Asymptotic Conformal Symmetry and Gravity Localisation in Brane Worlds K.S. Stelle Imperial College London Stringy Geometry Workshop Johannes Gutenberg-Universit¨ at, Mainz September 16, 2015 A. Salam & E. Sezgin, Phys.Lett. B147 (1984) 47 M. Cvetiˇ c, H. L¨ u & C.N. Pope, Nucl. Phys. B600 (2001) 103 M. Cvetiˇ c, G. Gibbons & C.N. Pope, Nucl. Phys. B677 (2004) 164 T. Pugh, E. Sezgin & K.S.S., JHEP 1102 (2011) 115 B. Crampton, C.N. Pope & K.S.S., JHEP 1412 (2014) 035; 1408.7072 1 / 29
The universe as a membrane The idea of formulating the cosmology of our universe on a brane embedded in a higher-dimensional spacetime dates back, at least, to Rubakov and Shaposhnikov. Phys. Lett. B125 (1983), 136 Attempts in a supergravity context to achieve a localization of gravity on a brane embedded in an infinite transverse space were made by Randall and Sundrum (RS II) Phys. Rev. Lett. 83 (1999) 4690 and by Karch and Randall JHEP 0105 (2001) 008 using patched-together sections of AdS 5 space with a delta-function source at the joining surface. This produced a “volcano potential” for the effective Schr¨ odinger problem in the direction transverse to the brane, giving rise to a bound state concentrating gravity in the 4D directions. 3 2 1 -4 -2 2 4 -1 -2 2 / 29
General problems with localization Attempting to embed such models into a full supergravity/string-theory context have proved to be problematic, however. Splicing together sections of AdS 5 is clearly an artificial construction which does not make use of the natural D-brane or NS-brane objects of string or supergravity theory. These difficulties were studied more generally by Bachas and Estes JHEP 1106 (2011) 005 , who traced the difficulty in obtaining localization within a string or supergravity context to the behavior of the warp factor for the 4D subspace. In the Karch-Randall spliced model, one obtains a “kink” in the warp factor at the junction: f 4 10 8 6 4 2 y � 10 � 5 0 5 10 3 / 29
The problem with string-theory attempts to localize gravity on a brane subspace as found by Bachas and Estes, e.g. , for a Janus discontinuous-dilaton solution, is that there is no similar “bump” in the warp factor for the 4D subgeometry: f 4 10 8 6 4 2 X � 5 0 5 In consequence, there is no concentration of gravity on the 4D subspace of such a model. Bachas and Estes raised the possibility that this difficulty could be generic for asymptotically maximally symmetric geometries of the embedding spacetime. c.f. also Freedman, Gubser, Pilch & Warner, Adv. Theor. Math. Phys. 3 (1999) 363 4 / 29
One interpretation of the patched AdS constructions is in terms of an effective Schr¨ odinger problem, in which the kink in the warp factor produces a bound state for the transverse part of the gravitational wavefunction. Trying to do this without an artificially generated kink runs into a key difficulty in attempts to obtain massless gravity in a lower-dimensional brane subspace when the transverse space is infinite. Here’s a sketch: Given an eigenvalue − λ for a normalizable wavefunction ξ of the √ ˆ transverse wave operator e − 2 A g ab ∂ b ) (where e 2 A is the ge 4 A ˆ g ( ∂ a √ ˆ warp factor of the 4d subspace), and provided one may integrate by parts , one may write � � λ || ξ || 2 = − d d − 4 y ξ ( ∂ a � ge 4 A ˆ g ab ∂ b ξ ) → d d − 4 y � ge 4 A | ∂ξ | 2 ˆ ˆ If one is looking for a transverse wavefunction ξ with λ = 0, corresponding to massless gravitational excitations in the 4d subspace, it would seem therefore that ξ has to be constant, which would be inconsistent with it being normalizable in an infinite transverse space. 5 / 29
Another approach: Salam-Sezgin theory and its embedding Abdus Salam and Ergin Sezgin constructed in 1984 a version of 6D minimal (chiral, i.e. (1,0)) supergravity coupled to a 6D 2-form tensor multiplet and a 6D super-Maxwell multiplet which gauges the U(1) R-symmetry of the theory. Phys.Lett. B147 (1984) 47 This Einstein-tensor-Maxwell system has the bosonic Lagrangian 4 g 2 e σ F µν F µν − 1 6 e − 2 σ G µνρ G µνρ − 1 1 1 2 ∂ µ σ∂ µ σ − g 2 e − σ L SS = 2 R − G µνρ = 3 ∂ [ µ B νρ ] + 3 F [ µν A ρ ] Note the positive potential term for the scalar field σ . This is a key feature of all R-symmetry gauged models generalizing the Salam-Sezgin model, leading to models with noncompact symmetries. For example, upon coupling to yet more vector multiplets, the sigma-model target space can have a structure SO ( p , q ) / ( SO ( p ) × SO ( q )). 6 / 29
The Salam-Sezgin theory does not admit a maximally symmetric 6D solution, but it does admit a (Minkowski) 4 × S 2 solution with the flux for a unit-strength U (1) monopole turned on in the S 2 directions η µν dx µ dx ν + a 2 ( d θ 2 + sin 2 θ d φ 2 ) ds 2 = A m dy m = ( n / 2 g )(cos θ ∓ 1) d φ σ = σ 0 = const , B µν = 0 e σ 0 g 2 = 2 a 2 , n = ± 1 This construction has been used in the SLED ↔ Supersymmetry in Large Extra Dimensions proposal for dilution of the cosmological constant in the two extra S 2 dimensions, leaving a naturally small residue in the four x µ dimensions. Aghababaie, Burgess, Parameswaran & Quevedo, Nucl. Phys. B680 (2004) 389 et seq. 7 / 29
H (2 , 2) embedding of the Salam-Sezgin theory A way to obtain the Salam-Sezgin theory from M theory was given by Cvetiˇ c, Gibbons & Pope. Nucl. Phys. B677 (2004) 164 This employed a reduction from 10D type IIA supergravity on the space H (2 , 2) , or, equivalently, from 11D supergravity on S 1 × H (2 , 2) . The H (2 , 2) space is a cohomogeneity-one 3D hyperbolic space which can be obtained by embedding into R 4 via the condition µ 2 1 + µ 2 2 − µ 2 3 − µ 2 4 = 1. This embedding condition is SO(2 , 2) invariant, but the embedding R 4 space has SO(4) symmetry, so the isometries of this space are just SO(2 , 2) ∩ SO(4) = SO(2) × SO(2) . The cohomogeneity-one H (2 , 2) metric is 3 = cosh 2 ρ d ρ 2 + cosh 2 ρ d α 2 + sinh 2 ρ d β 2 . ds 2 Since H (2 , 2) admits a natural SO (2 , 2) group action, the resulting 7D supergravity theory has maximal (32 supercharge) supersymmetry and a gauged SO (2 , 2) symmetry, linearly realized on SO (2) × SO (2). Note how this fits neatly into the general scheme of extended Salam-Sezgin gauged models. 8 / 29
The Kaluza-Klein spectrum Reduction on the non-compact H (2 , 2) space from ten to seven dimensions, despite its mathematical consistency, does not provide a full physical basis for compactification to 4D, however. The chief problem is that the truncated Kaluza-Klein modes form a continuum instead of a discrete set with mass gaps. Moreover, the wavefunction of “reduced” 4D states when viewed from 10D or 11D includes a non-normalizable factor owing to the infinite H (2 , 2) directions. This infinite transverse volume also has the consequence that the resulting 4D Newton constant vanishes. Accordingly, the higher-dimensional supergravity theory does not naturally localize gravity in the lower-dimensional subspace when handled by ordinary Kaluza-Klein methods. 9 / 29
Bound states and mass gaps Crampton, Pope & K.S.S., JHEP 1412 (2014) 035; 1408.7072 An approach to solving the non-localization problem of gravity on the 4D subspace of the ground-state Salam-Sezgin (SS) solution is to look for a normalizable transverse-space wavefunction with a mass gap before the onset of the continuous massive Kaluza-Klein spectrum. This could be viewed as analogous to an effective field theory for a system confined to a metal by a nonzero work function. General study of the fluctuation spectra about brane solutions shows that the mass spectrum of the spin-two fluctuations about a brane background is given by the spectrum of the scalar Laplacian in the transverse embedding space of the brane Csaki, Erlich, Hollowood & Shirman, Nucl.Phys. B581 (2000) 309; Bachas & Estes, JHEP 1106 (2011) 005 1 �� � − det g (10) g MN (10) F = ∂ M (10) ∂ N F � − det g (10) 1 (4) + g 2 △ θ,φ, y ,ψ,χ + g 2 △ rad ) = H SS ( 4 △ rad = ∂ 2 2 ∂ (cosh 2 ρ ) − 1 warp factor; H SS = ∂ρ 2 + tanh(2 ρ ) ∂ρ 10 / 29
The directions θ, φ, y , ψ & χ are all compact, and one can employ ordinary Kaluza-Klein methods for reduction on them by truncating to the invariant sector for these coordinates, i.e. by making an S-wave reduction. To handle the noncompact radial direction ρ , one needs to expand in eigenmodes of △ rad . The ansatz for 4D metric fluctuations simply replaces η µν in the 10D metric by η µν + h µν ( x , ρ ), where one may take ∂ µ h µν = η µν h µν = 0 � ∞ � h λ i d λ h λ h µν ( x , ρ ) = µν ( x ) ξ λ i ( ρ ) + µν ( x ) ξ λ ( ρ ) Λedge i in which the ξ λ i are discrete eigenmodes and the ξ λ are continuous Kaluza-Klein eigenmodes of the scalar Laplacian △ rad ; their eigenvalues give the Kaluza-Klein masses m 2 = g 2 λ in 4D from (10) h λ µν = 0 using △ θ,φ, y ,ψ,χ h λ µν ( x , ρ ) = 0: △ rad ξ λ ( ρ ) = − λξ λ ( ρ ) (4) h λ ( g 2 λ ) h λ µν ( x ) = µν ( x ) 11 / 29
The Schr¨ odinger problem One can rewrite the △ rad eigenvalue problem as a Schr¨ odinger equation by making the substitution � Ψ λ = sinh(2 ρ ) ξ λ after which the first derivative term is eliminated and the eigenfunction equation takes the Schr¨ odinger equation form − d 2 Ψ λ d ρ 2 + V ( ρ )Ψ λ = λ Ψ λ where the potential is 1 V ( ρ ) = 2 − tanh 2 (2 ρ ) 12 / 29
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