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Massive AdS Gravity from Theory

Costas BACHAS (ENS, Paris)

Conference on "Geometry and Strings’’

Schloss Ringberg, July 23 - 27, 2018 Based on two papers with Ioannis Lavdas

arXiv: 1807.00591 arXiv: 1711.11372

Earlier work with John Estes arXiv: 1103.2800

If I have time, I may also comment briefly on arXiv: 1711.06722 with Bianchi & Hanany Plan of Talk

1. Foreword

2. g-Mass from holography

3. Representation merging

4. Review of N=4 AdS4/CFT3

5. g-Mass operator

6. `Scottish Bagpipes’

7. 3 rewritings & bigravity

8. Final remarks 1 Foreword

An old question: Can gravity be `higgsed’ (become massive) ?

Extensive (recent & less recent) literature:

Pauli, Fierz, Proc.Roy.Soc. 1939 . . . . Nice reviews: Hinterblicher 1105.3735; de Rham 1401.4173 Schmidt-May & von Strauss 1512.00021

The question is obviously interesting, since any sound IR modification of General Relativity could have consequences for cosmology

[degravitating dark energy? `mimicking ’ ? . . . ] The main messages of this talk :

— Massive AdS Gravity is part of the string-theory landscape

— A quasi-universal, quantized formula for the mass

Setting is 10d IIB sugra, and holographic dual CFTs

If 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 time, I will comment on gauged 4d 9 Have little to say about the effective 4d theory around these string-theory vacua

But note that in certain sense massive AdS gravity is an `easier’ case:

m 0

the limit 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 g is smooth !

van Dam-Veltman-Zakharov discontinuity 69AAACz3icjVHLSsNAFD2Nr1pfVZdugkVwVRIRdFl047IF+4CmSDKd1qF5kUzUUhS3/oBb/SvxD/QvvDOmoBbRCUnOnHvPmbn3erEvUmlZrwVjbn5hcam4XFpZXVvfKG9utdIoSxhvssiPko7nptwXIW9KIX3eiRPuBp7P297oVMXbVzxJRRSey3HMe4E7DMVAMFcS5ThhJB1+Q+ek5kW5YlUtvcxZYOeggnzVo/ILHPQRgSFDAI4QkrAPFyk9XdiwEBPXw4S4hJDQcY5blEibURanDJfYEX2HtOvmbEh75ZlqNaNTfHoTUprYI01EeQlhdZqp45l2Vuxv3hPtqe42pr+XewXESlwS+5dumvlfnapFYoBjXYOgmmLNqOpY7pLprqibm1+qkuQQE6dwn+IJYaaV0z6bWpPq2lVvXR1/05mKVXuW52Z4V7ekAds/xzkLWgdV26rajcNK7SQfdRE72MU+zfMINZyhjiZ5x3jEE56NhnFt3Bn3n6lGIdds49syHj4AZzeUPA== No need for strong non-linearities of Vainshtein screening

Kogan, Mouslopoulos, Papazoglou ’00 Porrati ‘00 2 G-mass from Holography

Consider AdSAAAC83icjVHLSsNAFD3GV31XXboJFsFVSVTQpVoQlxXbKrSlJOm0DubFZKKU0r9w507c+gNu9R/EP9C/8M6YglpEZ0hy5txzTubOuLHPE2lZr2PG+MTk1HRuZnZufmFxKb+8UkuiVHis6kV+JM5dJ2E+D1lVcumz81gwJ3B9duZellT97IqJhEdhRfZi1gycbsg73HMkUa18sd8QgXnQPh20dsyGmj7rSMG7F9IRIrpWlJaUjiqD1nYrX7CKlh7mKLAzUEA2ylH+BQ20EcFDigAMISRhHw4SmnXYsBAT10SfOEGI6zrDALPkTUnFSOEQe0nvLq3qGRvSWmUm2u3RX3x6BDlNbJAnIp0grP5m6nqqkxX7W3ZfZ6q99ejrZlkBsRIXxP7lGyr/61O9SHSwp3vg1FOsGdWdl6Wk+lTUzs0vXUlKiIlTuE11QdjTzuE5m9qT6N7V2Tq6/qaVilVrL9OmeFe7pAu2f17nKKhtFW2raJ/sFPYPs6vOYQ3r2KT73MU+jlFGlbJv8IgnPBupcWvcGfefUmMs86zi2zAePgD6aqD4 CFT 4 $ 3 For a primary spin-2 operator: m2L 2 = ( 3) 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 4

(like stress tensor TAAACyXicjVHLSsNAFD2Nr1pfVZdugkVwVRIRdFl0I7ip0BfUUpLptMbmZTIRa+nKH3CrPyb+gf6Fd8YpqEV0QpIz595zZu69bux7qbCs15wxN7+wuJRfLqysrq1vFDe3GmmUJYzXWeRHSct1Uu57Ia8LT/i8FSfcCVyfN93hqYw3b3mSelFYE6OYdwJnEHp9jzmCqEatO3bcSbdYssqWWuYssDUoQa9qVHzBJXqIwJAhAEcIQdiHg5SeNmxYiInrYExcQshTcY4JCqTNKItThkPskL4D2rU1G9JeeqZKzegUn96ElCb2SBNRXkJYnmaqeKacJfub91h5yruN6O9qr4BYgSti/9JNM/+rk7UI9HGsavCoplgxsjqmXTLVFXlz80tVghxi4iTuUTwhzJRy2mdTaVJVu+yto+JvKlOycs90boZ3eUsasP1znLOgcVC2rbJ9cViqnOhR57GDXezTPI9QwRmqqJP3NR7xhGfj3Lgx7oz7z1QjpzXb+LaMhw/WMJGl ab )

ab

If conserved, @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 aT =0, the representation is short

& an algebraic manipulation gives =3 = m =0 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 ) g

canonical So to get G-mass we must allow 3d energy-momentum to leak out

Two options consistent with 3d conformal invariance:

— Couple to another 3d CFT: one massless & one massive (bigravity models)

— Leaking out to a 4d (or higher ?) defect CFT to get a `single’ massive-graviton theory

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.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 Need weak leakage

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 gap ✏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 1 ⌧

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 ✏ 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 Will focus on defect CFT, but bigravity is closely related

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 Bulk 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 SU(n) super Yang Mills 4 =4 AAACyHicjVHLSsNAFD2Nr1pfVZdugkWom5KIoMuiG3FV0bSFWiSZTutgmoRkopTixh9wq18m/oH+hXfGKahFdEKSM+fec2buvUESikw6zmvBmpmdm18oLpaWlldW18rrG80szlPGPRaHcdoO/IyHIuKeFDLk7STl/jAIeSu4OVbx1i1PMxFHF3KU8O7QH0SiL5gvifLOvWq0e1WuODVHL3sauAZUYFYjLr/gEj3EYMgxBEcESTiEj4yeDlw4SIjrYkxcSkjoOMc9SqTNKYtThk/sDX0HtOsYNqK98sy0mtEpIb0pKW3skCamvJSwOs3W8Vw7K/Y377H2VHcb0T8wXkNiJa6J/Us3yfyvTtUi0cehrkFQTYlmVHXMuOS6K+rm9peqJDkkxCnco3hKmGnlpM+21mS6dtVbX8ffdKZi1Z6Z3Bzv6pY0YPfnOKdBc6/mOjX3bL9SPzKjLmIL26jSPA9Qxwka8Mhb4BFPeLZOrcS6s0afqVbBaDbxbVkPHzz+kJQ= N

1/2 Bnry or Interface 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 BPS Gaiotto-Hanany-Witten theories

Osp(4 4) SO(2, 3) SO(4) Unbroken superconformal symmetry: 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 | ⇥ R

will NOT do (decompactification) 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 YM 0 Weak leakage ? ! scarce bulk a.o.t. bnry instead n2/F˜ 1 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 3 ⌧ degrees of freedom Idea of using defect CFT due to Karch + Randall `00, ‘01

Modelled with thin AdS4 in AdS5

Here: proper embedding of the idea;

Thin-brane approximation fails, so KK scale is 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 4 NOT 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 5 (problem of scale separation in flux vacua)

Porrati; Duff, Liu, Sati transparent AdS bnry Other approaches: Kiritsis; K+Niarchos multitrace coupling of two CFTs Aharony, Clark, Karch

I will discuss their relation in the end 3 Representations

AdS Higgsing converts a short rep into a long rep of SO(2, 3)

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

At the unitarity threshold 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 s +1 ! [s] [s] [s 1] 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 ! s+1 s+2

[2] [2] [1] For spin 2 : 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 3+✏ 3 4 Goldstone = massive vector !

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 =4 Osp(4 4)

With these fields should belong to reps of 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 N |

Dolan 0811.2740 These are classified completely Cordova, Dumitrescu, Intriligator 1612.00809 Reps come in four series:

(j,˜j) (j,˜j) (j,˜j) (j,˜j) L[s] A1[s] ˜ A2[0] B1[0]

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 ˜ ˜ AAAC7XicjVHLSsNAFD2N7/qqunQTLEKlUhIRdFl141LB1kKtJUmnddo0CZmJIKW/4M6duPUH3OpviH+gf+GdMQUfiE5Icu6595yZO9eNfC6kZb1kjLHxicmp6Zns7Nz8wmJuabkqwiT2WMUL/TCuuY5gPg9YRXLps1oUM6fv+uzU7R2o/OkliwUPgxN5FbFG3+kEvM09RxLVzBX2mrZZF43zQaG7aZ5J7reY2d0YNgei2C2O4qI9bObyVsnSy/wJ7BTkka6jMPeMM7QQwkOCPhgCSMI+HAh66rBhISKugQFxMSGu8wxDZEmbUBWjCofYHn07FNVTNqBYeQqt9mgXn96YlCbWSRNSXUxY7WbqfKKdFfub90B7qrNd0d9NvfrESlwQ+5duVPlfnepFoo1d3QOnniLNqO681CXRt6JObn7qSpJDRJzCLcrHhD2tHN2zqTVC967u1tH5V12pWBV7aW2CN3VKGrD9fZw/QXWrZFsl+3g7X95PRz2NVayhQPPcQRmHOEKFvK/xgEc8GaFxY9wadx+lRibVrODLMu7fAUXEngs= s+j+j+1 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 j+j+1 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 j+j ˜

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 >s+ j + j +1 s>0 absolutely long short short protected

Higgsing of the graviton amounts to

✏ 0 (0;0) ! (0;0) (1;1) L[0] A2[0] B1[0] , 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 1+✏ ! 1 2

massive supergraviton Goldstone mode graviton supermultiplet 112 bosonic d.o.f. (6 vectors, 2 scalars) & spin 3/2 fermion The spin-3/2 Goldstone multiplet is not part of the 4d gauged-sugra spectrum

So this corner of the landscape is not accessible by gauged sugra

That Higgsing is compatible with susy is not automatic.

For instance 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 =4 forbids the Higgsing of normal gauge symmetries N B [0](1;0) (0;1) because gauge fields, in 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 1 or B [0] , are protected 1 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 1 1

Louis, Triendl ’14 Corodova et al ‘16

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 =4 susy allows e-m to leak out but not flavor charge N 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 AdS /CFT 4 Review of =4 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 4 3 N

I will be brief — have talked about this before, only stress some features that we will need below.

CFT side Brane engineering using D3-D5-NS5 : Hanany, Witten ‘96

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 d Gaiotto, Witten ‘08

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 d SU(n)

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 d SU(n0)

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 Quiver

U(n ) gauge group 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 1

n . . . . biF-hyper

N 1 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 n2 n1 n0 n

F-hypers

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 0 mN 1 m2 m1

U(m ) 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 1 flavor group

In `good’ IR SCFT the interface depends only on discrete data

1/2

masses = FI terms = CS terms = 0 ; 3d YM couplings [mass] 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 !1

NB: magnetic quiver by exchanging D5 & NS5 has its own flavor symmetry Osp(4 4) The general type-IIB solution with 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 symmetry was found by | D’Hoker, Estes, Gutperle arXiv: 0705.0022 ; 0705.0024

The map to the Gaiotto-Witten interface CFTs was derived in

Assel, CB, Estes, Gomis arXiv: 1106.4253 ; 1210.2590

The geometry has the fibered form 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 M where, in order to 4 ⇥w 6 M =(S Sˆ ) ⌃ realize the R symmetry, 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 6 2 2 w ⇥ ⇥

ˆ All backgrounds can be written in terms of two harmonic functions h,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 h

which have singularities on the boundary of ⌃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

the explicit expressions are 2 2 2 2 2 ˆ2 2 2 ds10 = L4 dsAdS + f ds 2 + f dsˆ2 +4⇢ dzdz¯ 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 4 S S

ˆ ˆW 2 W 2 ˆW 2 ˆ L 8 = 16 UU ,f8 = 16 h8 U , fˆ8 = 16 hˆ8 U , ⇢8 = UU ,e4 = U , 4 2 3 ˆ 3 4ˆ4 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 W h h U U U

ˆ 2 2 ˆ ˆ ˆ 2 ˆ2 ˆ 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 =2hh @ h h W, =2hh @ h h W, and W = @ @ (hh) . 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 U | z | U | z | z z¯

There are also 3-form and 5-form fluxes, sourced by the 5-branes and D3-branes whose expressions we will not need today

The explicit harmonic functions with ⌃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 the infinite strip are D5

1 2 ··· ⌃ AdS / S5 50 ⇡/2 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 5 2 AdS / S Z 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 5 Z2 ⇥ ⇥

ˆ ˆ ˆ ··· 3 2 1

NS5

N i⇡ z a

h = i↵ sinh(z ) log tanh + 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 a + c.c. 4 2 2 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 a=1 X ✓ ◆ Nˆ ˆ ˆ ˆ z b

h =ˆ↵ cosh(z ) ˆb log tanh 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 + c.c. 2 2

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 ! Xb=1 Of particular interest is the supersymmetric Janus solution:

no 5-brane charges, only the (gauge coupling) varies with 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 z

ˆ

parameters: ↵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 , ↵ˆ, , L5, , ! 1 1

The detailed Janus geometry enters in the calculation of the mass,

but most other features are irrelevant; indeed what I describe

below should carry over to other solutions (lower 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 , other 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 ? ) N 5 Mass operator

The spectrum of spin-2 excitations from any 2-derivative gravity action depends only on geometry (not on scalar fields & fluxes)

Csaki, Erlich, Hollowood, Shirman hep-th/0001033 CB, Estes arXiv: 1103.2800 6 2 2 2 i j For any warped compactification ds 10 = L 4 ( y ) ds AdS 4 + gij(y) dy dy

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 i,j=1 the mass-eigenstate wavefunctions factorize: X

spin-2 wave operator

h (x, y)= (y) (x) 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 µ⌫ µ⌫ AdS (2) µ⌫ = µ⌫ 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

2 2

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 = ( 3)

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 M 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 +2=m (y)L (y) 6 AdS 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 4 4

invariant mass Reducing the 10d linearized Einstein equations & norm leads to:

2 2 L4 4 ij := @i L pgg @j =( + 2) M pg 4

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

6 2 = d y pgL ⇤ h 1| 2i 4 1 2 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 ZM6

2

AAACzXicjVHLSsNAFD2Nr1pfVZdugkVwVZIi6LLoxo1YwT6wVplMpzU0L5KJUGrd+gNu9bfEP9C/8M6YglpEJyQ5c+49Z+be60Sem0jLes0ZM7Nz8wv5xcLS8srqWnF9o5GEacxFnYdeGLcclgjPDURdutITrSgWzHc80XQGRyrevBVx4obBuRxGouOzfuD2XM4kURejS84882R8VbkulqyypZc5DewMlJCtWlh8wSW6CMGRwodAAEnYA0NCTxs2LETEdTAiLibk6rjAGAXSppQlKIMRO6Bvn3btjA1orzwTreZ0ikdvTEoTO6QJKS8mrE4zdTzVzor9zXukPdXdhvR3Mi+fWIkbYv/STTL/q1O1SPRwoGtwqaZIM6o6nrmkuivq5uaXqiQ5RMQp3KV4TJhr5aTPptYkunbVW6bjbzpTsWrPs9wU7+qWNGD75zinQaNStq2yfbZXqh5mo85jC9vYpXnuo4pj1FAn7wCPeMKzcWqkxp1x/5lq5DLNJr4t4+EDBhSS4A==

For direct-product reductions is the Laplace operator on 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 M 6

Integrating by parts gives 2 = d6y pgL4 (gij@ ?@ ) h |M | i 4 i j 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 ZM6 2

AAACzXicjVHLSsNAFD2Nr1pfVZdugkVwVZIi6LLoxo1YwT6wVplMpzU0L5KJUGrd+gNu9bfEP9C/8M6YglpEJyQ5c+49Z+be60Sem0jLes0ZM7Nz8wv5xcLS8srqWnF9o5GEacxFnYdeGLcclgjPDURdutITrSgWzHc80XQGRyrevBVx4obBuRxGouOzfuD2XM4kURejS84882R8VbkulqyypZc5DewMlJCtWlh8wSW6CMGRwodAAEnYA0NCTxs2LETEdTAiLibk6rjAGAXSppQlKIMRO6Bvn3btjA1orzwTreZ0ikdvTEoTO6QJKS8mrE4zdTzVzor9zXukPdXdhvR3Mi+fWIkbYv/STTL/q1O1SPRwoGtwqaZIM6o6nrmkuivq5uaXqiQ5RMQp3KV4TJhr5aTPptYkunbVW6bjbzpTsWrPs9wU7+qWNGD75zinQaNStq2yfbZXqh5mo85jC9vYpXnuo4pj1FAn7wCPeMKzcWqkxp1x/5lq5DLNJr4t4+EDBhSS4A== is (hermitean and) non-negative M

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sha1_base64="G30nvnJkyKBlKcCymNnrqTxJgV4=">AAACtXicjVLLSgMxFD0dX7VWrWs3g0VwVTJudCnowmUF+4BaZCZNa+y8TDJCKf6AWz9O/AP9C2/iCGoRzTAzJ+fec5KbmyiPpTaMvVS8peWV1bXqem2jXtvc2m7UuzorFBcdnsWZ6kehFrFMRcdIE4t+rkSYRLHoRdNTG+/dC6Vlll6aWS6GSThJ5Vjy0BDVvm40WYu54S+CoARNlCNrPOMKI2TgKJBAIIUhHCOEpmeAAAw5cUPMiVOEpIsLPKBG2oKyBGWExE7pO6HZoGRTmltP7dScVonpVaT0sU+ajPIUYbua7+KFc7bsb95z52n3NqN/VHolxBrcEPuX7jPzvzpbi8EYx64GSTXljrHV8dKlcKdid+5/qcqQQ06cxSOKK8LcKT/P2Xca7Wq3Zxu6+KvLtKyd8zK3wJvdJfU3+NnNRdA9bAWsFVwwVLGLPRxQG49wgnO00SHLER7x5J15t97dxz3wKuWF2MG34el34YWM3A==sha1_base64="SdqQ5EIOhFnWqsdtQBok6AC/AaU=">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sha1_base64="mktZiHtOKW2gd6ChqfL3X6lyMG8=">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 When 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 6 is compact and L 4 finite massless graviton with constant wavefunction (universality9 of 4d gravity) 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 6 Bagpipes manifolds

The `slightly’ non-compact manifolds that give massive gravity have the shape of ‘Scottish Bagpipes’

D3-brane 5 L throat

L 5 1 L ⌧

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 bag

L ⇠ bag

5-brane

They are obtained by taking with other parameters held fixed

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 , ↵ˆ 0 ! The limit is smooth in supergravity but not in string theory Here comes now the key idea:

To find the lowest-lying 4d graviton, replace the eigenvalue- by a minimization problem

+2= min d6y pgL4 (gij@ ?@ ) with d6y pgL2 2 =1. 0 4 i j 4 | | 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 ZM6 ZM6

If we were to truncate the pipes the (massless) graviton would be

1/2 6 2 2 1/2 0(y)= d y pgL := bag = V6 L 4 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 h 4i 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 ✓ZM6 ◆

bag

In the presence of the pipes

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 bag in the bag, and goes to zero at the '

the bottom of the pipes where 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 4 is minimal. The problem is now a minimization problem in the Janus-throat geometry with boundary conditions:

6 4 ij ? bag in matching region, 0 +2 min d y pgL4 g @i @j with ' throats ! ( 0 at infinity . 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 Z

Inserting the Janus solution gives

3 2 ⇡ 1 d at x = x , +2 = min L 8 dx (x) with (x) bag c 0 4 5 dx " xc G # ! ( 0 at x = , AAAEZnicjVHbbtNAEJ2kgRYDJQUhHngZESGlIo3s0NK+RKrgAR54KBK9SNnE8mWTrOKb7DU4svwv/BXiD8pfMLtxpEIVYC3bM+fMObOz6yaByKRp/mg0t1p37m7v3DPuP3i4+6i99/gii/PU4+deHMTpletkPBARP5dCBvwqSbkTugG/dBfvFH/5haeZiKPPcpnwcejMIjEVniMJsvca3xCRBSTwHdtEfDVAhkN6S5aGzJ1iKKLKZkkmqIpP5cjADatkiZi8RhZTOzys8KN9NCmR9fCkInajDJmIpF0WtldNKJzKJbJ5thAJHlj9ozBEvyAP7e85Ab6vusX+aitdLH21sVVHv6iQpWI2l/s4KQcViVbpmFr8rT+xTH/rofGrkPNKC1jNoWqj+iKTMRrM5TMRlR4de1YZirO10HVmVa3RuSNVWgyRhsMeYwZDs+Zv1GClmxfD1fQMe9SBR/7a3253zL6pF94OrDroQL3O4vZ3YOBDDB7kEAKHCCTFATiQ0TMCC0xICBtDSVhKkdA8hwoM0uZUxanCIXRB3xlloxqNKFeemVZ71CWgNyUlwkvSxFSXUqy6oeZz7azQTd6l9lR7W9Lfrb1CQiXMCf2Xbl35vzo1i4QpnOgZBM2UaERN59UuuT4VtXO8MZUkh4QwFfvEpxR7Wrk+Z9SaTM+uztbR/LWuVKjKvbo2h59ql3TB1p/XeTu4GPQts299Ouycvq2vegeewwvo0n0ewyl8gDM4B6+53Txovmkeb123dltPW89Wpc1GrXkCv60W/gInSgHd Z ✓ ◆ 1

cosh 2x + cosh 2 (x):= G cosh 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 bag param ✓ ◆

pipe param =

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 bag 1

The problem can be integrated analytically with the result 1 I(x, a) 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 =Rez 0(x, a) bag 1

2 I( ,a) 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 = cosh

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 '  1

a3 pa +1+pa 1 tanh x a2 tanh x I(x, a) = log 2 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 2(a2 1)3/2 pa +1 pa 1 tanh x (a2 1) [(a + 1) (a 1) tanh x ] 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

3 3⇡ 8 2 0 +2 = L5 bag J(a)

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 4

J()

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 Janus correction factor

3 2 1 3a 2 3a J(a) = log a + a 1 (a2 1)3/2 (a2 1)

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 h p i

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 is decompactifion limit !1 no continuous Higgsing from this CB, Estes ‘11 Cartoon illustration:

5

¯ 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 5 S

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 M 4 ⇥ 6 ⇥

L (x) 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 4

(x) 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 0

(III)

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 L

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 bag ⇠

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 bag

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 5

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 x xc 7 Rewritings and bigravity

cf Karch-Randall (L /L )2 ⇠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 5 4 3⇡3L 8 m2 L2 = 5 J(cosh ) (L /L )8 geometric: g 4 AAAC1XicjVHLSsQwFD3W97vq0k1xEMbN2MqILgfduHCh4DiCjkObiRrsiyQVBpmduPUH3OoviX+gf+FNrOAD0ZS2J+fec5J7b5THQmnffx5wBoeGR0bHxicmp6ZnZt25+UOVFZLxJsviTB5FoeKxSHlTCx3zo1zyMIli3oout028dcWlEll6oHs5byfheSrOBAs1UR3XPVEi8aq7nfXV3U595XSz41b8mm+X9xMEJaigXHuZ+4QTdJGBoUACjhSacIwQip5jBPCRE9fGNXGSkLBxjj4mSFtQFqeMkNhL+p7T7rhkU9obT2XVjE6J6ZWk9LBMmozyJGFzmmfjhXU27G/e19bT3K1H/6j0SojVuCD2L91H5n91phaNM2zaGgTVlFvGVMdKl8J2xdzc+1SVJoecOIO7FJeEmVV+9NmzGmVrN70NbfzFZhrW7FmZW+DV3JIGHHwf509wuFYL/FqwX680tspRj2ERS6jSPDfQwA720CTvK9zjAY9Oy+k7N87te6ozUGoW8GU5d28rrZRm 5 4 4V L2 ⇥ 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 6 bag ⇠ h 4i

quantized eff coupling

3 n2 2 m2 L2 = 4 J(cosh ) gravitational: g 4 16⇡2 L2 ⇥ AAADoHicjVFdaxQxFL3T8aOuX1t99CV0ESrIMrMW2xeh1Zcigi24bbFpl0w2nQ2bmaSZTKEM8ztF/AP6L7zJTkEtq2aYmXPPPefm3iQzSlYuSb5GK/Gt23furt7r3X/w8NHj/tqTw0rXlosx10rb44xVQslSjJ10ShwbK1iRKXGUzd/5/NGlsJXU5Sd3ZcRpwfJSnkvOHFKT/rfibDRpqC1I3tKX5MNk82xEemTZouQNJaR5RVBbohIZjeVJ+poa6eN2ubdB8ZwZw8IeCx+hipW5Et3G1IZo0VDG8pa0f2kG3U4Wogrw/QbluprRqVCOUTOTL3qT/iAZJmGRmyDtwAC6ta/7X4DCFDRwqKEAASU4xAoYVPicQAoJGOROoUHOIpIhL6CFHnprVAlUMGTn+M0xOunYEmNfswpujrsofC06CTxHj0adRex3IyFfh8qeXVa7CTV9b1f4z7paBbIOZsj+y3et/F+fn8XBOWyHGSTOZALjp+NdlTqciu+c/DKVwwoGOY+nmLeIeXBenzMJnirM7s+Whfz3oPSsj3mnreGH7xIvOP3zOm+Cw9EwTYbpweZg52131avwDNZhA+9zC3ZgD/ZhDDzajfLIRBfxerwXf4wPFtKVqPM8hd9W/PknVIzEkg== h 4ibag

˜ : generalized

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 ˜ d 2 2 3⇡ CFT: m L = J(cosh ) free energy g 4 ˜ ⇥ 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 32F3 Giombi + Klebanov ‘14 Simple to extend to models of bigravity:

quiver 1 quiver 2

n Field Theory

low rank messenger

bag 1 bag 2 Gravity thin Janus throat One massless and one massive graviton

3 n2 2 2 m2 L2 = 4 + 4 0 J(cosh ) g 4 16⇡2 L2 L2 ⇥ AAAEGnicjVHLbtNAFL2ueZTwSsuSzYgIqQgU2Ula0gVSBRuEWBSpaStlkmjsTJxR/BiNx0iV5T/pn7Bjh9giIZbsePwEdyYOKou0TJTk3HPPuY+ZQMYi1573zdlwr12/cXPzVuP2nbv37je3to/zrFAhH4RZnKnTgOU8FikfaKFjfioVZ0kQ85Ng8crkT95zlYssPdJnko8SFqViJkKmkZpsOUckGXcmJVUJiSr6jLyd9MYd0iDrDiUvKCFll6A2RSUyGTYg/h6VwsTVJd6Yz/SQlIgWTEpmWy3tmGNpFPO6P1U2Ws4VsKgiFSFPLxnrYs2yg7NRqUTCqyvK/60/XulNI6pENNej9d1QolGdW/hmh4ZZPqdTHmtG5Vw8aUyaLa/ted1ur0MM2O3v7lvQ7e/vER+BOS2oz2HW/AoUppBBCAUkwCEFjTgGBjl+huCDBxK5EZTIKUTC5jlU0EBvgSqOCobsAn8jjIY1m2JsaubWHWKXGL8KnQQeoydDnUJsuhGbL2xlw66rXdqaZrYz/A/qWgmyGubIXuVbKf/XZ3bRMIO+3UHgTtIyZruwrlLYWzGTkwtbaawgkTN4inmFOLTO1T0T68nt7uZumc3/sErDmjistQX8NFPiA69ekawHx52277X9d73Wwcv6qTfhITyCHXzP53AAr+EQBhA6H5zvzi/nt3vufnQ/uZ+X0g2n9jyAf4775Q8Zwex5 h 4ibag h 4ibag0

Similar to double-trace (one-loop) deformation formula of Aharony, Clark, Karch ‘06

But:

the background is exactly conformal (no RG running)

Integrating-in the messengers restores local geometry at the `expense’ of quantization of coupling 8 Last remarks

Massive AdS gravity is a corner of the string-theory landscape

The graviton mass is a (non-protected) quantized observable in these models. Can one compute it from CFT and match ?

Other examples ? Effective low-E theory ? Minkowski ? Thank You notation of Cordova et al =4Multiplet String mode Gauged SUGRA N scalar monopole (0;0) A2[0]1 Graviton yes harmonics on S2

(1;0) B1[0]1 D5 gauge bosons yes

(0;1) B1[0]1 NS5 gauge bosons yes

1 (R>1;0) Open F-strings, R 2 `a `b + N B1[0]R 2 | | only R =2 Closed strings, R N 2 superpotential 1 ˆ ˆ (0;R0>1) Open D-strings, R0 2 `aˆ `ˆb + N B1[0]R 2 | | only R0 =2 0 Closed strings, R0 N 2

(R 1;R0 1) missing (1;1) B1[0] Kaluza Klein gravitini (R,R0 N) no R+R0 2 superpotential

(R>0;R0>0) A2[0] Kaluza Klein (R,R0 N) no 1+R+R0 2

(R;R0) A1[j>0] Stringy excitations no 1+j+R+R0

Table 1: The short =4superconformal multiplets in the notation of ref. [12], possible dual N single-string excitations in the Type IIB solutions of [4][5], and the fate of these excitations after the gauged-supergravity truncation. In the middle column N = 1, 2, is the set of non-zero natural { ···} Fornumbers. more A yes/no on entrythe inBPS the third spectrum column indicates see that the CB, excitation Bianchi, survives/does Hanany not survivearXiv: 1711.06722

in gauged supergravity. Among the R =2or R0 =2modes some may (but need not) survive.

11