Introduction to AdS/CFT

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1 Introduction to AdS/CFT Who? From? Where? When? Nina Miekley University of Würzburg Young Scientists Workshop 2017 July 17, 2017 (Figure by Stan Brodsky)

2 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).

3 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)

4 Intuitive motivation What is meant by holography? Gravity S Area boundary S Volume

5 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)

6 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)

7 Anti-de Sitter space (AdS) spatial direction radius

8 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

9 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

10 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

11 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

12 Example: N = 4 SYM theory AdS 5 S 5 z

13 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

14 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)

15 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)

16 How does this work? boundary bulk radial direction

17 Generalisations Finite temperature finite temperature black hole z z thermodynamics temperature entropy... black hole thermodynamics Hawking temperature Bekenstein entropy...

18 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

19 Summary Holographic principle: gravity theory in region A conventional theory on A. Gravity conventional theory

20 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

21 Thank you for your attention