Group field theories generating polyhedral complexes Johannes Th¨ urigen w.i.p. with D. Oriti, J. Ryan Max-Planck Institute for Gravitational Physics, Potsdam (Albert-Einstein Institute) July 17, 2014 Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 1 / 13
Dynamics of LQG: Group field theory (GFT) Standard GFT: QFT of a field on a group φ : G × D − → R � � � � � � [ d g ] φ ( g 1 ) K ( g 1 , g 2 ) φ ( g 2 ) + [ d g ] V i { g e } E i S [ φ ] = λ i φ ( g e ) , i ∈ I e ∈ E i Perturbative expansion of state sum = sum over spin foams � � � ( − λ i ) V i � � 1 D φ e − S [ φ ] = Z GFT = A (Γ) C (Γ) Γ i ∈ I Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 2 / 13
GFT for arbitrary graphs? Apparent difference between canonical and covariant approaches to LQG: States in canonical LQG labeled by graphs of arbitrary valence Covariant theories developed in simplicial setting Generalization of spin foams to arbitrary valence exists [KKL ’10] Slightly more challenging for GFT Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 3 / 13
GFT for arbitrary graphs? Apparent difference between canonical and covariant approaches to LQG: States in canonical LQG labeled by graphs of arbitrary valence Covariant theories developed in simplicial setting Generalization of spin foams to arbitrary valence exists [KKL ’10] Slightly more challenging for GFT Goal here: Define the general class of combinatorial complexes relevant for LQG and Spin Foams Discuss equivalent diagrammatic representations for them Present two types of GFTs generating those in the perturbative sum Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 3 / 13
GFT for arbitrary graphs? Apparent difference between canonical and covariant approaches to LQG: States in canonical LQG labeled by graphs of arbitrary valence Covariant theories developed in simplicial setting Generalization of spin foams to arbitrary valence exists [KKL ’10] Slightly more challenging for GFT Goal here: Define the general class of combinatorial complexes relevant for LQG and Spin Foams Discuss equivalent diagrammatic representations for them Present two types of GFTs generating those in the perturbative sum Caveat: Focus here on (dual) polyhedral 2-complexes (just polygons) Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 3 / 13
Mathematical background: Abstract polyhedral complexes ”Combinatorial complexes = complexes of abstract polytopes” An abstract n -polytope is a poset P [McMullen,Schulte ’02] Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 4 / 13
Mathematical background: Abstract polyhedral complexes ”Combinatorial complexes = complexes of abstract polytopes” An abstract n -polytope is a poset P [McMullen,Schulte ’02] (P1) P contains a least and greatest face f − 1 , f n . Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 4 / 13
Mathematical background: Abstract polyhedral complexes ”Combinatorial complexes = complexes of abstract polytopes” An abstract n -polytope is a poset P [McMullen,Schulte ’02] 12345 (P1) P contains a least and greatest face f − 1 , f n . 1234 125 235 345 415 (P2) Each maximal chain has length n + 1 12 23 34 41 15 25 35 45 1 2 3 4 5 ∅ Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 4 / 13
Mathematical background: Abstract polyhedral complexes ”Combinatorial complexes = complexes of abstract polytopes” An abstract n -polytope is a poset P [McMullen,Schulte ’02] 12345 (P1) P contains a least and greatest face f − 1 , f n . 1234 125 235 345 415 (P2) Each maximal chain has length n + 1 12 23 34 41 15 25 35 45 (P3) P is strongly connected (sequence of faces) 1 2 3 4 5 ∅ Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 4 / 13
Mathematical background: Abstract polyhedral complexes ”Combinatorial complexes = complexes of abstract polytopes” An abstract n -polytope is a poset P [McMullen,Schulte ’02] (P1) P contains a least and greatest face f − 1 , f n . (P2) Each maximal chain has length n + 1 12345 (P3) P is strongly connected (sequence of faces) 1234 125 235 345 415 (P4) Homogeneity degrees k i = 2, i = 1 , .., n − 1 12 23 34 41 15 25 35 45 (for f < g of dimension i -1 and i +1 there are exactly 1 2 3 4 5 k i i -faces h in P such that f < h < g ) ∅ Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 4 / 13
Mathematical background: Abstract polyhedral complexes ”Combinatorial complexes = complexes of abstract polytopes” An abstract n -polytope is a poset P [McMullen,Schulte ’02] (P1) P contains a least and greatest face f − 1 , f n . (P2) Each maximal chain has length n + 1 12345 (P3) P is strongly connected (sequence of faces) 1234 125 235 345 415 (P4) Homogeneity degrees k i = 2, i = 1 , .., n − 1 12 23 34 41 15 25 35 45 (for f < g of dimension i -1 and i +1 there are exactly 1 2 3 4 5 k i i -faces h in P such that f < h < g ) ∅ Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 4 / 13
Mathematical background: Abstract polyhedral complexes ”Combinatorial complexes = complexes of abstract polytopes” 12345 An abstract n -polytope is a poset P [McMullen,Schulte ’02] 1234 125 235 345 415 (P1) P contains a least and greatest face f − 1 , f n . 12 23 34 41 15 25 35 45 (P2) Each maximal chain has length n + 1 1 2 3 4 5 (P3) P is strongly connected (sequence of faces) ∅ (P4) Homogeneity degrees k i = 2, i = 1 , .., n − 1 5 (for f < g of dimension i -1 and i +1 there are exactly k i i -faces h in P such that f < h < g ) 4 3 1 2 Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 4 / 13
Mathematical background: Abstract polyhedral complexes ”Combinatorial complexes = complexes of abstract polytopes” 12345 An abstract n -polytope is a poset P [McMullen,Schulte ’02] 1234 125 235 345 415 (P1) P contains a least and greatest face f − 1 , f n . 12 23 34 41 15 25 35 45 (P2) Each maximal chain has length n + 1 1 2 3 4 5 (P3) P is strongly connected (sequence of faces) ∅ (P4) Homogeneity degrees k i = 2, i = 1 , .., n − 1 5 (for f < g of dimension i -1 and i +1 there are exactly k i i -faces h in P such that f < h < g ) 4 3 more general than piecewise linear (e.g. 11-cell) 1 2 Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 4 / 13
Mathematical background: Abstract polyhedral complexes ”Combinatorial complexes = complexes of abstract polytopes” 12345 An abstract n -polytope is a poset P [McMullen,Schulte ’02] 1234 125 235 345 415 (P1) P contains a least and greatest face f − 1 , f n . 12 23 34 41 15 25 35 45 (P2) Each maximal chain has length n + 1 1 2 3 4 5 (P3) P is strongly connected (sequence of faces) ∅ (P4) Homogeneity degrees k i = 2, i = 1 , .., n − 1 5 (for f < g of dimension i -1 and i +1 there are exactly k i i -faces h in P such that f < h < g ) 4 3 more general than piecewise linear (e.g. 11-cell) dual polytope: invert partial order 1 2 Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 4 / 13
Mathematical background: Abstract polyhedral complexes ”Combinatorial complexes = complexes of abstract polytopes” An abstract n -polytope is a poset P ∅ [McMullen,Schulte ’02] 5 4 3 2 1 (P1) P contains a least and greatest face f − 1 , f n . 45 35 25 15 41 34 23 12 (P2) Each maximal chain has length n + 1 415 345 235 125 1234 (P3) P is strongly connected (sequence of faces) 12345 (P4) Homogeneity degrees k i = 2, i = 1 , .., n − 1 125 235 (for f < g of dimension i -1 and i +1 there are exactly k i i -faces h in P such that f < h < g ) 345 415 more general than piecewise linear (e.g. 11-cell) dual polytope: invert partial order 1234 Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 4 / 13
Mathematical background: Abstract polyhedral complexes ”Combinatorial complexes = complexes of abstract polytopes” An abstract n -polytope is a poset P ∅ [McMullen,Schulte ’02] 5 4 3 2 1 (P1) P contains a least and greatest face f − 1 , f n . 45 35 25 15 41 34 23 12 (P2) Each maximal chain has length n + 1 415 345 235 125 1234 (P3) P is strongly connected (sequence of faces) 12345 (P4) Homogeneity degrees k i = 2, i = 1 , .., n − 1 125 235 (for f < g of dimension i -1 and i +1 there are exactly k i i -faces h in P such that f < h < g ) 345 415 more general than piecewise linear (e.g. 11-cell) dual polytope: invert partial order boundary is a closed manifold → generalize! 1234 Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 4 / 13
Mathematical background: Abstract polyhedral complexes ”Combinatorial complexes = complexes of abstract polytopes” ∅ An abstract polyhedral n -complex is a poset P 5 4 3 2 1 (P1) P contains least and greatest face f − 1 , f n +1 . 45 35 25 15 41 34 23 12 415 345 235 125 1234 (P2) Each maximal chain has length n + 2 12345 (P3) ( P is strongly connected) 125 235 (P4’) Homogeneity degrees k i = 2, i = 1 , .., n − 1 345 more general than piecewise linear (e.g. 11-cell) 415 dual polytope branching possible! 1234 Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 4 / 13
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