2. Introduction to AdS/CFT

Óscar Dias

RESEARCHRESEARCHRESEARCH SSSTTTAAAGGG CENTERCENTERCENTER

Ernest Rutherford

References on Introduction to AdS/CFT: [1] https://arxiv.org/abs/0709.1523v1 ( Chapter 1-3 ) [2] https://arxiv.org/abs/0712.0689 ( Chapters 8 & 9 ) [3] https://arxiv.org/abs/1101.0618 ( Chapter 4 & 5 ) [4] https://arxiv.org/abs/1106.6073 [5] https://arxiv.org/abs/1612.07324 ( Chapter 1 )

School on AdS/CMT Correspondence, ICTP-SAIFR/IFT-UNESP, Brasil ➙ Large-N expansion of SU(N) ←→ genus (quantum) expansion of theory • QCD: with gauge group SU(3) but NO expansion parameter (low E)

=> NO perturbative analysis at low E [ for E >>1 => gqcd is small: perturbative QCD ]

• ’t Hooft’s large-Nc limit ( generalisation of QCD ): Figure 2: Double-line notation. replace gauge group by SU(Nc), take limit Nc → ∞ (# colours) & perform an expansion in 1/Nc

2 2 a • d.o.f.: ~ Nc gluon fields (Αµ)i j, Nc Nf << Nc quark fields q i (Nf # quark flavours).

2 • When Nc → ∞, dynamics dominated by gluons: Nc Nf ~0<< Nc Figure 2: Double-line notation.

• Organize expansion in Feymann diagrams (FD):

q-¯q-g vertex : g Figure 3: Vertices in double-line notation. YM 3-gluon vertex: gYM Figure 3: Vertices in double-line notation.

Figure 2: Double-line notation. Gluon self-energy FD: N ,g 0 !1 YM ! 2 Nc so that = gYMN finite gYM gYM Figure 4: Gluon self-energy diagram in double-line notation. λ → t’Hoof coupling

4 Figure 4: Gluon self-energy diagram in double-line notation.

4 Figure 3: Vertices in double-line notation.

Figure 4: Gluon self-energy diagram in double-line notation.

4 Feynman diagrams naturally organise in a double series expansion in powers of 1/N and λ:

• Planar diagrams: ‘drawn without crossing lines’ → all scale as N2

#vertices #loops 2 gYM N = gYMN 1 . (#vertices) # loops -1 = 2 N

• Non-planar diagrams: supressed by 1 / N2 wrt planar Figure 5: Some planar diagrams.(when N → ∞)

ℓ−1 2 power of λ ,withℓ the numberFigure of loops.5: Some The planarNc scaling diagrams. is not the same for all diagrams, however. For example, the three-loop diagram in fig. 6 scales as λ2,andisthussuppressed 2 with respect− to those in fig. 5 by a power of 1/N ;thereaderisinvitedtodrawother power of λℓ 1,withℓ the numberFigure of loops. 5: Some The planarN 2 scaling diagrams.c is not the same for all diagrams, diagrams suppressed by higher powers of 1/N 2.Thedic fference between the diagrams in however. For example, the three-loop diagram inc fig. 6 scales as λ2,andisthussuppressed fig. 5 andℓ− that1 of fig. 6 is that the former are planar,2 2 i.e., can be ‘drawn without crossingFigure 6: Anon-planardiagram. powerwith respect of λ ,with to thoseℓ the in number fig. 5 by of a loops. power The of 1N/Nc scalingc ;thereaderisinvitedtodrawother is not the same for all diagrams, lines’, whereas the latter is not. We thus see that2 diagrams are classified2 by their topology, however.diagrams For suppressed example, by the higher three-loop powers diagram of 1/Nc in.Thedi fig. 6 scalesfferenceas λ between,andisthussuppressed the diagrams in and that non-planar diagrams are suppressed in the2 large-Nc limit. withfig. 5 respect and that to of those fig. 6 in is fig. that 5 the by former a power are of planar, 1/Nc ;thereaderisinvitedtodrawotheri.e., can be ‘drawn without crossing The topological classification of diagrams,2 which leads to the connection with string diagramslines’, whereas suppressed the latter by higher is not. powers We thus of see 1/N thatc .Thedi diagramsfference are classified between by the their diagrams topology, in theory, can be made more precise by associating a Riemann surface to each Feynman fig.and 5 that and non-planar that of fig. diagrams 6 is that the are former suppressed are planar, in the large-i.e., canN limit. be ‘drawn without crossing diagram, as follows. In double-line notation, each line in a cFeynman diagram is a closed lines’,The whereas topological the latter classification is not. We of thus diagrams, see that which diagrams leads a tore classified the connection by their with topology, string loop that we think of as the boundary of a two-dimensional surface or ‘face’. The Riemann andtheory, that can non-planar be made diagrams more precise are suppressed by associating in the a large- RiemannNc limit. surface to each Feynman diagram,The topological as follows. classification In double-line of notation, diagrams, each which line leads in a toFeynman the connection diagram with is a closed ,loop that can we be think made of as more the boundary precise by of associating a two-dimensional5 a Riemann surface sur orface ‘face’. to each The Feynman Riemann diagram, as follows. In double-line notation, each line in a Feynman diagram is a closed loop that we think of as the boundary of a two-dimensional surface or ‘face’. The Riemann 5

5 Figure 7: ARiemannsurfaceassociatedtoaplanardiagram.

surface is obtained by gluing together these faces along their boundaries as indicated by the Feynman diagram. In order to obtain a compact surface we add ‘the point at infinity’ to the face associated to the external line in the diagram. This procedure is illustrated for a planar diagram in fig. 7, and for a non-planar diagram in fig. 8. In the first case we obtain a sphere, and the same is true for any planar diagram.Inthesecondcasewe

obtain a torus. It turns out that the power of Nc associated to a given Feynman diagram χ is precisely Nc ,whereχ is the Euler number of the corresponding Riemann surface. For acompact,orientablesurfaceofgenusg with no boundaries we have χ =2 2g.Thus − for the sphere χ =2andforthetorusχ =0.Wethereforeconcludethattheexpansion of any gauge theory amplitude in Feynman diagrams takes the form

∞ ∞ = N χ c λn , (3) A c g,n !g=0 n!=0

where cg,n are constants. We recognise the first sum as the loop expansioninRiemann

6 • We can make contact with string theory: fundamental objects are open and closed strings whose vibration modes are quantised and describe elementary particles including the

• In order to make contact with string theory first note that:

String theory has 2 free parameters, 2 — string length: ls ( α΄= ls ) Mass or Tension of the string ~ 1 / ls — string coupling: gs measures interactions between strings • Topological classificationFigure 6: Anon-planardiagram. of diagrams, leads to connection with string theory: each Feynman diagram ←→ Riemann surface

Planar diagram ←→ Riemann surface: sphere with genus g = 0, Euler # χ = 2-2g = 2 2N 2 1 (#vertices) # loops -1 ⇠ 2 N 1 (#vertices) Figure 7: ARiemannsurfaceassociatedtoaplanardiagram. = 2 N surface is obtained by gluing together these faces along their boundaries as indicated by the Feynman diagram. In order to obtain a compact surface we add ‘the point at infinity’ to the face associated to the external line in the diagram. This procedure is illustrated for a planar diagram in fig. 7, and for a non-planar2 diagram in fig. 8. In the first case we obtain a sphere, and the same is true⇠ for any planar diagram.InthesecondcaseweNon-planar diagram ←→ Riemann surface: obtain a torus. It turns out that the power of Nc associated to a given Feynman diagram χ χ torus with genus g = 1, Euler # χ = 2-2g = 0 ( g > 0, <2 ) is precisely NcFigure,where 8:χARiemannsurfaceassociatedtoanon-planardiagram.is the Euler number of the corresponding Riemann surface. For acompact,orientablesurfaceofgenusg with no boundaries we have χ =2 2g.Thus − 1 1 for the sphere• Expansionχ =2andforthetorus of any gaugeχ theory=0.Wethereforeconcludethattheexpansion amplitude in FD: = c N n surfacesof any gauge for a theory closed amplitude string theory in Feynman with diagrams coupling takes constant the formgs 1/Nc.Notethattheg,n 2 ∼ A expansion parameter is therefore 1/Nc .Aswewillseelater,thesecondsumisassociatedg=0 n=0 ′ ∞ ∞ X X 1 to the so-calledΣ : (quantum)α -expansion loop in expansion the stringχ theory.in Riemannn surf. for a closed string theory with coupling const g = Nc cg,nλ , (3) gs ‘ ~ ’ A The above analysis holds for any!g=0 gaugen!=0 theory with Yang-Millsfieldsandpossibly ⇠ N ⇠ 2 matter in theΣn adjoint: (string representation, loop) α΄-expansion since the(α΄= latter ls : string is described length by ) fields with two colour ( understand later ) where cg,n are constants. We recognise the first sum as the loop expansioninRiemann indices. In order to illustrate the eff2ect of the inclusion of quarks, or more generally Later: ↵0 1/ 2 p 2 p 3 p=3 of matter in the fundamental representation,⇠ which is described by fields with only oneg (2⇡) g ` g 6 YM ⌘ s s ⌘ N ) s ⇠ N colour index, consider the two diagrams in fig. 9. The bottom diagram differs from the top diagram solely in the fact that a gluon internal loop has been replaced by a quark loop. This leads to one fewer free colour line and hence to one fewer power of Nc.Since the flavour of the quark running in the loop must be summed over,italsoleadstoan additional power of Nf.Thusweconcludethatinternalquarkloopsaresuppressedby powers of Nf/Nc with respect to gluon loops. In terms of the Riemann surface associated to a Feynman diagram, the replacement of a gluon loop by a quarkloopcorrespondstothe introduction of a boundary, as illustrated in fig. 10 for the diagrams of fig. 9. The power χ of Nc associated to the Feynman diagram is still Nc ,butinthepresenceofb boundaries the Euler number is χ =2 2g b.Thismeansthatinthelarge-N expansion (3) we − − c must also sum over the number of boundaries, and so we now recognise it as an expansion for a theory with both closed and open strings. The open strings are associated to the 1/2 boundaries, and their coupling constant is gop Nf gs = Nf/Nc. ∼ The main conclusion of this section is therefore that the large-Nc expansion of a gauge theory can be identified with the genus expansion of a string theory. Through this identi- fication the planar limit of the gauge theory corresponds to the classical limit of the string theory. However, the analysis above does not tell us how to construct explicitly the string dual of a specific gauge theory. We will see that in some cases this can be ‘derived’ by thinking about the physics of D-.

7 CONCLUSION: Large-Nc expansion of a gauge theory ←→ genus expansion of a string theory. In this identification: planar limit of gauge theory ←→ classical limit of string theory

Classical string theory corresponds to: 1 exercise gs ‘ ~ ’ 0 (to suppress genus (quantum) loop corrections) ⇠ N ⇠ ! Low energy Classical string theory if:

4 2 1 ` = ↵0 0 (to suppress string loop corrections) s ⇠ ! E` 0 better later s ! Low energy Classical string theory =

if we identify: 16⇡G (2⇡)7g2`8 10 ⌘ s s

Pokémon Low energy Classical string theory = Supergravity

Next, look independently into 2 perspectives:

➙ Type II string theory & Dp-branes

➙ Type II supergravity & p-branes ➙ Type II string theory & Dp-branes & gauge theory

• Quantum mechanics: dynamics of point particles Proper time (dτ2 = - ds2) is an invariant

length of worldline t xa(σ)

a b Spp = m d⌧ = m gabdx dx z }| { x1 Z Z p x2,3…

• String theory (ST): quantum theory of interacting relativistic 1-dim objects (strings) with tension T

area of world sheet t xa(σ0,σ1) S = d2 det g, str T Zz }| { 1 p 2 = , ↵0 `s (fundamental string length) T 2⇡↵0 ⌘ σ0,1 a b 2-dim g induced on world sheet: g↵ = @↵x @x gab x1

x2,3… • ST also has (non-perturbative) Dp- solutions where open strings can end (BCs):

• Quantisation of closed strings in a fixed background: close string spectrum describes fluctuations of spacetime (graviton)

• Quantisation of open strings ending on Dp-branes (BC): open string spectrum describes fluctuations of the D-brane

• Single Dp-brane open string spectrum:

fermonic i massless gauge field Aa , 9-p scalars X + super partners + infinite tower massive fields ( m 1 / ` ) s ⇠ s

Describe excitations of Dp Describe excitations (δ shape, δy) along worlvolume of Dp along transverse directions (Goldstone modes ) f v = t + v =const: t , r r ) % & ingoing null rays )

• For classical electrodynamics ( at low E ):

µ ⌫ µ a 2 4 µ⌫ µ K = SK=µK m =dg⌧µ⌫Ke KAadx=0. 2 Ford xF Schw.:µ⌫ F K = K @xµ = @t H 4e | | HZ Z H Z - Spp Sint (e ,) Sbulk f | {z } | {z } | {z } • Action vfor= a t Dp-brane:+ v =const: t , r ingoing null rays r ) % & ) f S = S + S + S u = t u brane=const: t interaction, r bulkoutgoing null rays r ) exc. open strings %open-closed% exc.) closed str.

| {z2 } | {z2 } | {z } • Dirac-Born-Infeld 2action for singler D3-brane2 (p=3)dr 2 2 2 2 ds = 1+ dt + 2 + r (d✓ +sin ✓d ) (tension: Q/Vp) L2 1+ r ✓ ◆ L2 p+1 2 p (p+1) 1 SDBI = Dp d xe det (g↵ +2⇡`sF↵) +2 fermions , Dp =(2⇡) `s gs T 2 L T Z gµ⌫q gµ⌫ = ⌦ (r)transv.gµ⌫ =def. gµ⌫ z }| { e g ! r2 s 2 ⇠ g = ⌘ + 2⇡`2 @ Xi@ Xi 2 ↵ ↵ 4 s ↵ 2 L 2 2 e2 2 2 2 L • Expansionds = aroundds ls →= 0: dtexercise+ L (d✓ +sin ✓d ) + +subleading terms r2 r4 ⇥ 1Einstein4 Static1 Universeµ⌫ 1⇤ i µ i SDBIe Sbrane = d Fµ⌫ F + @µX @ X + fermions + (↵0) ! |2⇡gs {z4 2} O 2 AdS/CFT Z ✓ ◆ • When ls → 0 also have: 7 2 8 16⇡G10 (2⇡) gs `s 0 Grav. interaction 0 Sint 0,Sbulk Sfree closed str. ⌘ ! ) ! ) ! ! in Mink10

L2 ds2 = dz2 + G (z,x)dxadxb , AdS ,oddd z2 ab . d+1 ⇥ (0) d (d) ⇤ d (d) Gab z 0 = gab (x)+ + z gab (x)+ with Tab(x) gab (x) ! ··· ··· h i⌘16⇡GN

Then partition functions of 2 theories should agree: • ZCFT []=Zstring N , 1 @AdS . !1 Ssugra[] Z [] e h i CFT '

= e', ⇢1/2 (Cooper pair charge 2e, mass m), ' is macroscopic SC phase (1) | | | | ⇠ C e e

4 f v = t + v =const: t , r r ) % & ingoing null rays )

µ ⌫ µ µ K = KµK = gµ⌫K K =0. For Schw.: K = K @xµ = @t | |H H H f v = t + v =const: t , r ingoing null rays r ) % & ) f u = t u =const: t , r outgoing null rays r ) % % )

r2 dr2 ds2 = 1+ dt2 + + r2(d✓2 +sin2 ✓d2) L2 1+ r2 ✓ ◆ L2 L2 g g = ⌦2(r) g = g µ⌫ ! µ⌫ µ⌫ r2 µ⌫ L2 e L4 ds 2 = ds2 = dt2 + L2(d✓2 +sin2 ✓d2) + +subleading terms r2 r4 ⇥ Einstein Static Universe ⇤ e | {z } 2 AdS/CFT • When ls → 0 also have: 7 2 8 16⇡G10 (2⇡) gs `s 0 Grav. interaction 0 Sint 0,Sbulk Sfree closed str. ⌘ ! ) ! ) ! ! in Mink10 G E G 0 = 10 0 E 0 low E expansion 10 L2 Newton 7 ds!2 = )dz2 + G (z,x)dxardxb ,! , AdS! ,odd) d z2 ab . d+1 • Low-energy Action for single Dp-brane: ⇥ (0) d (d) ⇤ d (d) Gab z 0 = gab (x)+ + z gab (x)+ with Tab(x) gab (x) Dp! ··· ··· h i⌘16⇡GN S low E = SDBI + Sbulk + (↵0) O exc. open str. closed str. in Mink 10 (G10E) O Then| partition{z } functions| {z }of 2 theories| {z should} agree: • Low-energy Action for stack of N coincident D3-branes: ^ • ^ ZCFT []=Zstring N , 1 I @AdS . !1 Aµ (Aµ) ,I,J =1Ssugra, [] N ZJ [] e h i ! CFT ' ··· Xi (Xi)I N N matrices ! J & ⇥

ms r/`s 0 ' 1/2 r 0 = e , ⇢ (Cooper pair charge 2e, mass m), ' is macroscopic⇠ SC! phase!(1) | | | | ⇠ C @µ Dµ=@µ+i[Aµ,.] e ! Vint 1 µ⌫ i µ i e i j 2 DBI brane =Tr0 Fµ⌫ F DµX D X + ⇡ gs X ,X 1 + fermions L ! L 4⇡ gs Bz }| i { z i,j }| {C B X X ⇥ ⇤ C B 4 C @ A • Low-energy Action for stack of N coincident D3-branes:

Dp 1 2 =Tr F F µ⌫ D XiDµXi + ⇡ g Xi,Xj +f. + + L low E 04⇡ g µ⌫ µ s 1 Lclstr.Mink10 O(↵0) s i i,j X X ⇥ ⇤ @ PokémonA

2 2 p 2 p 3 But = if g (2⇡)g [ g (2⇡) g ` ] Lbrane LSYM YM ⌘ s YM ⌘ s s

Dp 1 1 2 =Tr F F µ⌫ D XiDµXi + Xi,Xj +f. + + L low E 02 g2 µ⌫ µ 2 YM 1 Lclstr.Mink10 O(↵0) YM i i,j X X ⇥ ⇤ @ A -LSYM | {z } • CONCLUSION: Low-energy Action for stack of N coincident D3-branes is

SU(N) N=4 Super-Yang-Mills (SYM) theory in 4 dimensions + Mink10 closed strings propagating freely in Minkowski 10 SYM is a non-abelian SU(N) gauge theory (like QCD) but with conformal invariance

(i.e. gYM is scale-independent and does not change with E) ➙ Type II supergravity & p-branes the other perspective • Type II supergravity (with some fields switched-off)

(E) 1 1 1 1 (p+1) 1 10 µ 2 2 (3 p) 2 III = d xp g R @µ@ gs e (dA(p+1)) 16⇡G10 2 2(p + 2)! Z ⇣ ⌘ • 3-brane solution ( p = 3): 2 3 1 1 dr 2 2 2 2 2 2 2 2 2 ds = H fdt +dx + H + r d⌦S5 , dx = dxi k f k ⇣ ⌘ ✓ ◆ i=1 1 1 X e = g ,A= g 1 H dt dx dx dx , s (4) s ^ 1 ^ 2 ^ 3 4 L where H =1+ 4 1+Newton r ⇠ Mink10 => L sets lengthscale of grav. effects:

r L H 1 Mink • ) ⇠ )⇠ 10 L4 r L H Horizon throat • ⌧ ) ⇠ r4 ) L

• Charge of 3-brane solution: Figure 12: Spacetime around D3-branes.

V3 V3⌦5 4 Q3 = ?dA(4) = L G 5 4⇡G 10 ZS 10

Figure 13: Tadpole-like diagrams whose sum leads to an effective geometry for closed strings.

10 • Identify 3-brane with D3-brane: 3= T ··· 16⇡G10 V3⌦5 4 4 N 1 7 2 8 Q3 QD3 L = N 3 V3 L = (3+1) (2⇡) gs `s ⌘ () 4⇡G10 T () 4⌦5 (2⇡)3` g z }|s s{ z }| { L4 4 =4⇡ gsN Grav. radius in string units , `s

Pokémon If L ` g N 1: ⌧ s , ⇠ s ⌧ 3-brane is zero thickness object in Mink ⇠ ⇠ 10 Mink10 If L ` g N 1: s , ⇠ s 3-brane is massive grav. backreacts on spacetime ) Mink10

L

Figure 12: Spacetime around D3-branes.

Figure 13: Tadpole-like diagrams whose sum leads to an effective geometry for closed strings.

10 dt2 +d⇢2 ds2 = dt2 +d⇢2 + ⇢2d'2 = ⇢2 +d'2 (26) @AdS ⇢2 ✓ ◆

3 L2 = L2 (27) IR 4

⇡L2 ✓⇤ sin ✓ ✓⇤ sin ✓ S =lim IR d✓ H(✓)G(✓) d✓ (28) imp ⇡ 2 2 ✓⇤ 2G cos ✓ cos ✓ ! 2 N ✓ 0 0 ◆ Z p Z

S = ⇢ d3x ( ✓)2 . (29) s r Z

1 R E ⇢ dR n2 ⇢ n2 log IR cutoff (30) ⇠ s R ⇠ s R Z ✓ core ◆

(0) A' = A' + J' z + ... , (31)

+ xx xx 1 4 i !t ik x J = G A with G = i d xe · ✓(t) J (t, x),J (0, 0) h xi R x R x x Z1 ⌦ ↵

This is given by Ohm’s law: J = E is the AC conductivity h xi x Fourier GG G G G G G G G GA In the gauge At =0onehas: Ex = @tAx i !Ax Jx = i !Ax • transf. ) h i ➙ Decoupling (low-energy, Maldacena) limit Gxx(!, 0) (!)= R Kubo formula: relation between a transport coecient &aGreen’s function i ! r L ` 0 with , , g N kept fixed s ! `2 `2 ⇠ s ⇢ s s Want low E string exc. as measured by Obs free closed str. in Mink10 • 1 ) ⇠ • But arbitrary E string exc. deep in the throat can survive:

` E : energy (in string units) of string excitation at a fixed r s r 1/4 energy measured by an Obs : E = p g00Er = H(r) Er 1 1 r deep in the throat, r L `sE `sEr 0 ⌧ ) 1 ⇠ L ! => interacting closed strings in bottom of throat Mink r 10 finite • Maldacena lim is a near-horizon lim: 2 = `s 0 r 0 `s | ! ) ! throat geometry decouples from asymptotic9 region and yields

r2 3 L2 ds2 = dt2 + dx2 + dr2 + L2d⌦2 L2 i r2 5 " i=1 ! # 2 2 X2 5 = ds + L ds 5 AdS5 S AdS5 S L ⇥ 20 20 5 5 5 RAdS5 = 2 ,RS = 2 RAdS5 S = RAdS5 + RS =0 L L ! ⇥ Figure 12: Spacetime around D3-branes.

Figure 13: Tadpole-like diagrams whose sum leads to an effective geometry for closed strings.

10 ➙ Two equivalent descriptions of the system

Closed string perspective,L ` g N 1: Mink • s , ⇠ s 10 3-brane is massive grav. backreacts on spacetime )

=> free closed str. in Mink10 + 5 interacting closed strings excitations in AdS5 x S 5 ( IIB supergravity in AdS5 x S ) AdS S5 L 5 ⇥

Figure 12: Spacetime around D3-branes. Open string perspective,L ` g N 1: • ⌧ s , ⇠ s ⌧ 3-brane is zero thickness object in Mink ⇠ ⇠ 10 => free closed str. in Mink10 + interacting open strings excitations on Dp worlvolume

( =4 SYM in R1,3 ) N Mink10

Two perspectives should be equivalent descriptions of the sameFigure physics, 13: Tadpole-like diagrams whose sum leads to an effective geometry for closed strings.

and type IIB supergravity on Mink10 is present in both perspectives … so identify 2 other theories! 10 ➙ The AdS/CFT conjecture Mink10

IIB supergravity in AdS5 × S5 is dual to N=4 SYM in R1,3 AdS S5 • SYM is a non-abelian SU(N) gauge theory (like QCD) L 5 ⇥ but with conformal invariance

(i.e. gYM is scale-independent and does not change with E): TrivialFigure 12: Spacetime around D3-branes. hence SYM is a CFT () match

• Relaxing the low energy limit:

N = 4 SYM theory in R1,3 with gauge group SU(N)

and YM coupling gYM Mink is equivalent to 10 SYM type IIB string theory with string length ls and Figure 13: Tadpole-like diagrams whose sum leads to an effective geometry for closed strings. 5 coupling constant gs on AdS5 × S with radius of 10 curvature L and N units of flux F(5) on S5 =4 SYM theory IIB theory on AdS S5 N () 5 ⇥ Strongest form: any N and Quantum string theory: g =0, ` /L =0 () s 6 s 6 Strong form: N ,any Classical string theory: gs 0, `s/L =0 Weak form: N !1, 1 () Classical supergravity: g !0, ` /L 6 0 !1 () s ! s !

Table 1: Di↵erent forms of the AdS/CFT correspondence.

=4 SYM theory IIB theory on AdS S5 N () 5 ⇥ Strongest form: any N and Quantum string theory: g =0, `s =0 () s 6 L 6

= e', ⇢1/2 (Cooper pair charge 2e, mass m), ' is macroscopic SC phase (1) | | | | ⇠ C e e

2 4 1 2 1 2 Fs(r,T)=Fn(r,T)+↵ + + ( ih 2eA) + B (2) | | 2 | | 2m | r | 2µ0

1 F (r,T)=F (r,T)+↵ 2 + 4 + ( ih 2eA) 2 (3) s n | | 2 | | 2m | r |

Minimize F , + , ⇤ ⇤ + ⇤ : (4) s ! ! =4 SYM theory IIB theory on AdS S5 N () 5 ⇥ ➙ Matching parameters ( the Pokémons ) Strongest form: any N and Quantum string theory: g =0, ` /L =0 Minimize Fs,()Fs/ =0: s 6 s 6 (5) Strong form: N ,any Classical string theory: g 0, ` /L =0 Free parameters of field theory!1 {gYM , N} are mapped() to free parameters { gs , L/ ls s} !on the strings 6 theory via: Weak form: N , 1 Classical supergravity: gs 0, `s/L 0 !1 () 4 ! ! 2 Dp SY M L g 2⇡g = 1 = 2 2 YM s low E 4 =4⇡ gsN =2 Q3 QD3 ⌘ N (( Lih 2eAL) + ↵ + ` =0 ( ⌘ (6) Table2m 1:| Di ↵erentr forms of| the AdS/CFT| s | correspondence. =4 SYM theory IIB theory on AdS S5 N () 5 ⇥ Strongest form: any N and Quantum string theory: g =0, `s =0 () s 6 L 6 B = µ J µ J = 2A (London gauge: A = 0) & Minimize F , A A + A : (7) r⇥ 0 ) 0 r r⇥ s !

' 1/2 = e , 2 ⇢C (Cooper pair charge 2e, mass m), ' is macroscopic SC phase (1) B = µ0J | |µ0J = | |A⇠ (London gauge: A = 0) & Minimize Fs, Fs/A =0: (8) r⇥ ) e r r⇥ e =4 SYM theory IIB theory on AdS S5 N () 5 ⇥ 1 1 `s Strong form: N ,any2 4 Classical string theory:2 gs 2 0, L =0 Fs(r,T)=Fn!1(r,T)+↵ + ()+ ( ih 2eA) + B! 6 (2) | | 2 | | 2m | r | 1 2µ0 gs ‘ ~ ’ 0 • ⇠ N ⇠ ! gYM 0 : ( planar lim of SYM ) 1 • ! t’Hooft limit (to suppress genus (quantum) loops) N 1 4 2 2 !14 L 2 = gYMN Ffinites(r,T)=Fn(r,T)+↵ + + ( ih 42eA)gsN finite (3) • | | 2 | | 2m | r• `s ⇠ | ⇠

Minimize F , + , ⇤ ⇤ + ⇤ : (4) s ! !

Minimize Fs, Fs/ =0: (5)

1 ( ih 2eA) 2 + ↵ + 2 =0 (6) 2m | r | | |

B = µ J µ J = 2A (London gauge: A = 0) & Minimize F , A A + A : (7) r⇥ 0 ) 0 r r⇥ s !

B = µ J µ J = 2A (London gauge: A = 0) & Minimize F , F /A =0: (8) r⇥ 0 ) 0 r r⇥ s s

=4 SYM theory IIB theory on AdS S5 N () 5 ⇥ Strong form: N ,any Classical string theory: g 0, `s =0 !1 () s ! L 6

1 ➙ Matching parameters

=4 SYM theory IIB theory on AdS S5 N () 5 ⇥ Strongest form: any N and Quantum string theory: g =0, ` /L =0 () s 6 s 6 Strong form: N ,any Classical string theory: gs 0, `s/L =0 Weak form: N !1, 1 () Classical supergravity: g !0, ` /L 6 0 !1 () s ! s !

Table 1: Di↵erent forms of the AdS/CFT correspondence.

1 g 0 : ( planar lim of SYM ) gs ‘ ~ ’ 0 • YM ! • ⇠ N ⇠ ! = e', ⇢1/2 (Cooper pair charge 2e, mass m), ' is macroscopic SC phase (1) | | | | ⇠ C t’Hooft limit (to suppress genus (quantum) loops) 2e `4 1 = gYMN 1 s 0 • N e4 SYM is strongly coupled !1 • L ⇠ ! ) (to suppress string loops)

2 4 1 2 1 2 Fs(r,T)=Fn(r,T)+↵ + + ( ih 2eA) + B (2) | | 2 | | 2m | r | 2µ0

1 F (r,T)=F (r,T)+↵ 2 + 4 + ( ih 2eA) 2 (3) s n | | 2 | | 2m | r |

Minimize F , + , ⇤ ⇤ + ⇤ : (4) s ! !

Minimize Fs, Fs/ =0: (5)

1 ( ih 2eA) 2 + ↵ + 2 =0 (6) 2m | r | | |

B = µ J µ J = 2A (London gauge: A = 0) & Minimize F , A A + A : (7) r⇥ 0 ) 0 r r⇥ s !

B = µ J µ J = 2A (London gauge: A = 0) & Minimize F , F /A =0: (8) r⇥ 0 ) 0 r r⇥ s s

e J = [⇤ ( ih 2eA) + c.c.] (9) m r

1 ➙ Strong / weak coupled duality !

=4 SYM theory IIB theory on AdS S5 N () 5 ⇥ Weak form: N , 1 Classical supergravity: g 0, `s 0 !1 () s ! L !

1 gYM 0 : ( planar lim of SYMe ) gs ‘ ~ ’ 0 • ! J = [⇤ ( ih 2eA) + c.c.] • ⇠ N ⇠ ! (9) m t’Hooftr limit (to suppress genus (quantum) loops) 2 `4 1 = gYMN 1 N s • 4 0 SYM is strongly coupled !1 • L ⇠ ! ) (to suppress string loops) 4 1 ab a 2 S = d x p g F F 2(D )(D )† 2V ( ) (10) 2 ab a | | Z  pointlike regime of string theory: 5 IIB supergravity on weakly curved AdS5 × S 4 6 1 ab a 2 S = d x p g R + FabF 2(Da)(D )† 2V ( ) (11) L2 2 | | Z ⇥ ⇤ Strong / weak coupled duality !

4 6 1 ab a 2 S = d x p g R + F F 2(D )(D )† 2V ( ) (12) L2 2 ab a | | Z ⇥ ⇤

↵ z=0 = + + (13) + | r r ···

2 2 ⌘ µ V (⌘)=⌘ µ 1 , ⌘ = † (14) 4 V ✓ 0 ◆

3 S S  d x † (15) bdry ! bdry O O Z

2 2 2 µBF <µ <µUnit (16)

= e', ' = in' (17) | | e e

F ab = g J b (18) ra c 2 ➙ Symmetries on both sides do match

Global symmetries of Isometries of 1,3 N = 4 SYM theory in R IIB string theory on AdS5 × S5 • diffeomorphisms = gauge transf. • large gauge transf. that leave the asymptotic form of the metric invariant: SO(2, 4) × SO(6) M change • Conf(1, 3): ie conformal group of Mink1,3 • SO(2, 4) ~ Conf(1, 3): Dilatation + Poincare +special conf. transf. isometry group of AdS5 • SO(6)~SU(4): R-symmetry (see review Maldacena) • SO(6)~SU(4): isometry group of S5. • Fermionic symmetries: (…) • Fermionic symmetries: (…)

SYM is CFT Also symmetry in grav.: => Invariant to Dilatation: 2 r2 µ ⌫ L2 2 ds = 2 ⌘µ⌫ dx dx + 2 dr invariant if xµ xµ AdS5 L r xµ xµ & r r ! ! !

(x, ⌦)= `(x)Y`(⌦) X`

Pokémon QFT tattoo @ Mink1,3 AdS5 ⇠

AdS5 ➙ UV / IR connection. RG flow z = 0 : boundaryAdS (UV)

2 2 2 2 5 ds = dsAdS5 + L dsS 2 2 2 r µ ⌫ L 2 2 2 L RG flow = ⌘µ⌫ dx dx + dr + L d⌦ r = L2 r2 5 . z 1 ✓ ◆ z L2 ⇠ EYM Bulk = ⌘ dxµdx⌫ + dz2 + L2d⌦2 z2 µ⌫ 5 ⇣ ⌘ z : Poincar´eHorizon (IR) const-z slice of AdS5 is isometric to Mink1,3. {xµ } coord. QFT. !1

warp factor L2/z2 L z SYM: E ,d in the bulk: d = d ,E= E • { YM YM} ! z YM L YM Same E but di↵erent z di↵erent E . • ) YM 1 Fix EL 1. SYM process with EYM Bulk process localised at: z ⌘ $ ⇠ EYM UV: E z 0 , IR: E 0 z • YM !1, ! YM ! , !1

• z-direction is identified with renormalization group (RG) scale in the gauge theory: [ Wilson… 60’s ]

Want properties at length scale z >>ε ? Integrate out short-distance dof a effective theory at length scale z. Repeat process if interested on z’>>z a effective theory at length scale z’ 1,3 Defines a renormalization group (RG) flow a continuous family of EFTs in Mink labeled by RG scale z.

1,3 May visualize continuous family of EFTs in Mink as a single theory in a (4 + 1)-dim spacetime with the RG scale z being a radial coord. Computational Methods in Holography The Gauge- Duality

The Gauge-Gravity Duality ➙ gauge/gravity duality is a geometrization of the RG evolution of a QFT

z = 0 : boundary (UV) Boundary AdS !d!2,1

AdSd RG flow 1 Bulk z ⇠ EYM Bulk IR UV z z : Poincar´eHorizon (IR) IR UV !1 z " L2 u (u=r )

Wilson renormalization group (RG): integrating out short distance! d.o.f.Motivation: the coupling constant g of the theory changes under scale transformations according RG flow ODE which is local in the RG scale u. t’Hooft original arguments: the maximum entropy in a region • That is, given an eqn, β(g)=0, we canof always space form is the the Ricciarea flow of eqn its: boundary@g(u,inPlanckunits.) u = (g(u)) Weinberg-Witten: graviton can’t@ beu a composite particle. u ~ (fictitious) flow `time’. Ricci flowRG is equationsthe one-parameterfor couplingfamily of couplingsg are localg(u) . in the energy scale u: Solve it as as a geometric evolution eqn for g (after some initial guess) . The system evolves towards a fixed point g that solves β(g)=0u @( CFTug( u) )=(g(u)) . • A QFT is defined by an UV fixed point and a RG flow: SYM is a fixed point living at UV boundary. 3/22 • Dilatation is exact isometry of AdS => RG flow is trivial for AdS/SYM. • For generic asymptotically AdS: Dilatation is only asymptotic isometry => non-trivial RG towards UV SYM