Þ’3ðg7; g3; g8; g9Þ’4ðg9; g6; g2; g10Þ’5ðg10; g8; g5; g1Þ

þ �Z½dg�10 �’5ðg1; g5; g8; g10Þ �’4ðg10; g2; g6; g9Þ �’3ðg9; g8; g3; g7Þ �’2ðg7; g6; g5; g4Þ �’1ðg4; g3; g2; g1Þ: (70)


104051-16

Just as in 3d, the above structures have a natural sim-plicial interpretation. The field ’‘ðg1; . . . ; g4Þ represents athree-simplex (tetrahedron), its four arguments being asso-ciated to its boundary triangles. The interaction encodesthe combinatorics of five such tetrahedra glued pairwisealong common triangles to form a four-simplex. The ki-netic term encodes the gluing of four-simplices alongshared three-simplices.

The group Fourier transform giving the metric represen-tation is easily extended [15] to functions of (several copiesof) SOð4Þ � SUð2Þ � SUð2Þ=Z2,

’‘ðx1; . . . ; x4Þ Z½dg�4’‘ðg1; . . . ; g4Þeg1ðx1Þ . . . eg4ðx4Þ;

xi 2 soð4Þ � R6: (71)

The plane waves eg � soð4Þ � R6 ! Uð1Þ are defined as

the product of SUð2Þ plane waves, defined in Sec. II, usingthe decompositions g ¼ ðg�; gþÞ and x ¼ ðx�; xþÞof the group and soð4Þ algebra elements into left and rightcomponents:

egðxÞ ¼ ei Trx�g�ei Trxþgþ : (72)

The ? product is the Fourier dual of the convolutionproduct of SUð2Þ introduced in Sec. II. The variables xare geometrically interpreted as bivectors that the standardlattice BF theory assigns to triangles, in each tetrahedron.Just as in 3d, the gauge invariance condition (69) is dual,upon a Fourier transform, to a closure constraint

Cðx1; . . . ; x4Þ ¼ �ðP4i¼1 xiÞ of the four field variables,

imposed by a noncommutative delta function defined asin (13).

By extending the 3d symmetry analysis to the 4d case,we will consider the action of rotations and translations ofthe quantum double10 DSOð4Þ on the scalar fields ’‘. Theaction of the double on fields over the group is the same wepresented in Sec. III A. Thus, an element f ��, with f 2CðSOð4ÞÞ and � 2 SOð4Þ, acts on a function� 2 CðSOð4ÞÞas

�ðgÞ ! �ð��1g�Þ; �ðgÞ ! fðgÞ�ðgÞ (73)

and dually on its group Fourier transform �ðxÞ by conju-gacy and translation of the Lie algebra variable x.

As in the Boulatov case, we easily check that the onlygauge covariant action of rotations which leaves the inter-action term invariant is the diagonal rotation: In the metric

formulation, gauge covariance simply means that a rotationpreserves the closure �ðP4

i¼1 xiÞ of the bivectors.The realization of the translation symmetries is analo-

gous to 3d, except that now they act at the edges of thesimplices patterned by the fields, rather than the vertices.The transformations are thus generated by four soð4Þ trans-lation parameters "e, where e labels the ten edges of theinteraction four-simplex, diagrammatically represented byits dual diagram in Fig. 9. Each edge of this four-simplex isrepresented by an subdiagram called an ‘‘edge graph’’.Thus, if e‘‘0 denotes the unique edge that does not belongto the tetrahedra ‘ or ‘0, the edge graph associated to e‘‘0 isobtained by removing all the lines which contain strands ofcolor ‘ or ‘0. The edge graph of e34 is pictured in Fig. 10:its three lines represent the three triangles 1, 3, 4 sharinge34.To define the action of a translation of the edge e34, we

equip the lines of the edge graph with an orientation, asdrawn in Fig. 10. Using this convention, each line has an‘‘incoming’’ and an ‘‘outcoming’’ external strand. A trans-lation of e34, generated by �34 2 soð4Þ, acts nontriviallyonly in the strands of the edge graph. In the metric repre-sentation, it shifts the corresponding variables x‘i by ��34,whether the strand i comes in or out of ‘:

xi � xi � �34 if i is outgoing;

xi � xi þ �34 if i is incoming;(74)

FIG. 9. 4d GFT vertex.

FIG. 10. Vertex diagram for the edge (34).

10Note that a priori we could choose a bigger quantum group,like a deformation of the Poincare group in six dimensions. Theclassification of quantum symmetries for noncommutativespaces has been only partially completed in 4d [44].Deformations of symmetries for higher-dimensional spaceshave still to be explored. In our case, the choice of the quantumgroup of interest is dictated by the kinematical phase space of 4dBF theory and by its known discrete classical symmetries, whichwe want to encode at the GFT level.


104051-17

in a way that preserves the closure �ðP4i¼1 x

‘i Þ of each tetrahedron. More precisely, the translationT �34 of the edge e34 acts

on the dual fields as

T �34x’1ðx1; x2; x3; x4Þ ¼ w�34’1ðx1 � �34; x2; x3; x4 þ �34Þ;T �34x’2ðx4; x5; x6; x7Þ ¼ w�34’2ðx4 � �34; x5 þ �34; x6; x7Þ;T �34x’5ðx10; x8; x5; x1Þ ¼ w�34’5ðx10; x8; x5 � �34; x1 þ �34Þ;

T �34x’‘ ¼ ’‘ if ‘ ¼ 3; 4:

(75)

The same field transformation is expressed in a moreexplicit way (without star product) in the group represen-tation, as follows:

T �34x’1ðg1; g2; g3; g4Þ ¼ eg�11

g4ð�34Þ’1ðg1; g2; g3; g4Þ;

T �34x’2ðg4; g5; g6; g7Þ ¼ eg�14

g5ð�34Þ’2ðg4; g5; g6; g7Þ;

T �34x’5ðg10; g8; g5; g1Þ ¼ eg�15

g1ð�34Þ’5ðg10; g8; g5; g1Þ;

T �34x’‘ ¼ ’‘ if ‘ ¼ 3; 4: (76)

We see that this transformation matches the intuitioncorresponding to translating bivectors [soð4Þ Lie algebraelements] associated to the triangles of the four-simplexdual to the GFT interaction vertex, by means of Lie algebravalued generators associated to its edges. This matchesalso the action of diffeomorphisms on the bivectors ofdiscrete BF theory [recall that the transformations wehave defined take the closure condition (metric compati-bility) into account].11 It can be checked by direct calcu-lation that the GFT action (70) is invariant under the abovefield transformations.

In fact, one verifies, as in the 3d case, that both kineticand vertex functions themselves are left invariant—beforeintegration. A way to make this invariance manifest is toextract from, say, the vertex function in group variables theconservation laws for the holonomies associated to edgesof the four-simplex dual to the GFT vertex. Just as inSec. III E, the explicit integration over the group elementsh‘ in the vertex gives

Vðg; g0Þ ¼Z Y5

i¼1

½dh‘�Y10i¼1

�ðg‘i h‘h�1‘0 ðg‘

0i Þ�1Þ

¼ �ðG12Þ�ðG13Þ�ðG15Þ�ðG23Þ�ðG25Þ�ðG35Þ;(77)

where

G12 ¼ g8g�19 g09g

�110 g

010g

0�18 ;

G13 ¼ g5g�16 g06g

�110 g

010g

0�15 ;

G15 ¼ g7g�16 g06g

0�19 g9g

0�17 ;

G23 ¼ g�12 g02g

�110 g

010g

0�11 g1;

G25 ¼ g0�13 g3g

�12 g02g0�1

9 g9;

G35 ¼ g4g�12 g02g

0�16 g6g

0�14 :

We recognize here the Gij as the holonomies around the

edges ðijÞ. The delta functions in (77) encode the flatnessconditions which, as expected from the canonical analysisof discrete BF theory, constrain the connection variables asa result of the diffeomorphism symmetry.Note that the holonomies associated to the edges ði4Þ are

missing. This is analogous to the 3d case where the trans-lations of the four vertices of the tetrahedron are not allindependent; only three of them are. Also, in the 4d case,the translations of the edges are not all independent, just asthe continuum symmetry can be shown to be reducible(cf. the Appendix): this is due to the Bianchi identitiessatisfied by the boundary connection represented by the fieldvariables. In fact, one can prove that translating a vertex, i.e.,translating all edges sharing this vertex, leaves invariant theinteraction term and, by extension, the integrand of theFeynman amplitude. The true symmetry is, therefore, repre-sented by the above edge translationsmodulo the translationsof the edges following a vertex translation.We thus see that, for (the GFT model describing) 4d BF

theory, everything proceeds in parallel with the 3d case, theonly new ingredient being the reducibility of the resultingsymmetry. However, the strategy used here to define theaction of diffeomorphisms in GFT can, in principle, beextended to the physically more interesting case of 4dgravity GFT models, obtained by constraining the topo-logical one [5,6]. In general, we expect that the impositionof the simplicity constraints will break the full symmetriesof Ooguri’s GFT. It will be interesting to determinewhether there is an eventual remnant symmetry and, ifnot, whether the vertex translations become then the rele-vant, if only approximate, symmetry [29]. In this case, sucha symmetry could admit a good geometric interpretation astranslations of the vertices of a geometric four-simplex inan embedding 4d flat space, as we expect from diffeo-morphisms in discrete gravity [7,10].

11Unlike the 3d case, however, we have no geometric descrip-tion in terms of translating the edges of a four-simplex embeddedin four-dimensional flat space. This is only to be expected, giventhat we are dealing with a nongeometric theory and thus withnongeometric four-simplices.


104051-18

VI. ADDITIONAL INSIGHTS

We now discuss additional insights that the newly iden-tified GFT symmetry provides, concerning various aspectsof the GFT formalism itself. While these are somewhatsecondary results, we believe they confirm the importanceof the new symmetry and suggest that further progress canbe triggered by its identification.

The necessity of coloring. The introduction of coloringin GFT models in [23] has already been proven useful instudies of the topological properties of the Feynman dia-grams generated by such models [24,25,45]—in particular,for the automatic removal of complexes with some types ofextended singularities that are instead generated by thenoncolored models. Most important, it has been crucialfor the proof that the 3d GFT we have studied admits atopological expansion of its Feynman diagrams such thatmanifold configurations of trivial topology dominate thesum for large values of the representation cutoff [28].These important results have important implications forthe program of GFT renormalization and for defining aGFT generalization of the notion of the (double) scalinglimit of matrix models [39], and thus for the understandingof the continuum limit. No obvious physical or geometricrelevance, however, had been discovered, until now, for thesame coloring. Our results show, on the other hand, thatcoloring is a necessary feature of GFT models for 3dgravity and general BF theories. In fact, coloring is anecessary ingredient in the definition of the GFT diffeo-morphism symmetry we have identified and discussed inthis paper. More precisely, it can be shown that removingthe coloring leads to the immediate breaking of the sym-metry and that only a restricted translation of the verticesof the tetrahedron dual to the GFT vertex can be defined asa field transformation leaving the noncolored action invari-ant, such as the one identified in [21]. This symmetry,however, although being a particular combination of thesymmetry transformation we have studied, does not have aclear simplicial gravity interpretation. Given the interpre-tation of our GFT symmetry as the counterpart of discretediffeomorphisms in simplicial gravity path integrals, theimportance of coloring from the physical/geometricalpoint of view becomes instead manifest. In its light, werecognize the colored Boulatov GFT model as the correctGFT description of 3d quantum gravity.

A braided group field theory formalism? In this paper,we have studied the issue of diffeomorphism symmetrywithin the standard (colored) group field theory formalism.In particular, the algebra of fields we have worked with hasbeen assumed to have trivial braiding [20,22]; i.e., the mapbetween the tensor product of two fields and the one withopposite ordering is given by the trivial flip map. At thesame time, however, the symmetry we have identified inthe GFT action corresponds, as we have stressed, to aquantum group symmetry acting on this space of fields.As such, its action on the space of fields would naturally

induce, when these are defined as elements in its represen-tation category, a nontrivial braiding structure [46]. Thisalso results in a corresponding braided statistics [47]. Mostimportant, it can be shown that the use of the inducedbraiding map in the algebra of fields is necessary, if thesymmetry is to be preserved at the quantum level[36,37,46], for example, so that the correct Ward identitiesfor n-point functions follow from the existence of thesymmetry at the level of the action. We will discuss brieflybelow whether this is necessary on physical grounds in ourcontext and what the properties of the n-point functions arein our trivially braided context. In any case, the aboveconsiderations suggest, at least from a mathematical andfield theoretic perspective, to consider a generalization ofthe GFT formalism, beyond the one as noncommutatvefield theories, achieved in [15], to a braided noncommuta-tive group field theory (see also [21] for further argumentsin this direction). The first issues to tackle, in this direction,are 1) what is the correct braiding among GFT fieldsintertwining our quantum group symmetry, if it exists atall; and 2) what are the physical consequences of theimplementation of a nontrivial braiding and of the resultingquantum Ward identities, from the point of view of sim-plicial quantum gravity, loop quantum gravity, and spinfoam models.Constraints on GFT model building. Another useful role

that symmetries play in usual quantum field theories is thatthey help constraining the allowed field interactions. Infact, the requirement that the GFT interactions preservethe quantum group symmetry we identified as discretediffeomorphisms rules out some GFT interactions thatcould be considered, a priori, as admissible.We have already discussed above how removing the

coloring form the GFT fields, i.e., considering the originalBoulatov formulation with a single field, breaks the sym-metry. This can also be understood as a special case of alarger set of possible GFT interactions within the coloredGFT formulation, that we now see to be ruled out bysymmetry considerations. The colored model we workedwith, for 3d gravity, was based on four different fields ’l,with l ¼ 1, 2, 3, 4, and the only interaction term was of theform ’1’3’2’4 (plus complex conjugate), with standardtetrahedral combinatorics of arguments. The single-colorBoulatov interaction corresponds to terms of the type’l’l’l’l. A quick calculation shows that, not only suchterms, but any interaction being more than linear in any ofthe colored fields (e.g., ’1’1’3’4 or ’1’2’3’2) is notinvariant under our GFT diffeomorphism symmetry. Weare then left with interactions that involve linearly all the dGFT fields (in models generating d-dimensional simplicialstructures). The ordering of such fields can be chosen atwill (in our trivially braided context).We can, however, also ask whether the ordering of the

(group or noncommutative) arguments of such fields in theinteraction term can be chosen at will. Different orderings,


104051-19

in fact, have been considered in the literature (see [39]). Incolored models, the order of the arguments of the field isconsidered as fixed and does not play any special role (theorientability of the resulting Feynman diagrams is alreadyensured by the complex structure and by the requirementsof same-color propagation only [45,48]). Regarding theinterplay between the ordering of arguments and symme-try, the situation is slightly trickier. It can be seen easilythat, for any given choice of ordering, there exists a (set of)transformation(s) acting on the dþ 1 fields leaving theaction invariant and corresponding to diffeomorphisms,in the sense we have discussed. The very definition of thetransformations retains the imprint of the chosen orderingof field arguments. At the same time, however, it can beshown that such a transformation would not, in general,leave invariant a vertex defined by a different ordering noran action involving a sum over different orderings. Thismeans, for example, that the GFT field itself cannot bedefined to be invariant under permutations of its argu-ments, as this imposition would break its covariance underthe diffeomorphism transformation and then the invarianceof the action. It must be said, however, that a possible wayout of this restriction could be, once more, an appropriatebraiding that relates fields defined with different orderingsof their arguments and, possibly, intertwines our symmetry.We leave this for future work.

Last, one could consider defining both higher-orderinteraction terms, i.e., terms of order higher than dþ 1involving colored fields, with various choices of pairing offield arguments, as well as other terms still of order dþ 1,but defined by nontetrahedral combinatorics of arguments.Our symmetry severely constrains model-building of thistype. We have not performed yet a complete analysis.However, we have considered some examples. One inter-esting example of an alternative interaction term, the so-called ‘‘pillow’’ term, has been introduced in [49] andstudied further in [43]. It has the following form (in itscolored version):

þ ��

4!

Y6i¼1

Zdgi½�1ðg1; g2; g3Þ�2ðg3; g4; g5Þ

��3ðg4; g2; g6Þ�4ðg6; g5; g1Þ�: (78)

So, it is given by the same type of vertex function, i.e., aproduct of delta functions on the group, enforcing theidentification of edge variables among four triangles, asin the standard tetrahedral term. However, the combinato-rial pattern is now different and corresponds to two pairs oftriangles glued to one another along two edges in each pairand along one single edge between the two pairs. Theinterest in the addition of such a term lies in the fact thatit turns the (noncolored) Boulatov model into a Borelsummable one [with some restrictions on the couplingconstant � and with a different (worse) scaling behavior].It can be proven, however, that this term is not invariant

under GFT diffeos and thus is not an admissible modifica-tion of the action of the model, in the colored case.We stress again that the above considerations would be

modified by the introduction of a nontrivial braidingamong fields, with a corresponding generalization of theGFT formalism. However, not knowing the correct braid-ing structure, it is impossible to be more definite aboutwhat the modifications would be.

VII. CONCLUSIONS

Using the recently introduced noncommutative metricformulation of group field theories, we have identified a setof GFT field transformations, forming a global quantumgroup symmetry of the GFT action and corresponding totranslations of the vertices of the simplices dual to the GFTinteraction vertex, in a flat space embedding. The analysisof the action of these transformations at the level of theGFT Feynman amplitudes, which are given, in this metricformulation, by simplicial gravity path integrals, showsthat the transformations we identified correspond to (thediscrete analogue of) diffeomorphisms for fixed simplicialcomplex satisfying manifold conditions and leave the sameamplitudes invariant thanks to discrete Bianchi identities,whose GFT origin we are now able to exhibit. Moreover,for open Feynman diagrams dual to simplicial manifoldswith boundaries, we have shown that the same transforma-tions enforce the flatness of the boundary connection andthus encode the simplicial version of the canonical gravityconstraints, as expected.While we focus on the case of 3d Riemannian gravity,

we also show how our results generalize straightforwardlyto BF theories in higher dimensions. Thus, our results, onthe one hand, match those obtained, concerning discretediffeomoprhisms, in the context of simplicial gravity (e.g.,Regge calculus) and, on the other hand, improve them byboth embedding them within a more general context andrephrasing them in purely (quantum) field theoretic lan-guage. An immediate advantage of this embedding is theclear way in which we can now link to one or anothervarious aspects of diffeomorphism invariance in spin foammodels, canonical loop quantum gravity, and simplicialgravity, previously discussed in the literature and nowunderstood to be all consequences and manifestations ofthe same GFT field symmetry: the symmetry of the Reggeaction and the simplicial Bianchi identities (manifest in themetric representation of GFTs), the canonical constraintsof loop quantum gravity (adapted to a simplicial complex,best seen in the group picture), and the algebraic identitiessatisfied by nj symbols and at the root of the topologicalinvariance of state sum (spin foam) models (obtained fromthe GFT symmetry in representation space).Our analysis also provides some new insights on the

GFT formalism itself. These include the need for coloringin the GFT formalism, from the point of view of simplicialgravity symmetries; the possible role of braiding in this


104051-20

class of models, and thus in simplicial gravity path inte-grals and spin foam models, and the potential interest in abraided group field theory formalism; the issue of Wardidentities and the relation of the same with canonicalquantum gravity constraints and recursion relations for 6jand 10j symbols; and the use of the GFT symmetries weidentified for constraining the possible interaction termsthat can be added to the standard GFT vertex.

We believe that the GFT symmetry we identify can alsoplay a useful role concerning ongoing work on GFT renor-malization and, possibly, for the extraction of continuumgravity from GFT models.

ACKNOWLEDGMENTS

We thank V. Bonzom, S. Carrozza, B. Dittrich, E.Livine, R. Oeckl, and A. Perez for useful comments anddiscussions. D. O. gratefully acknowledges financial sup-port from the A. von Humboldt Stiftung.

APPENDIX: BF ACTION AND ITS SYMMETRIES

In this appendix, we recall the standard basic facts aboutthe symmetries associated to the BF action.

We work with a d-dimensional manifold M, equippedwith a principal bundle associated with the semisimple LiegroupG. The Lie algebra ofG is noted as g and is equippedwith a nondegenerate Killing form which we note as tr. A isthe connection, i.e., a one-form with value in g, of theprincipal bundle, and we note as F ¼ dAþ A ^ A thecurvature two-form of the connection A. dA is the covariantderivative, defined in terms of the connection A. We nowintroduce to B a ðd� 2Þ form with value in a Lie algebra g.The BF action is built using the Killing form tr on g:

SBF ¼Z

trðB ^ FÞ: (A1)

The equations of motion are

dAB ¼ 0; F ¼ 0: (A2)

The action is invariant under both translation of the B fieldand the gauge transformations. The infinitesimal gaugetransformations are given by

A ! Aþ �LXA ¼ Aþ dAX ¼ Aþ dX þ ½A; X�;

B ! Bþ �LXB ¼ Bþ ½B; X�: (A3)

X 2 g is a scalar field with value in g. The B field is,therefore, transforming under the adjoint action of G. The

curvature F is also transforming under the adjoint action,and it is thus easy to check that the action is invariant underthese gauge transformations.Thanks to the Bianchi identity dAF ¼ 0, the action is

also left invariant if we translate the B field by dA�, where� is a ðd� 3Þ form with value in g:

A ! Aþ �T�A ¼ A;

B ! Bþ �T�B ¼ Bþ dA� ¼ Bþ d�þ A ^�:

(A4)

There is, however, a possible redundancy for the transla-tions if d 4. Indeed, assuming d 4, consider the d� 4form V with value in g; then, � and �0 ¼ �þ dAV gen-erate on shell the same transformation, since dA� ¼ dA�

0,due to d2AV ¼ ½F; V�. This last term is zero on shell.The BF action is clearly invariant under the diffeomor-

phisms, since it is purely topological. Let us considerexplicitly the (infinitesimal) action of the diffeomor-phisms. Considering a vector field , the infinitesimalaction of the diffeomorphisms is given by the Lie deriva-tive L. We have, therefore,

B ! LB ¼ dðBÞ þ ðdBÞ;A ! LA ¼ dðAÞ þ ðdAÞ;

(A5)

where we have introduced the interior product , whichsatisfies, in particular,

ð!1^!2Þ¼ ð!1Þ^!2þð�1Þp!1^ð!2Þ; (A6)

with !1 and !2, respectively, a p and a q form. Thesetransformations can actually be related to the previoustransformations (A3) and (A4). We have that

ðdABÞ ¼ ðdBÞ þ ðA ^ BÞ¼ ðdBÞ þ ½ðAÞ; B� � A ^ ðBÞ;

ðFÞ ¼ ðdAÞ þ ðA ^ AÞ ¼ ðdAÞ þ ½ðAÞ; A�:(A7)

Taking X ¼ B and� ¼ A, we can reexpress the actionof the diffeomorphisms as

LB ¼ �LA

Bþ �TB

Bþ ðdABÞ;LA ¼ �L

AAþ �T

BAþ ðFÞ:

(A8)

This means that on shell (A2), the diffeomorphism action isequivalent to the translation (A4) and gauge transformation(A3).


104051-21

[1] M. Gaul and C. Rovelli, Lect. Notes Phys. 541, 277(2000); L. Smolin et al., in The Structural Foundationsof Quantum Gravity, edited by D. Rickles (OxfordUniversity Press, New York, 2006), p. 196.

[2] S. A. Hojman, K. Kuchar, and C. Teitelboim, Ann. Phys.(N.Y.) 96, 88 (1976)

[3] D. Oriti, Approaches to Quantum Gravity (CambridgeUniversity Press, Cambridge, England, 2009).

[4] R. Loll, Living Rev. Relativity 1, 13 (1998).[5] J. C. Baez, Classical Quantum Gravity 15, 1827 (1998); D.

Oriti, Rep. Prog. Phys. 64, 1703 (2001); A. Perez,Classical Quantum Gravity 20, R43 (2003); D. Oriti,Ph.D. thesis, University of Cambridge, 2003.

[6] L. Freidel, Int. J. Theor. Phys. 44, 1769 (2005); D. Oriti, inFoundations of Space and Time, edited by G. Ellis and J.Murugan (Cambridge University Press, Cambridge,England, 2011).

[7] B. Dittrich, arXiv:0810.3594.[8] J. Ambjorn, J. Jurkiewicz, and R. Loll, Phys. Rev. D 72,

064014 (2005).[9] M. Rocek and R.M. Williams, Z. Phys. C 21, 371 (1984).[10] B. Bahr and B. Dittrich, Classical Quantum Gravity 26,

225011 (2009).[11] L. Freidel and D. Louapre, Nucl. Phys. B662, 279 (2003).[12] J.W. Barrett and L. Crane, Classical Quantum Gravity 14,

2113 (1997).[13] V. Bonzom, E. R. Livine, and S. Speziale, Classical

Quantum Gravity 27, 125002 (2010).[14] F. David, Nucl. Phys. B257, 45 (1985); P. Ginsparg,

arXiv:hep-th/9112013.[15] A. Baratin and D. Oriti, Phys. Rev. Lett. 105, 221302

(2010).[16] M. P. Reisenberger and C. Rovelli, Classical Quantum

Gravity 18, 121 (2001).[17] L. Freidel and E. Livine, Classical Quantum Gravity 23,

2021 (2006).[18] E. R. Livine, Classical Quantum Gravity 26, 195014

(2009).[19] A. Baratin, B. Dittrich, D. Oriti, and J. Tambornino,

arXiv:1004.3450.[20] S. Majid, Foundations of Quantum Group Theory

(Cambridge University Press Cambridge, England, 1995).[21] F. Girelli and E. Livine, Classical Quantum Gravity 27,

245018 (2010).[22] R. Oeckl, Commun. Math. Phys. 217, 451 (2001).[23] R. Gurau, arXiv:0907.2582.[24] J. Ben Geloun, J. Magnen, and V. Rivasseau, Eur. Phys. J.

C 70, 1119 (2010).[25] R. Gurau, Ann. Henri Poincare 11, 565 (2010).

[26] V. Bonzom and M. Smerlak, Lett. Math. Phys. 93, 295(2010); V. Bonzom and M. Smerlak, arXiv:1008.1476.

[27] R. De Pietri and C. Petronio, J. Math. Phys. (N.Y.) 41,6671 (2000).

[28] R. Gurau, arXiv:1011.2726.[29] A. Baratin, F. Girelli, and D. Oriti (unpublished).[30] J. Ambjorn, B. Duhuus, and T. Jonsson, Mod. Phys. Lett.

A 6, 1133 (1991); M. Gross, Nucl. Phys. B, Proc. Suppl.25, 144 (1992); N. Sasakura, Mod. Phys. Lett. A 6, 2613(1991).

[31] G. Ponzano and T. Regge, Spectroscopic and GroupTheoretical Methods in Physics, edited by F. Bloch(North-Holland, New York, 1968).

[32] L. Freidel and D. Louapre, Classical Quantum Gravity 21,5685 (2004).

[33] A. Baratin and D. Oriti (unpublished).[34] L. Freidel and D. Louapre, arXiv:gr-qc/0410141.[35] K. Noui, Classical Quantum Gravity 24, 329

(2007).[36] R. Oeckl, Nucl. Phys. B581, 559 (2000); A. P.

Balachandran, T. R. Govindarajan, G. Mangano, A.Pinzul, B. A. Qureshi, and S. Vaidya, Phys. Rev. D 75,045009 (2007).

[37] A. P. Balachandran, T. R. Govindarajan, G. Mangano, A.Pinzul, B. A. Qureshi, and S. Vaidya, Phys. Rev. D 75,045009 (2007).

[38] Y. Sasai and N. Sasakura, J. High Energy Phys. 06 (2009)013.

[39] L. Freidel, R. Gurau, and D. Oriti, Phys. Rev. D 80,044007 (2009).

[40] K. Noui and A. Perez, Classical Quantum Gravity 22,1739 (2005).

[41] H. Ooguri, Mod. Phys. Lett. A 7, 2799 (1992).[42] M. Dupuis and E. Livine, Classical Quantum Gravity 27,

135003 (2010).[43] J. Magnen, K. Noui, V. Rivasseau, and M. Smerlak,

Classical Quantum Gravity 26, 185012 (2009); E. Joung,J. Mourad, and K. Noui, J. Math. Phys. (N.Y.) 50, 052503(2009).

[44] S. Zakrzewski, arXiv:hep-th/9412099; P. Podles and S. L.Woronowicz, Commun. Math. Phys. 178, 61 (1996).

[45] R. Gurau, Classical Quantum Gravity 27, 235023(2010).

[46] Y. Sasai and N. Sasakura, Prog. Theor. Phys. 118, 785(2007).

[47] R. Oeckl, J. Geom. Phys. 39, 233 (2001).[48] F. Caravelli, arXiv:1012.4087.[49] L. Freidel and D. Louapre, Phys. Rev. D 68, 104004

(2003).


104051-22

http://dx.doi.org/10.1007/3-540-46634-7_11

http://dx.doi.org/10.1007/3-540-46634-7_11

http://dx.doi.org/10.1016/0003-4916(76)90112-3

http://dx.doi.org/10.1016/0003-4916(76)90112-3

http://dx.doi.org/10.1088/0264-9381/15/7/004

http://dx.doi.org/10.1088/0034-4885/64/12/203

http://dx.doi.org/10.1088/0264-9381/20/6/202

http://dx.doi.org/10.1007/s10773-005-8894-1

http://arXiv.org/abs/0810.3594



http://dx.doi.org/10.1007/BF01581603

http://dx.doi.org/10.1088/0264-9381/26/22/225011

http://dx.doi.org/10.1088/0264-9381/26/22/225011

http://dx.doi.org/10.1016/S0550-3213(03)00306-7

http://dx.doi.org/10.1088/0264-9381/14/8/011

http://dx.doi.org/10.1088/0264-9381/14/8/011

http://dx.doi.org/10.1088/0264-9381/27/12/125002

http://dx.doi.org/10.1088/0264-9381/27/12/125002

http://dx.doi.org/10.1016/0550-3213(85)90335-9

http://arXiv.org/abs/hep-th/9112013



http://dx.doi.org/10.1088/0264-9381/18/1/308

http://dx.doi.org/10.1088/0264-9381/18/1/308

http://dx.doi.org/10.1088/0264-9381/23/6/012

http://dx.doi.org/10.1088/0264-9381/23/6/012

http://dx.doi.org/10.1088/0264-9381/26/19/195014

http://dx.doi.org/10.1088/0264-9381/26/19/195014


http://dx.doi.org/10.1088/0264-9381/27/24/245018

http://dx.doi.org/10.1088/0264-9381/27/24/245018

http://dx.doi.org/10.1007/s002200100375


http://dx.doi.org/10.1140/epjc/s10052-010-1487-z

http://dx.doi.org/10.1140/epjc/s10052-010-1487-z

http://dx.doi.org/10.1007/s00023-010-0035-6

http://dx.doi.org/10.1007/s11005-010-0414-4

http://dx.doi.org/10.1007/s11005-010-0414-4


http://dx.doi.org/10.1063/1.1290053

http://dx.doi.org/10.1063/1.1290053


http://dx.doi.org/10.1142/S0217732391001184

http://dx.doi.org/10.1142/S0217732391001184

http://dx.doi.org/10.1016/S0920-5632(05)80015-5

http://dx.doi.org/10.1016/S0920-5632(05)80015-5

http://dx.doi.org/10.1142/S0217732391003055

http://dx.doi.org/10.1142/S0217732391003055

http://dx.doi.org/10.1088/0264-9381/21/24/002

http://dx.doi.org/10.1088/0264-9381/21/24/002

http://arXiv.org/abs/gr-qc/0410141

http://dx.doi.org/10.1088/0264-9381/24/2/005

http://dx.doi.org/10.1088/0264-9381/24/2/005

http://dx.doi.org/10.1016/S0550-3213(00)00281-9





http://dx.doi.org/10.1088/1126-6708/2009/06/013

http://dx.doi.org/10.1088/1126-6708/2009/06/013



http://dx.doi.org/10.1088/0264-9381/22/9/017

http://dx.doi.org/10.1088/0264-9381/22/9/017

http://dx.doi.org/10.1142/S0217732392004171

http://dx.doi.org/10.1088/0264-9381/27/13/135003

http://dx.doi.org/10.1088/0264-9381/27/13/135003

http://dx.doi.org/10.1088/0264-9381/26/18/185012

http://dx.doi.org/10.1063/1.3131682

http://dx.doi.org/10.1063/1.3131682

http://arXiv.org/abs/hep-th/9412099

http://dx.doi.org/10.1007/BF02104908

http://dx.doi.org/10.1088/0264-9381/27/23/235023

http://dx.doi.org/10.1088/0264-9381/27/23/235023

http://dx.doi.org/10.1143/PTP.118.785

http://dx.doi.org/10.1143/PTP.118.785

http://dx.doi.org/10.1016/S0393-0440(01)00016-X




PHYSICAL REVIEW D 104051 (2011) Diffeomorphisms in...

Documents

Transcript of PHYSICAL REVIEW D 104051 (2011) Diffeomorphisms in...