Higher spin gauge theories and their CFT duals

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1 Higher spin gauge theories and their CFT duals 2 AdS Vasiliev AdS/CFT 4 Vasiliev 3 O(N) 3 Vasiliev 2 W N 1 AdS/CFT g µν Vasiliev AdS [1] AdS/CFT anti-de Sitter (AdS) (CFT) 1997 Maldacena [2] 15 AdS/CFT AdS AdS/CFT AdS/CFT AdS/CFT Klebanov-Polyakov 4 Vasiliev 3 O(N) [3] Giombi-Yin [4, 5] Gaberdiel-Gopakumar [6] 3 Vasiliev 2 N N [7] 3 AdS/CFT 2 3 1

2 ϕ µ1...µ s s s = 1 abelian Yang-Mills A µ s = 2 g µν s = 1, 2 F µ1...µ s ϕ µ1...µ s (µ1 λ ϕ µ2...µ s)λ + (µ1 µ2 ϕ µ3...µ s)λ λ = 0 (1) [8] δϕ µ1...µ s = (µ1 ξ µ2...µ s), ξ λ λµ 3...µ s = 0 (2) S = 1 ( d D xϕ µ 1...µ s F µ1...µ 2 s 1 ) 2 η (µ 1 µ 2 F µ3...µ s )λ λ (3) ϕ λσ λσµ 5...µ s = 0 (4) (2) [9] Vasiliev AdS s = 2, 3,..., [1] 3 Chern-Simons [10] Chern-Simons 3 3 SL(2) SL(2) Chern-Simons [11, 12] S = S CS [A] S CS [Ã], S CS[A] = k CS 4π tr (A da + 23 ) A A A k CS AdS l G k CS = l/4g SL(2) Lie J a (a = 1, 2, 3) A = A a µj a dx µ Chern-Simons δa = dλ + [A, λ], δã = d λ + [Ã, λ] (6) e a µ ω µ,a,b e a µ = l 2 (Aa µ Ãa µ), ω µ,a,b = 1 2 ϵ abcω c µ, ω c µ = 1 2 (Ac µ + Ãc µ) (7) Chern-Simons (5) Einstein-Hilbert (6) (5) 2

3 3 Chern-Simons G G G = SL(N) sl(2) sl(n) ( N ) sl(n) = sl(2) g (s) (8) g (s) sl(2) (2s 1) sl(2) sl(2) g (s) s G = SL(N) s = 2, 3,,..., N Gaberdiel-Gopakumar [6] [13] G = SL(N) N [7] [13] N = 2 G = SL(N + 1 N) N Chern-Simons Chern-Simons G Wess-Zumino-Novikov-Witten (WZNW) AdS/CFT AdS Hamiltonain [14, 15] (G = SL(2)) (Virasoro) Brown-Henneaux [16] (G = SL(N)) Virasoro W N [17, 14] W N [18] (G = SL(N + 1 N)) N = 2 W N [7, 19, 20] AdS/CFT s=3 3 AdS/CFT AdS/CFT Klebanov-Polyakov [3] 4 AdS [1] 3 O(N) O(N) s J = N h a (µ1 µs )h a + (9) a=1 ϕ µ1...µ s 3

4 U(N) s tr[φ l 1 Φ l2 Φ], s = i l i (10) s s Klebanov-Polyakov [3] [4, 5] [21, 22] Gaberdiel-Gopakumar [6] [13] G = SL(N) Chern-Simons N M 2 = 1 + λ 2 AdS W W N SU(N) k SU(N) 1 SU(N) k+1 (11) N t Hooft λ = N k + N (12) λ [6] [23] 1 t Hooft [6] [24, 25] WD N [18] [7] [13] N = 2 G = SL(N + 1 N) Chern-Simons N 1/2 λ AdS N = (2, 2) W N = (2, 2) W N CP N [26, 27] SU(N + 1) k SO(2N) 1 SU(N) k+1 U(1) N(N+1)(k+N+1) (13) [28] t Hooft (12) N, k λ [29] 4

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