Massive Ads (Super)Gravity & Holography Corfu

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Massive AdS (Super)Gravity & Holography

Costas BACHAS Corfu (ENS, Paris) September 2019 Based on arXiv: 1711.11372 ; 1807.00591 with Ioannis Lavdas

& arXiv: 1905.05039

Collaborators on closely related work:

Benjamin Assel, Massimo Bianchi, John Estes, Jaume Gomis

Amihay Hanany & Bruno Le Floch A Higgs mechanism for Gravity ?

Fierz - Pauli 1939 . . . .

Van Dam, Veltman, Zakharov 1970 Vainshtein 1972 Boulware, Deser 1973 A question with a long history: . . . .

Arkani-Hamed, Georgi, Schwartz 2002 ’t Hooft; Chamseddine, Mukhanov; . . .

de Rham, Gabadadze, Tolley 2011 Hassan, Rosen 2011 cf. (spin-1) gauge theory:

1 µ⌫ 2 µ µ =tr F F + v (A U †@ U)(A U †@ U) 4g2 µ⌫ µ µ

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 

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 : three [for SU(2)] scalar Stueckelberg ﬁelds

Breakdown of perturbative unitarity at E v = m/g 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 ⇠

Restore with Higgs particle:

v + h = U 0 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 ✓ ◆ For (spin-2) gravity:

massless vector + scalar Stueckelberg ﬁelds : massive vector

Geometric interpretation from bimetric theories (two Universes + interaction):

2 4 2 4 p Sbm mPlanck d xpg[R(g)+]+MPlanck d X G[R(G)+⇤]

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 ⇠ Z Z

2 2 4 µ⌫ ˆ + m mPlanck d xpgf(g Gµ⌫ ) 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 Z

Gˆ = G (X)@ XM @ XN

where 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 µ⌫ MN µ ⌫ Mixing two massless gravitons : one massless & one massive

In the limit MAAAC3nicjVHLSsNAFD3G9zvqStwEi+BKUhF0WXTjRqhgW8GUmkynOjQvJhOhFHHnTtz6A271c8Q/0L/wzpiCWkQnJDlz7j1n5t4bpKHIlOu+jlijY+MTk1PTM7Nz8wuL9tJyPUtyyXiNJWEiTwM/46GIeU0JFfLTVHI/CkLeCLoHOt644jITSXyieilvRv5FLDqC+Yqolr161Op7MnKqoR+z7rXjqcTxRNxRvZZdcrdcs5xhUC5ACcWqJvYLPLSRgCFHBI4YinAIHxk9ZyjDRUpcE33iJCFh4hzXmCFtTlmcMnxiu/S9oN1Zwca0156ZUTM6JaRXktLBBmkSypOE9WmOiefGWbO/efeNp75bj/5B4RURq3BJ7F+6QeZ/dboWhQ72TA2CakoNo6tjhUtuuqJv7nypSpFDSpzGbYpLwswoB312jCYzteve+ib+ZjI1q/esyM3xrm9JAy7/HOcwqG9vlQkf75Qq+8Wop7CGdWzSPHdRwSGqqJH3DR7xhGfr3Lq17qz7z1RrpNCs4NuyHj4AGmWZgw== the massless one decouples Planck !1 and we are left with massive gravity.

universe UNIVERSE

XM (xµ)

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

Four Stueckelberg scalars

In consistent effective theory one of these should drop out. This happens in the dRGT [3 parameter] action

Ultimate breakdown scale in Minkowski : E =(m2m )1/3 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 ? Planck

LIGO quotes 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 10 eV  Tension with GW data: 6

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 10 eV ? ⇠

Analog of ‘Higgs particle’ ? Need quantum (UV) embedding

Have found one (but) for AdS (super)gravity

technical control ? Massive AdS supergravities

Particle states in AdSAAAC1HicjVHLSsNAFD2Nr1ofjbp0EyyCIJREBF1W3bisaB/QlpKk0xqaF5OJUGpX4tYfcKvfJP6B/oV3xhTUIjohyZlzz7kz914n9r1EmOZrTpubX1hcyi8XVlbX1ov6xmY9iVLuspob+RFvOnbCfC9kNeEJnzVjzuzA8VnDGZ7JeOOG8cSLwisxilknsAeh1/dcWxDV1YvjNg+Mk97lpDvu7VuTrl4yy6ZaxiywMlBCtqqR/oI2eojgIkUAhhCCsA8bCT0tWDARE9fBmDhOyFNxhgkK5E1JxUhhEzuk74B2rYwNaS9zJsrt0ik+vZycBnbJE5GOE5anGSqeqsyS/S33WOWUdxvR38lyBcQKXBP7l2+q/K9P1iLQx7GqwaOaYsXI6twsS6q6Im9ufKlKUIaYOIl7FOeEXeWc9tlQnkTVLntrq/ibUkpW7t1Mm+Jd3pIGbP0c5yyoH5QtwheHpcppNuo8trGDPZrnESo4RxU1NfNHPOFZq2u32p12/ynVcplnC9+W9vAB4gKVJA== d +1 form representations of the

conformal group SO(d) SO(2,d) 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 R ⇥ energy ang.mom.

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 Lowest-lying (in 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 =3 ): spin s , energy [s] Corresponding unitary representation denoted 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

Massless particles for s> : 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 = s +1

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 2 [s] 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 is a short representation At the unitarity threshold, 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 s +1 !

[s] [s] [s 1] 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 ! s+1 s+2

Long Short Stueckelberg

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

Massless graviton + `Goldstone’ Massive graviton

Porrati 2001, 2003 Supersymmetry:

Conformal group Superconformal group

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 6  e.g. SO(2, 3) Osp(4 N) 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 ! | Nahm 1978

Goldstone multiplet Goldstone supermultiplet includes spin 3/2

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 4 Absolutely protected massless supergravitons  =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 [short spin-2 supermultiplets] 1 ) N max

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  2

Cordova, Dumitrescu, Intriligator 1612.00809 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 D>5 1 ruled out N> max

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 2

Table 1: The AdSD supergravities for which the massless graviton is in an absolutely protected representation. We list the number of supersymmetries, the superconformal algebra g, and the absolutely protected representation in the notation of ref.[9]. The list includes all cases in D>5 dimensions, and all cases with more than half-maximal supersymmetry for reasons explained in the main text. The table has redundancies: when the massless graviton is protected for given (D, 0) it is also protected for all (D, > 0). N N N be clear in the following section. A quick mnemonic for string theorists is that Higgsing is only possible if the dual SCFTd can couple as a defect to a higher-dimensional bulk theory without breaking its own superconformal symmetries. This is indeed impossible whenever 6 NQ > 8 or for d =5, 6. Let me stress however that these no go statements are purely kinematical and do not depend on any details of the Brout-Englert-Higgs (BEH) mechanism. The only assumptions are superconformal invariance and unitarity. It should be also noted that the existence of marginal superconformal deformations is neither a necessary nor a suﬃcient condition for the Higgsing of the supergraviton. Indeed superconformal manifolds exist for =1, 2, 4 in d =4but only for 2 in d =3[26]. N N  On the other hand, Higgsing of the supergraviton is forbidden for =4in AdS whereas N 5 it is possible for =4in AdS . N 4 6Our counting of supersymmetries in the table is the standard counting on the ﬁeld theory side, i.e. N is the number of spinor supercharges of the SCFTd. Superconformal symmetries double this number on the gravity side. The two maximal massive supergravities in particular, dual to =4SCFT3 and =2 N N SCFT4,correspondtoAdSD vacua of gauged N =4supergravity in D =4and D =5dimensions. In both cases the number of Poincaré supercharges of the SCFT is NQ =8.

5 in the following section. The cases =2, d =3and =1, d =4are not ruled out as easily seen with explicit N N examples. The marginal or relevant deformations for =1, d =4are superpotential N deformations in chiral multiplets with U(1) charge 2 r>2 . One can form a marginal R 3 deformation (r = 2) from the product of two chiral multiplets with r =1. A famous example is the Klebanov-Witten theory [28] in which the bifundamental chiral ﬁelds have R charge 1 r = 2 , so there are gauge-invariant operators with r =1. The product of two such operators can be used to deform two previously decoupled Klebanov-Witten theories. The case =2, N d =3is even simpler, since quartic superpotentials are classically marginal. They can be used to make two decoupled theories interact. For instance =4SCFTs have a plethora N of marginal =2deformations of this kind (see e.g. [29]). N These conclusions are summarizedkinematically in table 2 . Also listed in this table are the multiplets that provide the extra polarizationallowed states of a massiveMsugras supergraviton in all the cases where Higgsing is a priori allowed. Consistency requires that such Stueckelberg multiplets should be available whenever two decoupled theories are made to interact. We will now see that this is indeed always the case.

Susy Multitrace Massless graviton Stueckelberg (0;0) (4;0) (2;2) =2 no A A¯ [0; 0] B B¯ [0; 0] A B¯ [0; 0] cc AdS 2 2 2 1 1 4 2 1 3 5 N (0) (1) =1 yes A A¯ [1; 1] LA¯ [1; 0] cc N 1 1 3 2 7/2 0,0) (2,2) =4 no A2[0]1 B1[0]2 N (0) (2) AdS4 =3 no A1[1]3/2 A2[0]2 N (0) (1) =2 yes A A¯ [2] LA¯ [1] cc N 1 1 2 1 5/2 =1 yes A [3] L[2] N 1 5/2 3

Table 2: The AdS supergravities for which the graviton multiplet can obtain a mass by gauging a global symmetry of the dual SCFT. The alternative ‘multi-trace mechanism’ is only possible for N 4 supercharges. The superconformal algebras are su(2, 2 ) in d =4and osp( 4) in d =3. Q  |N N| The two right-most columns list the massless graviton and the Stueckelberg multiplets in the notation N 1 N 1 of ref. [9]. The massive graviton multiplets contain 2 Q bosonic and 2 Q fermionic states.

3.3 Stueckelberg multiplets Since the reader may not be familiar with the representation-theoretic notation of ref. [9], let me discuss the case of =1supersymmetry in d =4in more familiar language. In this N case the conserved and traceless energy-momentum tensor sits in a real vector superﬁeld ↵˙ ↵ R↵↵˙ , subject to the conditions D¯ R↵↵˙ = D R↵↵˙ =0. This superﬁeld includes in addition

9 Case of N=4, D=4 supergraviton :

✏ 0 (0;0) ! (0;0) (1;1) L[0] A2[0] B1[0] , AAADbXicjVHbbtQwEJ1suJQAZQHxBEIWq4q2oFW8qgRSBSrlhQceisS2ldbLynHd3ahOHNkOUEX5H/4I8QfwxC8wNlluFQJHScZnzjnj8WSVyq1L009RLz53/sLFlUvJ5StXV6/1r9/Yt7o2Qo6FVtocZtxKlZdy7HKn5GFlJC8yJQ+yk+c+f/BWGpvr8rU7reS04PMyP84FdwjN+h9eTtLprKEPmKxsrnTZkjfNerqdbrSEEEZYwrTXS9cwW3ClyJJImNOEpC1p2HuTzxeOG6PfNWxhKy7kZkNF0bZokqwxZM9/UtA18c7k2Wzki9MfBZmuVG3J7ox6fIQ43aYbLSMPSTLrD9JhGhY5G9AuGEC39nT/IzA4Ag0CaihAQgkOYwUcLD4ToJBChdgUGsQMRnnIS2ghQW2NLIkMjugJfue4m3RoiXvvaYNaYBWFr0ElgTXUaOQZjH01EvJ1cPbo37yb4OnPdor/rPMqEHWwQPRfuiXzf3W+FwfH8Dj0kGNPVUB8d6JzqcOt+JOTX7py6FAh5uMjzBuMRVAu75kEjQ29+7vlIf85MD3q96Lj1vDFnxIHTP8c59lgfzSk6ZC+2hrs7HajXoHbcA/WcZ6PYAdewB6MQUSr0Vb0JHra+xrfiu/Ed79Te1GnuQm/rfj+NyJjxRk= 1+✏ ! 1 2

massive supergraviton Goldstone mode graviton supermultiplet 112 bosonic d.o.f. (6 vectors, 2 scalars) & 4 spin-3/2 fermion

(0;0) (0;0) (1;0) (0;1) (0;0) bosons : [0] [0] [1] [2] 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 1 2 2 3 1 1 1 1 ( 2 ; 2 ) ( 2 ; 2 ) fermions : [1/2] 3 [3/2] 5

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 2 2

(1;1) (1;1) (1;0) (0;1) (2;0) (0;2) (1;0) (0;1) (1;1) (0;0) bosons : [0] [1] [0] 2 3 3 (0;0) (1;0) (0;1) (1;1) (0;0) (0;0) [1] [0] [0] , 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 4 4 5

3 1 1 3 1 1 1 1 3 1 1 3 1 1 1 1 ( 2 ; 2 ) ( 2 ; 2 ) ( 2 ; 2 ) ( 2 ; 2 ) ( 2 ; 2 ) ( 2 ; 2 ) 2( 2 ; 2 ) ( 2 ; 2 ) fermions : [1/2] 5 [3/2] 7 [1/2] 7 [1/2] 9

AAAEO3icpVFNb9NAEJ3YQItbIIUjlxURUnopdgpKoZcKLhyLRNpKcYjszaZYtb3urhepWvmn8TsQEjcugLhyZ3ZZKw0taiXWsv3mzbz52EmrPJN1GH7ueP6Nm7dWVm8Ha+t37t7rbtw/kFwJykaU51wcpYlkeVayUZ3VOTuqBEuKNGeH6ckr4z/8wITMePm2PqvYpEiOy2ye0aRGarrRUUTHoiBzJgokZPOCkIDEp6cqmRF7xjp6MmgmU62fxRxTkUHTvNN9vd1auzpq4SaJeZUrSfoLbncRuRlc6l/Im9Y/1tuu6HCp6OWqwKmWuh3+d7ete0Cu7Pd84efX6Hja7YVboT3kIogc6IE7+7z7CWKYAQcKCgpgUEKNOIcEJD5jiCCECrkJaOQEosz6GTQQoFZhFMOIBNkT/B6jNXZsibbJKa2aYpUcX4FKAo9RwzFOIDbViPUrm9mw/8qtbU7T2xn+U5erQLaG98hepWsjr6szs9Qwhx07Q4YzVZYx01GXRdlbMZ2Tc1PVmKFCzuAZ+gViapXtPROrkXZ2c7eJ9X+zkYY1NnWxCr6bLnHB0d/rvAgOBlsR4jdPe3sv3apX4SE8gj7ucwh78Br2YQS088XzvDVv3f/of/V/+D//hHodp3kAS8f/9RtaMA7/ 2 2 2 2 A technical challenge :

Construct N=4, D=4 massive AdS supergravity 1 m

AAAC43icjVHLSsNAFD3G97vqUpDBIrgqiQi6rLpxqWitYEpJ0mkNnTyYTAQp3blzJ279Abf6L+If6F94Z0xBLaITkpw595wzc2f8VISZsu3XEWt0bHxicmp6ZnZufmGxtLR8liW5DHgtSEQiz30v4yKMeU2FSvDzVHIv8gWv+90DXa9fcZmFSXyqrlPeiLxOHLbDwFNENUtrLGKuEKznuAnpmMuFaPZcGbG91km/3yyV7YptBhsGTgHKKMZRUnqBixYSBMgRgSOGIizgIaPnAg5spMQ10CNOEgpNnaOPGfLmpOKk8Ijt0rdDs4uCjWmuMzPjDmgVQa8kJ8MGeRLSScJ6NWbquUnW7G/ZPZOp93ZNf7/IiohVuCT2L99A+V+f7kWhjV3TQ0g9pYbR3QVFSm5ORe+cfelKUUJKnMYtqkvCgXEOzpkZT2Z612frmfqbUWpWz4NCm+Nd75Iu2Pl5ncPgbKviED7eLlf3i6uewirWsUn3uYMqDnGEGmXf4BFPeLa4dWvdWfefUmuk8Kzg27AePgAIEJr9 ⌧ `AdS

NB1: The extra spin-3/2 Goldstones NOT part of 4d gauged supergravity But part of (gauged) N=8 sugra CB, Severin Luest in progress

NB2: Massive N=4 supergravity in Minkowski with higher-R terms by

Ferrara, Kehagias, Dieter Luest 2018 Holographic duals

DICTIONARY: CFT gravity

energy-momentum tensor graviton h t 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 µ⌫

AAACy3icjVHLSsNAFD2Nr1pfVZdugkVwVRIRdFl040aoYB9QS0mm0zo2L5KJUGOX/oBb/S/xD/QvvDOmoBbRCUnOnHvOnbn3upEnEmlZrwVjbn5hcam4XFpZXVvfKG9uNZMwjRlvsNAL47brJNwTAW9IIT3ejmLu+K7HW+7oVMVbtzxORBhcynHEu74zDMRAMEcS1c7kpJeJm0mvXLGqll7mLLBzUEG+6mH5BVfoIwRDCh8cASRhDw4SejqwYSEirouMuJiQ0HGOCUrkTUnFSeEQO6LvkHadnA1or3Im2s3oFI/emJwm9sgTki4mrE4zdTzVmRX7W+5M51R3G9PfzXP5xEpcE/uXb6r8r0/VIjHAsa5BUE2RZlR1LM+S6q6om5tfqpKUISJO4T7FY8JMO6d9NrUn0bWr3jo6/qaVilV7lmtTvKtb0oDtn+OcBc2Dqk344rBSO8lHXcQOdrFP8zxCDWeoo6Hn+IgnPBvnRmLcGfefUqOQe7bxbRkPH+fvkuE= ij

mass 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 dimension 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

( 3) = m2`2 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 AdS

i conserved @ t =0 massless

AAAC2nicjVHLSsNAFD3Gd31FxZWbwSK4klQE3QhFNy4VrBZsLZN01LFpEiYTQYobd+LWH3CrHyT+gf6Fd8YR1CI6IcmZc+85M/feMItlroPgZcAbHBoeGR0bL01MTk3P+LNzh3laqEjUojROVT3kuYhlImpa6ljUMyV4N4zFUdjZMfGjS6FymSYH+ioTzS4/S+SpjLgmquUvNDKutOTxiWS61ZMX12yLBS2/HKwGdrF+UHGgDLf2Uv8ZDbSRIkKBLgQSaMIxOHJ6jlFBgIy4JnrEKULSxgWuUSJtQVmCMjixHfqe0e7YsQntjWdu1RGdEtOrSMmwTJqU8hRhcxqz8cI6G/Y37571NHe7on/ovLrEapwT+5fuM/O/OlOLxik2bQ2SasosY6qLnEthu2Juzr5UpckhI87gNsUV4cgqP/vMrCa3tZvecht/tZmGNfvI5RZ4M7ekAVd+jrMfHK6tVgjvr5er227UY1jEElZonhuoYhd7qJF3Dw94xJPX8G68W+/uI9UbcJp5fFve/TvtGJd5 ij @it = V Stueckelberg leaking 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 ij j BIMETRIC DUAL :

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 = + + L Lcft LCFT L

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 (decoupled) t ,T separately conserved =0 = 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 ij ij L )

=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 two massless gravitons )

tij + Tij total conserved p 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 c + C

`small’ coupling =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 ) Ctij cTij

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 =3+✏, ✏ 1 p 2 2 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 C c + c C ⌧

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 limit of massive gravity !1 TWO `MECHANISMS’ :

1. Double trace = tr(o)tr( ) `quantum’ in gravity 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 L O

CFT perturbation theory: Kiritsis 2006 Aharony, Clark, Karch 2006 2 1 1 ✏ =# ( + ) Kiritsis, Niarchos 2008 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 c C

Relevant/marginal deformations =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 N max

) 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  4

Superpotential deforms 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 W = CB 1905.05039

Stueckelberg superﬁeld 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 D D ↵ ↵ 2. Gauge mediation gauge common global symmetry

quiver 1 quiver 2

low-n or weakly-coupled U(n) messenger gauge ﬁeld

1 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 =2,d=4 Allowed by N = max

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 AAAC33icjVHLSsNAFD3GV62vqjvdDBbBlSQi6EYounElFWwrtKVM4liDeTGZiKUU3LkTt/6AW/0b8Q/0L7wzpuAD0QlJzpx7zpm5M24S+Kmy7ZcRa3RsfGKyMFWcnpmdmy8tLNbTOJOeqHlxEMsTl6ci8CNRU74KxEkiBQ/dQDTci31db1wKmfpxdKx6iWiHvBv5Z77HFVGd0vIh22V9pxWTiG0OWL8lQxbyqwFjnVLZ3rDNYD+Bk4My8lGNS89o4RQxPGQIIRBBEQ7AkdLThAMbCXFt9ImThHxTFxigSN6MVIIUnNgL+nZp1szZiOY6MzVuj1YJ6JXkZFgjT0w6SVivxkw9M8ma/S27bzL13nr0d/OskFiFc2L/8g2V//XpXhTOsGN68KmnxDC6Oy9Pycyp6J2zT10pSkiI0/iU6pKwZ5zDc2bGk5re9dlyU381Ss3quZdrM7zpXdIFO9+v8yeob244hI+2ypW9/KoLWMEq1uk+t1HBAaqoUfY1HvCIJ4tbN9atdfchtUZyzxK+DOv+HckZmIc= 2 N =4,d=3

(conserved current) (free vector) Stueckelberg in 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 multiplet ⌦ String theory embedding

The embedding uses a well-studied class of

N = 4 AdS /SCFT 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 4 3 dual pairs

On CFT side: `good linear-quiver’ theories with IR ﬁxed points

Hanany, Witten 1996 Gaiotto, Witten 2008

The quiver describes the gauge group and hypermultiplet content

masses = FI terms = 0; gauge couplings ﬂow to 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 1 no other free parameters The ﬁeld content of the (Lagrangian) theory is determined by a quiver diagram:

A-type quiver bifundamental hypers nN 1 . . . . n2 n1

fundamental hypers

mN 1 m2 m1

Gauge group: U(n1) U(n2) U(nN 1) ⇥ ⇥ ···⇥

Flavor symmetry: [U(m1) U(m2) U(mN 1)] /U(1) ⇥ ⇥ ···⇥ On gravity side: Exact IIB solutions and holographic dictionary known

D’Hoker, Estes, Gutperle 2007 Aharony, Berdichevsky, Berkooz, Shamir 2011

Assel, CB, Estes, Gomis 2011

Gauge-group ranks = f(brane charges) continuous in supergravity

Step 1: Compute the geometry in the degeneration limit 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 0 !

quiver 1 quiver 2

Step 2: Compute the spin-2 Kaluza-Klein spectrum cf CB, Estes 2011 Graviton mass

1 2 2 2 1 1 ✏ m `AdS = e↵ ( + )

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 ' 3 c C

n2 tanh3 2 = e↵ 4⇡4 tanh 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 | |

dilaton variation of n2 d.o.f. mediator :

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

two universes 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 c ⇠ d.o.f. universe

mAAAC1XicjVHLSsNAFD2Nr1pfUZdugkVwVVIRdFl147KifUBbSpJOa+jkwWRSKKU7cesPuNVfEv9A/8I7YwpqEZ2Q5My595yZe68bcz+Rtv2aMxYWl5ZX8quFtfWNzS1ze6eeRKnwWM2LeCSarpMw7oesJn3JWTMWzAlczhru8ELFGyMmEj8Kb+Q4Zp3AGYR+3/ccSVTXNAOrzTjvTtoisM5619OuWbRLtl7WPChnoIhsVSPzBW30EMFDigAMISRhDgcJPS2UYSMmroMJcYKQr+MMUxRIm1IWowyH2CF9B7RrZWxIe+WZaLVHp3B6BSktHJAmojxBWJ1m6XiqnRX7m/dEe6q7jenvZl4BsRK3xP6lm2X+V6dqkejjVNfgU02xZlR1XuaS6q6om1tfqpLkEBOncI/igrCnlbM+W1qT6NpVbx0df9OZilV7L8tN8a5uSQMu/xznPKgflcqEr46LlfNs1HnsYR+HNM8TVHCJKmrkPcIjnvBsNIypcWfcf6YauUyzi2/LePgAEd2VnA== `AdS can be made parametrically small 1 3 mmPlanck Breakdown of Stueckelberg theory: E? ⇠ ` 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 ✓ AdS ◆

Spins < 2 do not help (?)

Tower of spin-2 or more must come down in 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 0 limit !

Non BPS spin-2 modes (in Janus throat) shown indeed to condense

visible in CFT ? bootstrap ?

Similar phenomenon for double-trace coupling ? Conclusions & Outlook

Massive AdS N=4, D=4 supergravity can be UV completed (part of the landscape)

Construct the Lagrangian; deform N=8 ?

Embedding for D=5 ? (class-S theories)

Minkowski ? Phenomenology ?