126. From quarks to strings. Migdal-Makeenko equation and AdS-CFT correspondence

Although Lubos wants to see my answer to the poll , I decided to finish my analysis of the recent Polyakov’s paper today.

Page 6. In order to justify my picture I have used intuition coming from the loop equation, while Klebanov and Maldacena appealed to the D brane picture of the gauge fielauds. Both points of view are useful but neither of them lead to the quantitative derivation of gauge/string duality.

Comment. He is talking about the Migdal-Makeenko loop equation for the expectation value of the Wilson loop . After Migdal’s groundbreaking proof of the fact that this equation describes a free motion of the contour in the large limit (1981, if I am not wrong (?), even before the Polyakov’s action), not much  progress has been reported in this direction. The main reason is that the Migdal-Makeenko equations are formulated on the lattice, and no their continuum limit is known (it is not clear how to perform the renormalization procedure for loops).

Page 6. But then one has to take a major step and postulate that the collection of the D branes can be replaced by their mean gravitational field. This is a little like replacing the famous cat by its smile. While this is most probably correct, it is not clear how to justify this result.

He is talking about AdS/CFT now, AdS is the near horizon geometry of the stack of N D3-branes, where N is taken to be large. What is he concerned about? He would probably want to see the following: take a CFT, the theory on the worldsheet, and solve the string theory in terms of oscillators as Green, Schwarz and Witten or Polchinski teach us D-brane is then a coherent state of the stringy oscillators. How to show that if you take N stacked D-branes, only gravity degrees of freedom (that is, oscillators corresponding to a massless spin 2 particle) survive?

Page 7. In the case of the loop equations , we argue that since the Wilson loop is zigzag invariant ( the back and forth parts of the Wilson loop cancel), the open string must have the finite number of states (or vertex operators), corresponding to the states of the gauge theory. As we explained above, this requirement implies warping needed to remove all higher states from the spectrum. The approach based on the loop equations starts from the first principles. However, the equation itself is singular and requires elaborations. Rychkov and I tried to use the operator product expansions on the contour in order to find a non-singular version of the equation. We only scratched the surface of highly non-trivial technical problems. The problem of reproducing gauge perturbation theory from the string theory side remains unsolved (and extremely important).

Comment: here is the list of papers that might be useful to understand the paragraph above

1. “Loop dynamics and the AdS/CFT correspondence” by Polyakov and Rychkov.
2. “Gauge fields strings duality and the loop equation” by Polyakov and Rychkov.
3. “String theory and quark confinement” by Polyakov.

Zigzag symmetry he is talking about is the following interesting thing. We want to construct a string theory which would reproduce behavior of Wilson loops in YM.  Standard non-critical string theory (when we are starting from the Nambu-Goto action to find the Louville type action after integrating out ‘s) is not the one we need. Although it is symmetric under diffeomorphisms of the world sheet

(1)

with

(and the Wilson loop is invariant w.r.t. the transformations (1), too), we need invariance w.r.t. any diffeomorphisms (even with changing sign of [Unparseable or potentially dangerous latex formula. Error 1 ] YM) actually strongly resemble equations of motion for the principal chiral field in D=2. The latter can be integrated by inverse scattering problem approach (i.e., the Lax pair can be found). As it (always?) happens in 2D, classical integrability means quantum integrability, and the quantum principal chiral field theory was solved exactly by Bethe anzatz. Classical Migdal-Makeenko equations can be solved exactly for the case . Does it automatically mean that quantum Migdal-Makeenko theory is exactly solvable as well?

On page 7. Today, due to the work of many people, we know that the dilatation operator of the super – Yang- Mills theory is represented by a completely integrable spin chain. It is superficially different from the integrability which I envisaged 30 years ago. However they must be related, since the AdS5 string sigma model is integrable and the boundary of the world sheet, being mapped onto the Wilson loop, must produce the integrals in the loop space. Establishing this fact is one of the yet unsolved problems.

Comment. By sigma model here he means exactly this – an analogue of sigma model with AdS target space. This latter can be probably exactly solved by Bethe anzatz (in the same way the sigma model is solved). Generally, its beta-function is positive (depends on the sign of the curvature of the target space). If the target space is compact (one needs to factor AdS by a discrete subgroup with compact fundamental domain), the spectrum at strong coupling is descrete, and we know that there is an RG fixed point described by a CFT. I think, Polyakov wants to say that this CFT is what we need in AdS/CFT.

On page 7. Even more important is to find the gauge theory for the de Sitter space. I conjectured that the large N gauge theories have a fixed point at the complex gauge coupling corresponding to the radius of convergence of the planar graphs. Presumably this point is described by a non-unitary CFT corresponding to the intrinsically unstable de Sitter space.

He is talking about his recent paper on de Sitter space and eternity. Eventually, we will hopefully discuss that paper as well.