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 string theory • QCD: gauge theory 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 string theory,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 graviton
• 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-planar 2 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-branes.
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 = Supergravity
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-brane 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 '