Euclidean Wilson loops and

Riemann theta functions

M. Kruczenski

Purdue University

Based on: arXiv:1104.3567 (w/ R. Ishizeki, S. Ziama)

Great Lakes 2011

Summary

●Introduction

String / gauge theory duality (AdS/CFT)

Wilson loops in AdS/CFT

Theta functions associated w/ Riemann surfaces

Main Properties and some interesting facts.

● Minimal area surfaces in Euclidean AdS3

Equations of motion and Pohlmeyer reduction

Theta functions solving e.o.m.

(*) Formula for the renormalized area.

● Closed Wilson loops for g=3

Particular solutions, plots, etc.

● Conclusions

AdS/CFT correspondence (Maldacena, GKP, Witten) Gives a precise example of the relation betweenstrings and gauge theory.

Gauge theory

N = 4 SYM SU(N) on R4

Aμ , Φi, Ψa

Operators w/ conf. dim.

String theory

IIB on AdS5xS5

radius RString states w/ E

R

g g R l g Ns YM s YM 2 2 1 4; / ( ) /

N g NYM , 2fixed

λ large → string th.λ small → field th.

Wilson loops in the AdS/CFT correspondence (Maldacena, Rey, Yee)Euclidean, Wilson loops with constant scalar =Minimal area surfaces in Euclidean AdS3

Closed curves:

circular lens-shapedBerenstein Corrado Fischler MaldacenaGross Ooguri, Erickson Semenoff ZaremboDrukker Gross, Pestun

Drukker Giombi Ricci Trancanelli

Multiple curves:

Drukker Fiol

concentric circles

Euclidean, open Wilson loops:

Maldacena, Rey Yee parallel lines Drukker Gross Ooguri cusp

Many interesting and important results for Wilson loops with non-constant scalar and for Minkowski Wilson loops (lots of recent activity related to light-like cusps and their relation to scattering amplitudes).

As shown later, more generic examples for Euclidean Wilson loops can be found using Riemann theta functions.

Babich, Bobenko. (our case)Dorey, Vicedo. (Minkowski space-time)Sakai, Satoh. (Minkowski space-time)

Theta functions associated with Riemann surfaces

Riemann surface:

a1

a3 a2

b3

b1

b2

a1

b1

a1

b1

a2

a2

a3

a3

b2

b2

b3

b3

Holomorphic differentials and period matrix:

Theta functions:

Theta functions with characteristics:

Simple properties:Symmetry:

Periodicity

Antisymmetry

and

Special functions

Algebraic problems:

Roots of polynomial in terms of coefficients.

Square root: quadratic equations (compass and straight edge or ruler) sin sin(/2) [sin sin(/3)]

Exponential and log: generic roots, allows solutions of cubic and quartic eqns.

Theta functions: Solves generic polynomial.

Differential Equations

sin, cos, exp: harmonic oscillator (Klein-Gordon).

theta functions: sine-Gordon, sinh-Gordon, cosh-Gordon.

Trisecant identity:

Derivatives:

cosh-Gordon:

Minimal Area surfaces in AdS3

Equations of motion and Pohlmeyer reduction

We can also use:

X hermitian can be solved by:

Global and gauge symmetries:

The currents:

satisfy:

Up to a gauge transformation (rotation) A is given by:

Then:

Summary

Solve

plug it in A, B giving:

Solve:

Theta functions solve e.o.m.

Hyperelliptic Riemann surface

Finally we write the solution in Poincare coordinates:

Renormalized area:

Subtracting the divergence gives:

Example of closed Wilson loop for g=3

Hyperelliptic Riemann surface

Zeros

Shape of Wilson loop:

Shape of dual surface:

Computation of area:

Using previous formula

Direct computation:

Circular Wilson loops , maximal area for fixed length. (Alexakis, Mazzeo)

Map from Wilson loop into the Riemann surface

Zeros determine shape of the WL. can be written as:

Conclusions

We argue that there is an infinite parameter family of closed Wilson loops whose dual surfaces can be found analytically. The world-sheet has the topologyof a disk and the renormalized area is found as a finite one dimensional contour integral over the world-sheet boundary.

We showed specific examples for g=3.

Integrability properties of Euclidean Wilson loops deserve further study.