From a curved-space reconstruction theorem to a 4d Spinfoam model with a Cosmological Constant

Transcription

1 From a curved-space reconstruction theorem to a 4d Spinfoam model with a Cosmological Constant Hal Haggard Bard College Collaboration with Muxin Han, Wojciech Kamiński, and Aldo Riello July 7th, 2015 Loops 15 Erlangen, Germany math-ph/ , gr-qc/

2 Λ = l 2 P = m 2

3 The model: Z CS (S 3 ) ; Γ 5 j, i = DADĀ A,Ā] ( eics[s3 Γ 5 j, i ) A, Ā, with SL(2, C) Chern-Simons action and complex couplings (t, t) CS[S 3 A, Ā] = t tr (A da + 2 ) 8π S 3 3 A A A + ( t tr Ā dā 8π + 2 ) S 3 3Ā Ā Ā. The semiclassical limit: j, t while j/ t cnst, and arg(t) = cnst. The result: Z CS (S 3 ; Γ 5 j, i ) [ d.s.l. N + e i( t a tθ t ΛV4 Λ ) + N e i( t a tθ t ΛV4 Λ ) ].

4 One The result Two Reconstructing a curved 4-simplex Three Curved spinfoams

5 The semiclassical limit is ( ) t := 12π 1 Λl 2 γ + i P l P 0, j, with a phys γl 2 Pj = cnst l P 0 means t, which corresponds to CS classical flat limit, however j makes the Wilson graph operator stand out and act as a distributional source for (A, A), thus avoiding flatness Semiclassical limit = study of flat connections on the graph complement S 3 \ Γ

6 The graph complement S 3 \ Γ is obtained by removing a tubular neighborhood of Γ from S 3 Here and below Γ is the graph dual to the 4-simplex boundary Γ S 3 Γ tubular neighborhood of Γ Zoom

7 Planar projection of Γ The boundary of S 3 \ Γ is a genus 6 surface

8 There are two types of holonomies in S 3 \ Γ: transverse H b (a) longitudinal G ba where a, b,... label the graph vertices [connects with Bahr, Dittrich, Geiller] We need to specify the exact paths, called a choice of framing for Γ longitudinal paths run on the top of the tubes G ba 5 Top view of the tubular neighborhood

9 Equations of motion The connection on the graph complement is flat, hence holonomies along contractible paths are trivial: closures bh b (a) = 1l a parallel transports G ba H b (a)g ab = H a (b) 1 around 5 out of the 6 independent faces G ac G cb G ba = 1l a b while, around the last independent face : G 34 G 42 G 23 = H 1 (3)

10 In 3D, we defined simple paths to determine a geometrically meaningful curved Gram matrix. The geometrical dot product ˆn 1 ˆn 3 is well defined at vertex 4, but we have to rotate ˆn 4 to give ˆn 2 ˆn 4 meaning at 4. The Gram matrix is 1 ˆn 1 ˆn 2 ˆn 1 ˆn 3 ˆn 1 ˆn 4 1 ˆn Gram = 2 ˆn 3 ˆn 2 O 1ˆn 4 1 ˆn 3 ˆn 4. sym 1

11 Trivializing the G s, [ ] G ab = ga 1 g b, except G 42 = g4 1 g 3 H 31 g3 1 g 2, we have, in terms of H ab := g a H ab g 1 a, H ba = H 1 ab except H42 = H 1 13 H 1 24 H 13. Gives a complete understanding of the 4D Gram matrix and a 4-simplex reconstruction:

12 The CS phase space is P = M flat (Σ, SL(2, C)) Natural complex coordinates are obtained via a trivalent decomposition of the graph and considering: H m ( xm x 1 m ) and G m ( ym hence u m := log x m and v m := 2π log y m y 1 m ) The Atiyah-Bott symplectic structure h induces the canonical Poisson brackets 4π M TrδA δa {u m, v n } = δ m,n and {ū m, v n } = δ m,n.

13 To do WKB, we relate (u, v) to simplicial geometries The 4-simplex reconstruction theorem shows that u ab = i Λ 6 a ab + 2πin ab v ab = h 4π Θ ab + i h 4π φ ab + i h 2 m ab where n ab, m ab Z are lifting ambiguities. At 4-vertex a, (u a, v a ) encodes shape of tet a with areas {a ab } b parity related solution with: (ṽ a, ṽ ab ) = (v a, v ab )

14 The WKB approximation for simplicial geometries is Z(u, ū M ) Z α e i Λh R( 12πi )(Σa abθ ab ΛV4 Λ)+ i Λh R( 6 )Σm ab a + Z α e i the Regge action of simplicial General Relativity with a cosmological constant [Regge 1961; Barrett, Foxon 1994; Bahr, Dittrich 2010] the two branches of opposite parity ( 3d QG, mini-superspace QC) arbitrary term depending on the choice of lift v := log y + 2πim

15 One The result Two Reconstructing a 4-simplex Three Curved spinfoams

16 At the level of a single building block, the EPRL amplitude of the 3d spin-network boundary state ψ Γ is holonomy i of A B F[A] 2l Z EPRL (ψ Γ ) := DBDA e 2 (fγ ) ( ) P ψ Γ (G[A]) = fγ ψ Γ (1l) SL(2, C) spin connection dual to 4-simplex boundary Γ B is the bivector field [B = e e on geometric states] f γ is the Dupuis-Livine map, it embeds ψ into spacetime S 3 *[one dimensional lower drawing]

17 Philosophy for ΛRegge: construct a manifold out of homogeneously curved building blocks & (d 2)-dimensional defects At the quantum level, the homogenous curvature is implemented via BF Λ 6 BB dynamics, and defects are created as in the flat case Λ-GR = BF Λ 6 BB + geometricity constraints

18 For boundary connection functionals, ΛBF in the bulk is equivalent to CS on the boundary i B F[A] 2l Z(ψ Γ ) := DBDA e 2 Λ 6 B B ( ) P f γ ψ Γ (G[A]) 3i F[A] F[A] 4Λl = DA e 2 (fγ ) P ψ Γ (G[A]) = DA e 3πi Λl 2 P CS[A] ( fγ ψ Γ ) (G[A]), where the Chern-Simons functional is Baez CS[A] := 1 da A + 2 4π S 3 3 A A A

19 Twisting the previous construction by using the γ-holst action gives Z ΛEPRL (ψ Γ ) := DADA e i t 2 CS[A]+i t 2 CS[ A ] ( f γ ψ Γ ) (G[A, A]) where (A, A) are the self- and antiself-dual parts of A ( ) and t := 12π 1 Λl 2 γ + i is the complex CS level P Note Z ΛEPRL involves only quantities living on the boundary ΛEPRL = SL(2, C)-CS evaluation of a specific Wilson graph operator

20 Two immediate consequences: ( ) t := 12π 1 Λl 2 γ + i P The CS level t is complex, no (known) quantum group structure associated to the graph evaluation Fairbairn & Meusburger, Han Invariance of the amplitude under large gauge transformations A A g implies R(t) Z, i.e. 12π Λ 4πR 2 Λ γl2 PN Kodama, Randono, Smolin, Wieland

21 Three interesting limits: ( ) t := 12π 1 Λl 2 γ + i P Semiclassical ΛRegge limit: j, t while j/ t cnst, and arg(t) = cnst. Vanishing cosmological constant Λ 0: t, & CS is projected onto its classical solutions flat EPRL q-deformed Lorentzian Barrett-Crane amplitude: when γ, the EPRL graph operator Barrett & Crane s, while t becomes ir, giving q = exp ( l 2 ) P /R2 Λ Noui & Roche

22 SL(2, C) Chern-Simons theory implements BF- Λ6 BB, leads to a quantized cosmological constant, and has a rich semiclassical limit Conjecture: the curved Minkowski theorem holds in general study of flat connections on Riemann surfaces closely related to study of discrete, curved polyhedra. Provide an enriched context for understanding the role of quantum groups in cosmological spacetimes. Λ