A sky without qualities

Transcription

1 A sky without qualities New boundaries for SL(2)xSL(2) Chern-Simons theory Bo Sundborg, work with Luis Apolo Stockholm university, Department of Physics and the Oskar Klein Centre August 27, 2015 B Sundborg (Stockholm university) A sky without qualities August 27, / 32

2 The sky without qualities 1 1 Hubble Extreme Deep Field (full resolution) by NASA; ESA;... B Sundborg (Stockholm university) A sky without qualities August 27, / 32

3 The sky without qualities Formalismen är alltid smartare än du. B Sundborg (Stockholm university) A sky without qualities August 27, / 32

4 The sky without qualities The formalism is always smarter than you. B Sundborg (Stockholm university) A sky without qualities August 27, / 32

5 Outline The sky without qualities Introduction Diffeomorphism invariant boundaries Background News on the Chern-Simons formulation News on gravity in Asymptotia Outroduction B Sundborg (Stockholm university) A sky without qualities August 27, / 32

6 Introduction Introduction I Chance for realistic solvable quantum gravity: Black holes with varying horizon size/entropy, BTZ Essentially a topological theory, since no local dofs Chance to study details important to higher spin theory SL(2) as special case of SL(N) Conical singularities, black holes, gauge invariance vs diffeomorphisms B Sundborg (Stockholm university) A sky without qualities August 27, / 32

7 Introduction Introduction II But hopes have dwindled: No (consensus) solution Can we expect to understand SL(N)? (Higher spin.) if we don t understand SL(2)? B Sundborg (Stockholm university) A sky without qualities August 27, / 32

8 Introduction Introduction III Therefore Re-examine the foundations Clue: The boundary term is where the action is Adapt to higher spin generalisation Maintain contact with standard gravity B Sundborg (Stockholm university) A sky without qualities August 27, / 32

9 Background Background General background Boundaries are important in low dimensions BTZ black holes Asymptotia Chern-Simons background Basics Double gauge groups G G Off shell boundary theory B Sundborg (Stockholm university) A sky without qualities August 27, / 32

10 Background General background Boundaries are important in low dimensions Boundaries become more important when massless fields are present. Long range effects of massless fields are more important in low dimensions. Care is required whenever there are potential IR divergences. B Sundborg (Stockholm university) A sky without qualities August 27, / 32

11 Background General background BTZ black holes In 2+1 dimensions with negative cosmological constant there are famous BTZ black holes: ds 2 = sinh 2 ρ[r + dt r dφ] 2 + dρ 2 + cosh 2 ρ[r dt r + dφ] 2 r + is the outer horizon radius and r is the inner horizon radius. B Sundborg (Stockholm university) A sky without qualities August 27, / 32

12 Background General background BTZ black holes In 2+1 dimensions with negative cosmological constant there are famous BTZ black holes: ds 2 = sinh 2 ρ[r + dt r dφ] 2 + dρ 2 + cosh 2 ρ[r dt r + dφ] 2 r + is the outer horizon radius and r is the inner horizon radius. Note At ρ = 0 there is a serious coordinate singularity joining two exterior solutions at their outer horizon. B Sundborg (Stockholm university) A sky without qualities August 27, / 32

13 Background General background Asymptotia One way to define gravity for non-compact spacetimes is by asymptotic conditions g µν = r 2 h (0) µν + O(r 0 ), g rµ = O(r 3 ), g rr = r 2 + O(r 4 ), where r is a radial coordinate. For a fixed h (0) µν the above conditions are standard Brown-Henneaux boundary conditions. If instead h (0) µν is allowed to vary they represent free boundary conditions. And the asymptotic symmetries are different. B Sundborg (Stockholm university) A sky without qualities August 27, / 32

14 Background General background Asymptotia One way to define gravity for non-compact spacetimes is by asymptotic conditions g µν = r 2 h (0) µν + O(r 0 ), g rµ = O(r 3 ), g rr = r 2 + O(r 4 ), where r is a radial coordinate. For a fixed h (0) µν the above conditions are standard Brown-Henneaux boundary conditions. If instead h (0) µν is allowed to vary they represent free boundary conditions. And the asymptotic symmetries are different. Free boundary are only compatible with a well-defined action principle if the Brown-York energy momentum tensor vanishes on the boundary: 0 = Tµν BY 2 h (0) where K µν is the extrinsic curvature. δs δh = 1 ( ) Kµν Kh (0)µν µν + h µν, κ B Sundborg (Stockholm university) A sky without qualities August 27, / 32

15 Background Chern-Simons background: Basics Achúcarro-Townsend-Witten gauge theory description A composite metric Diffeomorphisms and gauge transformations Role of degenerate metrics Are there solutions with degenerate metrics? Typical BTZ! B Sundborg (Stockholm university) A sky without qualities August 27, / 32

16 Background Chern-Simons background: Basics Achúcarro-Townsend-Witten gauge theory description Fields A a = (w a + e a ), Ā a = (w a e a ), Composite metric g µν = 1 4 tr { (A Ā) µ(a Ā) } ν There is nothing in gauge theory that forbids locally degenerate metrics. Degeneration locus typically codimension 1. Action I CS;bulk [A, Ā] = I CS[A] I CS [Ā] I CS [A] = tr (A da + 2 ) 3 A A A. Σ B Sundborg (Stockholm university) A sky without qualities August 27, / 32

17 Background Chern-Simons background: Basics Diffeomorphisms and gauge transformations Diffeomorphism are equivalent on shell to special gauge transformations δa = Du, δā = Du with parameter u = ρ a P a = v µ e a µp a where P a generates AdS translations and v µ parametrizes diffeomorphisms. Degeneration The map between gauge transformations and diffeomorphisms degenerate iff the metric degenerates. B Sundborg (Stockholm university) A sky without qualities August 27, / 32

18 Background Chern-Simons background: Basics Are there solutions with degenerate metrics? A = 0 = Ā, but a bit trivial, and may not agree with boundary conditions... BTZ black hole metric in the form ds 2 = sinh 2 ρ[r + dt r dφ] 2 + cosh 2 ρ[r dt r + dφ] 2 degenerates at the outer horizon joining two exterior solutions. B Sundborg (Stockholm university) A sky without qualities August 27, / 32

19 Background Chern-Simons background: Basics Are there solutions with degenerate metrics? A = 0 = Ā, but a bit trivial, and may not agree with boundary conditions... BTZ black hole metric in the form ds 2 = sinh 2 ρ[r + dt r dφ] 2 + cosh 2 ρ[r dt r + dφ] 2 degenerates at the outer horizon joining two exterior solutions. But the gauge potential A 0+ = 1 l (r + r ) sinhρ ( dt l ± dφ) A 1± = 1 l (r + r ) coshρ ( dt l ± dφ) A 2± = ±dρ is perfectly regular. B Sundborg (Stockholm university) A sky without qualities August 27, / 32

20 Background Chern-Simons background: Basics Are there solutions with degenerate metrics? A = 0 = Ā, but a bit trivial, and may not agree with boundary conditions... BTZ black hole metric in the form ds 2 = sinh 2 ρ[r + dt r dφ] 2 + cosh 2 ρ[r dt r + dφ] 2 degenerates at the outer horizon joining two exterior solutions. But the gauge potential A 0+ = 1 l (r + r ) sinhρ ( dt l ± dφ) A 1± = 1 l (r + r ) coshρ ( dt l ± dφ) A 2± = ±dρ is perfectly regular. The Chern-Simons description is more different than a different formulation. B Sundborg (Stockholm university) A sky without qualities August 27, / 32

21 Background Chern-Simons background: Double gauge groups G G Chern-Simons Wess-Zumino-Witten CFT (WZW) Arcioni-Blau-O Loughlin2003 off shell derivation: double gauge groups Leads to off shell non-chiral G WZW on boundary! (On shell in bulk) This is our intent! (On shell bulk relation to ordinary gravity, want off shell boundary relation for path integral.) B Sundborg (Stockholm university) A sky without qualities August 27, / 32

22 Background Chern-Simons background: Double gauge groups G G Chern-Simons Wess-Zumino-Witten CFT (WZW) I Action with boundary terms g (R + 2) = ICS [A] I CS [Ā] tr ( A Ā), Σ Σ The equations of motion 0 = F = da + A A, 0 = F = dā + Ā Ā, imply that all solutions are locally of the form, A = g 1 dg, Ā = ḡ 1 dḡ, Locally pure gauge, but not globally due to topology and fall off conditions. B Sundborg (Stockholm university) A sky without qualities August 27, / 32

23 Background Chern-Simons background: Double gauge groups G G Chern-Simons Wess-Zumino-Witten CFT (WZW) II Only boundary degrees of freedom. Their action: Non-chiral WZW action S W ZW = k { 1 η µν tr [ (G 1 µ G)(G 1 ν G) ] + 1 tr ( G 1 dg ) } 3, 4π 2 Σ 3 Σ with G = gḡ 1. Remarks Many different derivations, typically using boundary eoms (notably Coussaert-Henneaux-Van Driel) for combining two chiral WZW into one non-chiral Involve specification of boundary terms, typically breaking boundary diffeomorphisms (η µν ) B Sundborg (Stockholm university) A sky without qualities August 27, / 32

24 Background Chern-Simons background: Double gauge groups G G Arcioni-Blau-O Loughlin2003 off shell derivation I Special boundary terms and Polyakov-Wiegmann identities lead to Non-chiral WZW action S W ZW = k { 1 η µν tr [ (G 1 µ G)(G 1 ν G) ] + 1 tr ( G 1 dg ) } 3, 4π 2 Σ 3 Σ with G = gḡ 1 Remarks Their derivation only uses bulk equations of motion, not boundary (and they give a completely off-shell version). Again involves boundary terms, breaking boundary diffeomorphisms (η µν ) Generalized derivation to follow, for Arcioni et al just replace γ γ µν η µν B Sundborg (Stockholm university) A sky without qualities August 27, / 32

25 Background Chern-Simons background: Double gauge groups G G Arcioni-Blau-O Loughlin 2003 off shell derivation II Given solutions to the Chern-Simons equations A = g 1 dg, Ā = ḡ 1 dḡ, using a Wess-Zumino-Witten action W [g] = k { 1 γ γ µν ( g 1 µ g ) ( g 1 ν g ) + 1 ( g 1 dg ) } 3, 4π 2 Σ 3 Σ and the notation Γ µν ± = ɛ µν ± γ γ µν, the gravitational action with appropriate boundary terms becomes S W ZW = W [g] + W [ḡ 1 ] k Γ µν ( g 1 µ g ) ( ḡ 1 ν ḡ ) = W [gḡ 1 ]. 4π Σ where the final equality is a Polyakov-Wiegmann identity. B Sundborg (Stockholm university) A sky without qualities August 27, / 32

26 Background Chern-Simons background: Double gauge groups G G This is our intent This is our intent Metric gravity Chern-Simons formulation is an on shell relation To start with a bulk off shell formulation is not needed Physical degrees of freedom in the boundary: for the path integral we want an off shell boundary theory B Sundborg (Stockholm university) A sky without qualities August 27, / 32

27 News on the Chern-Simons formulation News on the Chern-Simons formulation No prior boundary geometry The boundary term Conditions on the boundary Other boundary conditions Consequences Virasoro constraints A string interpretation of 3d Gravity B Sundborg (Stockholm university) A sky without qualities August 27, / 32

28 News on the Chern-Simons formulation No prior boundary geometry: the boundary term Write the Cherns-Simons formulation with appropriate boundary terms, which do not specify a particular boundary geometry: S CS = k { ICS [A] I CS 4π [Ā] + J α,β[a, Ā]}, where k = 1/4G N and the boundary term J α,β [A, Ā] reads J α,β [A, Ā] = (2α 1) Arcioni-Blau-O Loughlin: Σ tr(a Ā) ± β 2 α = 0 to get local Lorentz invariance at the boundary. Σ [ (A γ γ µν tr Ā ) ( µ A Ā ) ν]. Also β = 1 2α = 1 for a regular action in the metric formalism. γ µν = η µν. B Sundborg (Stockholm university) A sky without qualities August 27, / 32

29 News on the Chern-Simons formulation No prior boundary geometry: conditions on the boundary We let γ µν be arbitrary. No prior geometry. This differs from previous approaches. Term in action: Σ γ γ µν tr [ ( A Ā) µ ( A Ā ) ν ] Integrate out γ µν This will favour no geometry. And give constraints. Note on Weyl invariance Cf the string action: same dependence on 2d metric and only two Virasoro constraints due to Weyl invariance (γ µν e φ γ µν ). B Sundborg (Stockholm university) A sky without qualities August 27, / 32

30 News on the Chern-Simons formulation No prior boundary geometry: other boundary conditions Gauge fix γ µν η µν, return to diffeomorphism constraints later! Vary action: δs CS = k { tr ( δa F + δā 2π F ) 2 tr ( e δa + e + δā ) }, Σ Σ A well-defined action principle is obtained for δa + = δā = 0, which together with factorisation G = gḡ 1 leads to the boundary conditions Boundary conditions A + = Ā = 0. B Sundborg (Stockholm university) A sky without qualities August 27, / 32

31 News on the Chern-Simons formulation Consequence: Virasoro conditions Given the A + = Ā = 0 conditions and the constraints Choosing the γ µν = η µν gauge, 0 = δ γ γ µν tr [ ( A δγ Ā) ( µ A Ā ) ] ν αβ Σ 0 = tr [ ( A Ā) ( A Ā ) ] = tr [ A A ], 0 = tr [Ā+ Ā + ] and inserting the solutions A = g 1 dg, Ā = ḡ 1 dḡ 0 = tr [ g 1 gg 1 g ] = tr [ gg 1 gg 1] = tr [ J J ], 0 = tr [ J+ J+ ], where J and J + are chiral conserved SL(2) currents. We recognise the look of Virasoro operators in a WZW CFT. Here they form Virasoro constraints. B Sundborg (Stockholm university) A sky without qualities August 27, / 32

32 News on the Chern-Simons formulation Consequence: A string interpretation of 3d gravity What is string theory? A conformal field theory Virasoro constraints c = c crit What is gravity? A boundary CFT Virasoro constraints c = c W ZW B Sundborg (Stockholm university) A sky without qualities August 27, / 32

33 News on the Chern-Simons formulation Consequence: A string interpretation of 3d gravity What is string theory? A conformal field theory Virasoro constraints c = c crit What is gravity? A boundary CFT Virasoro constraints c = c W ZW Target space AdS 3 Antisymmetric tensor Boundary action Chern-Simons invariant B Sundborg (Stockholm university) A sky without qualities August 27, / 32

34 News on the Chern-Simons formulation Consequence: A string interpretation of 3d gravity What is string theory? A conformal field theory Virasoro constraints c = c crit What is gravity? A boundary CFT Virasoro constraints c = c W ZW Target space AdS 3 Antisymmetric tensor Boundary action Chern-Simons invariant Multiple strings? String interactions? Multiple boundaries? Boundary topology change? B Sundborg (Stockholm university) A sky without qualities August 27, / 32

35 News on the Chern-Simons formulation Consequence: A string interpretation of 3d gravity What is string theory? A conformal field theory Virasoro constraints c = c crit What is gravity? A boundary CFT Virasoro constraints c = c W ZW Target space AdS 3 Antisymmetric tensor Boundary action Chern-Simons invariant Multiple strings? String interactions? Multiple boundaries? Boundary topology change? B Sundborg (Stockholm university) A sky without qualities August 27, / 32

36 News on gravity in Asymptotia News on gravity in Asymptotia Metric perspective on boundaries without prior geometry Free boundary conditions Why does it work? Metric interpretation of diffeomorphism invariance and Virasoro constraints B Sundborg (Stockholm university) A sky without qualities August 27, / 32

37 News on gravity in Asymptotia Metric perspective on boundaries without prior geometry Free boundary conditions Q: Guess the boundary conditions which do not specify a boundary geometry! A: Free boundary conditions do not fix a geometry. B Sundborg (Stockholm university) A sky without qualities August 27, / 32

38 News on gravity in Asymptotia Metric perspective on boundaries without prior geometry Free boundary conditions But Q: Guess the boundary conditions which do not specify a boundary geometry! A: Free boundary conditions do not fix a geometry. The boundary terms come with non-standard coefficients. The requirement of a vanishing Brown-York tensor T BY µν for free boundary conditions gives 3 equations rather than 2 expected from Virasoro constraints. B Sundborg (Stockholm university) A sky without qualities August 27, / 32

39 News on gravity in Asymptotia Metric perspective on boundaries without prior geometry Why does it work? For arbitrary coefficients of the boundary terms the variational principle of the generalised action S GR = 1 κ is not well defined. { } 1 g (R + 2) + α h K + β h. 2 Σ Σ Σ However, if 2α + β = 1 the bulk equations of motion and the vanishing of T BY µν is enough! B Sundborg (Stockholm university) A sky without qualities August 27, / 32

40 News on gravity in Asymptotia Metric perspective on boundaries without prior geometry Why does it work? For arbitrary coefficients of the boundary terms the variational principle of the generalised action S GR = 1 κ is not well defined. { } 1 g (R + 2) + α h K + β h. 2 Σ Σ Σ However, if 2α + β = 1 the bulk equations of motion and the vanishing of T BY µν is enough! The standard gravity action corresponds to α = 1, β = 1, while our Chern-Simons case corresponds to α = 0, β = 1. The Virasoro constraints make the traceless part of the Brown-York tensor vanish, and the chirality of the group elements g and ḡ makes the trace vanish. B Sundborg (Stockholm university) A sky without qualities August 27, / 32

41 News on gravity in Asymptotia Metric interpretation of diffeomorphism invariance and Virasoro constraints Metric interpretation We can choose an arbitrary metric on the boundary, modulo topological obstructions. The subleading terms of the metric will be fixed by the Virasoro constraints. In fact, for the choice of a simple flat metric, the metric will be characterised by the conformal factor. The conformal factor is governed by a wave equation. The gravity solutions are given in an unfamiliar conformal gauge, and global issues are not yet fully investigated... B Sundborg (Stockholm university) A sky without qualities August 27, / 32

42 Outroduction Conclusions and speculations Strings Quantisation Higher spin B Sundborg (Stockholm university) A sky without qualities August 27, / 32