QUANTUM GEOMETRY: ITS DYNAMICS, SYMMETRIES AND

Transcription

1 QUANTUM GEOMETRY: ITS DYNAMICS, SYMMETRIES AND STATISTICS IN THE GROUP FIELD THEORY FORMALISM 1 Daniele Oriti Max Planck Institutefor Gravitational Physics (Albert Einstein Institute) ILQGS AEI, Golm, Germany, EU 10/05/ Based on work done with A. Baratin, S. Carrozza, F. Girelli, M. Raasakka, and by many others 1 / 25

2 PLAN OF THE TALK Introduction GFT and its three representations 3d gravity: simplicial gravity path integrals and spin foam models diffeomorphism symmetry in GFT simplicial diffeos GFT symmetry three representations of GFT invariance the invariance of GFT amplitudes the issue of braiding the vertex formulation of 3d GFT and its use braiding and braided statistics in (3d) GFT conclusions and outlook 2 / 25

3 INTRODUCTION LQG has identified what could be the fundamental kinematical degrees of freedom of quantum geometry and many mathematical tools to deal with them group field theories are combinatorially non-local field theories on (Lie) groups or algebras which bring the fundamental degrees of freedom of quantum geometry, as identified by canonical loop quantum gravity in the general framework of matrix/tensor models for simplicial quantum gravity they define (potentially) a quantum field theory (2nd quantization) of spin networks of simplicial geometry thus an alternative definition of the complete dynamics of geometry (and topology) of quantum space, from the microscopic, pre-geometric, quantum regime to the macroscopic, geometric, (semi-)classical one that can now be studied using (almost) standard methods and ideas from quantum (and statistical) field theory 3 / 25

4 INTRODUCTION GFT field(s) φ = a 2nd quantized spinnet vertex or a 2nd quantized simplex classical dynamics S(φ) (and corresponding equations of motion) quantum dynamics = R [dφ] e S(φ) quantum dynamics: perturbation theory around no-space vacuum φ = 0: = [dφ] e S(φ) = X A(Γ) Γ Γ = possible interaction/evolution process of spin networks/simplices = cellular complex of arbitrary topology and complexity A(Γ) = spin foam model or, equivalently, simplicial gravity path integral within this general framework, the tasks are then: define interesting models for quantum gravity, i.e. S(φ) and thus A(Γ) understand quantum geometry (d.o.f., symmetries,...) in S(φ) and in A(Γ) make sense of QFT: space of states, symmetries, statistics, observables, quantum (SD) eqns, conserved quantities,... make sense of QFT: combinatorial structure of Feynman diagrams Γ, sum over topologies, perturbative renormalization define non-perturbative (quantum) dynamics (sum over diagrams, continuum/thermodynamic limit, thermodynamic phases) make contact with continuum semi-classical gravity and extract (new) physics 4 / 25

5 COLORED GFT FOR 3D EUCLIDEAN GRAVITY 4 fields ϕ l for l = 1,.., 4 function on SO(3) 3, subject to gauge invariance: h SO(3), ϕ l (hg 1, hg 2, hg 3 ) = ϕ l (g 1, g 2, g 3 ) action S[ϕ l ] = S kin [ϕ l ] + S int [ϕl]: S kin [ϕ l ] = [dg i ] 3 4X ϕ l (g 1, g 2, g 3 )ϕ l (g 1, g 2.g 3 ), l=1 S int [ϕ l ] = λ [dg i ] 6 ϕ 1 (g 1, g 2, g 3 )ϕ 2 (g 3, g 4, g 5 )ϕ 3 (g 5, g 2, g 6 )ϕ 4 (g 6, g 4, g 1 ) +λ [dg i ] 6 ϕ 4 (g 1, g 4, g 6 )ϕ 3 (g 6, g 2, g 5 )ϕ 2 (g 5, g 4, g 3 )ϕ 1 (g 3, g 2, g 1 ) spin network representation obtained by Peter-Weyl expansion ϕ l (g 1, g 2, g 3 ) = X C j 1,j 2,j 3 m 1,m 2,m 3 φ j 1,j 2,j 3 l,n 1,n 2,n 3 D j 1 m1 n 1 (g 1 )D j 2 m2 n 2 (g 2 )D j 3 m3 n 3 (g 3 ) field spin network vertex 5 / 25

6 COLORED GFT FOR 3D EUCLIDEAN GRAVITY non-commutative triad (flux) representation with continuous variables x su(2) R 3 (A. Baratin, DO, arxiv: [hep-th]), (A. Baratin, B. Dittrich, DO, J. Tambornino, arxiv: [hep-th]) use group Fourier transform of the fields (E. Livine, L. Freidel, S. Majid, K. Noui, E. Joung, J. Mourad) bϕ l (x 1, x 2, x 3 ) := [dg i ] 3 ϕ l (g 1, g 2, g 3 ) e g1 (x 1 )e g2 (x 2 )e g3 (x 3 ), with plane-waves e g : su(2) R 3 U(1) (g=e θ n τ and x= x τ): e g(x) := e itrxg, non-commutative product dual to convolution product on the group: (e g e g )(x):=e gg (x), gauge invariance condition becomes closure constraint for x i P ϕ l = bc bϕ l, bc(x 1, x 2, x 3 ) := δ 0 (x 1 +x 2 +x 3 ), P ϕ l = [dh] ϕ l (hg 1, hg 2, hg 3 ) with δ x(y) := [dh] e h -1(x)e h (y) [d 3 y] (δ x f)(y) = f(x) x i = closed edges vectors of a triangle in R 3 field geometric simplex because of duality between group convolution and -product, GFT action in triad variables maintains same combinatorial structure 6 / 25

7 COLORED GFT FOR 3D EUCLIDEAN GRAVITY Feynman diagrams Γ built from 3-stranded propagators re-routed at interaction vertices, and are dual to simplicial complexes Feynman amplitudes A Γ equivalently written in group, representation or algebra variables Y Y A Γ = dh l δ `H Y Y Y «f (h l ) = dh l δ h l = l f l f = X Y Y j j τ d 1 j τ 2 j τ 3 je j τ {j e} e τ 4 j τ 5 j τ 6 l ff = f Y [dh l ] Y [d 3 x e] e i P e Tr xehe e f l e construction extended with same data to simplicial complexes with boundary 7 / 25

8 SUMMARY SO FAR GFT defines quantum dynamics for spin networks or simplices full dynamics defined by sum over all gluings of colored tetrahedra Γ, respecting the coloring and the boundary data, (thus arbitrary topological manifolds as well as pseudo-manifolds) with amplitude A Γ spin foam amplitudes are exactly dual to (non-commutative) simplicial gravity path integrals non-commutative representation corresponds to quantization by Duflo map for phase spaces T G flux representation of LQG role of non-commutative variables can be analyzed in simpler case: particle on SO(3) (DO, M. Raasakka, arxiv: [hep-th]): group Fourier transform gives natural momentum representation dynamics expressed by non-commutative phase space path integral -product encodes correct quantum corrections semi-classical analysis of dynamical amplitudes simplified in non-commutative variables triad representation of dynamics brings geometry to the forefront, so useful for: analyze gravitational symmetries in GFT (ad spin foams) diffeomorphism symmetry construct models for 4d gravity based on BF theory classical constraints on bivectors and connection straightforward to quantize as non-commutative delta functions insertions in GFT action insertions become simply constraints in BF simplicial path integral with clear geometric meaning translation as constraints on quantum states achieved by direct change of representation V. Bonzom, E. Livine, arxiv: [gr-qc]; V. Bonzom, arxiv: [gr-qc],arxiv: [gr-qc]; A. Baratin, DO, arxiv: [hep-th]; A. Baratin, DO, to appear 8 / 25

9 DIFFEOMORPHISMS IN DISCRETE (QUANTUM) GRAVITY understanding implementation diffeomorphism symmetry crucial for any QG model constrain fundamental quantum dynamics and emergent classical one guidance in extraction of effective gravitational dynamics QG models based on discrete structures continuum diffeo symmetry generically broken need to identify discrete (exact or approximate) symmetry on pre-geometric data extensive studies in Regge calculus B. Dittrich, arxiv: [gr-qc]; B. Bahr, B. Dittrich, arxiv: [gr-qc]: diffeomorphisms translations of vertices of triangulation (in local flat embedding in R d ) invariance of Regge action exact in 3d without cosmological constant (flat space) invariance only approximate in 4d recovered in continuum limit invariance of action related to Bianchi identities at vertices of triangulation closest analysis to our setting: 3d Ponzano-Regge spin foam model (actually, simplicial BF path integral) L. Freidel, D. Louapre, gr-qc/ / 25

10 DISCRETE DIFFEOMORPHISMS IN SIMPLICIAL 3D BF PATH INTEGRAL closest analysis to our setting: 3d spin foam models (actually, simplicial BF path integral): GFT amplitude A Γ, for triangulation, for discrete triad {x e} e and connection {h l} l Γ: Y Y Y Y = dh l d 3 x e e is (xe,h l ) = dh l d 3 x e e i P e Tr(xeHe(h l )), l e S (x e, h l) = discrete version of S(B, A)= R Tr B F(A), for triad B and connection A SO(3) gauge invariance + translation symmetry for su(2)-valued scalars X and φ (equivalent on-shell to diffeos): B B + d Aφ A A B [B, X] A A + dx + [A, X], l e F F + [F, X], discrete symmetry of action S (x e, h l) due to discrete Bianchi identity at vertices v : consider link L v of v: link l bounding faces f e dual to the edges e v meeting at v (in GFT: boundary surface of bubble ) if is triangulated manifold, L v is a 2-sphere v for any ordering of edges e v, Q e v (ke v ) 1 H e k e v = 1, for some ke v := ke v (hl) G ( = parallel transport between a fixed vertex in L v to the reference vertex of f e from which the holonomy H e computed) in terms of the projections P e := TrH e τ, and for (complicated) functions Ue v (Pe) and Ωv e X (Pe): (k e v ) 1 (U v e Pe + [Ωv e, Pe])ke v = 0, (1) e v invariance of discrete action under x e x e + U v e εe v [Ωv e, εe v ], with εe v (ǫv) = kv e ǫv(kv e ) 1 for translation vectors ǫ v su(2) in frame of L v and ε e v for same translation vector in frame of fe need to identify same transformations, symmetry and identity in GFT formulation 10 / 25

11 BASICS ON DSO(3) ACTION ON FUNCTIONS ON SO(3) AND THEIR DUAL Drinfeld double DSO(3) =C(SO(3)) CSO(3) (CSO(3) = group algebra, C(SO(3)) = algebra of functions) is a deformation of 3d Euclidean Poincaré group ISO(3) representation on functions on the group C(SO(3)): rotations act by adjoint action on the variable and translations act by multiplication: φ(g) φ(λ 1 g):=φ(λ 1 gλ), φ(g) f(g)φ(g), φ C(SO(3)) in particular, one can consider f ε(g) = e g(ε), labelled by ε su(2) R 3 upon group Fourier transform this corresponds to dual action φ(x) b φ(x b + ε) analogue of Poincaré transformations on functions on flat space, here replaced by su(2), with momentum space replaced by SO(3) deformed action on tensor product of fields, due to non-trivial co-product on C(SO(3)), f(g 1 g 2 ) = f (1) (g 1 )f (2) (g 2 ) = f(g 1 g 2 ), f C(SO(3)): φ 1 (g 1 )φ 2 (g 2 ) f(g 1 g 2 )φ 1 (g 1 )φ 2 (g 2 ) using e g1 g 2 (ε)=(e g1 e g2 )(ε), one can check that bφ 1 (x 1 ) b φ 2 (x 2 ) b φ 1 (x 1 + ε) ε b φ2 (x 2 + ε) need to upgrade products of functions to tensor products introduce braiding map standard GFT corresponds to trivial braiding map: B 12 : C(SO(3)) C(SO(3)) C(SO(3)) C(SO(3)) φ(g 1 ) φ(g 2 ) φ(g 2 ) φ(g 1 ) 11 / 25

12 GFT SYMMETRIES - A. BARATIN, F. GIRELLI, DO, ARXIV: [HEP-TH] symmetries of GFT model for 3d Euclidean gravity = subset of DSO(3) 4, one for each vertex of a tetrahedron (GFT interaction, number of field colors ) available symmetry transformations restricted by gauge covariance/closure P)/C: P (T ϕ l ) = T (P ϕ l ) rotational symmetry: ϕ l (g 1, g 2, g 3) ϕ l (Λ 1 1,l g1, Λ 1 2,l g2 Λ 1 3,l g3), restricted to single rotation Λ l :=Λ invariance under local changes of frame in each tetrahedron and in each triangle translation (diffeo) symmetry: transformations generated by four su(2)-translation parameters ε v, one per vertex of tetrahedron take portion of GFT interaction relative to vertex, e.g., v 3, obtained by removing strands of color 3 translation of v 3 generated by ε 3 su(2) acts non-trivially only on strands of vertex graph in metric representation, it shifts x l =3 i by ±ε 3 according to orientation: x l i x l i + ε 3 if i outgoing x l i x l i ε 3 if i incoming. 12 / 25

13 GFT SYMMETRIES - TRANSLATIONS/DIFFEOMORPHISMS (A. BARATIN, F. GIRELLI, DO, ARXIV: [HEP-TH]) T ε3 bϕ 1 (x 1, x 2, x 3 ) := ε3 bϕ 1 (x 1 ε 3, x 2, x 3 + ε 3 ) T ε3 bϕ 2 (x 3, x 4, x 5 ) := ε3 bϕ 2 (x 3 ε 3, x 4 + ε 3, x 5 ) T ε3 bϕ 4 (x 6, x 4, x 1 ) := ε3 bϕ 4 (x 6, x 4 ε 3, x 1 + ε 3 ) T ε3 bϕ 3 (x 5, x 2, x 6 ) := bϕ 3 (x 5, x 2, x 6 ) in the group representation: T ε3 ϕ 1 (g 1, g 2, g 3 ) := e g -1 1 g 3 (ε 3 ) ϕ 1 (g 1, g 2, g 3 ) T ε3 ϕ 2 (g 3, g 4, g 5 ) := e g -1 3 g (ε 4 3 ) ϕ 2 (g 3, g 4, g 5 ) T ε3 ϕ 4 (g 6, g 4, g 1 ) := e g -1 4 g (ε 1 3 ) ϕ 4 (g 6, g 4, g 1 ) T ε3 ϕ 3 (g 5, g 2, g 6 ) := ϕ 3 (g 5, g 2, g 6 ) transformation extended to complex conjugate fields ϕ l using e g(ε) = e g -1(ε). geometric meaning of transformation is manifest: when translanting a vertex, one translates the edge vectors sharing this vertex, taking into account orientation gauge covariance of T is manifest: the shift of edge-variables preserves the closure of each triangle; the arguments g 1 j g k of the plane-waves are gauge invariant these transformations leave GFT action (more: the integrands) invariant symmetry is reducible: fields transform trivially under global translation of four vertices 13 / 25

14 INVARIANCE OF GFT VERTEX AND ITS QUANTUM GEOMETRY (A. BARATIN, F. GIRELLI, DO, ARXIV: [HEP-TH]) non-commutative triad representation vertex function given by: V(x l i, xl i ) = 4Y 6Y [dh l ] (δ x l e i hl h 1 )(x l l=1 i=1 l i ) fixing ordering of variables to that defined by interaction polynomials, this is lifted to a tensor product in C(SO(3)) 12 invariant under the (non-commutative) translations the function V(x l i, xl i ) imposes the variables x l i, interpreted as edge-vectors in different frames, to match the metric of Euclidean tetrahedron. symmetry expresses invariance of the matching condition under a translation of each of the vertices in an embedding of this tetrahedron in R 3 group representation vertex function is: V(g l i, gl i ) = 4Y 6Y dh l l=1 i=1 δ((g l i ) 1 h l h 1 l g l i ) its invariance under translations of the vertex v 3 means that, for all ǫ 3 su(2): e Gv3 (ε 3)V(g l i, gl i ) = V(g l i, gl i ) where G v3 = (g 1 1 ) 1 g 1 3 (g2 3 ) 1 g 2 4 (g4 4 ) 1 g 4 1 translation invariance reflects conservation rule G v3 = 1 14 / 25

15 INVARIANCE OF GFT VERTEX AND ITS QUANTUM GEOMETRY (...group representation, continued...) the group field variables g l i encode boundary holonomies along paths connecting the center of each triangle l to its edges. G v3 is the holonomy along a loop circling the vertex v 3 of the tetrahedron. symmetry under translation of each vertex says boundary connection is flat Hamiltonian and vector constraints, i.e. canonical counterpart of diffeomorphisms constraint on tetrahedral wave-function constructed from the GFT field spin representation the vertex function takes the form of SO(3) 6j-symbols: X Y i l m l V j i i m l = Y δ i nl n l, n i l = Y i i i i {m l i } l δ n l i, n l i j ff j1 j 2 j 3 j 4 j 5 j 6 (2) the flatness constraint (WdW equation) on boundary connection becomes algebraic (recursion) identity for 6j-symbols 15 / 25

16 INVARIANCE OF GFT VERTEX AND ITS QUANTUM GEOMETRY (...spin representation, continued...) act on vertex with translation ε 3, with bχ j = R dgχ j (g)e g(ε) X Y i l m l {m l i } i l V j i m l i nl i = X k i,j d k1 d k3 d k4 d j bχ j (ε 3) = = X k i,j j ff j ff j ff j ff d k1 d k3 d k4 d j bχ j j1 j (ε 2 j 3 j1 j 5 j 4 j3 j 6 j 4 k1 k 3 j 2 3) j k 1 k 3 j k 4 k 1 j k 3 k 4 k 4 j 5 j 6 which implies ε: j ff j1 j 2 j 3 = j 4 j 5 j 6 = X j ff j ff j ff j ff d k1 d k3 d k4 d j bχ j j1 j (ε) 2 j 3 j1 j 5 j 4 j3 j 6 j 4 k1 k 3 j 2 j k 1 k 3 j k 4 k 1 j k 3 k 4 k 4 j 5 j 6 k i,j algebraic WdW equation on tetrahedral state 16 / 25

17 GFT TRANSFORMATIONS AT SIMPLICIAL LEVEL (GFT AMPLITUDES) we saw that discrete action e is (x e,h l ) := e i P e TrxeHe (for manifold ) is invariant under x e x e + U v eε e v [Ω v e, ε e v], with ε e v(ǫ v) = k v eǫ v(k v e) 1 for vectors ǫ v su(2) in frame of L v and ε e v(ǫ v) for same translation in frame of f e thanks to Bianchi identity, in terms of P e := TrH e τ, and for functions Ue v(pe) and Ωv e (Pe): X (kv e ) 1 (Ue v Pe + [Ωv e, Pe])ke v = 0, (3) e v this is the GFT translation symmetry w.r.t. the non-commutative translations fix x e for all edges e which do not touch vertex v function of remaining n v variables in su(2) choosing an ordering, one can lift this function to an element of tensor product C(SO(3)) e v of n v copies of C(SO(3)) act with non-commutative translation x e x e + ε e v (ǫv), εe v (ǫv) = kv e ǫv(kv e ) 1 (4) (k v e = parallel transport from fixed vertex in Lv to reference vertex of face fe) function gets transformed into a -product of functions of ǫ v: Y e itrxehe Y e itr(xe+εe v )He (ǫ v) e v e e such non-commutative translation acts on action term by multiplication by plane wave: h Q e itr ǫv e v (ke v ) 1 He k v i e = 1, which is trivial due to the Bianchi identity 17 / 25

18 BRAIDED DIAGRAMS, ALGEBRAIC DERIVATION, N-POINT FUNCTIONS given GFT symmetry, one would like to derive Bianchi identity as above from GFT amplitudes in purely algebraic way involves writing GFT amplitudes in the framework of braided quantum field theory and separate strands associated to bubbles (Bianchi id) into loops (discrete action): requires a general rule (braiding) for crossing and swapping strands in a bubble diagram we have been able to identify such rule only in one simple case we expect it to exist for all spherical diagrams Bianchi identities for manifold we expect Bianchi identity to be broken (or twisted) for non-manifold diagrams a general treatment requires a better definition of braiding at GFT level, and probably the introduction of non trivial braiding map among GFT fields braided statistics trivial braiding for GFT fields does not intertwine the GFT diffeomorphism symmetry symmetry is broken at full quantum level GFT n-point functions are not covariant 18 / 25

19 GFT IN VERTEX VARIABLES (BARATIN-GIRELLI-DO, [HEP-TH]), (S. CARROA, DO, [HEP-TH]) analysis of diffeos at GFT level suggests that: fundamental degrees of freedom of 3d colored GFT can be associated to vertices if one wants to define fields in terms of representations of symmetry (quantum) group, they have to be expressed in vertex variables, where this group acts naturally the step from edge to vertex variables can be made (for field ϕ(g 1, g 2, g 3 ) = ϕ(hg 1, hg 2, hg 3 ), and G u := g -1 2 g 1, G v := g -1 1 g 3, G w := g -1 3 g 2): ψ l (u, v, w) = dg vdg wϕ l (G -1 v, Gw, 1l)e (G vg w) -1(u)e G v (v)e Gw (w) = dg udg vdg wδ(g ug vg w)ϕ l (G -1 v, G w, 1l)e Gu (u)e Gv (v)e Gw (w) = dg udg vdg wδ(g ug vg w)ψ l (G u, G v, G w)e Gu (u)e Gv (v)e Gw (w) = dg udg vdg w dε ε ψ l (G u, G v, G w)e Gu (u + ε)e Gv (v + ε)e Gw (w + ε) = dε ε ψl b (u + ε, v + ε, w + ε). ψ l (g -1 2 g 1, g -1 1 g 3, g -1 3 g 2) ϕ l (g 1, g 2, g 3 ) triangle in R 3 three edge vectors that close three positions of vertices up to translation GFT diffeo transformations now act naturally on vertex variables 19 / 25

20 GFT IN VERTEX VARIABLES - ACTION can rewrite colored GFT action as: S kin [ ψ] = X [d 3 v i ] 2 ψl (v 1, v 2, v 3 ) ψ l (v 1, v 2, v 3 ), l S int [ ψ] = λ [d 3 v i ] 3 ψ1 (v 2, v 3, v 4 ) ψ 2 (v 4, v 3, v 1 ) ψ 3 (v 4, v 1, v 2 ) ψ 4 (v 1, v 3, v 2 ) + c.c. or also: S[ψ] = X 3Y 3Y [ dg i d G i ]K(G i ; G i )ψ l (G 1, G 2, G 3 )ψ l ( G 1, G 2, G 3 ) l i=1 i=1 + λ [ Y dg l l ]V(Gl l )ψ234 1 ψ2 431 ψ3 412 ψ c.c. l l l = K(G i, G i ) = δ(g 1 G 2 G 3 )δ(g 1 G -1 1 )δ(g -1 2 G 2 )δ(g -1 3 G 3 ), V(G l l ) = δ(g1 2 G1 3 G1 4 )δ(g2 4 G2 3 G2 1 )δ(g3 4 G3 1 G3 2 )δ(g4 1 G4 3 G4 2 ) δ(g 4 2 G3 2 G1 2 )δ(g4 3 G1 3 G2 3 )δ(g1 4 G3 4 G2 4 ) this corresponds to the interactions of four non-commutative φ 3 QFTs on su(2) l = l = l = 2 20 / 25

21 GFT IN VERTEX VARIABLES - AMPLITUDES (S. CARROA, DO, ARXIV: [HEP-TH]) evaluation of Feynman amplitudes: cut-off: δ Λ (G), rescaling of fields and coupling different combinatorial structure from usual GFT Feynman diagrams: subgraph of color l is graph of non-commutative φ 3 scalar QFT on su(2) dual to triangulation of boundary of bubble around vertex of color l, encoding its topology overall amplitude: φ 3 graphs encoding structure of bubbles, glued through propagators l = 1 l = 4 l = 2 l = 3 quantum amplitude is: 0 A Γ [dg] 3N Y Y [δ Λ (1l)] 2 2g b V b b B l v V b δ Λ 11 0 Y (G f v )ǫf v Y f b v f F l δ Λ factorized in bubbles plus gluing convenient for analyzing structure of divergences example of result: optimal bounds proving suppression of pseudo-manifolds: A Γ (λλ) N 2 [δ Λ (1l)] 2 2P b B l g b 0 Y G f AA v v f 21 / 25

22 BRAIDING AND BRAIDED STATISTICS IN GFT - - A. BARATIN, S. CARROA, F. GIRELLI, DO, M. RAASAKKA, TO APPEAR because GFT diffeos act naturally on vertex variables, these are also convenient for studying the issue of braiding change coloring notation: ψ l (g 1, g 2, g 3 ) ψ c1 c 2 c 3 (g 1, g 2, g 3 ) with l c 1, c 2, c 3 {1, 2, 3, 4} translations act as: j egi (ǫ)ψ ˆT c(ǫ) ψ c1 c 2 c 3 (g 1, g 2, g 3 ) = c1 c 2 c 3 (g 1, g 2, g 3 ), if c i = c, ψ c1 c 2 c 3 (g 1, g 2, g 3 ), otherwise. want to find braiding map B among GFT fields that intertwine GFT translations: B (ˆT c(ǫ)) = (ˆT c(ǫ)) B with co-product: (ˆT c(ǫ)) = ˆT c(ǫ) ˆT c(ǫ) the braiding map B that satisfy this intertwining property is given by the general rule: B ψ ci c j c k (g i, g j, g k )ψ c i c (h i, h j c j, h k ) = ψ k c i c (g jh j c i g 1 k j, h j, g i h k g 1 i )ψ ci c j c k (g i, g j, g k ), if c i = c k, c j = c i, for example: B`ψ 421 (g 1, g 2, g 3 ) ψ 123 (h 1, h 2, h 3 ) = ψ 123 (g 1 3 h 1 g 3, g 2 h 2 g 1 2, h 3 ) ψ 421 (g 1, g 2, g 3 ) 22 / 25

23 BRAIDING AND BRAIDED STATISTICS IN GFT - - A. BARATIN, S. CARROA, F. GIRELLI, DO, M. RAASAKKA, TO APPEAR from this, one can also deduce (more complicated) braiding for fields ϕ in edge variables the above braiding is the one naturally inherited from the representation theory of DSO(3) this braiding intertwines the GFT translation/diffeo symmetry n-point function for braided GFT are covariant at full quantum level fields in GFT interaction can now be written as tensor products in any order, then related by braiding map one can go beyond: interpret each field (in vertex variables) as a tensor product of three representations of DSO(3) find a canonical way of writing the GFT interaction in terms of tensor products of such representations identify a braiding map among them that intertwines the diffeo symmetry and gives also the braiding of GFT fields stage is now set for: obtain Bianchi identities from invariance of GFT amplitudes under GFT diffeos in purely algebraic language study Ward identities on n-point functions following from GFT diffeo invariance at quantum level construct Fock space for GFT states, thus for LQG spin networks in 2nd quantization, and rewrite dynamics in such formalism 23 / 25

24 CONCLUSIONS AND OUTLOOK Conclusions GFTs define a tentative but complete quantum dynamics for spin networks or simplices just as LQG, they can be written in three main representations: non-commutative triad/flux, connection, spin network spin foam amplitudes are dual to (non-commutative) simplicial gravity path integrals in this representation, construction of 4d gravity model is geometrically transparent one can identify the GFT analogue of simplicial diffeomorphism transformations this GFT symmetry makes possible to relate nicely the consequences of diffeo invariance in quantum geometry: simplicial vertex translations, WdW equation on cylindrical functions, algebraic recursion relations for spin foam amplitudes one also elucidates the GFT origin and role of Bianchi identities GFT diffeos are quantum group transformations, so suggest a role of braiding and braided statistics in GFT and quantum geometry the appropriate braiding can be identified Outlook clarify further algebraic nature of Bianchi identities role of braiding in GFT amplitudes, dependence on topology of diagrams, scaling limits Ward identities and conserved quantities following from GFT diffeo symmetry GFT Schwinger-Dyson equations (where all the quantum dynamics is contained) Fock space of GFT states (and of LQG), reformulate dynamics in 2nd quantization extend analysis to 4d gravity models (broken diffeos, recovering of diffeos, braiding, etc)...and much much more / 25

25 Thank you for your attention! 25 / 25