introduction to [0.5em] ads/cft...introduction to ads/cft who? nina miekley from?university of...
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Introduction toAdS/CFTWho? Nina Miekley
From? University of WürzburgWhere? Young Scientists Workshop 2017When? July 17, 2017
(Figure by Stan Brodsky)
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Intuitive motivationWhat is meant by holography?A theory of gravity containsblack holes with entropyBlack holeentropy SBH = 1
4GNAreaHorizon.
More generally, the entropy in a spacetime region Ais bounded byBekensteinbound S(A) ≤ 14GN
Area(∂A),Ñ bound on # DoFs in A ∝ S(A).
(Bekenstein 1972, 1981)
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Intuitive motivationWhat is meant by holography?In contrast, in a QFT the entropy scales as
S(A) ∝ Volume(A).Holographic
principleFor theory of quantum gravity, the degrees offreedom 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)
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Intuitive motivationWhat is meant by holography?Gravity
boundaryS ∝ Volume
S ∝ Area
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AdS/CFT dualityAdS/CFT theory in Anti-de Sitter ÎÏ conformal field theory
� realisation of the holographic principle� origin in string theory
Duality two theories describe same physics
� same dynamics� one-to-one map between DoFs� vastly different theoriesI different spacetime dimensionI different coupling regimesI . . .� example:particle-wave duality
particle
waves
(Maldacena, 1997)
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AdS/CFT dualityAnti-de Sitter space (AdS)� solution to the vacuum Einstein equations with Λ < 0space with constant negative curvature R < 0� d + 1 dimensional AdS space embedded in d + 2dimensions asEmbedding
−(x0)2 + d∑i=1
(x i )2 − (xd+1)2 = const. < 0
� symmetry group: orthogonal group of Rd ,2
SO(d , 2)
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AdS/CFT dualityAnti-de Sitter space (AdS)
spatial directionradius
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AdS/CFT dualityAnti-de Sitter space (AdS)metric ds2 ∝ 1
z2
(dz2 − dt2 + d~x2)
� radial coordinate z� metric singular at z = 0 Ñ boundary ∂AdS
radialcoordinate
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AdS/CFT dualityConformal field theory� conformal symmetryI extension of Poincare symmetryI coordinate transformation such that
gµν(x)Ï e2σ (x) · gµν(x)I preserves angles and causality� no length-scale, i.e. onlydimensionless coupling constants� vanishing β-functions
↓
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AdS/CFT dualityComparison of Symmetries� SO(d , 2) symmetry of AdS ↔ conformal symmetry
How can we understand this?I coordinate transformation
↓conformal transformation on boundaryds2
AdS ∝1z2
(dz2 + gµνdxµdxν
)I AdS metric fixes boundary metric only up to conformalfactor:Boundarymetric ds2
∂AdS = limzÏ0
ω(z , t, ~x) ds2AdS |z=const.
Ï ‘defining function’ ω(z , t, ~x),second order zero at the boundary
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AdS/CFT dualityExample: N = 4 SYM theory, d = 4� supersymmetric Yang-Mills theory� adjoint DoFs with gauge group SU(N)
field content one gauge fieldfour Weyl fermionssix (real) scalars ÊÏ in adjoint rep.� dimensionless coupling-constant gYM
� N = 4: number of supersymmetriesl
� string theory onbulkgeometry AdS5 × S5
� compactify S5 Ï effective theory in five dimensions� symmetry SO(6) ÎÏ additional N = 4 SUSY
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AdS/CFT dualityExample: N = 4 SYM theory
z
AdS5
⊗
S5
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AdS/CFT dualityExample: N = 4 SYM theorydictionary:
g2YM ∝ string coupling
λ = g2YMN ∝ string length−4
strongestform N, λ arbitrary ÎÏ quantum string theorystrong form N Ï∞,
λ arbitrary ÎÏ classical string theory,only tree level diagramsweak form N Ï∞, λ large ÎÏ point-particle limit,classical field theory
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AdS/CFT dualityHow does this work?I promised you a one-to-one map:Field-operatormapCFT operator O
x ÏλxO(x)Ïλ−∆O(x)
⇐Ñ AdS field φ withmass m
m2 ∝ ∆(∆− d )near-boundary expansion:
φ(z , t, ~x) ∼ φ(0)↑source of O
(t, ~x) · zd−∆ + 〈O(t, ~x)〉↑expectation value
· z∆ + · · ·
(Witten, 1998; Gubser, Klebanov, Polyakov 1998)
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AdS/CFT dualityHow does this work?We identify the partition functions on both sidesGKP-Wittenrelation ZCFT [φ(0)] = ZAdS [φ]= expiSAdS [φ],where φ(0) is the source of the operator O.
Correlationfunctions 〈O1(x1) · · · On(xn)〉 = 1in
δnZAdS
δφ1(0)(x1)φn(0)(xn)(Witten, 1998; Gubser, Klebanov, Polyakov 1998)
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AdS/CFT dualityHow does this work?
radial direction
boundary bulk
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GeneralisationsFinite temperature� finite temperature ÎÏ black hole
zÍÑ
z
� thermodynamicsI temperatureI entropyI . . .
ÎÏ
� black holethermodynamicsI Hawking temperatureI Bekenstein entropyI . . .
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Generalisations� different dimensions:conformal field theory in d -dimensions
ltheory in Anti-de Sitter space in d + 1 dimensions� fundamental degrees of freedom� string theoryI highly symmetric theoriesI additional compact space� alternative:tailor gravity background to desired properties
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SummaryHolographicprinciple: gravity theory in region A
l‘conventional’ theory on ∂A.
AdS/CFT: CFT in d dimensionsltheory on AdS in d + 1 dimen-sions
� one-to-one map between DoFs,same dynamics� symmetries match
Gravity
‘conventional’ theory
� strong coupled field theoriesÏ probes interesting new regime
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SummaryHolographicprinciple: gravity theory in region A
l‘conventional’ theory on ∂A.AdS/CFT: CFT in d dimensions
ltheory on AdS in d + 1 dimen-sions� one-to-one map between DoFs,same dynamics� symmetries match
AdS
CFT� strong coupled field theoriesÏ probes interesting new regime
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Thank you for yourattention