Gravity localization & AdS4 / CFT3

Transcription

1 Gravity localization & AdS4 / CFT3 C. Bachas STRINGS 2012 (Munich) based on: CB, J. Estes, arxiv: [hep-th] B. Assel, CB, J. Estes, J. Gomis, [hep-th] 1207.xxxx [hep-th]

2 Standard Hypothesis: 4D Graviton is massless couples universally to all forms of matter & radiation These properties are automatic in any compactification of string theory (or higher-dim theory of gravity)... [Note: I will stick with classical 2-derivative supergravities]

3 KK reduction for spin 2: Consider warped-(a)ds vacuum, 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

4 Consistent reduction of (spin-2) 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 equations (λ = m 2 +2k)

5 Linearizing the Einstein equations R MN 1 2 g MNR = T MN leads to a universal Schrödinger problem in the 6D transverse space: depends only on geometry, not on matter fields Brandhuber, Sfetsos; Csaki, Erlich, Hollowood, Shirman; CB, Estes e 2A [ĝ] ( a [ĝ]ĝ ab e 4A b ) ψ = m 2 ψ mass operator M Using the standard norm ψ 2 = [dy]e 2A ψ ψ one finds :

6 ψ, M 2 ψ = [dy]e 4A a ψ a ψ M 2 0 and M 2 =0 ψ 0 = constant A massless 4D graviton requires [dy] e 2A < (automatic for smooth compactification) Its couplings are universal: [dy] e 2A ψ 0 φ i φ j = ψ 0 δ ij Can it be otherwise? - 4D graviton must be massive - transverse space non-compact

7 A model for this was proposed ten years ago by Karch & Randall: Thin AdS4 brane in AdS5 bulk Randall, Sundrum; Dvali, Gabadadze, Porrati;... I KR = 1 2κ 2 5 d 4 x dy g ( R + 12 ) L 2 + λ d 4 x [g] 4, AdS5 radius brane tension ds 2 = L 2 cosh 2 ( y0 y L ) ḡ µν dx µ dx ν + dy 2 AdS4 foliation Brane radius of curvature: l = L cosh(y 0 /L) = L 1 κ 2 5 λl/6 L can be tuned so that

8 Two slices of AdS5 glued along a AdS4 brane: y = y = y 0 y =0 y =0 y = y 0 y = AdS 5 AdS 5 (cut away green slices and glue)

9 Warp factor as function of transverse coordinate: 10 f y AdS5 boundary would be horizon (Randall-Sundrum II) 3-brane

10 Spectrum : one nearly-constant, nearly massless mode vanishes near bottom of warp factor wells l 2 m 2 0 3L2 2l 2 two infinite towers of nearly AdS5 modes l 2 m 2 (2n + 1)(2n + 4) n =0, 1, supported at bottom of wells Their wavefunctions are exponentially suppressed at the brane position, so they are hidden from 4D gravity: [dy]e 2A ψ 0 φ hφ h ψ 0

11 No conventional effective 4D theory, but physics does look 4D at L r l 8πG N κ 2 5/L as in usual KK Newton s law: V Newton + V G Nm 1 m 2 r (1 + γ L2 r 2 + ) unlike standard KK Warning: this is not a model for the Universe, since Λ eff < 0

12 But any long-distance modification of Einstein gravity is (potentially) interesting Can the KR model be embedded in string theory? (holographic duals to conformal domain walls of CFT4) S 3 Boundary in global coordinates

13 Two beautiful developments got us started: D Hoker, Estes & Gutperle 07 found the general local form of type-iib supergravity solutions with OSp(2, 2 4) SO(2, 3) SO(3) SO(3) supersymmetry. This is the symmetry of conformal 1/2-BPS domain walls in N=4 D=4 super-yang-mills theory, which were classified by Gaiotto & Witten 08 DeWolfe, Freedman, Ooguri;... To address our question, we established the precise dictionary between these two works. Bonus: some new lessons for AdS4/CFT3.

14 The Gaiotto-Witten domain walls support linear-quiver N=4 CFT3s bi-fundamental hypers coupling to bulk 4D superym N 1 N 2 Nˆp 1 M 1 M 2 Mˆp 1 fundamental hypermultiplets U(N 1 ) U(N 2 ) U(Nˆp 1 ) U(M 1 ) U(M 2 ) U(Mˆp 1 ) U( ˆM 1 ) U( ˆM 2 ) U( ˆM p 1 ) gauge group manifest global symmetry hidden global symmetry

15 Engineered on D3-branes suspended/intersecting NS5- and D5- branes N 2 N 1 Hanany-Witten M 2 M 1 Mirror symmetry exchanges NS5 with D5 (Intriligator, Seiberg; de Boer, Hori, Ooguri, Oz;...) (Almost) invariant description in terms of linking numbers: i th D5 l i = (net # of D3-branes ending on it from left) + (# of NS5 to its right) j th NS5 ˆl j = (net # of D3-branes ending on it from right) + (# of D5 to its left)

16 Focus on 3D gauge theories (coupling to 4D YM: slight complication) Gaiotto-Witten conjectured that these flow to non-trivial SCFT whenever # hypers 2N a for each gauge group (complete Higgsing) These conditions translate to ˆl1 ˆl 2 ˆlˆp > 0 (labelled from right to left) while also l 1 l 2 l p > 0 (labelled from left to right) Move all NS5 to the left of all D5 = l ˆlj i = = N two partitions ρ & ˆρ

17 Supersymmetry (s-rule) implies the (partial ordering) constraints ρ T > ˆρ ( # of boxes in first n columns of ρ > # boxes in first n rows of ˆρ n ) e.g. > arise in many contexts related to Nahm s equations Nakajima; Kronheimer; CB, Hoppe, Pioline;... Summarize: non-trivial N=4 SCFT3 T ˆρ ρ (SU(N)) conjectured for any pair of partitions obeying these constraints

18 The IIB solutions are AdS4 x S2 x S2 fibrations over a Riemann surface basis Σ h 1 h 2 Modulo an SL(2,R) rotation: functions and.. Local solutions are determined by two harmonic 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,, ρ 8 = N 1N 2 W 2 h 4 1 h4 2 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.

19 There are also p-form backgrounds: F 5, H 3, F 3 expressed in terms of the dual harmonic functions: h 1 = i(a 1 Ā1) h D 1 = A 1 + Ā1 h 2 = A 2 + Ā2 h D 2 = i(a 2 Ā2) and the (normalized) volume forms of the (pseudo)spheres: ω 0123 AdS 4 ω 45 ω 67 S 2 1 S 2 2 Note: the constant parts of Re(A 1 ) and Im(A 2 ) are pure gauge.

20 3-forms : H (3) + if (3) = ω 45 db 1 + i ω 67 db 2 where: b 1 =2ih 1 h 1 h 2 ( h 1 h2 h 1 h 2 ) N 1 +2h D 2 b 2 =2ih 2 h 1 h 2 ( h 1 h2 h 1 h 2 ) N 2 2h D 1 5-form : F (5) = 4 f 4 4 ω 0123 F +4f 2 1 f 2 2 ω 45 ω 67 ( 2 F), where f 4 4 F = dj 1 with j 1 =3C +3 C 3D + i h 1h 2 W ( h 1 h 2 h 1 h 2 ) C = A 1 A 2 A 2 A 1 D = Ā1A 2 + A 1 Ā 2

21 The surface Σ can have boundaries which are interior points of the 10D geometry. For this, require either h 1 to have a Dirichlet condition and h 2 a Neumann condition, or vice versa. One 2-sphere then shrinks to zero, and the geometry is (locally) AdS 4 S 2 D 2 Asymptotic regions arise as singularities on boundary: Σ S 2 2 shrinks to point here S2 1 shrinks to point here h 1 N h 2 D D5 AdS 5 S 5 NS5 h 2 N h 1 D

22 Local form of NS5 singularity: (For D5-brane: exchange roles of A 1 and A 2 ) Σ A 1 πˆl + I S 2 2 S 3 A 2 ˆM logw + w =0 ˆM ˆl = 1 (4π) 4 ˆM = 1 16π 2 S 3 S 2 1 S 3 H 3 (F 5 + C 2 H 3 ) # NS5 branes D3-brane Page charge

23 Local form of AdS 5 S 5 : Σ I S 2 1 S 2 2 S 5 A 1 = 1 w (a 1 + b 1 w + ) A 2 = 1 w (a 2 + b 2 w + ) w =0 n = 1 (4π) 4 F 5 = (a 2b 1 a 1 b 2 ) S 2π 5 D3 charge Limit a 1,a 2 0 gives a smooth capping-off of geometry Aharony, Berdichevsky, Berkooz, Shamir; ABEG

24 Solution for linear quivers h 1 = h 2 = [ iα sinh(z β) [ ˆα cosh(z ˆβ) q γ a ln a=1 ˆq b=1 ˆγ b ln ( ( iπ tanh 4 z δ )) ] a + c.c. 2 ( tanh ( z ˆδ ))] b 2 + c.c. ˆM 1, ˆl 1 ˆM2, ˆl 2... n, φ n, φ... M 2, l 2 M 1, l 1 Parameter count: {M a, l a, ˆM b, ˆl b } {γ a, δ a, ˆγ b, ˆδ b }

25 May have zero, one or two asymptotic more leads to conical singularities. AdS 5 S 5 regions, All parameters in first case are discrete: no N=4 moduli. Global symmetries of CFT3 realized as gauge symmetries on 5-branes Can show that linking numbers define two partitions obeying the Young-tableau inequalities One-to-one correspondence with Gaiotto-Witten SCFT3s

26 Solution for circular quivers z w iπ 2 C C q t A 1 = i A 2 = p γ a ln a=1 ˆp b=1 ˆγ b ln ( ) ϑ1 (q ν a ) ϑ 2 (q ν a ) ( ) ϑ1 (q ˆν b ) ϑ 2 (q ˆν b ) + ϕ 1 + iϕ 2 i ν a = z δ a 2π i ˆν b = z ˆδ b 2π + i 4

27 Annulus modulus K = # of winding D3 branes Cannot couple to 4D sym, i.e. add asymptotic AdS 5 S 5 regions Several calculations/observations concerning this rich set of holographic dualities, that I have no time to discuss here. Just mention some: Free energy on S3 from sugra = from CFT: checked that for T (SU(N)) F = 1 2 N 2 lnn + O(N 2 ) and some related theories non-trivial Assel, Estes, Yamazaki using results of Benvenuti & Pasquetti; Nishioka,Tachikawa, Yamazaki

28 Large K corresponds to t 0 ; T-dual to ABJM-like theories. SL(2, Q) transformations generalize mirror symmetry in classical limit... Let s return now to the original question

29 To create the hierarchy l L must look at degeneration limits AdS 4 AdS 5 Factorize 5-brane singularities Pinch annulus t } Worm-branes (Janus throats) Cap-off AdS 5 S 5 α, ˆα 0 (small rank for bulk D=4 SYM group)

30 This is possible, provided there are both NS5s and D5s (to stabilize dilaton) ; The lowest AdS4 spin-2 mode has a (tiny) mass and non-universal couplings. (Assel, CB, unpublished) But, as in other efforts to create classical hierarchies, spacetime decompactifies AdS 4 K 6 radius l narrow AdS 5 S 5 of radius L throat Gravity is never 4-dimensional!

31 10 f warp factor X 6 f X sphere radii

32 What we need: An AdS4/CFT3 pair for which the compact space stays small ( T µν and few other operators, then large gap) interesting, and hard problem by itself Couple CFT3 to SYM4 without destabilizing fixed point need understanding of geometric singularities

33 Summary Found a large class of type-iib backgrounds, that are duals of the Gaiotto-Witten N=4 SCFT3s, and of their circular-quiver extensions. [These are AdS4 compactifications with localized 5-branes] Degeneration limits correspond to couplings via small-rank gauge groups. Failed to localize 4D gravity, because the brane fattens to 10D spacetime. Given the difficulty in modifying gravity in IR, worth pursuing... as opposed to local multitrace couplings,e.g. Kiritsis, Niarchos; Aharony, Clark, Karch

34 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 CFT1 N small rank CFT2

35 AKT IΣ AEΛIOY KAΛΛIΣT ON ΦAOΣ A ray of sun is the best of lights (chorus in Antigone of Sophocles)