Group field theories a QFT for the building blocks of (quantum) space Feynman perturbative expansion around trivial vacuum λ N Γ Z X D ϕ D ϕ e i S λ ( ϕ , ϕ ) Z = = sym ( Γ ) A Γ Γ Feynman diagrams (obtained by convoluting propagators with interaction kernels) = = stranded diagrams dual to cellular complexes of arbitrary topology (simplicial case: simplicial complexes obtained by gluing d-simplices) Feynman amplitudes (model-dependent): equivalently: spin foam models (sum-over-histories of • spin networks ~ covariant LQG) Reisenberger,Rovelli, ’00 • lattice path integrals GFT as lattice quantum gravity: (with group+Lie algebra variables) dynamical triangulations + quantum Regge calculus A. Baratin, DO, ‘11
GFTs and Loop Quantum Gravity second quantized version of Loop Quantum Gravity but dynamics not derived from canonical quantization of GR (DO, 1310.7786 [gr-qc]) DO, J. Ryan, J. Thurigen, ‘14 (LQG spin network states ~ many-particles states, “particle” ~ spin network vertex)
GFTs and Loop Quantum Gravity second quantized version of Loop Quantum Gravity but dynamics not derived from canonical quantization of GR (DO, 1310.7786 [gr-qc]) DO, J. Ryan, J. Thurigen, ‘14 (LQG spin network states ~ many-particles states, “particle” ~ spin network vertex) 1 G 14 G 12 G 13 4 2 G 24 G 34 G 23 3
GFTs and Loop Quantum Gravity second quantized version of Loop Quantum Gravity but dynamics not derived from canonical quantization of GR (DO, 1310.7786 [gr-qc]) DO, J. Ryan, J. Thurigen, ‘14 (LQG spin network states ~ many-particles states, “particle” ~ spin network vertex) 1 3 g 1 g 1 1 1 2 g 1 G 14 G 12 1 g 2 3 g 4 G 13 2 g 4 2 g 4 2 4 2 2 G 24 1 g 4 g 3 2 G 34 G 23 2 g 3 g 3 1 g 3 3 3 3
GFTs and Loop Quantum Gravity second quantized version of Loop Quantum Gravity but dynamics not derived from canonical quantization of GR (DO, 1310.7786 [gr-qc]) DO, J. Ryan, J. Thurigen, ‘14 (LQG spin network states ~ many-particles states, “particle” ~ spin network vertex) 1 3 g 1 g 1 1 1 2 g 1 G 14 G 12 1 g 2 3 g 4 G 13 2 g 4 2 g 4 2 4 2 2 G 24 1 g 4 g 3 2 G 34 G 23 2 g 3 g 3 1 g 3 3 3 3 GFT Hilbert space = Fock space of open spin network vertices - contains any LQG state (all spin networks) any LQG observable has a 2nd quantised, GFT counterpart choice of LQG dynamics (Hamiltonian constraint operator) translates into choice of GFT action
GFTs and Loop Quantum Gravity second quantized version of Loop Quantum Gravity but dynamics not derived from canonical quantization of GR (DO, 1310.7786 [gr-qc]) DO, J. Ryan, J. Thurigen, ‘14 (LQG spin network states ~ many-particles states, “particle” ~ spin network vertex) 1 3 g 1 g 1 1 2 g 1 1 g 2 3 g 4 2 g 4 2 g 4 2 2 1 g 4 g 3 2 2 g 3 g 3 1 g 3 3 3 GFT Hilbert space = Fock space of open spin network vertices - contains any LQG state (all spin networks) any LQG observable has a 2nd quantised, GFT counterpart choice of LQG dynamics (Hamiltonian constraint operator) translates into choice of GFT action
GFTs and Loop Quantum Gravity second quantized version of Loop Quantum Gravity but dynamics not derived from canonical quantization of GR (DO, 1310.7786 [gr-qc]) DO, J. Ryan, J. Thurigen, ‘14 (LQG spin network states ~ many-particles states, “particle” ~ spin network vertex) GFT Hilbert space = Fock space of open spin network vertices - contains any LQG state (all spin networks) any LQG observable has a 2nd quantised, GFT counterpart choice of LQG dynamics (Hamiltonian constraint operator) translates into choice of GFT action
GFTs and Loop Quantum Gravity second quantized version of Loop Quantum Gravity but dynamics not derived from canonical quantization of GR (DO, 1310.7786 [gr-qc]) DO, J. Ryan, J. Thurigen, ‘14 (LQG spin network states ~ many-particles states, “particle” ~ spin network vertex)
GFTs and Loop Quantum Gravity second quantized version of Loop Quantum Gravity but dynamics not derived from canonical quantization of GR (DO, 1310.7786 [gr-qc]) DO, J. Ryan, J. Thurigen, ‘14 (LQG spin network states ~ many-particles states, “particle” ~ spin network vertex) QFT methods (i.e. GFT reformulation of LQG and spin foam models) useful to address physics of large numbers of LQG d.o.f.s, i.e. many and refined graphs (continuum limit)
Group Field Theory: crossroad of approaches Matrix models Simplicial gravity path integrals (e.g. quantum Regge calculus) LQG Spin foam models GFT Tensor models (causal) Dynamical Non-commutative geometry Triangulations
how GFT tackles open issues in QG
how GFT tackles open issues in QG • how to constrain quantisation and construction ambiguities in model building? • GFT perturbative renormalization —-> renormalizability of GFT for given discrete gravity path integral/spin foam amplitudes • GFT symmetries (at both classical and quantum level) Ben Geloun, ’11; Girelli, Livine, ’11; Baratin, Girelli, DO, ‘11 —-> in particular, those with geometric interpretation (e.g. diffeomorphisms) Kegeles, DO, ‘15
how GFT tackles open issues in QG • how to constrain quantisation and construction ambiguities in model building? • GFT perturbative renormalization —-> renormalizability of GFT for given discrete gravity path integral/spin foam amplitudes • GFT symmetries (at both classical and quantum level) Ben Geloun, ’11; Girelli, Livine, ’11; Baratin, Girelli, DO, ‘11 —-> in particular, those with geometric interpretation (e.g. diffeomorphisms) Kegeles, DO, ‘15 • how to define the continuum limit (of the LQG/SF dynamics or, equivalently, of discrete gravity path integral)? controlling quantum dynamics of more and more interacting degrees of freedom new analytic tools from QFT embedding • Non-perturbative GFT renormalization and phase diagram - what are the QG phases? which one is geometric? • Extraction of effective continuum dynamics in different phases (as in QFT for condensed matter systems….)
Part II: the continuum limit of GFTs
The problem of the continuum limit in QG new (non-geometric, non-spatio-temporal) physical degrees of freedom (“building blocks”) for space-time
The problem of the continuum limit in QG new (non-geometric, non-spatio-temporal) physical degrees of freedom (“building blocks”) for space-time new direction to explore: number of fundamental degrees of freedom
The problem of the continuum limit in QG new (non-geometric, non-spatio-temporal) physical degrees of freedom (“building blocks”) for space-time new direction to explore: number of fundamental degrees of freedom (quantum) continuum, geometric space-time should be recovered in the regime of large number N of non-spatio-temporal d.o.f.s
The problem of the continuum limit in QG new (non-geometric, non-spatio-temporal) physical degrees of freedom (“building blocks”) for space-time new direction to explore: number of fundamental degrees of freedom (quantum) continuum, geometric space-time should be recovered in the regime of large number N of non-spatio-temporal d.o.f.s continuum approximation very different (conceptually, technically) few QG d.o.f.s ! full Quantum Gravity (e.g. simple LQG spinnets) from classical approximation h N few QG d.o.f.s in classical approx. ! General Relativity ! (e.g. discrete/lattice gravity) (continuum spacetime)
The problem of the continuum limit in QG new (non-geometric, non-spatio-temporal) physical degrees of freedom (“building blocks”) for space-time new direction to explore: number of fundamental degrees of freedom (quantum) continuum, geometric space-time should be recovered in the regime of large number N of non-spatio-temporal d.o.f.s continuum approximation very different (conceptually, technically) few QG d.o.f.s ! full Quantum Gravity (e.g. simple LQG spinnets) from classical approximation N-direction (collective behaviour of many interacting degrees of freedom): continuum approximation h h-direction: classical approximation N few QG d.o.f.s in classical approx. ! General Relativity ! (e.g. discrete/lattice gravity) (continuum spacetime)
The problem of the continuum limit in QG new (non-geometric, non-spatio-temporal) physical degrees of freedom (“building blocks”) for space-time new direction to explore: number of fundamental degrees of freedom (quantum) continuum, geometric space-time should be recovered in the regime of large number N of non-spatio-temporal d.o.f.s continuum approximation very different (conceptually, technically) few QG d.o.f.s ! full Quantum Gravity (e.g. simple LQG spinnets) from classical approximation N-direction (collective behaviour of many interacting degrees of freedom): continuum approximation h h-direction: classical approximation N few QG d.o.f.s in classical approx. ! General Relativity ! “well-understood” in spin foam models and (e.g. discrete/lattice gravity) (continuum spacetime) discrete gravity
Problem of the continuum in QG: role of RG Renormalization Group is crucial tool for taking into account the physics of more and more d.o.f.s
Problem of the continuum in QG: role of RG Renormalization Group is crucial tool for taking into account the physics of more and more d.o.f.s • for our QG models, do not expect to have a unique continuum limit collective behaviour of (interacting) fundamental d.o.f.s should lead to different macroscopic phases, separated by phase transitions • for a non-spatio-temporal QG system (e.g. LQG in GFT formulation), which of the macroscopic phases is described by a smooth geometry with matter fields?
Problem of the continuum in QG: role of RG Renormalization Group is crucial tool for taking into account the physics of more and more d.o.f.s • for our QG models, do not expect to have a unique continuum limit collective behaviour of (interacting) fundamental d.o.f.s should lead to different macroscopic phases, separated by phase transitions • for a non-spatio-temporal QG system (e.g. LQG in GFT formulation), which of the macroscopic phases is described by a smooth geometry with matter fields? in specific GFT case: • treat GFT models as analogous to atomic QFTs in condensed matter systems • need to understand effective dynamics at different “GFT scales”: RG flow of effective actions & phase structure & phase transitions
Continuum limit of GFT (and LQG, discrete gravity etc) the issue: controlling quantum dynamics of more and more interacting degrees of freedom
Continuum limit of GFT (and LQG, discrete gravity etc) the issue: controlling quantum dynamics of more and more interacting degrees of freedom - control GFT quantum dynamics for boundary states involving (superpositions of) large graphs -compute- spin foam amplitudes for finer complexes and corresponding sum over complexes up to infinite refinement (infinite number of degrees of freedom), at least in simple approximations
Continuum limit of GFT (and LQG, discrete gravity etc) the issue: controlling quantum dynamics of more and more interacting degrees of freedom - control GFT quantum dynamics for boundary states involving (superpositions of) large graphs -compute- spin foam amplitudes for finer complexes and corresponding sum over complexes up to infinite refinement (infinite number of degrees of freedom), at least in simple approximations need control over parameter space of SF models (full theory space) expect different phases and phase transitions Koslowski, ’07; DO, ‘07 as result of quantum dynamics (what are the phases of LQG?)
Continuum limit of GFT (and LQG, discrete gravity etc) the issue: controlling quantum dynamics of more and more interacting degrees of freedom - control GFT quantum dynamics for boundary states involving (superpositions of) large graphs -compute- spin foam amplitudes for finer complexes and corresponding sum over complexes up to infinite refinement (infinite number of degrees of freedom), at least in simple approximations need control over parameter space of SF models AL (full theory space) DG vacuum vacuum ? (or BF vacuum) expect different phases KS vacuum and phase transitions Koslowski, ’07; DO, ‘07 GFT ? as result of quantum dynamics condensate (what are the phases of LQG?) phase Ashtekar, Lewandowski, ’94 Koslowski, Sahlmann, ’10 Dittrich, Geiller, ’14, ‘15 transitions ? Gielen, DO, Sindoni, ’13 Kegeles, DO, Tomlin, to appear
Part III: the FRG analysis of GFTs
GFT renormalisation - general scheme λ N Γ Z X D ϕ D ϕ e i S λ ( ϕ , ϕ ) Z = = sym ( Γ ) A Γ Γ S ( ϕ , ϕ ) = 1 Z [ dg i ] ϕ ( g i ) K ( g i ) ϕ ( g i ) + λ Z [ dg ia ] ϕ ( g i 1 ) .... ϕ (¯ g iD ) V ( g ia , ¯ g iD ) + c.c. 2 D !
GFT renormalisation - general scheme λ N Γ Z X D ϕ D ϕ e i S λ ( ϕ , ϕ ) Z = = sym ( Γ ) A Γ Γ S ( ϕ , ϕ ) = 1 Z [ dg i ] ϕ ( g i ) K ( g i ) ϕ ( g i ) + λ Z [ dg ia ] ϕ ( g i 1 ) .... ϕ (¯ g iD ) V ( g ia , ¯ g iD ) + c.c. 2 D ! general strategy: treat GFTs as ordinary QFTs defined on Lie group manifold use group structures (Killing form, topology, etc) to define notion of scale and to set up mode integration subtleties of quantum gravity context at the level of interpretation
GFT renormalisation - general scheme λ N Γ Z X D ϕ D ϕ e i S λ ( ϕ , ϕ ) Z = = sym ( Γ ) A Γ Γ S ( ϕ , ϕ ) = 1 Z [ dg i ] ϕ ( g i ) K ( g i ) ϕ ( g i ) + λ Z [ dg ia ] ϕ ( g i 1 ) .... ϕ (¯ g iD ) V ( g ia , ¯ g iD ) + c.c. 2 D ! general strategy: treat GFTs as ordinary QFTs defined on Lie group manifold use group structures (Killing form, topology, etc) to define notion of scale and to set up mode integration subtleties of quantum gravity context at the level of interpretation scales: defined by propagator: e.g. spectrum of Laplacian on G = indexed by group representations
GFT renormalisation - general scheme λ N Γ Z X D ϕ D ϕ e i S λ ( ϕ , ϕ ) Z = = sym ( Γ ) A Γ Γ S ( ϕ , ϕ ) = 1 Z [ dg i ] ϕ ( g i ) K ( g i ) ϕ ( g i ) + λ Z [ dg ia ] ϕ ( g i 1 ) .... ϕ (¯ g iD ) V ( g ia , ¯ g iD ) + c.c. 2 D ! general strategy: treat GFTs as ordinary QFTs defined on Lie group manifold use group structures (Killing form, topology, etc) to define notion of scale and to set up mode integration subtleties of quantum gravity context at the level of interpretation scales: defined by propagator: e.g. spectrum of Laplacian on G = indexed by group representations key difficulties: • need to have control over “theory space” (e.g. via symmetries) • main difficulty (at perturbative level): controlling the combinatorics of GFT Feynman diagrams to control the structure of divergences need to adapt/redefine many QFT notions: connectedness, subgraph contraction, Wick ordering, …..
GFT renormalisation - general scheme λ N Γ Z X D ϕ D ϕ e i S λ ( ϕ , ϕ ) Z = = sym ( Γ ) A Γ Γ S ( ϕ , ϕ ) = 1 Z [ dg i ] ϕ ( g i ) K ( g i ) ϕ ( g i ) + λ Z [ dg ia ] ϕ ( g i 1 ) .... ϕ (¯ g iD ) V ( g ia , ¯ g iD ) + c.c. 2 D ! general strategy: treat GFTs as ordinary QFTs defined on Lie group manifold use group structures (Killing form, topology, etc) to define notion of scale and to set up mode integration subtleties of quantum gravity context at the level of interpretation scales: defined by propagator: e.g. spectrum of Laplacian on G = indexed by group representations key difficulties: • need to have control over “theory space” (e.g. via symmetries) • main difficulty (at perturbative level): controlling the combinatorics of GFT Feynman diagrams to control the structure of divergences need to adapt/redefine many QFT notions: connectedness, subgraph contraction, Wick ordering, ….. most results for Tensorial GFTs
Tensorial GFTs (key insights from tensor models) locality principle and soft breaking of locality:
Tensorial GFTs (key insights from tensor models) locality principle and soft breaking of locality: � S ( ϕ , ϕ ) = t b I b ( ϕ , ϕ ) . tensor invariant interactions b ∈ B indexed by bipartite d-colored graphs (“bubbles”) ~ dual to d-cells with triangulated boundary � [ d g i ] 12 � ( g 1 , g 2 , g 3 , g 4 ) � ( g 1 , g 2 , g 3 , g 5 ) � ( g 8 , g 7 , g 6 , g 5 ) � ( g 8 , g 9 , g 10 , g 11 ) � ( g 12 , g 9 , g 10 , g 11 ) � ( g 12 , g 7 , g 6 , g 4 )
Tensorial GFTs (key insights from tensor models) locality principle and soft breaking of locality: � S ( ϕ , ϕ ) = t b I b ( ϕ , ϕ ) . tensor invariant interactions b ∈ B indexed by bipartite d-colored graphs (“bubbles”) ~ dual to d-cells with triangulated boundary kinetic term = e.g. Laplacian on G ⇥ � 1 � � d [ d g i ] 12 � ( g 1 , g 2 , g 3 , g 4 ) � ( g 1 , g 2 , g 3 , g 5 ) � ( g 8 , g 7 , g 6 , g 5 ) m 2 − ⇤ ∆ ⇥ propagator � ( g 8 , g 9 , g 10 , g 11 ) � ( g 12 , g 9 , g 10 , g 11 ) � ( g 12 , g 7 , g 6 , g 4 ) ⇥ =1
Tensorial GFTs (key insights from tensor models) locality principle and soft breaking of locality: � S ( ϕ , ϕ ) = t b I b ( ϕ , ϕ ) . tensor invariant interactions b ∈ B indexed by bipartite d-colored graphs (“bubbles”) ~ dual to d-cells with triangulated boundary kinetic term = e.g. Laplacian on G ⇥ � 1 � � d [ d g i ] 12 � ( g 1 , g 2 , g 3 , g 4 ) � ( g 1 , g 2 , g 3 , g 5 ) � ( g 8 , g 7 , g 6 , g 5 ) m 2 − ⇤ ∆ ⇥ propagator � ( g 8 , g 9 , g 10 , g 11 ) � ( g 12 , g 9 , g 10 , g 11 ) � ( g 12 , g 7 , g 6 , g 4 ) ⇥ =1 “coloring” allows control over topology of Feynman diagrams
Tensorial GFTs (key insights from tensor models) locality principle and soft breaking of locality: � S ( ϕ , ϕ ) = t b I b ( ϕ , ϕ ) . tensor invariant interactions b ∈ B indexed by bipartite d-colored graphs (“bubbles”) ~ dual to d-cells with triangulated boundary kinetic term = e.g. Laplacian on G ⇥ � 1 � � d [ d g i ] 12 � ( g 1 , g 2 , g 3 , g 4 ) � ( g 1 , g 2 , g 3 , g 5 ) � ( g 8 , g 7 , g 6 , g 5 ) m 2 − ⇤ ∆ ⇥ propagator � ( g 8 , g 9 , g 10 , g 11 ) � ( g 12 , g 9 , g 10 , g 11 ) � ( g 12 , g 7 , g 6 , g 4 ) ⇥ =1 “coloring” allows control over topology of Feynman diagrams require generalization of notions of “connectedness”, “contraction of high subgraphs”, “locality”, Wick ordering, …. taking into account internal structure of Feynman graphs, full combinatorics of dual cellular complex, results from crystallization theory (dipole moves)
TGFT renormalization example of Feynman diagram • building blocks: coloured bubbles, dual to d-cells with triangulated boundary • glued along their boundary (d-1)-simplices • parallel transports (discrete connection) associated to dashed (color 0, propagator) lines • faces of color i = connected set of (alternating) lines of color 0 and i “contraction of internal line” ~ dipole contraction
GFT Renormalization: “geometric” interpretation? consistent with cosmological interpretation of classical GFT fields and with results of GFT condensate cosmology (see later)
GFT Renormalization: “geometric” interpretation? • GFT “UV” cut-off N ~ J max • RG flow: J max ——-> small J • (perturbative) GFT renormalizability: UV fixed point as J max ——-> oo consistent with cosmological interpretation of classical GFT fields and with results of GFT condensate cosmology (see later)
GFT Renormalization: “geometric” interpretation? • GFT “UV” cut-off N ~ J max N • RG flow: J max ——-> small J • (perturbative) GFT renormalizability: UV fixed point as J max ——-> oo b b from LQG 1 4 from Regge calculus b 2 b i ∈ su (2) arguments of GFT field: gravity case: d=4 b | b | ~ J = irrep of SU(2) ~ “area of triangles” 3 consistent with cosmological interpretation of classical GFT fields and with results of GFT condensate cosmology (see later)
GFT Renormalization: “geometric” interpretation? • GFT “UV” cut-off N ~ J max N • RG flow: J max ——-> small J • (perturbative) GFT renormalizability: UV fixed point as J max ——-> oo b b from LQG 1 4 from Regge calculus b 2 b i ∈ su (2) arguments of GFT field: gravity case: d=4 b | b | ~ J = irrep of SU(2) ~ “area of triangles” 3 “geometric” interpretation of the RG flow? • RG flow from large areas to small areas? not quite • theory defined in non-geometric phase of “large” disconnected tetrahedra • flow of couplings to region of many interacting (thus, connected) “small” tetrahedra consistent with cosmological interpretation of classical GFT fields and with results of GFT condensate cosmology (see later)
GFT Renormalization: “geometric” interpretation? • GFT “UV” cut-off N ~ J max N • RG flow: J max ——-> small J • (perturbative) GFT renormalizability: UV fixed point as J max ——-> oo b b from LQG 1 4 from Regge calculus b 2 b i ∈ su (2) arguments of GFT field: gravity case: d=4 b | b | ~ J = irrep of SU(2) ~ “area of triangles” 3 “geometric” interpretation of the RG flow? • RG flow from large areas to small areas? not quite • theory defined in non-geometric phase of “large” disconnected tetrahedra • flow of couplings to region of many interacting (thus, connected) “small” tetrahedra • CAUTION in interpreting things geometrically outside continuum geometric approx • e.g. expect “physical” continuum areas A ~ < J > < n > • expect proper continuum geometric interpretation (and effective metric field) for < J > small, < n > large, A finite (not too small), and small curvature consistent with cosmological interpretation of classical GFT fields and with results of GFT condensate cosmology (see later)
GFT perturbative renormalisation
GFT perturbative renormalisation ] h 1 g 1 g � 1 step by step, towards renormalizable 4d gravity models: g � g 2 2 h 2 g � g 3 - scale indexed by group representations 3 h 3 - interplay between algebraic data and combinatorics of diagrams • calculation of some radiative corrections T. Krajewski, J. Magnen, V. Rivasseau, A. Tanasa, P. Vitale, ’10; A. Riello, ’13; Bonzom, Dittrich, ‘15 Ben Geloun, Bonzom, ’11; Ben Geloun, ‘13 • finiteness results in 3d simplicial models (Boulatov with Laplacian kinetic term) • renormalizable TGFT models (3d, 4d, and higher) - Laplacian + tensorial interactions Ben Geloun, Rivasseau, ’11 � S ( ϕ , ϕ ) = t b I b ( ϕ , ϕ ) . Carrozza, DO, Rivasseau, ’12. ‘13 -> with gauge invariance b ∈ B —> non-abelian ( SU(2) ) ——> SO(4) or SO(3,1) with simplicity constraints: first results on BC-like 4d models Lahoche, DO, ’15; Carrozza, Lahoche, DO, ‘16 ———> generic (and robust?) asymptotic freedom Ben Geloun, ’12; Carrozza, ‘14
GFT perturbative renormalisation ] h 1 g 1 g � 1 step by step, towards renormalizable 4d gravity models: g � g 2 2 h 2 g � g 3 - scale indexed by group representations 3 h 3 - interplay between algebraic data and combinatorics of diagrams • calculation of some radiative corrections T. Krajewski, J. Magnen, V. Rivasseau, A. Tanasa, P. Vitale, ’10; A. Riello, ’13; Bonzom, Dittrich, ‘15 Ben Geloun, Bonzom, ’11; Ben Geloun, ‘13 • finiteness results in 3d simplicial models (Boulatov with Laplacian kinetic term) • renormalizable TGFT models (3d, 4d, and higher) - Laplacian + tensorial interactions Ben Geloun, Rivasseau, ’11 � S ( ϕ , ϕ ) = t b I b ( ϕ , ϕ ) . Carrozza, DO, Rivasseau, ’12. ‘13 -> with gauge invariance b ∈ B —> non-abelian ( SU(2) ) ——> SO(4) or SO(3,1) with simplicity constraints: first results on BC-like 4d models Lahoche, DO, ’15; Carrozza, Lahoche, DO, ‘16 ———> generic (and robust?) asymptotic freedom Ben Geloun, ’12; Carrozza, ‘14 many important lessons (e.g. learnt to deal with combinatorics and topology of spin foam complex)
GFT perturbative renormalisation ] h 1 g 1 g � 1 step by step, towards renormalizable 4d gravity models: g � g 2 2 h 2 g � g 3 - scale indexed by group representations 3 h 3 - interplay between algebraic data and combinatorics of diagrams • calculation of some radiative corrections T. Krajewski, J. Magnen, V. Rivasseau, A. Tanasa, P. Vitale, ’10; A. Riello, ’13; Bonzom, Dittrich, ‘15 Ben Geloun, Bonzom, ’11; Ben Geloun, ‘13 • finiteness results in 3d simplicial models (Boulatov with Laplacian kinetic term) • renormalizable TGFT models (3d, 4d, and higher) - Laplacian + tensorial interactions Ben Geloun, Rivasseau, ’11 � S ( ϕ , ϕ ) = t b I b ( ϕ , ϕ ) . Carrozza, DO, Rivasseau, ’12. ‘13 -> with gauge invariance b ∈ B —> non-abelian ( SU(2) ) ——> SO(4) or SO(3,1) with simplicity constraints: first results on BC-like 4d models Lahoche, DO, ’15; Carrozza, Lahoche, DO, ‘16 ———> generic (and robust?) asymptotic freedom Ben Geloun, ’12; Carrozza, ‘14 many important lessons main open issues: • characterise better theory space (kinetic term, combinatorics of interactions, …) (e.g. learnt to deal with • deal with non-group structures (due to Immirzi parameter) combinatorics and topology of spin foam complex) understand in full the geometric interpretation of UV/IR and of RG flow
GFT perturbative renormalisation ] recent results: h 1 g 1 g � 1 step by step, towards renormalizable 4d gravity models: g � g 2 2 h 2 g � g 3 - scale indexed by group representations 3 h 3 - interplay between algebraic data and combinatorics of diagrams • calculation of some radiative corrections T. Krajewski, J. Magnen, V. Rivasseau, A. Tanasa, P. Vitale, ’10; A. Riello, ’13; Bonzom, Dittrich, ‘15 Ben Geloun, Bonzom, ’11; Ben Geloun, ‘13 • finiteness results in 3d simplicial models (Boulatov with Laplacian kinetic term) • renormalizable TGFT models (3d, 4d, and higher) - Laplacian + tensorial interactions Ben Geloun, Rivasseau, ’11 � S ( ϕ , ϕ ) = t b I b ( ϕ , ϕ ) . Carrozza, DO, Rivasseau, ’12. ‘13 -> with gauge invariance b ∈ B —> non-abelian ( SU(2) ) ——> SO(4) or SO(3,1) with simplicity constraints: first results on BC-like 4d models Lahoche, DO, ’15; Carrozza, Lahoche, DO, ‘16 ———> generic (and robust?) asymptotic freedom Ben Geloun, ’12; Carrozza, ‘14 many important lessons main open issues: • characterise better theory space (kinetic term, combinatorics of interactions, …) (e.g. learnt to deal with • deal with non-group structures (due to Immirzi parameter) combinatorics and topology of spin foam complex) understand in full the geometric interpretation of UV/IR and of RG flow
GFT non-perturbative renormalisation λ N Γ Z X D ϕ D ϕ e i S λ ( ϕ , ϕ ) Z = = sym ( Γ ) A Γ the GFT proposal: Γ controlling the continuum limit ~ evaluating GFT path integral (in some non-perturbative approximation) (computing full SF sum) Benedetti, Ben Geloun, DO, Martini, Lahoche, Carrozza, Douarte, …. Freidel, Louapre, Noui, Magnen, Smerlak, Gurau, Rivasseau, Tanasa, Dartois, Delpouve, …..
GFT non-perturbative renormalisation λ N Γ Z X D ϕ D ϕ e i S λ ( ϕ , ϕ ) Z = = sym ( Γ ) A Γ the GFT proposal: Γ controlling the continuum limit ~ evaluating GFT path integral (in some non-perturbative approximation) (computing full SF sum) two directions: • GFT non-perturbative renormalization and “IR” fixed points (e.g. FRG analysis - e.g. a la Wetterich Benedetti, Ben Geloun, DO, Martini, Lahoche, Carrozza, Douarte, …. • GFT constructive analysis Freidel, Louapre, Noui, Magnen, Smerlak, Gurau, Rivasseau, Tanasa, Dartois, Delpouve, ….. non-perturbative resummation of perturbative (SF) series variety of techniques: • intermediate field method • loop-vertex expansion • Borel summability
GFT non-perturbative renormalisation recent results: FRG for (tensorial) GFT models
GFT non-perturbative renormalisation recent results: (similar to matrix model but distinctively field-theoretic) FRG for (tensorial) GFT models Eichhorn, Koslowski, ‘14
GFT non-perturbative renormalisation recent results: (similar to matrix model but distinctively field-theoretic) FRG for (tensorial) GFT models Eichhorn, Koslowski, ‘14 • Polchinski formulation based on SD equations Krajewski, Toriumi, ‘14 • general set-up for Wetterich formulation based on effective action • analysis of TGFT on compact U(1)^d Benedetti, Ben Geloun, DO, ’14 ; Ben Geloun, Martini, DO, ’15, ’16, Benedetti, Lahoche, ’15; Douarte, DO, ‘16 • RG flow and phase diagram established 0.2 • analysis of TGFT on non-compact R^d • RG flow and phase diagram established 0.0 • analysis of TGFT on non-compact R^d with gauge invariance � 0.2 • RG flow and phase diagram established • analysis of TGFT on SU(2)^3 Carrozza, Lahoche, ‘16 m N � 0.4 generically (so far): � 0.6 two FPs (Gaussian-UV, Wilson-Fisher-IR) � 0.8 asymptotic freedom one symmetric phase � 1.0 0.00 0.01 0.02 0.03 0.04 one broken or condensate phase Λ N
FRG analysis of GFT models D. Benedetti, J. Ben Geloun, DO, ‘14
FRG analysis of GFT models D. Benedetti, J. Ben Geloun, DO, ‘14 Z regularised path integral: Z N [ J, J ] = e W N [ J,J ] = d φ d φ e � S [ φ , φ ] � ∆ S N [ φ , φ ]+Tr( J · φ )+Tr( J · φ ) k k k regulator cutting off IR modes (UV well-defined with appropriate choice of IR regulator) X φ P R N ( P ; P 0 ) φ P 0 ∆ S N [ φ , φ ] = Tr( φ · R N · φ ) = k k k P , P 0 R k ( p , p 0 ) = θ ( k 2 − Σ s p 2 s ) Z k ( k 2 − Σ s p 2 s ) δ ( p − p 0 ) (
FRG analysis of GFT models D. Benedetti, J. Ben Geloun, DO, ‘14 Z regularised path integral: Z N [ J, J ] = e W N [ J,J ] = d φ d φ e � S [ φ , φ ] � ∆ S N [ φ , φ ]+Tr( J · φ )+Tr( J · φ ) k k k regulator cutting off IR modes (UV well-defined with appropriate choice of IR regulator) X φ P R N ( P ; P 0 ) φ P 0 ∆ S N [ φ , φ ] = Tr( φ · R N · φ ) = k k k P , P 0 R k ( p , p 0 ) = θ ( k 2 − Σ s p 2 s ) Z k ( k 2 − Σ s p 2 s ) δ ( p − p 0 ) ( ⇢ � Γ N [ ϕ , ϕ ] = sup Tr( J · ϕ ) + Tr( J · ϕ ) � W N [ J, J ] � ∆ S N [ ϕ , ϕ ] effective action: k k k J,J
FRG analysis of GFT models D. Benedetti, J. Ben Geloun, DO, ‘14 Z regularised path integral: Z N [ J, J ] = e W N [ J,J ] = d φ d φ e � S [ φ , φ ] � ∆ S N [ φ , φ ]+Tr( J · φ )+Tr( J · φ ) k k k regulator cutting off IR modes (UV well-defined with appropriate choice of IR regulator) X φ P R N ( P ; P 0 ) φ P 0 ∆ S N [ φ , φ ] = Tr( φ · R N · φ ) = k k k P , P 0 R k ( p , p 0 ) = θ ( k 2 − Σ s p 2 s ) Z k ( k 2 − Σ s p 2 s ) δ ( p − p 0 ) ( ⇢ � Γ N [ ϕ , ϕ ] = sup Tr( J · ϕ ) + Tr( J · ϕ ) � W N [ J, J ] � ∆ S N [ ϕ , ϕ ] effective action: k k k J,J ∂ t Γ k = Tr [ ∂ t R k · ( Γ (2) + R k ) � 1 ] re t = log k . Wetterich equation: k
FRG analysis of GFT models D. Benedetti, J. Ben Geloun, DO, ‘14 Z regularised path integral: Z N [ J, J ] = e W N [ J,J ] = d φ d φ e � S [ φ , φ ] � ∆ S N [ φ , φ ]+Tr( J · φ )+Tr( J · φ ) k k k regulator cutting off IR modes (UV well-defined with appropriate choice of IR regulator) X φ P R N ( P ; P 0 ) φ P 0 ∆ S N [ φ , φ ] = Tr( φ · R N · φ ) = k k k P , P 0 R k ( p , p 0 ) = θ ( k 2 − Σ s p 2 s ) Z k ( k 2 − Σ s p 2 s ) δ ( p − p 0 ) ( ⇢ � Γ N [ ϕ , ϕ ] = sup Tr( J · ϕ ) + Tr( J · ϕ ) � W N [ J, J ] � ∆ S N [ ϕ , ϕ ] effective action: k k k J,J ∂ t Γ k = Tr [ ∂ t R k · ( Γ (2) + R k ) � 1 ] re t = log k . Wetterich equation: k Γ k = Λ [ ϕ , ϕ ] = S [ ϕ , ϕ ] re ϕ = h φ i . Γ k =0 [ ϕ , ϕ ] = Γ [ ϕ , ϕ ] , boundary conditions:
FRG analysis of GFT models D. Benedetti, J. Ben Geloun, DO, ‘14 Z regularised path integral: Z N [ J, J ] = e W N [ J,J ] = d φ d φ e � S [ φ , φ ] � ∆ S N [ φ , φ ]+Tr( J · φ )+Tr( J · φ ) k k k regulator cutting off IR modes (UV well-defined with appropriate choice of IR regulator) X φ P R N ( P ; P 0 ) φ P 0 ∆ S N [ φ , φ ] = Tr( φ · R N · φ ) = k k k P , P 0 R k ( p , p 0 ) = θ ( k 2 − Σ s p 2 s ) Z k ( k 2 − Σ s p 2 s ) δ ( p − p 0 ) ( ⇢ � Γ N [ ϕ , ϕ ] = sup Tr( J · ϕ ) + Tr( J · ϕ ) � W N [ J, J ] � ∆ S N [ ϕ , ϕ ] effective action: k k k J,J ∂ t Γ k = Tr [ ∂ t R k · ( Γ (2) + R k ) � 1 ] re t = log k . Wetterich equation: k Γ k = Λ [ ϕ , ϕ ] = S [ ϕ , ϕ ] re ϕ = h φ i . Γ k =0 [ ϕ , ϕ ] = Γ [ ϕ , ϕ ] , boundary conditions: computing the effective action solving the Wetterich equation amounts to solving the GFT path integral
FRG analysis of GFT models D. Benedetti, J. Ben Geloun, DO, ‘14 Z regularised path integral: Z N [ J, J ] = e W N [ J,J ] = d φ d φ e � S [ φ , φ ] � ∆ S N [ φ , φ ]+Tr( J · φ )+Tr( J · φ ) k k k regulator cutting off IR modes (UV well-defined with appropriate choice of IR regulator) X φ P R N ( P ; P 0 ) φ P 0 ∆ S N [ φ , φ ] = Tr( φ · R N · φ ) = k k k P , P 0 R k ( p , p 0 ) = θ ( k 2 − Σ s p 2 s ) Z k ( k 2 − Σ s p 2 s ) δ ( p − p 0 ) ( ⇢ � Γ N [ ϕ , ϕ ] = sup Tr( J · ϕ ) + Tr( J · ϕ ) � W N [ J, J ] � ∆ S N [ ϕ , ϕ ] effective action: k k k J,J ∂ t Γ k = Tr [ ∂ t R k · ( Γ (2) + R k ) � 1 ] re t = log k . Wetterich equation: k Γ k = Λ [ ϕ , ϕ ] = S [ ϕ , ϕ ] re ϕ = h φ i . Γ k =0 [ ϕ , ϕ ] = Γ [ ϕ , ϕ ] , boundary conditions: computing the effective action solving the Wetterich equation amounts to solving the GFT path integral Wetterich equation expanded in field powers, with all possible contractions; truncation matching classical action system of flow equations is generically non-homogeneous, because of combinatorial patterns of contractions for compact groups, it is also non-autonomous, due to hidden scale (size of group)
FRG analysis of a quartic abelian rank-d TGFT model d ✓ ◆ Z the model: X S [ φ , φ ] = (2 π ) d R × d [ dx i ] d i =1 φ ( x 1 , . . . , x d ) 4 s + µ φ ( x 1 , . . . , x d ) � s =1 G = R Z + λ 2 (2 π ) 2 d R × 2 d [ dx i ] d j ] d i =1 [ dx 0 φ ( x 1 , x 2 , . . . , x d ) φ ( x 0 1 , x 2 , . . . , x d ) φ ( x 0 1 , x 0 2 , . . . , x 0 d ) φ ( x 1 , x 0 2 , . . . , x 0 + sym d ) j =1 φ φ φ φ φ φ o� n φ φ φ φ φ φ
FRG analysis of a quartic abelian rank-d TGFT model d ✓ ◆ Z the model: X S [ φ , φ ] = (2 π ) d R × d [ dx i ] d i =1 φ ( x 1 , . . . , x d ) 4 s + µ φ ( x 1 , . . . , x d ) � s =1 G = R Z + λ 2 (2 π ) 2 d R × 2 d [ dx i ] d j ] d i =1 [ dx 0 φ ( x 1 , x 2 , . . . , x d ) φ ( x 0 1 , x 2 , . . . , x d ) φ ( x 0 1 , x 0 2 , . . . , x 0 d ) φ ( x 1 , x 0 2 , . . . , x 0 + sym d ) j =1 φ φ φ φ φ φ o� n φ φ φ φ φ φ Z X R ⇥ d [ dp i ] d p 2 Γ k [ ϕ , ϕ ] = i =1 ϕ 12 ...d ( Z k s + µ k ) ϕ 12 ...d s o� + λ k Z n R ⇥ 2 d [ dp i ] d j ] d i =1 [ dp 0 ϕ 12 ...d ϕ 1 0 2 ...d ϕ 1 0 2 0 ...d 0 ϕ 12 0 ...d 0 + sym 1 , 2 , . . . , d j =1 2
FRG analysis of a quartic abelian rank-d TGFT model d ✓ ◆ Z the model: X S [ φ , φ ] = (2 π ) d R × d [ dx i ] d i =1 φ ( x 1 , . . . , x d ) 4 s + µ φ ( x 1 , . . . , x d ) � s =1 G = R Z + λ 2 (2 π ) 2 d R × 2 d [ dx i ] d j ] d i =1 [ dx 0 φ ( x 1 , x 2 , . . . , x d ) φ ( x 0 1 , x 2 , . . . , x d ) φ ( x 0 1 , x 0 2 , . . . , x 0 d ) φ ( x 1 , x 0 2 , . . . , x 0 + sym d ) j =1 φ φ φ φ φ φ o� n φ φ φ φ φ φ Z X R ⇥ d [ dp i ] d p 2 Γ k [ ϕ , ϕ ] = i =1 ϕ 12 ...d ( Z k s + µ k ) ϕ 12 ...d s o� + λ k Z n R ⇥ 2 d [ dp i ] d j ] d i =1 [ dp 0 ϕ 12 ...d ϕ 1 0 2 ...d ϕ 1 0 2 0 ...d 0 ϕ 12 0 ...d 0 + sym 1 , 2 , . . . , d j =1 2 • divergences in Wetterich equation due to non-compactness of group manifold • non-locality of interactions prevents from using standard methods, e.g. local potential approx. • thermodynamic limit must be taken carefully ⌘ d ⇣ 2 π step 1: compactly configuration space to U(1)^d, with la V = l step 2: determine (non-standard) scaling of coupling constants step 3: take non-compact limit so to regularise the most divergent contributions to the RG flow
FRG analysis of a quartic abelian rank-d TGFT model 2 Z k = Z k l χ k � χ , µ k = µ k Z k l χ k 2 � χ , k l ξ k 4 � ξ scaling of couplings: λ k = λ k Z
FRG analysis of a quartic abelian rank-d TGFT model 2 Z k = Z k l χ k � χ , µ k = µ k Z k l χ k 2 � χ , k l ξ k 4 � ξ scaling of couplings: λ k = λ k Z (regularized) flow equations: 8 d − 1 d − 1 λ k l ξ k σ ⌘ k d − 1 ⌘ k d − 1 π l d − 1 + 2( d − 1) k ( d − 1) k π n h i h io 2 2 > η k = ( η k − χ ) + 2 l + + χ > > l d − 1 l 2 χ k 2(2 − χ ) (1 + µ k ) 2 ⇣ ⇣ l > d +1 d − 1 > Γ E Γ E > 2 2 > > > > > > > > > > d − 1 d − 1 ⌘ k d +1 k 3 2 k 3 ⌘ k d +1 d λ k l ξ k σ > l d − 1 + 4 π π > n h 2 i h 2 io > β ( µ k ) = − ( η − χ ) + 2 l + − η k µ k − (2 − χ ) µ k > > l 2 χ k 6 − 2 χ (1 + µ k ) 2 ⇣ ⇣ l d − 1 3 l > d +3 d +1 Γ E Γ E > > 2 2 < > > > 2 > d − 1 ⌘ k d +1 k 3 k l ξ k σ > 2 λ π l d − 1 + 4(2 d − 1) > n h 2 l + 2 δ d, 3 k 2 i > β ( λ k ) = ( η − χ ) > > l 2 χ k 6 − 2 χ (1 + µ k ) 3 ⇣ 3 > d +3 Γ E > > 2 > > > > d − 1 > ⌘ k d +1 l d − 1 + 2(2 d − 1) k 3 π > h 2 l + 2 δ d, 3 k 2 io > + 2 − 2 η k λ k − σλ k > > > ⇣ d +1 > Γ E > : 2 (49) non-autonomous, non-homogeneous; matches TGFT on U(1)^d
FRG analysis of a quartic abelian rank-d TGFT model 2 Z k = Z k l χ k � χ , µ k = µ k Z k l χ k 2 � χ , k l ξ k 4 � ξ scaling of couplings: λ k = λ k Z (regularized) flow equations: 8 d − 1 d − 1 λ k l ξ k σ ⌘ k d − 1 ⌘ k d − 1 π l d − 1 + 2( d − 1) k ( d − 1) k π n h i h io 2 2 > η k = ( η k − χ ) + 2 l + + χ > > l d − 1 l 2 χ k 2(2 − χ ) (1 + µ k ) 2 ⇣ ⇣ l > d +1 d − 1 > Γ E Γ E > 2 2 > > > > > > > > > > d − 1 d − 1 ⌘ k d +1 k 3 2 k 3 ⌘ k d +1 d λ k l ξ k σ > l d − 1 + 4 π π > n h 2 i h 2 io > β ( µ k ) = − ( η − χ ) + 2 l + − η k µ k − (2 − χ ) µ k > > l 2 χ k 6 − 2 χ (1 + µ k ) 2 ⇣ ⇣ l d − 1 3 l > d +3 d +1 Γ E Γ E > > 2 2 < > > > 2 > d − 1 ⌘ k d +1 k 3 k l ξ k σ > 2 λ π l d − 1 + 4(2 d − 1) > n h 2 l + 2 δ d, 3 k 2 i > β ( λ k ) = ( η − χ ) > > l 2 χ k 6 − 2 χ (1 + µ k ) 3 ⇣ 3 > d +3 Γ E > > 2 > > > > d − 1 > ⌘ k d +1 l d − 1 + 2(2 d − 1) k 3 π > h 2 l + 2 δ d, 3 k 2 io > + 2 − 2 η k λ k − σλ k > > > ⇣ d +1 > Γ E > : 2 (49) non-autonomous, non-homogeneous; matches TGFT on U(1)^d most divergent contributions finite for: ξ = 2 χ + ( d − 1) η k = 1 β ( Z k ) = 1 β ( Z k ) + χ and redefined anomalous dimension: g η 0 k = η k − χ : Z k Z k
FRG analysis of a quartic abelian rank-d TGFT model 2 Z k = Z k l χ k � χ , µ k = µ k Z k l χ k 2 � χ , k l ξ k 4 � ξ scaling of couplings: λ k = λ k Z (regularized) flow equations: 8 d − 1 d − 1 λ k l ξ k σ ⌘ k d − 1 ⌘ k d − 1 π l d − 1 + 2( d − 1) k ( d − 1) k π n h i h io 2 2 > η k = ( η k − χ ) + 2 l + + χ > > l d − 1 l 2 χ k 2(2 − χ ) (1 + µ k ) 2 ⇣ ⇣ l > d +1 d − 1 > Γ E Γ E > 2 2 > > > > > > > > > > d − 1 d − 1 ⌘ k d +1 k 3 2 k 3 ⌘ k d +1 d λ k l ξ k σ > l d − 1 + 4 π π > n h 2 i h 2 io > β ( µ k ) = − ( η − χ ) + 2 l + − η k µ k − (2 − χ ) µ k > > l 2 χ k 6 − 2 χ (1 + µ k ) 2 ⇣ ⇣ l d − 1 3 l > d +3 d +1 Γ E Γ E > > 2 2 < > > > 2 > d − 1 ⌘ k d +1 k 3 k l ξ k σ > 2 λ π l d − 1 + 4(2 d − 1) > n h 2 l + 2 δ d, 3 k 2 i > β ( λ k ) = ( η − χ ) > > l 2 χ k 6 − 2 χ (1 + µ k ) 3 ⇣ 3 > d +3 Γ E > > 2 > > > > d − 1 > ⌘ k d +1 l d − 1 + 2(2 d − 1) k 3 π > h 2 l + 2 δ d, 3 k 2 io > + 2 − 2 η k λ k − σλ k > > > ⇣ d +1 > Γ E > : 2 (49) non-autonomous, non-homogeneous; matches TGFT on U(1)^d most divergent contributions finite for: ξ = 2 χ + ( d − 1) η k = 1 β ( Z k ) = 1 β ( Z k ) + χ and redefined anomalous dimension: g η 0 k = η k − χ : Z k Z k now can take thermodynamic limit
FRG analysis of a quartic abelian rank-d TGFT model d − 1 8 2 π λ k η k 2 flow equations for couplings: h i η k = d − 1 + 1 > > > ⇣ ⌘ (1 + µ k ) 2 d � 1 > Γ E > > 2 > > > autonomous, > d − 1 > β ( µ k ) = − 2 d π λ k η k > 2 h i < d + 1 + 1 − η k µ k − 2 µ k ⇣ ⌘ (1 + µ k ) 2 d +1 Γ E still non-homogeneous > 2 > > > > d − 1 2 > 4 π λ η k > 2 h i > k β ( λ k ) = d + 1 + 1 − 2 η k λ k − (5 − d ) λ k > > > ⇣ ⌘ (1 + µ k ) 3 > d +1 Γ E : 2
FRG analysis of a quartic abelian rank-d TGFT model d − 1 8 2 π λ k η k 2 flow equations for couplings: h i η k = d − 1 + 1 > > > ⇣ ⌘ (1 + µ k ) 2 d � 1 > Γ E > > 2 > > > autonomous, > d − 1 > β ( µ k ) = − 2 d π λ k η k > 2 h i < d + 1 + 1 − η k µ k − 2 µ k ⇣ ⌘ (1 + µ k ) 2 d +1 Γ E still non-homogeneous > 2 > > > > d − 1 2 > 4 π λ η k > 2 h i > k β ( λ k ) = d + 1 + 1 − 2 η k λ k − (5 − d ) λ k > > > ⇣ ⌘ (1 + µ k ) 3 > d +1 Γ E : 2 d=3 d=4 0.0 0.0 - 0.2 - 0.2 - 0.4 μ N - 0.4 μ N - 0.6 - 0.6 - 0.8 - 0.8 0.000 0.005 0.010 0.015 0.020 λ N 0.000 0.005 0.010 0.015 0.020 λ N
FRG analysis of a quartic abelian rank-d TGFT model d − 1 8 2 π λ k η k 2 flow equations for couplings: h i η k = d − 1 + 1 > > > ⇣ ⌘ (1 + µ k ) 2 d � 1 > Γ E > > 2 > > > autonomous, > d − 1 > β ( µ k ) = − 2 d π λ k η k > 2 h i < d + 1 + 1 − η k µ k − 2 µ k ⇣ ⌘ (1 + µ k ) 2 d +1 Γ E still non-homogeneous > 2 > > > > d − 1 2 > 4 π λ η k > 2 h i > k β ( λ k ) = d + 1 + 1 − 2 η k λ k − (5 − d ) λ k > > > ⇣ ⌘ (1 + µ k ) 3 > d +1 Γ E : 2 d=3 d=4 0.0 0.0 general features independent of rank-d: - 0.2 - 0.2 Gaussian-UV FP, Wilson-Fisher-IR FP - 0.4 μ N asymptotic freedom - 0.4 μ N one symmetric phase - 0.6 one broken or condensate phase - 0.6 2nd non-G IR FP at negative coupling - 0.8 - 0.8 0.000 0.005 0.010 0.015 0.020 λ N 0.000 0.005 0.010 0.015 0.020 λ N
FRG analysis of a quartic abelian rank-d TGFT model similar model with gauge invariance (imposed in both kinetic and interaction terms): Z h i Z k Σ s p 2 Γ k [ ϕ , ϕ ] = d p ϕ ( p ) s + µ k ϕ ( p ) δ ( Σ p ) Z + λ k d p d p 0 ϕ 12 ... d ϕ 1 0 2 ... d ϕ 1 0 2 0 ... d 0 ϕ 12 0 ... d 0 δ ( Σ p ) δ ( Σ p 0 ) δ ( p 0 1 + p 2 + · · · + p d ) δ ( p 1 + p 0 2 + · · 2
FRG analysis of a quartic abelian rank-d TGFT model similar model with gauge invariance (imposed in both kinetic and interaction terms): Z h i Z k Σ s p 2 Γ k [ ϕ , ϕ ] = d p ϕ ( p ) s + µ k ϕ ( p ) δ ( Σ p ) Z + λ k d p d p 0 ϕ 12 ... d ϕ 1 0 2 ... d ϕ 1 0 2 0 ... d 0 ϕ 12 0 ... d 0 δ ( Σ p ) δ ( Σ p 0 ) δ ( p 0 1 + p 2 + · · · + p d ) δ ( p 1 + p 0 2 + · · 2 similar RG flow equations, different scaling dimensions of couplings: d − 2 8 1 2 d λ k π 2 n o η 0 η 0 ⌘ + k = > > k 3 (1 + µ k ) 2 ⇣ ⇣ ⌘ > d � 2 ( d − 1) d > Γ E Γ E 2 > > 2 2 > > > > d − 2 > 1 2 d λ k π > 2 n o < η 0 − ( η 0 ⌘ + β d 6 =4 ( µ k ) = − k + 2) µ k √ k (1 + µ k ) 2 ⇣ ⇣ ⌘ d − 1 d +2 d Γ E Γ E > 2 2 > > > > d − 2 2 > 2 λ 1 2 π > 2 n o > η 0 − 2 η 0 k ⌘ + β d 6 =4 ( λ k ) = k λ k + ( d − 6) λ k > > √ k (1 + µ k ) 3 > ⇣ ⇣ ⌘ d − 1 d +2 d > Γ E Γ E : 2 2
FRG analysis of a quartic abelian rank-d TGFT model similar model with gauge invariance (imposed in both kinetic and interaction terms): Z h i Z k Σ s p 2 Γ k [ ϕ , ϕ ] = d p ϕ ( p ) s + µ k ϕ ( p ) δ ( Σ p ) Z + λ k d p d p 0 ϕ 12 ... d ϕ 1 0 2 ... d ϕ 1 0 2 0 ... d 0 ϕ 12 0 ... d 0 δ ( Σ p ) δ ( Σ p 0 ) δ ( p 0 1 + p 2 + · · · + p d ) δ ( p 1 + p 0 2 + · · 2 d=4 similar RG flow equations, different scaling dimensions of couplings: ��� d − 2 8 1 2 d λ k π 2 n o η 0 η 0 ⌘ + k = > > k 3 (1 + µ k ) 2 ⇣ ⇣ ⌘ > d � 2 ( d − 1) d > Γ E Γ E 2 > > 2 2 > - ��� > > > d − 2 > 1 2 d λ k π > 2 n o < η 0 − ( η 0 ⌘ + β d 6 =4 ( µ k ) = − k + 2) µ k √ k (1 + µ k ) 2 ⇣ ⇣ ⌘ d − 1 d +2 d Γ E Γ E > 2 2 > - ��� > > μ � > d − 2 2 > 2 λ 1 2 π > 2 n o > η 0 − 2 η 0 k ⌘ + β d 6 =4 ( λ k ) = k λ k + ( d − 6) λ k > > √ k (1 + µ k ) 3 > ⇣ ⇣ ⌘ d − 1 d +2 d > Γ E Γ E : 2 2 - ��� - ��� ����� ����� ����� ����� ����� λ �
FRG analysis of a quartic abelian rank-d TGFT model similar model with gauge invariance (imposed in both kinetic and interaction terms): Z h i Z k Σ s p 2 Γ k [ ϕ , ϕ ] = d p ϕ ( p ) s + µ k ϕ ( p ) δ ( Σ p ) Z + λ k d p d p 0 ϕ 12 ... d ϕ 1 0 2 ... d ϕ 1 0 2 0 ... d 0 ϕ 12 0 ... d 0 δ ( Σ p ) δ ( Σ p 0 ) δ ( p 0 1 + p 2 + · · · + p d ) δ ( p 1 + p 0 2 + · · 2 d=4 similar RG flow equations, different scaling dimensions of couplings: ��� d − 2 8 1 2 d λ k π 2 n o η 0 η 0 ⌘ + k = > > k 3 (1 + µ k ) 2 ⇣ ⇣ ⌘ > d � 2 ( d − 1) d > Γ E Γ E 2 > > 2 2 > - ��� > > > d − 2 > 1 2 d λ k π > 2 n o < η 0 − ( η 0 ⌘ + β d 6 =4 ( µ k ) = − k + 2) µ k √ k (1 + µ k ) 2 ⇣ ⇣ ⌘ d − 1 d +2 d Γ E Γ E > 2 2 > - ��� > > μ � > d − 2 2 > 2 λ 1 2 π > 2 n o > η 0 − 2 η 0 k ⌘ + β d 6 =4 ( λ k ) = k λ k + ( d − 6) λ k > > √ k (1 + µ k ) 3 > ⇣ ⇣ ⌘ d − 1 d +2 d > Γ E Γ E : 2 2 - ��� again, general features independent of rank-d: Gaussian-UV FP (asymptotic freedom), Wilson-Fisher-IR FP - ��� symmetric phase + broken or condensate phase ����� ����� ����� ����� ����� 2nd non-G IR FP at negative coupling λ �
Part IV: effective continuum physics from GFTs
Quantum spacetime: the difficult path from microstructure to cosmology the issue: identify relevant phase for effective continuum geometry extract effective continuum dynamics and relate it to GR is GR a good effective description of LQG/SF/GFT in some approximation (in one continuum phase)?
Quantum spacetime: the difficult path from microstructure to cosmology the issue: identify relevant phase for effective continuum geometry extract effective continuum dynamics and relate it to GR is GR a good effective description of LQG/SF/GFT in some approximation (in one continuum phase)? Quantum Gravity problem: identify microscopic d.o.f. of quantum spacetime and their fundamental dynamics derive e ff ective (QG-inspired) models for fundamental (quantum) cosmology: explain features of early Universe, obtain testable QG predictions various models: loop quantum cosmology, ....
Quantum spacetime: the difficult path from microstructure to cosmology the issue: identify relevant phase for effective continuum geometry extract effective continuum dynamics and relate it to GR is GR a good effective description of LQG/SF/GFT in some approximation (in one continuum phase)? Quantum Gravity problem: identify microscopic d.o.f. of quantum spacetime and their fundamental dynamics derive e ff ective (QG-inspired) models for fundamental (quantum) cosmology: explain features of early Universe, obtain testable QG predictions various models: loop quantum cosmology, .... also work by: C. Rovelli, F. Vidotto (spin foam context); E. Alesci, F. Cianfrani (canonical LQG context); …..
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