Correlator of Wilson and t’Hooft loops at strong coupling in N=4 SYM theory

Andrew Zayakin works at LMU, Munich and ITEP, Moscow. His interests include non-perturbative physics of QCD, string theory and AdS/CFT correspondence. Dmitry.

This post is about my recent paper with Alexander Gorsky and Alexander Monin about a correlator of a Wilson and a ‘t Hooft loop. Before I proceed, I should explain what these objects are and why they are important to be studied. QCD possesses a consistent description in terms of “dual variables” – charges and monopoles. Reader familiar with the systematics of particle-like solutions in different theories would stop me at this very moment by pointing out there are no monopoles in QCD. True, there are no monopoles in the sense of e.g. Georgi-Glashow model. However, effectively there is such a thing as monopole, which is widely observed on lattice as a non-zero Abelian flux through a closed lattice surface. A lot is known on “thermodynamics” and “phenomenology” of these quasiparticles. They do not exist in the sense of theory spectrum. Still, they are an important tool of describing QCD. The QCD phase transition, which is an element of common lore, can easily be understood in terms of monopoles (Fig.1).

This diagram distinguishes between a phase of condensed, strongly correlated or dominating monopoles in each particular phase of QCD. In the transition area both monopoles and charges are determining the dynamics of the theory. In general, most interesting thermodynamic properties of the theory are supposed to take place in the region of the phase transition. Therefore, after we know something about monopoles and charges per se, one would ask himself what the dynamics of their interaction is.

A typical measure of charge or monopole mutual properties are Wilson lines. In particular, they allow one to introduce quark-quark potential by studying a correlator of two straight lines, and many other useful things non-perturbatively. A reasonable guess then would be to use Wilson lines as a measure of quark-monopole interaction. Therefore, a natural idea for someone interested in QCD properties next to phase transition point would be to consider a correlator of corresponding Wilson loops. actually the loop carrying magnetic charge is also known quite well in the literature as ‘t Hooft loop.

One could actually have done this on a lattice (actually, this was how this project started due to a discussion with our lattice-oriented colleagues). This however hasn’t been accomplished so far. Perturbative calculation is out of question, for a small coupling of the charge will automatically make monopole charge large. A resummation technique a la Erickson-Semenoff-Szabo-Zarembo may apply, but that would be a long story in its own turn. So the only way to calculate something well-defined would be to use duality approach.

There have already been some posts discussing some aspects of AdS/CFT here at NEQNET, as well as more particular models of AdS/QCD, so I will just briefly remind the reader a simplest formulation of duality: maximally supersymmetric gauge theory on a 4-dimensional boundary is equivalent to IIB string theory in the bulk, the equivalence requiring identification of currents in the boundary with bulk SUGRA fields’ limit towards boundary. When we calculate a Wilson or ‘t Hooft loop on the boundary, its AdS counterpart is a string two-dimensional surface in bulk, whose one-dimensional boundary is the contour of the loop. This recipe will be used by us as well. To find a correlator of a Wilson and a ‘t Hooft loop we must consider a configuration in AdS with two different boundaries, carrying electric and magnetic charges correspondingly.

Charge conservation would not allow to connect directly an electric string to a magnetic one. Thus an intermediate dyonic string world-surface emerges. Happily, there exists a vertex “charge, monopole, dyon”, or, more generically, vertex, which relates more generic dyons with each other. This vertex will allow the connection between electric, magnetic and dyonic string surfaces.

The question on whether monopole and charge degrees of freedom are mutually strongly correlated or not acquires a simple and beautiful explanation. Dependent on conditions which I discuss below, either configuration shown in Fig. 2

may be realized. The configuration on the left takes the advantage of the dyon vertex. The configuration on the left consists of free, non-correlated charge and monopole. Actually they are not absolutely non-interacting, since they are capable of exchanging SUGRA modes. Still, this is already at PT level and is not of foremost interest to us.

The choice between the two configurations is done dynamically, dependent on which of the two actions is the lower. Contribution by the other configuration to the partition function is then exponentially suppressed. The parameters on which the action depends are temperature, internal radius and external radius. I show in Fig. 3 the phase transition between the connected and disconnected phases dependent on radii ratio. Denoting them in the Figure as “interacting” and “inert” I mean the absence of non-perturbative correlation between the particles; they may still interact via exchange of supergravity modes.

The meaning of this transition is simple and physically quite clear: only when monopoles and charges are sufficiently close, they are non-perturbatively mutually strongly correlated. At low temperatures, this limiting ratio of radii is approximately 0.6. When , the loops are already uncorrelated. However, at high temperatures the critical ratio falls like , therefore, monopoles and charges may become correlated.

We have shown some temperature results, although we never said where the temperature came from. For that purpose the pure AdS space is generalized towards either a t-direction compactified version of the latter or a black-hole-in-AdS background. The transition between the two is known as Hawking-Page transition. Although we are far away from normal QCD with fundamental fermions, this phase transition maybe in some or other way related to QCD phase transition at zero chemical potential.

The results for the phase transition may lead us to a speculation on thermal behaviour of QCD, remembering again the caution that should be taken when extrapolating SUSY results upon a non-supersymmetric theory. We however remember here the standard lore that at finite temperature QCD and SUSY are closer in their dynamics, thus we hope that these results are actually more useful in a non-SUSY context than what one might think from the first sight.

The other issue we discuss in our paper, related to the existence of the vertex and different configurations of monopole/charge/dyon worldsheets is the generalization of the entanglement entropy.

This notion has been introduced long time ago but attracted a lot of attention recently because of its effective derivation in the holographic picture. Roughly speaking if we have a set of regions
divided by boundaries than the entanglement entropy is defined as the entropy seen by an observer in a region who does not communicate with the other regions. In the simplest case one has two regions A and B and introduce the vacuum density matrix . Then the reduced density matrix defines the entanglement entropy . The entanglement entropy is generically UV divergent but the UV divergent part of the entropy does not depend on the size of the region hence the finite -dependent contribution to the entanglement entropy can be, for instance, safely defined as the difference of the entropies at two different and .

The multicomponent regions has been investigated as well and the following generalization has been suggested, inspired by the one-dimensional case

where is the entropy of the single component and and are the right and left boundaries of the -th component. An important question concerns the property of the strong subadditivity which has been proven in holographic picture by Hirata. Another interesting feature of the system to study is the extensive mutual information

where

.

It was argued in literature that the extensivity does not generically hold which is triggered by nonvanishing tripartite information function

The holographic calculation of the entanglement entropy is practically identical to the Wilson loop calculation hence our mixed correlators suggest the natural generalization of the entanglement entropy when the charges are attributed to each boundary. That is, the entropy function for each interval takes values in lattice and has the following structure

for the -th interval. In the conformal case the calculation of the generalized entropy corresponds to the calculation of the partition function with nontrivial boundary conditions. One can define the generalized entropy by summing over the all boundary charges or introducing a kind of boundary chemical potentials for different charges.

Our recipe for the holographic calculation of the generalized entanglement entropy is very transparent. One has just to calculate of the area of the composite minimal surface as the function of the geometrical characteristics. Since all boundaries generically have electric and magnetic charges, the corresponding boundary contour has to be a boundary of the string worldsheet. Such connected composite surfaces may exist or not depending on the geometry of the boundary regions. Similar to canonical entanglement entropy, charged entanglement entropy is UV divergent but the UV divergent part is independent of the geometrical factors.

A natural question concerns the properties of the generalized entropy. The first one to be mentioned is the strong subadditivity which can be simply tested in the holographic picture. A comparison of the corresponding area indicates that for the simplest (0,1)-(1,0) correlator this property is satisfied, however, the analysis of the multiple loop correlators deserves a special consideration. The most interesting question related to the generalized entanglement entropy concerns its modular properties. Indeed, when we have a correlator of multiple dyonic loops, it takes values in with some integer and it would be very interesting to investigate the action of the -duality group on it, which could be related to the deconfinement phase transition.

Certainly there are other questions to come after this work. It would be interesting to recognize the phase transition in terms of the summation of the perturbative series in the spirit of Zarembo. However in the case under consideration the perturbation analysis is more involved since the interactions between electric and magnetic objects have to be summed up.

One of the most interesting questions concerns the action of the S-duality group on the generic correlators of the dyonic loops. A generic correlator of , dyonic loops has to possess interesting properties under the action of group. In particular, it would be interesting to investigate the modular properties of phase transition points of dyonic loop correlator.

Calculation of a correlator of several nonlocal observables has a lot in common with the calculation of the entanglement entropy. Our calculation suggests the natural generalization of the entanglement entropy notion to the case when the boundaries of the regions are charged under the -duality group. That is, generically the generalized entanglement entropy for the region with boundaries takes values in the group tensor product . Since the entanglement entropy at strong coupling is similar to the Bekenstein-Hawking black hole entropy the generalized entanglement entropy can be considered as an analogue of charged black hole entropy. We plan to discuss these issues elsewhere.