![Page 1: Minimal area surfaces and Wilson loops in the AdS/CFT ...Minimal area surfaces and Wilson loops in the AdS/CFT correspondence M. Kruczenski Purdue University Miami 2015 Based on: arXiv:1406.4945](https://reader036.vdocuments.net/reader036/viewer/2022081619/60f8ad07aa5a073c34565577/html5/thumbnails/1.jpg)
Minimal area surfaces and Wilson loops in the AdS/CFT correspondence
M. Kruczenski
Purdue University
Miami 2015
Based on: arXiv:1406.4945 [JHEP 1411 (2014) 065] + recent work w/ A. Irrgang, arXiv:1507.02787 and w/ Changyu Huang, Yifei He (to appear)
![Page 2: Minimal area surfaces and Wilson loops in the AdS/CFT ...Minimal area surfaces and Wilson loops in the AdS/CFT correspondence M. Kruczenski Purdue University Miami 2015 Based on: arXiv:1406.4945](https://reader036.vdocuments.net/reader036/viewer/2022081619/60f8ad07aa5a073c34565577/html5/thumbnails/2.jpg)
Summary ●Introduction and Motivation
Explore the (AdS/CFT) relation between Wilson loops and minimal area surfaces in AdS.
Wilson loops Strings Minimal surfaces
● Minimal area surfaces in hyperbolic space (EAdS3)
Examples: Surfaces ending on a circle, and other analytical solutions.
2
![Page 3: Minimal area surfaces and Wilson loops in the AdS/CFT ...Minimal area surfaces and Wilson loops in the AdS/CFT correspondence M. Kruczenski Purdue University Miami 2015 Based on: arXiv:1406.4945](https://reader036.vdocuments.net/reader036/viewer/2022081619/60f8ad07aa5a073c34565577/html5/thumbnails/3.jpg)
● Linear systems
Minimal area surfaces in flat space. Plateau problem and Douglas functional. Soap film problem.
● Non-linear systems
The problem is studied in terms of a Schrödinger equation defined on the Wilson loop.
● Conclusions
3
![Page 4: Minimal area surfaces and Wilson loops in the AdS/CFT ...Minimal area surfaces and Wilson loops in the AdS/CFT correspondence M. Kruczenski Purdue University Miami 2015 Based on: arXiv:1406.4945](https://reader036.vdocuments.net/reader036/viewer/2022081619/60f8ad07aa5a073c34565577/html5/thumbnails/4.jpg)
λ small → gauge th.
Gauge theory String theory
λ large → string th.
N=4 SYM Strings in AdS5xS5
S5 : X12+X2
2+…+X62 = 1
AdS5: Y12+Y2
2+…-Y52-Y6
2 =-1 4
![Page 5: Minimal area surfaces and Wilson loops in the AdS/CFT ...Minimal area surfaces and Wilson loops in the AdS/CFT correspondence M. Kruczenski Purdue University Miami 2015 Based on: arXiv:1406.4945](https://reader036.vdocuments.net/reader036/viewer/2022081619/60f8ad07aa5a073c34565577/html5/thumbnails/5.jpg)
Wilson loops: associated with a closed curve in space. Basic operators in gauge theories. E.g. qq potential.
Simplest example: single, flat, smooth, space-like curve (with constant scalar).
C
C
x
y
5
![Page 6: Minimal area surfaces and Wilson loops in the AdS/CFT ...Minimal area surfaces and Wilson loops in the AdS/CFT correspondence M. Kruczenski Purdue University Miami 2015 Based on: arXiv:1406.4945](https://reader036.vdocuments.net/reader036/viewer/2022081619/60f8ad07aa5a073c34565577/html5/thumbnails/6.jpg)
String theory: Wilson loops are computed by finding a minimal area surface (Maldacena, Rey, Yee)
Circle:
circular (~ Lobachevsky plane) Berenstein Corrado Fischler Maldacena Gross Ooguri, Erickson Semenoff Zarembo Drukker Gross, Pestun
6
![Page 7: Minimal area surfaces and Wilson loops in the AdS/CFT ...Minimal area surfaces and Wilson loops in the AdS/CFT correspondence M. Kruczenski Purdue University Miami 2015 Based on: arXiv:1406.4945](https://reader036.vdocuments.net/reader036/viewer/2022081619/60f8ad07aa5a073c34565577/html5/thumbnails/7.jpg)
Infinite parameter family of solutions: Ishizeki, Ziama, M.K.
7
EAdS3 Poincare coordinates
![Page 8: Minimal area surfaces and Wilson loops in the AdS/CFT ...Minimal area surfaces and Wilson loops in the AdS/CFT correspondence M. Kruczenski Purdue University Miami 2015 Based on: arXiv:1406.4945](https://reader036.vdocuments.net/reader036/viewer/2022081619/60f8ad07aa5a073c34565577/html5/thumbnails/8.jpg)
Also Euclidean surfaces in Minkowski signature: (w/ A. Irrgang)
8 AdS3 global coordinates
![Page 9: Minimal area surfaces and Wilson loops in the AdS/CFT ...Minimal area surfaces and Wilson loops in the AdS/CFT correspondence M. Kruczenski Purdue University Miami 2015 Based on: arXiv:1406.4945](https://reader036.vdocuments.net/reader036/viewer/2022081619/60f8ad07aa5a073c34565577/html5/thumbnails/9.jpg)
What do we want to find?
Map from unit circle to EAdS3 (H3). Minimizes area and has as boundary condition:
Z
X1 X2 Z=0 X=X(s)
|z|=1
![Page 10: Minimal area surfaces and Wilson loops in the AdS/CFT ...Minimal area surfaces and Wilson loops in the AdS/CFT correspondence M. Kruczenski Purdue University Miami 2015 Based on: arXiv:1406.4945](https://reader036.vdocuments.net/reader036/viewer/2022081619/60f8ad07aa5a073c34565577/html5/thumbnails/10.jpg)
Consider first easier case: flat space
In conformal gauge:
|z|=1
10
linear
![Page 11: Minimal area surfaces and Wilson loops in the AdS/CFT ...Minimal area surfaces and Wilson loops in the AdS/CFT correspondence M. Kruczenski Purdue University Miami 2015 Based on: arXiv:1406.4945](https://reader036.vdocuments.net/reader036/viewer/2022081619/60f8ad07aa5a073c34565577/html5/thumbnails/11.jpg)
Using standard results for the Laplace equation, we immediately compute the area as (Douglas functional)
However, we do not actually know , namely we did not yet found the conformal coordinates.
It turns out that the correct is the one that minimizes the functional .
Thus, the problem reduces to find the correct reparameterization 11
![Page 12: Minimal area surfaces and Wilson loops in the AdS/CFT ...Minimal area surfaces and Wilson loops in the AdS/CFT correspondence M. Kruczenski Purdue University Miami 2015 Based on: arXiv:1406.4945](https://reader036.vdocuments.net/reader036/viewer/2022081619/60f8ad07aa5a073c34565577/html5/thumbnails/12.jpg)
Minimal Area surfaces in EAdS3
Equations of motion
12
Non-linear
![Page 13: Minimal area surfaces and Wilson loops in the AdS/CFT ...Minimal area surfaces and Wilson loops in the AdS/CFT correspondence M. Kruczenski Purdue University Miami 2015 Based on: arXiv:1406.4945](https://reader036.vdocuments.net/reader036/viewer/2022081619/60f8ad07aa5a073c34565577/html5/thumbnails/13.jpg)
Method of solution: Pohlmeyer reduction
Solve And plug into
Then solve the linear problem:
13
![Page 14: Minimal area surfaces and Wilson loops in the AdS/CFT ...Minimal area surfaces and Wilson loops in the AdS/CFT correspondence M. Kruczenski Purdue University Miami 2015 Based on: arXiv:1406.4945](https://reader036.vdocuments.net/reader036/viewer/2022081619/60f8ad07aa5a073c34565577/html5/thumbnails/14.jpg)
Integrability There are an infinite number of conserved charges. How can we use them?
|z|=1
Initial value problem Boundary condition problem
14
![Page 15: Minimal area surfaces and Wilson loops in the AdS/CFT ...Minimal area surfaces and Wilson loops in the AdS/CFT correspondence M. Kruczenski Purdue University Miami 2015 Based on: arXiv:1406.4945](https://reader036.vdocuments.net/reader036/viewer/2022081619/60f8ad07aa5a073c34565577/html5/thumbnails/15.jpg)
15
n
C
0W!C 0 The Wilson loop operator creates a multi-
particle state/ boundary string state, we compute overlap with vacuum, zero charge component.
No radiation to infinity String disappears. No strings falling into the horizon
![Page 16: Minimal area surfaces and Wilson loops in the AdS/CFT ...Minimal area surfaces and Wilson loops in the AdS/CFT correspondence M. Kruczenski Purdue University Miami 2015 Based on: arXiv:1406.4945](https://reader036.vdocuments.net/reader036/viewer/2022081619/60f8ad07aa5a073c34565577/html5/thumbnails/16.jpg)
Minimal Area surfaces in EAdS3 (H3)
Can we recast the problem into finding the correct reparameterization , ? Yes, gives the necessary boundary conditions.
|z|=1
16
![Page 17: Minimal area surfaces and Wilson loops in the AdS/CFT ...Minimal area surfaces and Wilson loops in the AdS/CFT correspondence M. Kruczenski Purdue University Miami 2015 Based on: arXiv:1406.4945](https://reader036.vdocuments.net/reader036/viewer/2022081619/60f8ad07aa5a073c34565577/html5/thumbnails/17.jpg)
Boundary conditions in terms of boundary data . However, because of global conformal invariance there are many equivalent boundary data related by
Use Schwarzian derivative!
And indeed, the Schwarzian derivative gives the boundary data for and .
17
![Page 18: Minimal area surfaces and Wilson loops in the AdS/CFT ...Minimal area surfaces and Wilson loops in the AdS/CFT correspondence M. Kruczenski Purdue University Miami 2015 Based on: arXiv:1406.4945](https://reader036.vdocuments.net/reader036/viewer/2022081619/60f8ad07aa5a073c34565577/html5/thumbnails/18.jpg)
Derivation
The linear problem for the flat current j can be written in the angular direction as Taking the limit and redefining we obtain
With
Reconstructing the boundary shape involves finding two l.i. solutions and computing the ratio:
( Notice is anti-periodic.) χ θ( )
18
![Page 19: Minimal area surfaces and Wilson loops in the AdS/CFT ...Minimal area surfaces and Wilson loops in the AdS/CFT correspondence M. Kruczenski Purdue University Miami 2015 Based on: arXiv:1406.4945](https://reader036.vdocuments.net/reader036/viewer/2022081619/60f8ad07aa5a073c34565577/html5/thumbnails/19.jpg)
Computation of the Area (assumption, f no zeros)
The area is given by
From the cosh-Gordon eq. it follows that
19
![Page 20: Minimal area surfaces and Wilson loops in the AdS/CFT ...Minimal area surfaces and Wilson loops in the AdS/CFT correspondence M. Kruczenski Purdue University Miami 2015 Based on: arXiv:1406.4945](https://reader036.vdocuments.net/reader036/viewer/2022081619/60f8ad07aa5a073c34565577/html5/thumbnails/20.jpg)
20
Then:
Replacing the limiting behavior we get
![Page 21: Minimal area surfaces and Wilson loops in the AdS/CFT ...Minimal area surfaces and Wilson loops in the AdS/CFT correspondence M. Kruczenski Purdue University Miami 2015 Based on: arXiv:1406.4945](https://reader036.vdocuments.net/reader036/viewer/2022081619/60f8ad07aa5a073c34565577/html5/thumbnails/21.jpg)
21 In the case where f(z) has zeros (umbilical points)
a1
b1
c
d
A
AMSV
![Page 22: Minimal area surfaces and Wilson loops in the AdS/CFT ...Minimal area surfaces and Wilson loops in the AdS/CFT correspondence M. Kruczenski Purdue University Miami 2015 Based on: arXiv:1406.4945](https://reader036.vdocuments.net/reader036/viewer/2022081619/60f8ad07aa5a073c34565577/html5/thumbnails/22.jpg)
Finding q(s)
Rewriting the Schrödinger equation using the coordinate s:
We find that the periodic potential is:
Known
22
![Page 23: Minimal area surfaces and Wilson loops in the AdS/CFT ...Minimal area surfaces and Wilson loops in the AdS/CFT correspondence M. Kruczenski Purdue University Miami 2015 Based on: arXiv:1406.4945](https://reader036.vdocuments.net/reader036/viewer/2022081619/60f8ad07aa5a073c34565577/html5/thumbnails/23.jpg)
So, V0(s) and V1(s) are known but for V2(s) we need q(s).
Alternatively, requiring that the solutions must be (anti)-periodic for all l allows, in principle, to compute V2(s) given V0(s) and V1(s). Not an easy problem, but it is equivalent to finding the area.
Moreover, when this is done, for each value of λ such that |λ|=1 we have a different potential. The ratio of two solutions of the equation give a Wilson loop of different shape but the same expectation value. This implies the existence of a symmetry:
λ-deformations: symmetry not related to global symmetries of the theory but to integrability. 23
![Page 24: Minimal area surfaces and Wilson loops in the AdS/CFT ...Minimal area surfaces and Wilson loops in the AdS/CFT correspondence M. Kruczenski Purdue University Miami 2015 Based on: arXiv:1406.4945](https://reader036.vdocuments.net/reader036/viewer/2022081619/60f8ad07aa5a073c34565577/html5/thumbnails/24.jpg)
Important case: near circular Wilson loops
Near circular Wilson loops were studied initially by Semenoff and Young and subsequently by many authors (Drukker; Correa, Henn, Maldacena, Sever; Cagnazzo).
Recently Amit Dekel showed that the reparameterization q(s) can be found order by order. He was able to compute expansion to order eight in some cases.
24
![Page 25: Minimal area surfaces and Wilson loops in the AdS/CFT ...Minimal area surfaces and Wilson loops in the AdS/CFT correspondence M. Kruczenski Purdue University Miami 2015 Based on: arXiv:1406.4945](https://reader036.vdocuments.net/reader036/viewer/2022081619/60f8ad07aa5a073c34565577/html5/thumbnails/25.jpg)
25
New solutions( ) w/ C. Huang and Yifei He
We propose a periodic potential
such that all the solutions of the corresponding Schrödinger equation are anti-periodic. Indeed, setting , changing l only shifts s preserving the periodicity. Moreover, the change of variables
reduces the Schrödinger equation to the Mathieu equation:
![Page 26: Minimal area surfaces and Wilson loops in the AdS/CFT ...Minimal area surfaces and Wilson loops in the AdS/CFT correspondence M. Kruczenski Purdue University Miami 2015 Based on: arXiv:1406.4945](https://reader036.vdocuments.net/reader036/viewer/2022081619/60f8ad07aa5a073c34565577/html5/thumbnails/26.jpg)
26
with
We can find Floquet solutions
We need that happens for a discrete set
of eigenvalues , namely
![Page 27: Minimal area surfaces and Wilson loops in the AdS/CFT ...Minimal area surfaces and Wilson loops in the AdS/CFT correspondence M. Kruczenski Purdue University Miami 2015 Based on: arXiv:1406.4945](https://reader036.vdocuments.net/reader036/viewer/2022081619/60f8ad07aa5a073c34565577/html5/thumbnails/27.jpg)
27
The shape of the contour is given by the ratio of two linearly independent solutions
And the area by (using the same reasoning than before)
![Page 28: Minimal area surfaces and Wilson loops in the AdS/CFT ...Minimal area surfaces and Wilson loops in the AdS/CFT correspondence M. Kruczenski Purdue University Miami 2015 Based on: arXiv:1406.4945](https://reader036.vdocuments.net/reader036/viewer/2022081619/60f8ad07aa5a073c34565577/html5/thumbnails/28.jpg)
28
Example:
n=2
Analytical Numerical
A = -6.660397
![Page 29: Minimal area surfaces and Wilson loops in the AdS/CFT ...Minimal area surfaces and Wilson loops in the AdS/CFT correspondence M. Kruczenski Purdue University Miami 2015 Based on: arXiv:1406.4945](https://reader036.vdocuments.net/reader036/viewer/2022081619/60f8ad07aa5a073c34565577/html5/thumbnails/29.jpg)
29
New solutions ( ) Minkowski case
For n=0 it is the solution proposed by J. Toledo For it is a regular polygon (Alday-Maldacena). f0 →∞
![Page 30: Minimal area surfaces and Wilson loops in the AdS/CFT ...Minimal area surfaces and Wilson loops in the AdS/CFT correspondence M. Kruczenski Purdue University Miami 2015 Based on: arXiv:1406.4945](https://reader036.vdocuments.net/reader036/viewer/2022081619/60f8ad07aa5a073c34565577/html5/thumbnails/30.jpg)
30
Example:
n=2 n=4
![Page 31: Minimal area surfaces and Wilson loops in the AdS/CFT ...Minimal area surfaces and Wilson loops in the AdS/CFT correspondence M. Kruczenski Purdue University Miami 2015 Based on: arXiv:1406.4945](https://reader036.vdocuments.net/reader036/viewer/2022081619/60f8ad07aa5a073c34565577/html5/thumbnails/31.jpg)
31
For f0=0 it is a circle. The Mathieu equation reduces to the free particle equation (q=0). When f0 is small then q is small and one can use perturbation theory to compute the eigenvalues and from there the area. Alternatively one can use Dekel’s procedure. The results agree.
![Page 32: Minimal area surfaces and Wilson loops in the AdS/CFT ...Minimal area surfaces and Wilson loops in the AdS/CFT correspondence M. Kruczenski Purdue University Miami 2015 Based on: arXiv:1406.4945](https://reader036.vdocuments.net/reader036/viewer/2022081619/60f8ad07aa5a073c34565577/html5/thumbnails/32.jpg)
Conclusions
Wilson loops in terms of minimal area surfaces:
The Wilson loop can be computed if we know the reparameterization q(s). We have to study a certain Schrödinger equation with potential given by the shape.
The reparameterization q(s) can be found by imposing the vanishing of an infinite number of conserved charges. (The Wilson loop creates a string with the quantum numbers of the vacuum)
The system has a symmetry, λ-deformations that is not related to the global symmetries of the theory but to its integrability properties. 32
![Page 33: Minimal area surfaces and Wilson loops in the AdS/CFT ...Minimal area surfaces and Wilson loops in the AdS/CFT correspondence M. Kruczenski Purdue University Miami 2015 Based on: arXiv:1406.4945](https://reader036.vdocuments.net/reader036/viewer/2022081619/60f8ad07aa5a073c34565577/html5/thumbnails/33.jpg)
Future work
Relation to Y-sytem approach (J. Toledo).
Extend to higher dimensional spaces (AdSn). General integrable system
Compute quantum corrections.
We need to explore the field theory side.
33