125. From quarks to strings. On Liouville mode, instantons and confinement in abelian theories

Alexander Polyakov have released this week a preprint about history of string theory, which is also so full of non-trivial physical ideas that I decided to list some of them in this post as well as to include my comments (or rather my ramblings )

So, here we go.

On Page 2. It took some more months before I added to my work the abelian theory of quark confinement based on the idea of instantons. The result was quite stunning – in 3d the instantons ( which were magnetic monopoles) lead to the formation of the electric string for all  couplings, while in 4d the instantons where the closed loops of the monopoles trajectories and the confinement occurred after the coupling exceeded the critical one. A little later Gerard t’Hooft and Stanley Mandelstam arrived at the qualitative picture of dual superconductors , which is of course equivalent to the one I just described.

Comment: although nonabelian instantons were discovered soon (Belavin-Polyakov-Schwartz-Tyupkin), the picture did not work so well in nonabelian case. Abelian instantons are weakly interacting and rigid gas approximation works well for them. To sum over instantons in the partition function, you basically need to invent an analogue of Debye theory for the corresponding plasma – the mass gap that appears in the abelian confining theory is inverse Debye length in the plasma of abelian instantons. In the nonabelian theory the situation is much more complicated – instantons represent the liquid, i.e. strongly interact with each other. Situation is again simplified in SUSY theories, where instantons again start to interact weakly (=> Seiberg-Witten theory).

On pages 2-3. It made its appearance already in the Wilson work on the lattice gauge theory, in which the strong coupling expansion was described as a sum over random surfaces. These surfaces were the result of propagation of one dimensional objects- electric fluxes. The major difficulty was to find the continuous limit of this picture. But already on the qualitative level I found the picture very useful. It helped me to predict the deconfining transition, leading to the quark- gluon plasma. This transition takes place simply because the strings are melting, as can be seen from the Peierls argument.

Comment: One (wrong, but a kind of appealing) picture I have in mind is the following. Short strings with quarks on its ends can be considered as dipoles. As seen from large distances, dipoles are neutral, so the overall system is color neutral. At large temperatures dipoles want to dissociate, and color degrees of freedom become accessible. Why is it wrong?  Because in plasma of dipoles correlation length is infinite in the neutral phase, while it is finite in dissociation phase. Here it is vise versa: mass gap exits at low temperatures and is absent (deconfinement) at large temperatures. So, let me erase this wrong picture from my head zzzz…. done.

And proceed to correct one (I hope I did not confuse you too much) He thinks in terms of instantons of the abelian theory. In 2+1 compact QED, these instantons are point-like, while in 3+1 they become lines. We again construct a kind of Debye theory but for lines instead of particles now. The line of a length L gives the contrbution

to the partition function (here is the coupling) and, if the coupling is strong enough, combinatoric factor overcomes the action factor, and instanton lines should dissociate. That’s Peierls argument he mentions and that’s what he means here by strings -  instanton world lines.

On page 3. This picture of the strings describing the flux lines is often confused with the t’ Hooft picture which suggests that the string world sheet appears because the lines of Feynman diagrams become dense. In the normal gauge theory this certainly doesn’t happen.

Comment: He saying is that strings in AdS/CFT are flux tubes, not those “strings” that appear from the planar expansion in the YM perturbation theory, with worldsheets given by planar Feyman diagrams (and people confuse these two things even in the reviews of AdS/CFT!). He explains that planar diagrams only become dense (that is when we can interpret them as worldsheets of strings) at YM complex coupling. Recall that he is using this idea in this paper to find a physical meaning of dS/CFT.

On page 3. I also thought that the string representation may help to solve the 3d Ising model
by reducing it to the free fermionic strings.

Comment: See his book. 3D Ising is related to gauge theory through the Kramers-Wannier duality, and strong coupling limit of the latter should be described in terms of strings (as usual for the gauge theory). By the way, now I (a kind of) see how to make sense of AdS/CFT applications to condensed matter systems I critisized so bluntly

On page 4. But when I quantized it there was a surprise – an extra, longitudinal mode, which appears due to the quantum ? thickening? of the string. This new field is called the Liouville
mode.

Comment: He is talking about Polyakov’s action, where all are integrated out and only Liouville mode survives due to anomaly non-cancellation in non-critical number of dimensions. I’ve heard many times that Liouville mode corresponds to thickening of the string, but honestly was never able to understand why. Liouville mode comes from the overall rescaling of the metric on the worldsheet by a factor

,

i.e., it corresponds to dilatation of the world sheet. If the Liouville mode is not cancelled, world sheet either wants to stretch or to collapse, doesn’t it? Dear Smart Reader, if you understand this topic, I beg you, please explain it to me.

On page 4. Dynamics of 2d gravity is very rich and even now not completely explored. One of the problems was the field -dependent cut-off which one must use in order to preserve general covariance on the world sheet.

Comment: Naturally, you want to introduce a covariant UV cutoff to your theory (otherwise, if general covariance is broken in the UV, nothing will prefent it from breaking in the IR, too). Therefore, what you need to cut is the interval

,

i.e., you have to introduce the minimal . Here comes the field dependence from.

On page 5. Soon after the Liouville mode was discovered it became clear to many people including myself that its natural interpretation is that random surfaces in 4d are described by
the strings flying in 5d with the Liouville field playing the role of the fifth dimension. The precise meaning of this statement is that the wave function of the general string state depends on the four center of mass coordinates and also on the fifth, the Liouville one.

Comment: Oh, now he is talking about AdS/CFT and basically says that the Liouville field is the fifth coordinate of the AdS. How can it correspond to thickening of the string then??? An infinitely thin string is finely propagating on , vibrating, rotating and creating gravity in the target space

On page 5. In ’96 I came to the conclusion that in order to describe gauge theories this five dimensional space must be warped. The logic was as following. In gauge/ string duality the open strings describe the Wilson loop and the only allowed vertex operators in the open string sector are the ones corresponding to gluons ( and extra fields, if present). At the same time, in the closed string sector we have infinite number of states. So, all massive modes of the open string must go away. This can happen only if the ends of the open strings lie either at singularity or infinity and the metric is such that this region has infinite blue shift with respect to the bulk. In this case the masses of all but massless open string states go to infinity.

Comment: This is for Moshe Rozali, who cannot find where infinite blueshift is in the AdS space

To be continued.

Update: Peter Woit also discusses the preprint here but covers mostly the “historical” part.