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
5 4 2 Vasiliev AdS [1] 3 Chern- Simons AdS/CFT 4 Vasiliev 3 O(N) [3] 3 Vasiliev 2 W N [6] N 1/N 1/N AdS/CFT [1] M. A. Vasiliev, Consistent equation for interacting gauge fields of all spins in (3+1)- dimensions, Phys. Lett. B 243, 378 (1990). [2] J. M. Maldacena, The Large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2, 231 (1998) [hep-th/ ]. [3] I. R. Klebanov and A. M. Polyakov, AdS dual of the critical O(N) vector model, Phys. Lett. B 550, 213 (2002) [hep-th/ ]. [4] S. Giombi and X. Yin, Higher spin gauge theory and holography: The three-point functions, JHEP 1009, 115 (2010) [arxiv: [hep-th]]. [5] S. Giombi and X. Yin, Higher spins in AdS and twistorial holography, JHEP 1104, 086 (2011) [arxiv: [hep-th]]. [6] M. R. Gaberdiel and R. Gopakumar, An AdS 3 dual for minimal model CFTs, Phys. Rev. D 83, (2011) [arxiv: [hep-th]]. [7] T. Creutzig, Y. Hikida and P. B. Ronne, Higher spin AdS 3 supergravity and its dual CFT, JHEP 1202, 109 (2012) [arxiv: [hep-th]]. [8] C. Fronsdal, Massless fields with integer spin, Phys. Rev. D 18, 3624 (1978). [9] B. de Wit and D. Z. Freedman, Systematics of higher spin gauge fields, Phys. Rev. D 21, 358 (1980). [10] M. P. Blencowe, A consistent interacting massless higher spin field theory in D = (2 + 1), Class. Quant. Grav. 6, 443 (1989). 5
6 [11] A. Achucarro and P. K. Townsend, A Chern-Simons action for three-dimensional anti-de Sitter supergravity theories, Phys. Lett. B 180, 89 (1986). [12] E. Witten, (2+1)-dimensional gravity as an exactly soluble system, Nucl. Phys. B 311, 46 (1988). [13] S. F. Prokushkin and M. A. Vasiliev, Higher spin gauge interactions for massive matter fields in 3-D AdS space-time, Nucl. Phys. B 545, 385 (1999) [hep-th/ ]. [14] A. Campoleoni, S. Fredenhagen, S. Pfenninger and S. Theisen, Asymptotic symmetries of three-dimensional gravity coupled to higher-spin fields, JHEP 1011 (2010) 007 [arxiv: [hep-th]]. [15] A. Campoleoni, S. Fredenhagen and S. Pfenninger, Asymptotic W -symmetries in threedimensional higher-spin gauge theories, JHEP 1109 (2011) 113 [arxiv: [hep-th]]. [16] J. D. Brown and M. Henneaux, Central charges in the canonical realization of asymptotic symmetries: An example from three-dimensional gravity, Commun. Math. Phys. 104, 207 (1986). [17] M. Henneaux and S. -J. Rey, Nonlinear W as asymptotic symmetry of three-dimensional higher spin anti-de Sitter gravity, JHEP 1012, 007 (2010) [arxiv: [hep-th]]. [18] P. Bouwknegt and K. Schoutens, W symmetry in conformal field theory, Phys. Rept. 223, 183 (1993) [hep-th/ ]. [19] M. Henneaux, G. Lucena Gomez, J. Park and S. -J. Rey, Super- W asymptotic symmetry of higher-spin AdS 3 supergravity, JHEP 1206, 037 (2012) [arxiv: [hep-th]]. [20] K. Hanaki and C. Peng, Symmetries of holographic super-minimal models, arxiv: [hep-th]. [21] J. Maldacena and A. Zhiboedov, Constraining conformal field theories with a higher spin symmetry, arxiv: [hep-th]. [22] J. Maldacena and A. Zhiboedov, Constraining conformal field theories with a slightly broken higher spin symmetry, arxiv: [hep-th]. [23] M. R. Gaberdiel, R. Gopakumar, T. Hartman and S. Raju, Partition functions of holographic minimal models, JHEP 1108, 077 (2011) [arxiv: [hep-th]]. [24] C. Ahn, The large N t Hooft limit of coset minimal models, JHEP 1110, 125 (2011) [arxiv: [hep-th]]. [25] M. R. Gaberdiel and C. Vollenweider, Minimal model holography for SO(2N), JHEP 1108, 104 (2011) [arxiv: [hep-th]]. 6
7 [26] Y. Kazama and H. Suzuki, New N = 2 superconformal field theories and superstring compactification, Nucl. Phys. B 321, 232 (1989). [27] Y. Kazama and H. Suzuki, Characterization of N = 2 superconformal models generated by coset space method, Phys. Lett. B 216, 112 (1989). [28] K. Ito, Quantum Hamiltonian reduction and N = 2 coset models, Phys. Lett. B 259, 73 (1991). [29] C. Candu and M. R. Gaberdiel, Supersymmetric holography on AdS 3, arxiv: [hep-th]. 7
1 What is CFT? 1. 3 Strange duality conjecture (G) Geometric strange duality conjecture... 5
1 1994 9 6 1 What is CFT? 1 2 Wess-Zumino-Witten model 2 2.1 (R Representation theoretic formulation of WZW model.......... 2 2.2 (G Geometric formulation of WZW model.................. 4 2.3 (R=(G.....................................
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