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)

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.

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● 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

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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

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

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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

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Infinite parameter family of solutions: Ishizeki, Ziama, M.K.

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EAdS3 Poincare coordinates

Also Euclidean surfaces in Minkowski signature: (w/ A. Irrgang)

8 AdS3 global coordinates

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

Consider first easier case: flat space

In conformal gauge:

|z|=1

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linear

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

Minimal Area surfaces in EAdS3

Equations of motion

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Non-linear

Method of solution: Pohlmeyer reduction

Solve And plug into

Then solve the linear problem:

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Integrability There are an infinite number of conserved charges. How can we use them?

|z|=1

Initial value problem Boundary condition problem

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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

Minimal Area surfaces in EAdS3 (H3)

Can we recast the problem into finding the correct reparameterization , ? Yes, gives the necessary boundary conditions.

|z|=1

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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 .

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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( )

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Computation of the Area (assumption, f no zeros)

The area is given by

From the cosh-Gordon eq. it follows that

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Then:

Replacing the limiting behavior we get

21 In the case where f(z) has zeros (umbilical points)

a1

b1

c

d

A

AMSV

Finding q(s)

Rewriting the Schrödinger equation using the coordinate s:

We find that the periodic potential is:

Known

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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

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.

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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:

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with

We can find Floquet solutions

We need that happens for a discrete set

of eigenvalues , namely

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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)

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Example:

n=2

Analytical Numerical

A = -6.660397

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New solutions ( ) Minkowski case

For n=0 it is the solution proposed by J. Toledo For it is a regular polygon (Alday-Maldacena). f0 →∞

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Example:

n=2 n=4

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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.

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

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.

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