Topologically Massive Gravity with a Cosmological Constant

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1 Topologically Massive Gravity with a Cosmological Constant Derek K. Wise Joint work with S. Carlip, S. Deser, A. Waldron Details and references at arxiv: [hep-th] (or for the short story, , appearing in today s mailing) Quantum Gravity and Quantum Geometry, University of Nottingham, July 2008

Waldron Details and references at arxiv:0803.

2 Topologically Massive Gravity The topologically massive 2+1 gravity model has action I[g] = giving field equations: { d 3 x g R + 1 2µ εµνρ( Γ α µβ νγ β ρα Γα µγγ γ ) } νβ Γβ ρα } {{ } } {{ } Einstein Chern Simons G µν + 1 µ C µν = 0 where C µν is the Cotton tensor the conformal curvature in 3d. µ 2 = pure 2+1 GR: flat geometry µ 2 0 = conformally flat geometry But for 0 < µ 2 < there are local propagating modes! Linearized analysis: field redefinitions give massive gravitons obeying ( +µ 2 )ϕ = 0. The nonstandard Einstein Hilbert sign is required for positive energy.

curvature in 3d. µ 2 = pure 2+1 GR: flat geometry µ 2 0 = conformally flat geometry But for 0 < µ 2 < there are local propagating modes!

3 Adding a Cosmological Constant The simple addition of a cosmological constant: I[g] = { d 3 x g (R 2Λ) + 1 2µ εµνρ( Γ α µβ νγ β ρα Γα µγγ γ )} νβ Γβ ρα has some surprising effects! G µν + Λg µν + 1 µ C µν = 0 I ll consider only Λ < 0, and work in units for simplicity. Λ = 1 To describe these effects, we first need a review of: Light-front coordinates in AdS Scalar fields in AdS

G µν + Λg µν + 1 µ C µν = 0 I ll consider only Λ < 0, and work in units for simplicity.

4 Light-front method We work in a Poincaré patch in AdS 3, with two null coordinates x +, x (x ± := x ± t) and spacelike coordinate z, and metric ds 2 = g µν dx µ dx ν = 2dx+ dx + dz 2 z 2 At t = 0, space looks like: z = z = 0 But we often view x + τ as time and (x, z) as space. This setup is useful for studying fields in AdS backgrounds, e.g....

2dx+ dx + dz 2 z 2 At t = 0, space looks like: z = z = 0 But we often view x + τ as

5 Scalar fields in AdS 3 The action for a massive scalar field in AdS 3 background (Λ = 1) I = 1 d 3 x { g µ ϕg µν ν ϕ + m 2 ϕ 2}, 2 written in light-front coordinates, automatically takes Hamiltonian form: I = where dτ { φ, φ A, B ( 1 2 [ zφ] z 2 [ m ] ) } φ 2 dx dz 4 dx dz A B = B, A. φ 1 z ϕ is the light-front symplectic structure. The field equations are: [ z 2 m2 + 3/4 ] z 2 φ = 0 Or, after a Fourier transform in x ± : [ d 2 dz d z dz + ω2 ν2 ]( ϕ ) z 2 = 0 Bessel eq. w. ν 2 = m z

+ 3 ] ) } φ 2 dx dz 4 dx dz A B = B, A. φ 1 z ϕ is the light-front symplectic structure.

6 For ν 2 = m = (1/2) 2, (3/2) 2, (5/2) 2,... the Bessel solutions ( reduce to: ) ϕ slowly varying function of z exp(ik µ x µ ), which exhibit null propagation. Also, the Breitenlohner Freedman bound says we get a consistent field theory down to m 2 = 1. Restoring Λ, we get: null m 2 Breitenlohner Freedman null null Λ

7 Linearized Topologically Massive Gravity The full action for topologically massive gravity is { I = d 3 x γ (R 2Λ)+ 1 2µ εµνρ( Γ α µβ νγ β ρα+ 2 3 Γα µγγ γ )} νβ Γβ ρα. Any 1st order AdS metric fluctuation is equivalent, by a linearized diffeomorphism, to one in light-front gauge : ds 2 = 2dx+ dx + dz 2 z 2 + h ++ (dx + ) 2 + 2h + dx + dz + h dz 2. Up to quadratic order in the fluctuations h, h +, h ++, the action is: I = dτ { z4 µ X, ḣ z ([zx h] 2 + z3 2 µ [ X + h + µ + 2 z ] h Y ) dx dz with X h +, Y 2 h ++ The field equations for X and Y are algebraic; integrating them out and setting φ = z 3/2 h, we recover the standard AdS scalar action, with mass m 2 = (µ + 2) 2 1. }

8 Chiral Mass Splitting In fact, the components of the linearized Einstein tensor: H ρσ = [ G ρσ g ρσ ]LINEAR obey the AdS massive wave equation [ z 2 m2 + 3/4 ] z 2 ( 1 H ρσ ) = 0 z with mass depending on the mode s chirality: Field m 2 H ++ (µ 2) 2 1 H +z (µ 1) 2 1 H zz, H + µ 2 1 H z (µ + 1) 2 1 H (µ + 2) 2 1

2 m2 + 3/4 ] z 2 ( 1 H ρσ ) = 0 z with mass depending on the mode s chirality:

9 Or, restoring Λ: Notes: Field m 2 H ++ H +z H zz, H + H z H (µ 2 Λ) 2 + Λ (µ Λ) 2 + Λ µ 2 + Λ (µ + Λ) 2 + Λ (µ + 2 Λ) 2 + Λ 1. Not independent components all are derivable from the single scalar degree of freedom. 2. m 2 µ 2 for Λ = invariance under simultaneous µ µ and +.

10 Fixing µ (say µ = 1) and adjusting Λ, the modes have these masses: m 2 H Breitenlohner Freedman H -1.5 H + H ++ H Λ

11 Or, zooming out a bit m H Breitenlohner Freedman H -3 H H ++ H Λ -5.5

2 m 2-0.5 H Breitenlohner Freedman -1-1.

12 Asymptotics & Wave Packets The basic solutions for linearized curvature fluctuations are Bessel functions, as we saw for AdS scalars. For these, we find: Solutions are asymptotically AdS near the z = 0 boundary; obey Fefferman-Graham asymptotics at z = 0 only if µ 2 > Λ. However, the metric fluctuations diverge at z =. BUT, we can construct: A conserved symplectic current, yielding a Klein Gordon inner product on linearized solutions. A complete orthonormal system of linearized solutions w.r.t. this inner product. This lets us construct wave packets with arbitrary profile in particular, support away from z =.

However, the metric fluctuations diverge at z =.

13 Chern Simons Formulation The topologically massive gravity action (returning to Λ = 1) is: I[g] = { d 3 x g (R + 2) + 1 2µ εµνρ( Γ α µβ νγ β ρα Γα µγγ γ νβ Γβ ρα Since pure gravity can be written as a Chern Simons theory, and the second term is Chern Simons, an obvious question is: Can topologically massive gravity be given a pure Chern Simons formulation? )} Yes: with I TMG [e] = 1 2 (1 1 µ )I CS[ + A[e]] (1 + 1 µ )I CS[ A[e]] ± A µ a b = ω µ a b ± ε a bce µ c, where e is the dreibein, ω = ω[e] is the torsionless spin connection, and the coefficients 1(1 ± 1 2 µ ) are the central charges of the boundary conformal field theory. (Note: this theory isn t topological, because of the torsion constraint)

14 The critical value µ = 1 When the topological mass takes the value µ = 1, some interesting things happen: One of the coefficients in the Chern Simons formulation disappears, leaving: I TMG [e] = I CS [ A[e]] The linearized curvature fluctuations about AdS become exact : H µν = D (µ F ν), where D µ is the covariant derivative for the AdS background. F ν turns out obey precisely the equations for the dual field strength in topologically massive electromagnetism : I = 1 { g } d 3 x Fµν g µρ g νσ F ρσ + µ ε µνρ F µν A ρ, 4 So, we get an equivalence (at µ = 1) between topologically massive spin-1 and spin-2 theories!

the covariant derivative for the AdS background.

15 Some final remarks Topologically massive gravity is a surprisingly rich toy model, with all of the essential conceptual difficulties of quantizing full-fledged GR: it is diffeomorphism invariant it has local propagating modes (unlike 3d GR) But it s also simpler than 4d GR in some ways. In particular: it has only one propagating degree of freedom So far, relatively little is known about quantizing the full nonlinear theory. Advertisement: this community has many techniques for quantizing background independent field theories. Topologically massive gravity is an interesting test ground! Again, details and references at arxiv: [hep-th]

In particular: it has only one propagating degree of freedom So far, relatively little is known about quantizing the full nonlinear theory.

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