Covariant loop quantum gravity as a topological field theory with defects

Transcription

1 Covariant loop quantum gravity as a topological field theory with defects Wolfgang Wieland Perimeter Institute for Theoretical Physics, Waterloo (Ontario) ILQGS 5 April 2016

2 Basic idea There are hints that LQG can be understood as a topological field theory with defects. Which kind of defects: Curvature defects? Area defects? Defects that carry torsion as well? Goal of this talk: Study a class of models for discretised gravity in terms of connection variables, where all gravitational degrees of freedom are carried by a finite number of topological defects. 2 / 27

3 Hints from Loop Quantum Gravity LQG à la Aharonov Bohm [Bianchi], [Freidel, Geiller, Ziprick], [ww] Three-manifold Σ Γ = Σ Γ 1 with defects on one-skeleton of cellular decomposition of Σ. dµ AL(A)Ψ Γ[A]Φ Γ[A] = A Σ A Σ Γ x Σ Γ [ ] da( x) δ ( F [A] ) δ ( Ψ[A] ) Ψ FP[A] Ψ Γ[A]Φ Γ[A]. Area defects in LQG Electric field around a link γ dual to a face f Ẽi a ( x) = βl 2 P j f (j f + 1)n f i dt δ ( x, X(t) ) X a (t). t DG BF distribution [Geiller, Dittrich] Distributional states as excitations over F = 0. Isolated Horizons Spinnetwork functions create punctures on an isolated horizon, which is a null surface of topology S 2 R. The spin network intertwiners are blown up and turn into punctured two-spheres. *M Geiller and B Dittrich, A new vacuum for Loop Quantum Gravity Class. Quantum Grav. 32 (2015), arxiv: *E Bianchi, Loop Quantum Gravity a la Aharonov-Bohm, Gen. Rel. Grav. 46 (2014), arxiv: *ww, One-dimensional action for simplicial gravity in three dimensions, Phys. Rev. D 90 (2014), arxiv: *L Freidel, M Geiller, J Ziprick, Continuous formulation of the Loop Quantum Gravity phase space, Class. Quant. Grav. 30 (2013), arxiv: γ 3 / 27

4 LQG as a TFT with defects which defects? Roughly speaking, two different strategies in the literature (i) covariant theory with spacetime defects Start from cellular (simplicial) decomposition of spacetime M (discretization), remove the two-dimensional faces (triangles) and define M := M 2. The curvature vanishes in M, classical configuration space: moduli space Hom(π 1(M :G))/G of flat connections. Idea: Identify the non-contractible cycles in M with worldlines of LQG area defects. Write down the simplest action and study the coupled system: area defects (auxiliary particles) coupled to a topological field theory. (ii) Canonical theory with defects on spatial slices [Dittrich s program] Similar idea, but one dimension lower. The Dittrich Geiller BF distribution on a spatial slice Σ is seen as the unique ground state of a certain Hamiltonian. Excitations over this ground state are topological defects and carry curvature and torsion. Challenge: Find the correct gapped Hamiltonian, which should have a vastly degenerate vacuum containing all physical states of quantum gravity. 4 / 27

5 Outline of the talk Main message: BF action with simplicity constraints can be written as a wordline integral for area defects. These defects carry charges for curvature and torsion that can be naturally coupled to a topological field theory. Table of contents 1 Worldline action for LQG area defects 2 Coupling to a three-dimensional field theory on the horizon 3 Integrability conditions and the value of the cosmological constant 4 Outlook and conclusion 5 / 27

6 Worldline action for LQG area defects

7 Plebański principle The BF action is topological, and determines the symplectic structure of the theory: 1 ( S BF[Σ, A] = Σ αβ 1 αβ) M 16πG β Σ F αβ [A]. (1) }{{} Π αβ General relativity follows from the simplicity constraints added to the action: Σ αβ Σ µν ɛ αβµν. (2) With the solutions: Σ αβ = { ±e α e β, ± (e α e β ). (3) Notation: α, β, γ... are internal Lorentz indices. Σ α β is an so(1, 3)-valued two-form. A α β is an SO(1, 3) connection, with F α β = daα β + Aα µ Aµ β denoting its curvature. e α is the tetrad, diagonalizing the four-dimensional metric g = η αβ e α e β. 7 / 27

8 Simplicial discretization 8 / 27

9 Spinorial action for discretised BF theory Discretise S BF[Π, A] = M Π αβ F αβ as a sum over faces f. Worldline action for discretised BF S BF discrete [Z, A, a, b] = f:faces [ f ] ( ) π A D a ω A + b da + cc. (4) f Geometric interpretation: the spinors Z = ( π, Ā ωa ) diagonalise the simplicial fluxes. Dπ A = dπ A + A A Bπ B is the selfdual Ashtekar conenction. a is a C 0 connection from the ( π, Ā ωa ) (e z π, Ā ez ω A ) gauge symmetry. the Lagrange multiplier b imposes that a is flat in f, da = 0 *L Freidel and S Speziale, From twistors to twisted geometries, Phys. Rev. D 82 (2010), arxiv: *S Speziale and ww, Twistorial structure of loop-gravity transition amplitudes, Phys. Rev. D 86 (2012), arxiv: *ww, Hamiltonian spinfoam gravity, Class.Quant.Grav. 31 (2014), arxiv: *ww, New action for simplicial gravity in four dimensions, Class. Quant. Grav. 32 (2015), arxiv: *L Freidel, M Geiller, J Ziprick, Continuous formulation of the Loop Quantum Gravity phase space, Class. Quant. Grav. 30 (2013), arxiv: / 27

10 Geometric interpretation of the spinors SO(1, 3) BF-action S BF[Π, A] = SO(1, 3) fluxes over triangles f dual to the faces [Π f ] αβ [ = hf [A] ] α [ hf [A] ] β µ f M Π αβ F αβ (5) Spinors are eigenvectors of the selfdual component Π AB f ν Πµν (6) of Π αβ f. Π AB := 1 4 σa C[α σ CBβ]Π αβ = 1 2 ω(a π B) (7) C {0} gauge symmetry (requires gauging D (D a) of covariant derivative. (π A, ω A ) (e z π A, e z ω A ) (8) 9 / 27

11 Linear simplicity constraints Instead of discretizing the quadratic simplicity constraints we will use the linear simplicity constraints: Σ αβ Σ µν ɛ αβµν, (9) For a tetrahedron T e (dual to an edge e) there exist an internal future-oriented four-vector n α e such that the fluxes through the four bounding triangles f (dual to a face f: e f) annihilate n α e : f Σ αβ n β e = 0. (10) The spinorial parametrization turns the simplicity constraints into the following complex conditions: V f = i β + i πf A ωa f + cc.! = 0, (11a) W ef = n AĀ e π f A ωf! = 0. (11b) Ā 10 / 27

12 What is the geometric interpretation of n AĀ? We interpret the normal n AĀ = n α as the pull back of a one-form e α a to the boundary of f. n AĀ = e AĀat a. We add the constraints and arrive at the following action: [ ] S f [Z ζ, ϕ A, e, a, b] = π A(D a)ω A ζe A Āπ A ωā + cc. + f [ ( )] + b da + b i dā ϕ f β + i b + cc.. 11 / 27

13 What is the geometric interpretation of n AĀ? We interpret the normal n AĀ = n α as the pull back of a tetrad e α a to the boundary of f. n AĀ = e AĀat a. We add the constraints and arrive at the following action: [ ] S f [Z ζ, ϕ A, e, a, b] = π A(D a)ω A ζe A Āπ A ωā + cc. + discretised BF f [ ( )] + b da + b i dā ϕ f β + i b + cc.. 12 / 27

14 What is the geometric interpretation of n AĀ? We interpret the normal n AĀ = n α as the pull back of a tetrad e α a to the boundary of f. n AĀ = e AĀat a. We add the constraints and arrive at the following action: [ ] S f [Z ζ, ϕ A, e, a, b] = π A(D a)ω A ζe A Āπ A ωā + cc. + simplicity constraints f [ ( )] + b da + b i dā ϕ f β + i b + cc.. 13 / 27

15 What is the geometric interpretation of n AĀ? We interpret the normal n AĀ = n α as the pull back of a tetrad e α a to the boundary of f. n AĀ = e AĀat a. We add the constraints and arrive at the following action: [ ] S f [Z ζ, ϕ A, e, a, b] = π A(D a)ω A ζe A Āπ A ωā + cc. + Lagrange multipliers f [ ( )] + b da + b i dā ϕ f β + i b + cc.. 14 / 27

16 Coupling to a three-dimensional field theory on the horizon

17 Background fields in the action The action depends on background fields. [ ] S f [Z ζ, ϕ A, e, a, b] = π A(D a)ω A ζe A Āπ A ωā + cc. + f [ ( )] + b da + b i dā ϕ f β + i b + cc.. It seems natural to turn the one-forms A A Ba and e AĀa into dynamical fields as well. We propose an additional field theory for A A Ba and e AĀa. 16 / 27

18 Smearing out the discretisation f M e f S 2 t f S 2 link Smear out the system of edges e of the discretisation and replace them by a three-manifold M. Similar topological structure appears in the isolated horizon framework: Punctured two-spheres connected by spin network links Advantage: Removal of point-like interactions spinfoam vertices. 17 / 27

19 Simplest action on M : Chern Simons action for de-sitter connection Propose the following action for a topological field theory on M S M [A, e] = α [ A α β da β α + 2 ] 2 3 Aα µ A µ ν A ν α λ e α De α. M Equations of motion vanishing of torsion: De α = 0 (12a) local de Sitter curvature: F α β λ α eα e β = 0 (12b) M Comments: cosmological constant Λ = 3λ/α on M. S M is an SO(1, 4) resp. SO(2, 3) Chern Simons action depending on the sign Λ > 0 resp. Λ < 0 of the cosmological constant. Local SO(p, q) gauge symmetries are on-shell equivalent to diffeomorphisms Lorentz transformations. The limit α, ζ, ϕ, β 0 yields a recent model of Perez and Freidel. *L Freidel and A Perez, Quantum gravity at the corner (2015), arxiv: *W Donnelly and L Freidel, Local subsystems in gauge theory and gravity (2015), arxiv: *ww, Complex Ashtekar variables, the Kodama state and spinfoam gravity (2011), arxiv: *H Haggard, M Han, W Kaminski, A Riello, Four-dimensional Quantum Gravity with a Cosmological Constant from Three-dimensional Holomorphic Blocks, Phys Lett. B 752 (2016), arxiv: / 27

20 Integrability conditions and the value of the cosmological constant

21 Equations of motion The N defects on S 2 are sources for curvature and torsion T AĀ = De AĀ = + 1 2λ F AB Λ 6 ea C e B C = 1 2α N ( ζ i πi A ωāi cc. ) δ( xi), (13a) i=1 N π (A i ω B) i i=1 δ( x i). The spinors have a non-trivial dynamics along their trajectory (13b) with Hamiltonian D dt ωa = { t H, ω A}, D dt πa = { t H, π A}. (14) and Poisson brackets t H = t b a b π Aω A + t b e AĀb ζπ A ω Ā + cc. (15) { πa, ω B} = δ B A, { πā, ω B } = δ BĀ. (16) 20 / 27

22 Integrability conditions Bianchi identities DF αβ = 0, DT α = D 2 e α = F α β e β. (17) Imply integrability conditions (secondary constraints) Definite sign of Λ d ζ = t (a ā)ζ, dt (18a) Λ = 6ζ ζ < 0. (18b) 21 / 27

23 Deformed Gauss law The N defects on S 2 are sources for curvature and torsion D T AĀ = De AĀ = F AB Λ 6 ea C e B C = N i=1 N i=1 Ji AĀ δ( x i), Ji AB δ( x i). SO(1, 4) De Sitter connection A IJ a = 1 2 Aµν am IJ µν +e µ ap IJ µ, [ Mαβ, M µν ] = +4δ ρ [α δρ β] η ρ σ δσ [µ δ σ ν]m ρσ, [ Pµ, P ν ] = Λ 3 Mµν, [ Mµν, P α ] = 2ηα[µ P ν]. 22 / 27

24 Deformed Gauss law The N defects on S 2 are sources for curvature and torsion D T AĀ = De AĀ = F AB Λ 6 ea C e B C = N i=1 N i=1 Ji AĀ δ( x i), Ji AB δ( x i). Deformed Gauss law from nonabelian Stokes theorem ( 1 = Pexp 1 ( A αβ M αβ + 2e α ) ) P α = 2 D ( = PPexp 1 ( F αβ M αβ + 2T α ) ) P α = 2 D N ( = P exp 1 2 J αβ i M αβ Ji α P α ). (21) i=1 23 / 27

25 Conclusion

26 Spin-foam amplitudes = TFT with defects four-dimensional amplitudes through three-dimensional path integral Z = 3-geometries M D[A, e]e is M [A,e] D[Z ζ, ϕ a, b] f:faces e is f [Z ζ,ϕ A,e,a,b]. M f SO(1,4) Chern Simons action 25 / 27

27 Spin-foam amplitudes = TFT with defects four-dimensional amplitudes through three-dimensional path integral Z = 3-geometries M D[A, e]e is M [A,e] D[Z ζ, ϕ a, b] f:faces e is f [Z ζ,ϕ A,e,a,b]. M f insertion of charges along worldlines γ i = f i 26 / 27

28 Concluding remarks The discretised BF action can be written as a worldline action for spinors propagating along the edges of the discretisation. Gravity is a constrained topological theory (BF+simplicity) we assume the same holds for the discretised theory and add the discretised simplicity constraints to the worldline model. We proposed to blew up the one-dimensional edges and replaced them by an oriented three-manifold M. This has a number of advantages: (i) removal of point-like interactions (spinfoam vertices), (ii) same topological structure as in isolated horizon framework, (iii) the spinors couple naturally to a three-dimensional field theory on M. The simplest field theory that we can build out of the background fields e α and A α β on M is an G = SO(p, q) Chern Simons theory for the de Sitter group. Integrability conditions fix the gauge group to SO(2, 3) rather than SO(1, 4). AdS/CFT? 27 / 27