Euclidean Wilson loops and Riemann theta functions M. Kruczenski Purdue University Based on: arXiv: (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 between strings and gauge theory. Gauge theory N = 4 SYM SU(N) on R 4 A μ, Φ i, Ψ a Operators w/ conf. dim. String theory IIB on AdS 5 xS 5 radius R String states w/ fixed 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 AdS 3 Closed curves: circular lens-shaped Berenstein Corrado Fischler Maldacena Gross Ooguri, Erickson Semenoff Zarembo Drukker 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: a1a1 a3a3 a2a2 b3b3 b1b1 b2b2 a1a1 b1b1 a1a1 b1b1 a2a2 a2a2 a3a3 a3a3 b2b2 b2b2 b3b3 b3b3

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 AdS 3 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. z 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 topology of 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.