Can Gravity be Localized?

Transcription

1 Can Gravity be Localized? based on : CB, J. Estes, arxiv: [hep-th] B. Assel, CB, J. Estes, J. Gomis, 1106.xxxx also: O. Aharony, L. Berdichevsky, M; Berkooz, I. Shamir, arxiv: [hep-th] C. Bachas, ETH 06/11

2 Fields can be localized on (extended) solitons in QFT: there are many spin-0 and spin-1/2 examples Goldstone modes e.g. superconducting strings Can also localize abelian spin-1 fields Josephson junctions Renormalizability requires D 4 = d 3 which severely restrict the possibilities.

3 E In string theory D=10 and so more room UV completion can do some calculations without infinities non-abelian spin-1 gauge fields localized on D-branes but what about spin 2? Einstein s theory is much harder to tinker with...

4 This is closely related to the questions: Can the graviton have mass? Can it be a resonance? Are sectors hidden from gravity possible? Other IR modifications of Einstein equations? The subject has a long history, to which I will not try to do justice here... see also Slava Mukhanov s talk

5 In Minkowski spacetime, the answer seems to be NO An important obstruction is the vdvz discontinuity van Dam, Veltman, Zakharov 70 Notice that for the photon the answer is YES Indeed, the particle data group quotes the experimental bound: m γ < ev range > 10 9 km 1 light hour but could be finite!

6 To understand the difference, consider the linearized Lagrangian for a massive spin-1 particle: L = 1 4 ( µa ν ν A µ ) 2 m2 2 A µa µ + A µ j µ Introducing a spurious field A µ = A µ + 1 m µφ and taking m 0 gives: L = 1 4 ( µa ν ν A µ) 2 + A µj µ 1 2 µφ µ φ + 1 m µφj µ = L Maxwell 1 2 µφ µ φ The dangerous last term drops out, provided the e-m current is conserved, so that the extra scalar mode decouples.

7 Now repeat the exercise for a massive spin-2 field. The (ghost-free) massive Pauli - Fierz Lagrangian is: L PF = L EH m2 2 ( h νλ h νλ (h ρ ρ) 2) where L EH = 1 2 µh νλ µ h νλ + µ h νλ ν h µλ µ h µν ν h λ λ νh λ λ ν h ρ ρ + h µν T µν with µ T µν =0

8 Introduce again compensators to restore gauge invariance: h µν = h µν + 1 m ( µa ν + ν A µ )+ 2 m 2 µ ν φ invariant under δh µν = µ ξ ν + ν ξ µ δa µ = mξ µ + µ Λ mλ δφ = Λ Inserting in L PF gives a free massless spin-1 field, and a two-derivative φ h µν Lagrangian mixing and. PF was precisely devised for this!

9 h µν = h µν + η µν φ Redefining fields to remove the mixing ( ) finally gives: L PF = L EH 1 2 F µνf µν 3 µ φ µ φ + φ T ρ ρ The residual coupling is different for light, than for massive matter; thus the Pauli-Fierz theory does not give Einstein s theory when m 0 If we set Newton s law to its measured form, light bending = 3/4 of measured effect... so however tiny the mass, it is ruled out!

10 The story looks more promising in AdS: The vdvz discontinuity is absent if m gr < 1/L AdS Kogan - Mouslopoulos - Papazoglou; Porrati a simple model, possibly embed-able in string theory Karch-Randall Supersymmetry can protect the required hierarchy Of course, we don t seem to live in AdS spacetime! OK, take attitude that anything one can learn about IR gravity is interesting, and proceed.

11 KK reduction for spin 2 Interested in warped-(a)ds geometries, ds 2 = e 2A(y) ḡ µν (x) dx µ dx ν +ĝ ab (y) dy a dy b M 4 = AdS 4, M 4, ds 4 k = 1, 0, 1

12 Consider (consistent reduction to) metric perturbations ds 2 = e 2A (ḡ µν + h µν ) dx µ dx ν +ĝ ab dy a dy b, with h µν (x, y) =h [tt] µν (x) ψ(y) where ( (2) x λ) h [tt] µν = 0 and µ h [tt] µν =ḡ µν h [tt] µν =0. Pauli-Fierz (λ = m 2 +2k)

13 Linearize the Einstein equations R MN 1 2 g MNR = T MN to find the Schrodinger problem : e 2A [ĝ] ( a [ĝ]ĝ ab e 4A b ) ψ = m 2 ψ This is equivalent to a scalar-laplace equation in d dimensions : 1 ĝ ( M ĝ ĝ MN N ) h µν (x, y) = 0.

14 Important: the linearized equation depends only on the geometry, not on the detailed matter-fields that created it. Csaki, Erlich, Hollowood, Shirman CB, JE Localization of spin-2 can only come from geometry The wavefunction norm is ψ 2 d d 4 y [ĝ] e 2A ψ 2

15 The would-be massless graviton has ψ(y) = constant It is normalizable iff the transverse volume is finite Why can t the warp factor help? When it does, infinity is an apparent horizon, so -- geometry should be made geodesically complete -- or should supplement quantum theory with boundary conditions at horizon ( IR brane )

16 In the cases M 4 = M 4 or ds 4 the energy conditions show (at least in codim = 1) that the warp factor A is monotonic, so it cannot turn around to form an effective graviton trap But for M 4 = AdS 4 localization, and a tiny AdS graviton mass cannot be a priori ruled out.

17 Karch-Randall model Starting point is 5D Einstein action plus a thin 3-brane I KR = 1 2κ 2 5 d 4 x dy g ( R + 12 ) L 2 + λ d 4 x [g] 4, The solution is: ds 2 = L 2 cosh 2 ( y0 y L ) ḡ µν dx µ dx ν + dy 2, where y 0 = L arctanh ( ) κ 2 5 λl 6 It describes two (large) slices of AdS5 glued along a AdS4 brane with radius l 2 = e 2A(0) = L 2 cosh 2 ( y 0 L One can tune λl to make ). l L 1

18 AdS 5 AdS 5 Cut away green slices, then glue the white ones in a symmetric fashion. Gives two 4D boundaries glued across two 3D defects (domain walls).

19 10 f y Warp factor as l/l ( ) e 2A f4 2 = L 2 cosh 2 y0 y L is gradually tuned up

20 4D parameters: 8πG N κ 2 5/L as in usual KK V Newton + V G Nm 1 m 2 r (1 + γ L2 r 2 + ) so l L unlike standard KK Spectrum : - a nearly-constant, nearly massless mode - two towers of AdS5 modes m 2 0 3L2 2l 2 m 2 (2n + 1)(2n + 4) n =0, 1,

21 These masses are in units of the AdS4 radius so states with m 2 o(1) mediate long-range interactions. What saves the day is that the AdS5 states live at the bottom of the warp-factor well. Their wavefunctions are exponentially suppressed at the brane position Furthermore, ψ 0 ψ ψ universal so the nearly-massless graviton has non-universal couplings to the other fields!

22 The exact (super)gravity solutions Karch and Randall proposed to embed their model in IIB string theory, by inserting 5-branes in the AdS 5 S 5 geometry of D3-branes. The exact geometry of these configurations was discovered recently by D Hoker, Estes and Gutperle Try to understand whether graviton in these geometries is localized.

23 The solutions are AdS 4 S 2 S 2 fibrations over a surface. They depend on two harmonic functions global consistency conditions. h 1,h 2 subject to certain metric : ds 2 = f 2 4 ds 2 AdS 4 + f 2 1 ds 2 S f 2 2 ds 2 S ρ 2 dzd z, f 8 4 = 16 N 1N 2 W 2 f1 8 = 16 h 8 N 2 W 2 1,, N 3 1 f 8 2 = 16 h 8 2 N 1 W 2 N 3 2 dilaton : e 4φ = N 2 N 1 where : W = h 1 h2 + h 1 h 2 = (h 1 h 2 ), N 1 =2h 1 h 2 h 1 2 h 2 1W, N 2 =2h 1 h 2 h 2 2 h 2 2W. There are also 3-form and 5-form backgrounds, and 1/4 unbroken supersymmetry.

24 The solutions of interest have = infinite strip with h 1,h 2 obeying N or D conditions, possibly with isolated singularities on the boundary, e.g. NS5 AdS 5 S 5 AdS 5 S 5 D5 The harmonic functions for this choice are: h 1 = [ ( iα 1 sinh(z β 1 ) γ 1 ln tanh( iπ 4 z δ )] 1 ) 2 +c.c., h 2 = [ ( α 2 cosh(z β 2 ) γ 2 ln tanh( z δ )] 2 ) 2 +c.c..

25 Reduction of eigenmode equation: ψ(y a )=Y l1 m 1 Y l2 m 2 ψ l1 l 2 (z, z) Σ leads to a Laplace-Beltrami spectral problem on : 2h 1 h 2 (h 1 h 2 ) ψ 00 = (m 2 + 2) ψ 00, where ψ00 h 1 h 2 ψ 00. The norm is ψ 2 = Σ d 2 z Wh 1 h 2 ψ l1 l 2 2 = Σ d 2 z W h 1 h 2 ψl1 l 2 2 and the b.conditions for ψ 00 are Neumann.

26 The simplest Janus solution γ i =0 is a dilaton domain wall L 4 = 16 α 1 α 2 cosh φ e 2φ ± = α 2 α 1 e± φ radius of asymptotic AdS 5 S 5 dilaton jump The spectral equation reduces to a ODE with 4 regular singular points (Heun s equation) which can be solved with fast numerics The results are not particularly exciting:

27 There is one free parameter, (tanh φ 2 )2 ξ 1 [0, 1) m Ξ 1 AdS 5 S 5 φ = The only interesting limit is one in which a linear-dilaton dimension decompactifies, and the geometry becomes AdS 4 R φ w S5

28 φ = 0, 1, 4, 10 f f f X X X

29 The problem is that the dilaton has no (super)potential, so its domain wall spreads to infinite thickness. Adding one type of 5-branes does not help: the dilaton adjusts to ( ly) small or large value, so as to minimize 5-brane tension. The only interesting limit is one with both NS5 and D5 charges, and with Q 5 Q 3 1. Inspection of the geometry shows that this creates a bubble of (almost flat) R 1,9 in the central region.

30 10 f warp factor X 6 f X sphere radii

31 Actually the limit Q 3 0 is smooth, transverse space compactifies: AdS 5 S 5 AdS 4 D 6 the asymptotic regions go over to smooth caps These AdS 4 w M 6 solutions must be gravity duals to 3-dimensional (super)conformal field theories Which ones?

32 By studying the flat-space configurations, Gaiotto and Witten have proposed the existence of a class of interacting SCFTs in three dimensions that they called T ˆρ ρ (SU(N)) They are in 1-to-1 corrspondence with solutions of Nahm s equations: dx a dt = iɛ abc [X b,x c ] on the interval, with boundary conditions that are simple poles, X a J a t N-dimensional generators of SU(2)

33 This problem has been solved by Kronheimer and Nakajima One can associate a partition of N with each choice of the J a e.g. ρ : 12 = K & N have shown that solutions exist iff ρ T > ˆρ where these are the two partitions at the interval ends. (CB, J.Hoppe, B. Pioline 00) By computing these partitions from the 5-brane charges, could show that supergravity knows about this constraint!

34 The underlying gauge theories are described by linear quivers N 1 N 2 N 3 M 1 M 2 M 3 U(N 1 ) U(N 2 ) U(N3) The interesting limits ˆρ ρ T correspond to severing one (or more) link, by taking N i 0 This corresponds to factorizing the 5-brane singularities.... more on blackboard

35 Holographic comment on massive AdS gravity theories: h 1 h decoupling conserved e-m tensor 3 3+ɛ CFT spectrum defect CFT spectrum

36 Thank you