EXACT RESULTS FOR STATIC AND RADIATIVE FIELDS OF A QUARK IN N=4 SUPER YANG-MILLS

Blai Garolera - Universitat de Barcelona

B. Fiol, B.G. hep-th\1106.5418 B. Fiol, B.G, A. Lewkowycz hep-th\1202.5292

- Exact results in 4D QFT are extremely hard to obtain...

- The situation improves A LOT with additional symmetries: CFTs, SUSY, ...

N=4 SYM is both conformal and maximally supersymmetric!

- Various techniques can apply: Integrability, Localization, AdS/CFT, ...

Preview

- Exact results in 4D QFT are extremely hard to obtain...

- The situation improves A LOT with additional symmetries: CFTs, SUSY, ...

N=4 SYM is both conformal and maximally supersymmetric!

- Various techniques can apply: Integrability, Localization, AdS/CFT, ...

Preview

Preview

- We will focus on observables related with external probes of N=4 SU(N) SYM.

(～idealization of QCD with two parameters: λ and N )

- First, we will use AdS/CFT to obtain results valid at large λ and to all orders in 1/N.

- Then we will use localization results to provide exact expressions, valid for all λ and N.

Maxwell N=4 SU(N) SYM

Static particle

Accelerated particle

1|⇥x|4

q2< L(⌅x) >=

aµaµP =23q2 aµaµP = 8⇥f(�, N)

< L(⌅x) >= f(�, N)1

|⇥x|4

Preview

f(�, N) =�

64⇥2N

L2N�1(� �

4N ) + L2N�2(� �

4N )L1

N�1(� �4N )

Maxwell N=4 SU(N) SYM

Static particle

Accelerated particle

1|⇥x|4

q2< L(⌅x) >=

aµaµP =23q2 aµaµP = 8⇥f(�, N)

< L(⌅x) >= f(�, N)1

|⇥x|4

Preview

External probes in AdS/CFT

UV

IR

N=4 SU(N) SYM

IIB SUGRA in AdS5xS5

z

External probes in AdS/CFT

(z=0)

(z➝∞)

UV

IR

N=4 SU(N) SYM

IIB SUGRA in AdS5xS5

z

External probes in AdS/CFT

(z=0)

(z➝∞)

Φ(xμ, z)

<O(xμ)>

UV

IR

N=4 SU(N) SYM

IIB SUGRA in AdS5xS5

z

External probes in AdS/CFT

(z=0)

(z➝∞)

UV

IR

N=4 SU(N) SYM

IIB SUGRA in AdS5xS5

z

External probes in AdS/CFT

(z=0)

(z➝∞)

(z=zH)

T=TH

UV

IR

N=4 SU(N) SYM

IIB SUGRA in AdS5xS5

z

External probes in AdS/CFT

(z=0)

(z➝∞)

UV

IR

N=4 SU(N) SYM

IIB SUGRA in AdS5xS5

z

External probes in AdS/CFT

(z=0)

(z➝∞)

xμ(τ)

@AdS5

AdS5

z

external color source

F1

worldsheet horizon

Poincare horizon

x

µ

(t

ret

)

P =2�

aµaµ

��

< L(⇧x) >=1

16�2 |⇧x|4

��

Vqq = � 4�2

�4�

14

⇥L

��

��ln < W� >=

External probes in AdS/CFT

There are two types of corrections: (world-sheet fluctuations) (higher genus world-sheets)

1/�

�1/N

Fundamental rep:

@AdS5

AdS5

z

external color source

F1

worldsheet horizon

Poincare horizon

x

µ

(t

ret

)

P =2�

aµaµ

��

< L(⇧x) >=1

16�2 |⇧x|4

��

Vqq = � 4�2

�4�

14

⇥L

��

��ln < W� >=

External probes in AdS/CFT

There are two types of corrections: (world-sheet fluctuations) (higher genus world-sheets)

1/�

�1/N

Fundamental rep:

External probes in AdS/CFT

k-symmetric rep: @AdS5

AdS5

z

external color source

Poincare horizon

worldvolume horizon

D3/D5

< L(⌅x) >Sk=1

|⇥x|4k�

�

16⇥2

�1 +

k2�

16N2

k�

�

2⇥

�1 +

k2�

16N2P = aµaµ

ln < W (�) >=k�

�

2

�1 +

k2�

16N2+ 2N sinh�1 k

��

4N

Exact results for external probes

Exact results for external probes

SCT(E) A.C.

SCT(L)

r1 = R

r1 = 0

x

0

x

1

< W| >= 1 < W� >�= 1

Exact results for external probes

SCT(E) A.C.

SCT(L)

r1 = R

r1 = 0

x

0

x

1

< W| >= 1 < W� >�= 1

Conformal anomaly!

Exact results for external probes

The anomaly is localized in a single point in space-time

It is perturbatively captured by a matrix model:

< W⇥ >=1N

L1N�1

�� �

4N

⇥e

�8N

By means of localization techniques, it was proved that this result is correct and exact

Exact results for external probes

The two-point function of a circular Wilson loop and a chiral primary operator can also be computed with a matrix model

< W (⇥)�O2 >=⌅

2�

4N3

⇤L2

N�1

�� �

4N

⇥+ L2

N�2

�� �

4N

⇥⌅e

�8N

and from this we can extract the one-point function of the chiral primary operator in the presence of the probe

< O2 >J=< WO2 >

< W >=⇤

2�

4N2

L2N�1(� �

4N ) + L2N�2(� �

4N )L1

N�1(� �4N )

A key observation for our problem at hand is that in N=4 SYM, the Lagrangian and the stress-energy tensor belong to the same short multiplet, the supercurrent multiplet:

Exact results for external probes

O2 = Tr��{I�J}

⇥L � Q4O2 Tµ⇥ � Q2Q2O2

A key observation for our problem at hand is that in N=4 SYM, the Lagrangian and the stress-energy tensor belong to the same short multiplet, the supercurrent multiplet:

Exact results for external probes

O2 = Tr��{I�J}

⇥L � Q4O2 Tµ⇥ � Q2Q2O2

A key observation for our problem at hand is that in N=4 SYM, the Lagrangian and the stress-energy tensor belong to the same short multiplet, the supercurrent multiplet:

Exact results for external probes

O2 = Tr��{I�J}

⇥L � Q4O2 Tµ⇥ � Q2Q2O2

< L(�x) >�=�

64⇥2N

L2N�1(� �

4N ) + L2N�2(� �

4N )L1

N�1(� �4N )

1|�x|4

P =�

8⇥N

L2N�1(� �

4N ) + L2N�2(� �

4N )L1

N�1(� �4N )

1R2

QFT results EXACT for any λ and N.

If we take the large λ and large N limit we precisely recover our AdS/CFT results obtained with D3-branes, for k=1!

Exact results for external probes

These are very precise tests of the AdS/CFT correspondence, valid to all orders in 1/N!

Setting k=1 in our D-brane computation is not justified a priori... but it works! WHY?