1. Wilson Loops in AdS/CFT M. Kruczenski Purdue University Miami 2012 1 Based on work in collaboration with Arkady Tseytlin (Imperial College). To appear.

2. Summary ● Introduction String / gauge theory duality (AdS/CFT) Wilson loops in AdS/CFT = Minimal area surfaces in hyperbolic space ● How to describe in the field theory a string falling into the Poincare horizon Simple examples: point-like string and short strings. They correspond to inserting a WL in the T-dual theory. 2

3. ● Map between short strings and Wilson loops. The oscillations of a short string (flat space) falling into the horizon are mapped to the oscillations of the world- sheet dual to a Wilson loop. ● Conclusions 3

4. λ small → gauge th. Gauge theoryString theory λ large → string th. Strings live in curved space, e.g. AdS 5 xS 5 S 5 : Y 1 2 +Y 2 2 +…+Y 6 2 = 1 AdS 5 : X 1 2 +X 2 2 +…-X 0 2 -X -1 2 =-1 (hyperbolic space) 4

5. AdS/CFT correspondence (Maldacena) 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.

6. Minkowski AdS metric has a horizon at Hyperbolic space 2d: Lobachevsky plane, Poincare plane/disk 6 Poincare coordinates

7. 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 7

8. 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 8

9. Maldacena, Rey Yee parallel lines Drukker Gross Ooguri cusp 9

10. 10 Multiple curves: Drukker Fiol concentric circles Lens shaped: closed with cusps Drukker Giombi Ricci Trancanelli

11. New examples using Riemann Theta functions (w/ Sannah Ziama and R. Ishizeki) 11

12. Even more examples: Concentric curves 12

13. 13 Two main sets of coordinates for hyperbolic space: Poincare and global coordinates. The boundary of global coordinates is RxS 3 and the boundary of Poincare is R 3,1. The dual field theory lives in the boundary space, therefore string theory in global AdS is dual to gauge theory on RxS 3 and in Poincare dual to gauge theory on R 3,1. In Minkowski signature, Poincare coordinates have an extremal horizon and, in fact, only cover part of the space.

14. 14 Relation between global and Poincare patches

15. 15 In this talk we discuss other use for Wilson loops Consider the computation of a correlation function for the field theory living on S 3 Should map to a correlation function of the theory in flat space. However, Poincare coordinates only cover part of the space and also of the boundary. A string can ”escape” through the horizon!.

16. 16 The disappearing string should be represented by an operator in the field theory. Which operator?

17. 17 Consider the case of a small string

18. 18 ● Alday-Maldacena T-duality Relates AdS 5 xS 5 to itself, equivalently relates N =4 SYM to itself as a “momentum position” duality. Proposed by Alday-Maldacena and made more precise by Berkovits-Maldacena and Beisert-Ricci-Tseytlin-Wolf. It involves a novel fermionic form of T-duality. At the level we need here, it relates two classical solutions in AdS 5. The simplest form is in Poincare coordinates and in conformal gauge.

19. 19 Metric: Action: EOM:

20. 20 Simple but useful example:

21. 21 A point-like string becomes a straight WL of the T-dual theory.

22. 22 Consider now small fluctuations of the string. Since the string is a small fluctuation of the point-like string we can just study the string in flat space. Moreover, we can use light-cone gauge (since in flat space it is a type of conformal gauge) and the equations reduce to a set of harmonic oscillators. More formally, start from embedding coordinates: and expand around X 0 =1:

23. 23 The metric around this (any) point is approximately: Light-cone gauge: Most general solution for the other coordinates:  =0 corresponds to the horizon. Classically we need to specify the shape and velocity of the string on that surface. r =1..4

24. 24 Pictorially:

25. 25 There seems to be a problem. These fluctuations should be mapped to fluctuations of the Wilson loop which is in AdS and not in flat space ?!. Indeed, the action for the fluctuations around Z= , T=  is (notice interchange of  and  : which gives the equations: ?!

26. 26 But, in fact, it was already shown by A. Mikhailov that these equations are solved by So, indeed the equations are like in flat space. In fact using the rules of T-duality we find

27. 27 First change to Poincare coordinates and perform a rescaling (boost) by  the solution is T-duality gives:

28. 28 The expectation value of the Wilson loop is related to the area of the dual world-sheet

29. 29 The energy of the open string ending on the boundary is not conserved (we need do work on the quark to move it along a specified trajectory). The total energy change, however, for these solutions is: It vanishes from the level matching condition for the closed string. However if we add the left and right movers we get the energy of the closed string:

30. 30 The spin is also not conserved. The velocity and acceleration are evaluated at the boundary. The reason is that the spin defines a conserved current, any increase in spin is due to a flux from the boundary. The total spin absorbed by the open string is precisely equal to the spin of the original closed string.

31. Conclusions We review the duality between Wilson loops and minimal area surfaces in hyperbolic space. We then considered the case where a string crosses the Poincare horizon. Using T-duality we found that such string is represented by the insertion of a Wilson loop of the T-dual theory. In that way we found another interesting application of Wilson loops that can help understand the physics of horizons in the context of AdS/CFT and perhaps more in general. 31