Can Gravity be Localized? & new AdS4-SCFT3 duality

Transcription

1 Can Gravity be Localized? & new AdS4-SCFT3 duality C. Bachas, ENS - Paris Corfu Workshop, September 2011 based on : CB, J. Estes, arxiv: [hep-th] B. Assel, CB, J. Estes, J. Gomis, [hep-th] and in preparation also: O. Aharony, L. Berdichevsky, M. Berkooz, I. Shamir, arxiv: [hep-th]

2 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...

3 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!

4 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

5 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, i.e. the 4-derivative term drops out

6 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!

7 The story is more complicated than this, because strong coupling sets in at intermediate scales Hotly debated whether this can cure the discontinuity; (some) consensus that the issue cannot be settled without UV completion.

8 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.

9 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

10 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)

11 Linearizing the Einstein equations R MN 1 2 g MNR = T MN leads to 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.

12 Important: the linearized equation depends only on the geometry, not on the detailed matter-field backgrounds. Brandhuber, Sfetsos 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

13 The would-be massless graviton has ψ(y) = constant It is normalizable iff the transverse volume is finite For infinite transverse volume, either the spin-2 spectrum has a continuum, or the lowest-lying state must be massive. In the latter case we may talk about localization of the graviton provided m 0 can be made arbitrarily small, and in any case much smaller than the typical KK scale.

14 Wait a minute: Why can t the warp factor help? When it does, infinity is an apparent horizon, which can be reached in finite proper time. This can be shown for flat brane-worlds as follows: For a particle moving in a flat transverse dimension y : e 2A = C e 2A ẏ 2 The total proper time If we request As y goes to infinity, we need dy e 2A e A 0, so that ẏ e A 0 dτ = dt C 1 e 2A dy e A. must be infinite, for geodesic completeness. finite for a normalizable zero mode, then A νlog y with 1 > ν > 1/2 This is ruled out by the holographic c-theorem conditions in the flat-brane case QED. A 0 which follows from the energy Girardello et al, Freedman et al 98-99

15 Given an apparent horizon, we need to supplement the quantum theory with boundary conditions at the horizon ( IR brane ) This is an effective compactification.

16 For M 4 = AdS 4 the warp factor need not be monotonic. Thus it can approach zero and then turn around and diverge, so as to create a graviton trap. The almost constant lowest mode goes to zero near the warp-factor minimum, giving the graviton a tiny mass. The Karch-Randall model illustrates this point.

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 so that ). 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 e A 10 f y Warp factor as l/l ( ) e 2A f4 2 = L 2 cosh 2 y0 y e 2A 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 expressed 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 String-theory embedding [ ] NS5 D5 [ ] D3 [012 3] near-horizon 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.

23 The geometry of a 5-brane in the probe limit is AdS 4 S 2 S 5 5-brane

24 The exact geometry of these configurations was discovered some years ago by D Hoker, Estes and Gutperle Q: Is the graviton in these geometries localized? A: No; but it does obtain an arbitrarily-small mass in a curved 10d spacetime. The ensuing geometries are interesting for other reasons: - holographic duals of N=4 SCFT3 of Gaiotto-Witten - (first?) IIB compactifications without moduli

25 The EDG 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.

26 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..

27 Reduction of eigenmode equation: ψ(y a )=Y l1 m 1 Y l2 m 2 ψ l1 l 2 (z, z) leads to a Laplace-Beltrami spectral problem on : Σ 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. Too hard to solve analytically, except in the simplest case of the (dilaton domain wall) Janus solution where the equation reduces to Heun s equation.

28 Janus cannot localize gravity because the dilaton has no (super)potential, so its domain wall tends to spread 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 the 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 factorized AdS 4 K geometry in the central region. AdS 5 S 5 The regions are much more curved

29 10 f warp factor X 6 f X sphere radii

30 The 10d geometry looks like this: AdS 4 K radius l narrow AdS 5 S 5 throat radius L graviton mass L l 1

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.

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)

35 for example: D3 NS5 D5 N = 6 ; ρ = (2, 2, 1, 1) ; ˆρ = (3, 2, 1)

36 General result (by moving branes): supersymmetry ˆρ T ρ, and irreducibility ˆρ T > ρ. When the inequality is saturated, the quiver breaks down to disjoint pieces.

37 y x δ 1 δ δ q ˆδˆq.... ˆδ 2 ˆδ 1 h 1 = h 2 = [ iα sinh(z β) [ q γ a ln a=1 ˆα cosh(z ˆβ) ˆq b=1 ( ( iπ tanh 4 z δ )) ] a + c.c. 2 ˆγ b ln ( tanh ( z ˆδ b 2 ))] + c.c.

38 D3-brane Page charges in fivebrane stacks: Q inv(a) D3 = C a F 5 B 2 F 3 + C a F 3 B 2 z= =2 8 π 3 ( ˆαγ a sinh(δ a ˆβ) 2 γ a ˆq b=1 ˆγ b arctan(eˆδ b δ a ) ) ˆQ inv(b) D3 = Ĉ b F 5 + C 2 H 3 Ĉ b H 3 C 2 z= =2 8 π 3 ( α ˆγ b sinh(ˆδ b β) + 2 ˆγ b q γ a arctan(eˆδ b δ a ) ). a=1

39 N (a) D3 = N (a) D5 ˆq b=1 ˆN (b) NS5 2 π arctan(eˆδ b δ a ), ˆN (b) D3 = ˆN (b) NS5 q a=1 N (a) D5 2 π arctan(eˆδ b δ a ) Compute the linking numbers: l (a) N (a) D3 N (a) D5 and ˆl(b) ˆN (b) D3 ˆN (b) NS5. Prove ρ T > ˆρ using the fact that acrtanθ π/2

40 By counting parameters one can check that they are all quantized These are examples of IIB AdS vacua with no moduli

41 Another interesting limit ˆρ ρ T correspond to severing one (or more) link, by taking N i 0 This corresponds to factorizing the 5-brane singularities on the strip.

42 δ 1... δ I δi+1... δ q ˆδˆq.... ˆδ J+1 ˆδJ... ˆδ 1 δ 1... δ I δ I δ q ˆδˆq.... ˆδ J+1 ˆδJ... ˆδ 1

43 this is a string-theory wormhole

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

45 CONCLUSIONS (1) Embedding of Karch-Randall in string theory gives massive graviton in backgrounds. AdS 4 K (2) New isolated vacua, and stringy wormholes (3) Holographic duals for (large class of) N=4 d=3 SCFTs.

46 Thank you