Introduction to AdS/CFT Who? From? Where? When? Nina Miekley University of Würzburg Young Scientists Workshop 2017 July 17, 2017 (Figure by Stan Brodsky)
Intuitive motivation What is meant by holography? (Bekenstein 1972, 1981) Black hole entropy A theory of gravity contains black holes with entropy S BH = 1 4G N Area Horizon. Bekenstein bound More generally, the entropy in a spacetime region A is bounded by S(A) 1 Area( A), 4G N bound on # DoFs in A S(A).
Intuitive motivation What is meant by holography? In contrast, in a QFT the entropy scales as S(A) Volume(A). Holographic principle For theory of quantum gravity, the degrees of freedom in a region A can be described by a conventional theory living on its boundary A. holography: encoding the DoFs in a volume on its boundary quite general, no explicit construction of dual theory (Hooft, 1993; Susskind 1994)
Intuitive motivation What is meant by holography? Gravity S Area boundary S Volume
AdS/CFT Duality theory in Anti-de Sitter conformal field theory realisation of the holographic principle origin in string theory two theories describe same physics same dynamics one-to-one map between DoFs vastly different theories different spacetime dimension different coupling regimes... example: particle-wave duality particle waves (Maldacena, 1997)
Anti-de Sitter space (AdS) Embedding solution to the vacuum Einstein equations with Λ < 0 space with constant negative curvature R < 0 d + 1 dimensional AdS space embedded in d + 2 dimensions as (x 0 ) 2 + d (x i ) 2 (x d+1 ) 2 = const. < 0 i=1 symmetry group: orthogonal group of R d,2 SO(d, 2)
Anti-de Sitter space (AdS) spatial direction radius
Anti-de Sitter space (AdS) metric ds 2 1 z 2 ( dz 2 dt 2 + d x 2) radial coordinate z metric singular at z = 0 boundary AdS radial coordinate
Conformal field theory conformal symmetry extension of Poincare symmetry coordinate transformation such that g µν (x) e 2σ(x) g µν (x) preserves angles and causality no length-scale, i.e. only dimensionless coupling constants vanishing β-functions
Comparison of Symmetries Boundary metric SO(d, 2) symmetry of AdS conformal symmetry How can we understand this? coordinate transformation conformal transformation on boundary ds 2 AdS 1 z 2 ( dz 2 + ĝ µν dx µ dx ν) AdS metric fixes boundary metric only up to conformal factor: ds 2 AdS = lim z0 ω(z, t, x) ds 2 AdS z=const. defining function ω(z, t, x), second order zero at the boundary
field content bulk geometry AdS/CFT duality Example: N = 4 SYM theory, d = 4 supersymmetric Yang-Mills theory adjoint DoFs with gauge group SU(N) one four six gauge field Weyl fermions (real) scalars in adjoint rep. dimensionless coupling-constant g YM N = 4: number of supersymmetries string theory on AdS 5 S 5 compactify S 5 effective theory in five dimensions symmetry SO(6) additional N = 4 SUSY
Example: N = 4 SYM theory AdS 5 S 5 z
Example: N = 4 SYM theory dictionary: g 2 YM string coupling λ = gym 2 N string length 4 strongest form N, λ arbitrary quantum string theory strong form N, λ arbitrary classical string theory, only tree level diagrams weak form N, λ large point-particle limit, classical field theory
How does this work? Fieldoperator map I promised you a one-to-one map: CFT operator O x λx O(x) λ O(x) near-boundary expansion: AdS field φ with mass m m 2 ( d) φ(z, t, x) φ (0) source of O (t, x) z d + O(t, x) z + expectation value (Witten, 1998; Gubser, Klebanov, Polyakov 1998)
How does this work? GKP-Witten relation We identify the partition functions on both sides Z CFT [φ (0) ] = Z AdS [φ] = exp is AdS[φ], where φ (0) is the source of the operator O. Correlation functions O 1 (x 1 ) O n (x n ) = 1 i n δ n Z AdS δφ 1 (0) (x 1)φ n (0) (x n) (Witten, 1998; Gubser, Klebanov, Polyakov 1998)
How does this work? boundary bulk radial direction
Generalisations Finite temperature finite temperature black hole z z thermodynamics temperature entropy... black hole thermodynamics Hawking temperature Bekenstein entropy...
Generalisations different dimensions: conformal field theory in d-dimensions theory in Anti-de Sitter space in d + 1 dimensions fundamental degrees of freedom string theory highly symmetric theories additional compact space alternative: tailor gravity background to desired properties
Summary Holographic principle: gravity theory in region A conventional theory on A. Gravity conventional theory
Summary Holographic principle: gravity theory in region A conventional theory on A. AdS AdS/CFT: CFT in d dimensions theory on AdS in d + 1 dimensions one-to-one map between DoFs, same dynamics symmetries match strong coupled field theories probes interesting new regime CFT
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