Massive AdS Gravity from String 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 dark matter’ ? . ] 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 <latexit 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sha1_base64="mybtUkQUT/MZwyOi8nPg/PbM4YA=">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</latexit> g is smooth ! van Dam-Veltman-Zakharov discontinuity 69<latexit sha1_base64="hYm/eqY6JDpRDHLCrWE9GUmdBr0=">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</latexit> No need for strong non-linearities of Vainshtein screening Kogan, Mouslopoulos, Papazoglou ’00 Porrati ‘00 2 G-mass from Holography Consider AdS<latexit sha1_base64="FvYRYTpFuM5bjTRrOhZKbu/jeSQ=">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</latexit> CFT 4 $ 3 For a primary spin-2 operator: m2L 2 = ∆(∆ 3) <latexit 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sha1_base64="pnG9moH4tU5IlCXsYdAmACl6Zvk=">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</latexit> ab ) − ab If conserved, @<latexit sha1_base64="heXjNnBPawjreDCkeUu9iUz/65c=">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</latexit> aT =0, the representation is short & an algebraic manipulation gives ∆ =3 = m =0 <latexit sha1_base64="pYgDBq3Mim3r9nFfGuE2MXLome4=">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</latexit> ) 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 graviton (bigravity models) — Leaking out to a 4d (or higher ?) defect CFT to get a `single’ massive-graviton theory ∆<latexit sha1_base64="42Xm8I3AQa5qmXxkP+h7ZtTvcDg=">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</latexit> .<latexit sha1_base64="B08Rmmmr9ksp+8jWtHaiW8nsBT0=">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</latexit> .<latexit sha1_base64="B08Rmmmr9ksp+8jWtHaiW8nsBT0=">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</latexit> .<latexit sha1_base64="B08Rmmmr9ksp+8jWtHaiW8nsBT0=">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</latexit> Need weak leakage 4<latexit sha1_base64="z0DfbDolk8vyxYWaG2DQmkvsdkQ=">AAACxHicjVHLSsNAFD2Nr1pfVZdugkVwVRIp6LIoiMsWbC3UIkk6rUPTJMxMhFL0B9zqt4l/oH/hnXEKahGdkOTMufecmXtvmMVcKs97LTgLi0vLK8XV0tr6xuZWeXunLdNcRKwVpXEqOmEgWcwT1lJcxayTCRaMw5hdhaMzHb+6Y0LyNLlUk4z1xsEw4QMeBYqoZu2mXPGqnlnuPPAtqMCuRlp+wTX6SBEhxxgMCRThGAEkPV348JAR18OUOEGImzjDPUqkzSmLUUZA7Ii+Q9p1LZvQXntKo47olJheQUoXB6RJKU8Q1qe5Jp4bZ83+5j01nvpuE/qH1