Instantons in AdS/CFT duality

Pei-Ming Ho Department of Physics National Taiwan University

1 Yang-Mills theory

µ a gauge potential A = dx Aµ(x)τa

Lie algebra [τa, τb] = fabcτc

field strength F = dA + AA

1 µ ν a F = dx dx F (x)τa 2 µν

Fµν =[∂µ + Aµ, ∂ν + Aν]

2 Yang-Mills action 1 = Tr SYM 2 F ∗F gYM Z   Variation δA equation of motion. →

Hodge duality = electromagnetic duality

F F → ∗ 1 Fµν ǫ F ∗ ≡±2 µναβ αβ

3 (anti-) Self-dual = (anti-) Instanton

F = F ±∗

8π2 = SYM 2 Q ⇒ gYM

Instanton number = topological charge 1 Q = − Tr (FF ) Z 16π2 Z ∈ 2 Tr (FF ) = dTr AF AAA  − 3    δQ =0 4 Self-duality equation of motion • ⇒ YM instantons have vanishing • energy-momentum tensor. Conformal group SO(5, 1) acts on • the moduli space.

5 1-instanton solution 2 r 1 A =   U− dU r2 + L2     2 µ r = xµx (µ =1, , 4) · · · A U 1dU, r → − →∞ 4 a x ix σa U = − (i =1, 2, 3) r 0 1 0 i 1 0 σ =   σ =  −  σ =   1 1 0 2 i 0 3 0 1            − 

6 1-instanton solution

2 Fµν =4Z (x; x0,L)σµν

1 1 σ4a = σa, σ = [σa, σ ] 2 ab 4i b L Z(x; x0,L) ≡ L2 + x x 2 | − 0|

( ) µν 4 Tr Fµν F Z (x; x ,L). ∗ ∝ 0

7 AdS/CFT duality:

Type IIB superstring in AdS S5 5 × SU(N) super Yang-Mills theory in 4D. ≃

Maldacena [98]; Gubser, Klebanov, Polyakov [98]; Witten [98]

(g, R)IIB (gYM,N)YM: ↔ 2 2 2 √ 4πg = gYM, R /ls =4πg N.

8 AdS Euclidean AdS 5 → 5

5,1 EAdS5 embedded in R : 5 2 2 2 R = X0 Xi − i=1X

Poincar´epatch of EAdS5 : 2 R zXµ z = (z > 0), xµ = (µ =1, 4) X0 + X5 R · · ·

R2 ds2 = dz2 + dx2 z2  

9 Geometry of EAdS2

10 Duality

IIB superstring in EAdS 5 ≃ SU(N) super Yang-Mills on ∂EAdS R4 . 5 ≃

Isometry of EAdS SO(5, 1) 5 ≃ conformal symmetry of R4. ≃

YM instanton and its dual are good probes.

11 What is the dual of YM instanton?

Possibilities: 1D object with an endpoint on the boundary  •  AdS  This works only for 5 .    or   0D object in EAdS5   •   How to match the coordinates?   

12 Supergravity (Einstein frame)

1 1 2φ 1 αβγλ Rµν ∂µφ∂νφ + e ∂µχ∂νχ = F F ν − 2 2 6 αβγλµ

2φ µ 2 2φ µ µ e ∂ χ =0 φ + e ∂µχ∂ χ =0 ∇   ∇ φ = dilaton, χ = axion

Background: EAdS5 φ = 0 = χ   F = self-dual vol. form FF g.  → ∝ 

13 “Self-dual” ansatz

(τ τ ) = i(τ τ )∗ − ∞ ± − ∞ τ χ + ie φ ≡ −

φ φ χ χ = e− e− ∞ − ∞ ±  −  Supergravity equations become

1 αβγλ Rµν = F F ν 6 αβγλµ

2eφ =0 ∇ 14 S-duality ατ + βτ α β R τ τ ′ =   SL(2, ) → γτ + δτ γ δ ∈    

15 The “self-dual” ansatz is very special: The background metric is not affected. • It is self-dual like. • It preserves 1/2 SUSY. • The action becomes • 5 2φ µν S = d x ∂µ e χg ∂νχ − Z   2πn φ = (g = e ∞) g ∞ ∞

16 D-instanton solution

Chu, Ho, Wu: hep-th/9806103 2 4 2 φ (d + 2)(d +4d 2) e = c0 + c1 − d3(d2 + 4)3/2 d = SO(1, 5)-invariant length

ηABX X 1 2 A B = ( )2 + 2 d 2 z z0 x x0 ≡ R z0z  − | − | 

(z0, x0) = location of the D-instanton

17 Approaching to the boundary (z 0) ∼ eφ eφ 6c z4Z4(x; x ,z ) ≃ ∞ − 1 0 0

2φ 4 4 χ χ 6c1e− ∞z Z (x; x0,z0) ≃ ∞ ± z0 Z(x; x0,z0) ≡ z2 + x x 2 0 | − 0|

D´ej`avu!

18 D instanton = YM instanton 4πn L = z0 c1 = n = Q N2 Charge quantization due to QM

φ φ µν e e Tr FµνF − − − ∞ ↔

µν χ χ Tr Fµν∗F − ∞ ↔

19 Scaling (z, x) (λz, λx) is a symmetry of → 2 the EAdS metric ds2 = R (dz2 + dx2). 5 z2 R4 : (L, x) (λL, λx) preserves → 2 the YM instanton sol. A = r U 1dU r2+L2! − 4 a x ix σa where U = −r .

z = L ⇒ 0

20 Generalizations:

B-field background NC R4 • →

EAdS w. black hole R3 S1 • 5 → × Rey, Jun. 05

EAdS conformally compact • 5 → Einstein manifolds.

21 ADHM multi-instantons

For =4 SU(N) super YM theory: N The N k limit of k instanton moduli ≫ space has the geometry of a single copy of EAdS S5. (Instantons tend to stay on 5 × top of each other.) The SU(2) embeddings in SU(N) are mutually commuting.

Dorey, Hollowood, Khoze, Mattis, Vandoren: hep-th/9901128

Dorey, Hollowood, Khoze, Mattis: hep-th/0206063

22 For finite N: Corrections at the subleading term in the 1/N expansion Something must happen when

k = N/2 N/2+1. → Supergravity approximation of superstring is good only for large N. Quantum effect is important for finite N.

23 Quantum deformation of EAdS5 q-EAdS5

Jevicki, Ramgoolam: hep-th/9902059; Ho, Ramgoolam, Tatar:

hep-th/9907145; Ho, Li: hep-th/0004072, hep-th/0005268.

q-AdS2: Lie algebra of SL(2, R) ≃ algebra of the Cartesian coordinates.

q-AdS2 can be defined by the representation (j, α) with j =1/2 + iN and α is the VEV of χ.

24 Noncommutative instanton

Nekrasov, Schwarz: hep-th/9802068 ADHM construction

V = CN,W = Ck

B ,B Hom(V, V ), I,J Hom(W, V ) 0 1 ∈ † ∈

µr [B ,B†]+[B ,B†] + II J J ≡ 0 0 1 1 † − †

µc [B ,B ] + IJ ≡ 0 1

25 Instantons on R4 are parametrized by

solutions of (µr =0, µc = 0) modulo equivalence relations.

1 1 = µr− (0) + µc− (0) /U(V ) M   has conical singularities. M

26 Noncommutative R4 is defined by ζ [z , z¯ ]=[z , z¯ ] = 0 0 1 1 −2 ADHM on noncommutative R4

µr = ζ, µc =0

1 1 = µ−r (ζ) + µc− (0) /U(V ) M   Singularities resolved by the deformation.

(Deformation of µc =0 is equivalent.)

27 Conclusion

D-instanton YM instanton • ↔ isometry conformal symemtry • ↔ Moduli space for k =1 and large N • AdS/CFT notion of (q-AdS ) • → 5 N Generalizations •

28