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 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 time, I will comment on gauged 4d supergravity 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 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 No need for strong non-linearities of Vainshtein screening
Kogan, Mouslopoulos, Papazoglou ’00 Porrati ‘00 2 G-mass from Holography
Consider 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 CFT 4 $ 3 For a primary spin-2 operator: m2L 2 = ( 3) 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 4
(like stress tensor 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 ab )
ab
If conserved, @AAAC13icjVHLSsNAFD2N73esSzfBIrgqiQi6EYpuXFawWmm1TOJUh6ZJmEzEUoo7cesPuNU/Ev9A/8I7YwpqEZ2Q5My595yZe6+fhCJVrvtasMbGJyanpmdm5+YXFpfs5eJxGmcy4LUgDmNZ91nKQxHxmhIq5PVEctb1Q37id/Z1/OSay1TE0ZHqJfysyy4j0RYBU0S17GIzYVIJFrbY0Xmf+YNdt2WX3LJrljMKvByUkK9qbL+giQvECJChC44IinAIhpSeBjy4SIg7Q584SUiYOMcAs6TNKItTBiO2Q99L2jVyNqK99kyNOqBTQnolKR2skyamPElYn+aYeGacNfubd9946rv16O/nXl1iFa6I/Us3zPyvTtei0MaOqUFQTYlhdHVB7pKZruibO1+qUuSQEKfxBcUl4cAoh312jCY1teveMhN/M5ma1fsgz83wrm9JA/Z+jnMUHG+WPbfsHW6VKnv5qKexijVs0Dy3UcEBqqiR9w0e8YRn69S6te6s+89Uq5BrVvBtWQ8f/TGWww== 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 graviton (bigravity models)
— Leaking out to a 4d (or higher ?) defect CFT to get a `single’ massive-graviton theory