# Confinement- Deconfinement Phase Transition and Fractional Instanton Quarks in Dense Matter

###### Abstract

We present arguments suggesting that large size overlapping instantons are the driving mechanism of the confinement-deconfinement phase transition at nonzero chemical potential . The arguments are based on the picture that instantons at very large chemical potential in the weak coupling regime are localized configurations with finite size . At the same time, the same instantons at smaller chemical potential in the strong coupling regime are well represented by the so-called instanton-quarks with fractional topological charge . We estimate the critical chemical potential where transition between these two regimes takes place. We identify this transition with confinement- deconfinement phase transition. We also argue that the instanton quarks carry magnetic charges. As a consequence of it, there is a relation between our picture and the standard t’Hooft and Mandelstam picture of the confinement. We also comment on possible relations of instanton-quarks with “periodic instantons” , “ center vortices” , and “fractional instantons” in the brane construction. We also argue that the variation of the external parameter , which plays the role of the vacuum expectation value of a “Higgs” field at , allows to study the transition from a “Higgs -like” gauge theory (weak coupling regime, ) to ordinary QCD (strong coupling regime, ). We also comment on some recent lattice results on topological charge density distribution which support our picture.

## I Introduction

This talk is based on a number of original resultsTZ -JZ obtained with different collaborators at different times.

Color confinement, spontaneous breaking of chiral symmetry, the problem and the dependence are some of the most interesting questions in . Unfortunately, progress in the understanding of these problems has been extremely slow. At the end of the 1970’s A. M. Polyakov Po77 demonstrated charge confinement in . This was the first example where nontrivial dynamics was shown to be a key ingredient for confinement: The instantons (the monopoles in 3d) play a crucial role in the dynamics of confinement in . Instantons in four dimensional QCD were discovered 30 (!) years ago Belavin:1975fg . However, their role in remains unclear even today due to the divergence of the instanton density for large size instantons.

Approximately at the same time instanton dynamics was developed in two dimensional, classically conformal, asymptotically free models (which may have some analogies with ). Namely, using an exact accounting and resummation of the -instanton solutions in models, the original problem of a statistical instanton ensemble was mapped unto a -Coulomb Gas (CG) system of pseudo-particles with fractional topological charges (the so-called instanton-quarks) Fateev . The instanton-quarks do not exist separately as individual objects. Rather, they appear in the system all together as a set of instanton-quarks so that the total topological charge of each configuration is always an integer. This means that a charge for an individual instanton-quark cannot be created and measured. Instead, only the total topological charge for the whole configuration is forced to be integer and has a physical meaning. This picture leads to the elegant explanation of confinement and other important properties of the models Fateev . Unfortunately, despite some attempts Belavin , there is no demonstration that a similar picture occurs in gauge theories, where the instanton-quarks would become the relevant quasiparticles. Nevertheless, there remains a strong suspicion that this picture, which assumes that instanton-quarks with fractional topological charges become the relevant degrees of freedom in the confined phase, may be correct in .

On the phenomenological side, the development of the instanton liquid model (ILM) ILM ; shuryak_rev has encountered successes (chiral symmetry breaking, resolution of the problem, etc) and failures (confinement could not be described by well separated and localized lumps with integer topological charges). Therefore, it is fair to say that at present, the widely accepted viewpoint is that the ILM can explain many experimental data (such as hadron masses, widths, correlation functions, decay couplings, etc), with one, but crucial exception: confinement. There are many arguments against the ILM approach, see e.g. Horvath , there are many arguments supporting it shuryak_rev .

In this talk we present new arguments supporting the idea that the instanton-quarks along with instantons are the relevant quasiparticles in the strong coupling regime. In this case, many problems formulated in Horvath are naturally resolved as both phenomena, confinement and chiral symmetry breaking are originated from the same vacuum configurations, instantons, which may have arbitrary scales: the finite size localized lumps with integer topological charges, as well as set of fractionally -charged correlated objects sitting at arbitrary large distances from each other. In this picture when fractionally charged constituents propagate far away from each other, the confinement could be a natural consequence of a dynamics of these well correlated objects. We emphasize that along with instanton quarks there are ordinary instantons with integer topological charges. Indeed, if the instanton-quarks are close to each other they bound together and likely to form an ordinary instanton. If the instanton quarks far away from each other, the description in terms of fundamental instanton quarks is more appropriate. The precise probability for each configuration depends on the interplay between action and entropy. Such a feature when the well- localized instantons and de-localized instanton- quarks coexist may lead to the understanding why the chiral symmetry breaking phenomenon ( which, as ILM suggests shuryak_rev , is due to the well- localized instantons) and the confinement - deconfinement phase transitions (which is due to, the de-localized instanton quarks, according to the present proposal) are so close to each other. In our picture such a “conspiracy” is a simple reflection of the fact that both phenomena are due to the same configurations, instantons, which however can be in different configurations.

More importantly, we make some very specific predictions which can be tested with traditional Monte Carlo techniques, by studying QCD at nonzero isospin chemical potentialIsospin .

We start in Section II by reviewing recent work for QCD at large in the deconfined phase ssz , where the instanton calculations are under complete theoretical control, since the instantons are well-localized objects with a typical size .

We then discuss in Section III the dual representation of the low-energy effective chiral Lagrangian in the regime of small chemical potential where confinement takes place. We shall argue that the corresponding dual representation corresponds to a statistical system of interacting pseudo-particles with fractional topological charges which can be identified with instanton-quarks JZ suspected long ago Fateev ; Belavin .

Based on these observations we make a conjectureTZ formulated in Section IV that the transition from the description in terms of well localized instantons with finite size at large to the description in terms of the instanton quarks with fractional topological charges precisely corresponds to the deconfinement-confinement phase transition.

In Section V we explicitly calculate the critical chemical potential where this phase transition should occur. Our conjecture can be explicitly and readily tested in numerical simulations due to the absence of the sign problem at arbitrary value of the isospin chemical potential. If our conjecture turns out to be correct, it would be an explicit demonstration of the link between confinement and instantons.

In section VI we present some arguments explaining why the standard picture of confinement suggested long ago by t’Hooft and MandelstamHooft ( which is based on the condensation of the magnetic monopoles) is consistent with our interpretation of the confinement when the instanton quarks play the key role. Finally, Section VII is our conclusion where we argue that our picture of the confinement deconfinement phase transition can be tested on the lattice with traditional Monte Carlo techniques if one studies QCD at nonzero isospin (rather than baryon) chemical potential. We also comment on relations with different works. Finally, we make some comments on recent lattice results on topological density distribution.

## Ii Instantons at large

At low energy and large chemical potential, the is light and described by the Lagrangian derived in ssz :

(1) |

where the decay constant, and , and its velocity, ssz ; schaefer . We define baryon and isospin chemical potentials as . The nonperturbative potential is due to instantons, which are suppressed at large chemical potential.

The instanton-induced effective four-fermion interaction for 2 flavors, , is given by tHooft ; SVZ ,

We study this problem at nonzero temperature and chemical potential for , and we use the standard formula for the instanton density at two-loop order shuryak_rev

where

By taking the average of Eq. (II) over the state with nonzero vacuum expectation value for the condensate, one finds

where and , and is the gap ssz ; schaefer . Therefore the mass of the field is given by

(5) |

The approach presented above is valid as long as the field is lighter than , the mass of the other mesons in the system ssz , that is if

(6) |

This is exactly the vicinity where the Debye screening scale and the inverse gap become of the same order of magnitude ssz , and therefore, where the instanton expansion breaks down.

For reasons which will be clear soon, we want to represent the Sine-Gordon (SG) partition function (1, II) in the equivalent dual Coulomb Gas (CG) representation ssz ,

(7) | |||

Physical interpretation of the dual CG representation (7):

a) Since
is the total charge and it appears in
the action
multiplied be the parameter , one
concludes that is the total topological
charge of a given
configuration.

b) Each charge
in a
given configuration should be identified with an
integer topological charge well localized at the point . This,
by definition,
corresponds to a small instanton positioned at .

c) While the starting low-energy effective Lagrangian contains only
a colorless field we
have ended up with a representation of the partition function in which
objects carrying color (the instantons) can be studied.

d)
In particular, and
interactions (at very large distances) are exactly the same up to a
sign, order ,
and are Coulomb-like. This is in contrast with semiclassical expressions
when interaction is zero and interaction is order .

e) The very complicated picture of the bare and
interactions
becomes very simple for dressed instantons/anti-instantons
when all integrations over all possible sizes, color orientations
and interactions with background fields are properly accounted for.

f) As expected, the ensemble of small instantons can
not produce confinement.
This is in accordance with the fact that there is no confinement at
large .

## Iii Instantons at small

We want to repeat the same procedure that led to the CG representation in the confined phase at small to see if any traces from the instantons can be recovered. We start from the chiral Lagrangian and keep only the diagonal elements of the chiral matrix which are relevant in the description of the ground state. Singlet combination is defined as . The effective Lagrangian for the is

(8) | |||||

A Sine-Gordon structure for the singlet combination corresponds to the following behavior of the derivative of the vacuum energy in pure gluodynamics veneziano ,

(9) |

The same structure was also advocated in HZ from a different perspective. As in (7) the Sine-Gordon effective field theory (8) can be represented in terms of a classical statistical ensemble (CG representation) similar to (7) with the replacements , more precisely,

(10) |

The functional integral is trivial to perform and one arrives at the dual CG action,

(11) |

The fundamental difference in comparison with the previous case
(7) is that
while the total charge is integer, the individual charges are
fractional .
This is a
direct consequence of the dependence in the underlying
effective Lagrangian (8) before integrating out fields, see eq. (III).

Physical Interpretation of the CG representation (III) of
theory (8):

a) As before, one can identify
with the total topological charge of
the given configuration.

b) Due to the
periodicity of the theory, only configurations which contain an integer
topological number contribute to the partition function. Therefore,
the number of particles for each given configuration with
charges
must be proportional to .

c) Therefore, the number of integrations
over in CS representation exactly equals , where
is integer. This number exactly
corresponds to the number of zero modes in the -instanton
background. This is basis for the conjecture JZ that
at low energies (large distances) the fractionally charged species,
are the instanton-quarks suspected long ago Fateev .

d) For the gauge group,
the number of integrations would be equal to where
is the quadratic Casimir of the gauge group
( dependence in physical observables comes in
the combination ). This number
exactly corresponds
to the number of zero modes in the -instanton background for gauge
group .

e) The CG representation corresponding to eq.(8) describes
the confinement phase of the theory.

One obvious objection for such an identification of with the topological charge immediately comes in mind: it has long been known that instantons can explain most low energy QCD phenomenology ILM with the exception confinement; and we claim that confinement also arises in this picture: how can this be consistent? We note that quark confinement can not be described in the dilute gas approximation, when the instantons and anti-instantons are well separated and maintain their individual properties (sizes, positions, orientations), as it happens at large . However, in strongly coupled theories the instantons and anti-instantons lose their individual properties (instantons will “melt”) their sizes become very large and they overlap. The relevant description is that of instanton-quarks which can be far away from each other, but still strongly correlated. For such configurations the confinement is a possible outcome of the dynamics.

We should remark here that a precise
form of the potential (8) in the form of a single function
is not a crucial issue for discussions below. A combination of a number of terms,
may change the interactions of the instanton
quarks^{2}^{2}2We refer to ref. Diakonov:1999ae
where it is argued, based on analysis of two dimensional model,
that much more complicated
structure for the instanton quark interactions could result.. However,
the most important element here remains the same: the
behavior is well established result and remains untouched
even when more complicated terms are introduced. It will lead to the fractional charges
in Coulomb Gas representation (III,III) in big contrast
with weakly interacting phase at large (7) where only integer topological charges appear.

We should also comment at this point that our numerical estimates below are based exclusively on the instanton density at large while we approaching the critical value. In this region the potential is well established and unique (II). Therefore, our results below are not sensitive to the specific details of the potential (8) at small when some additional terms might be present.

## Iv Conjecture.

We thus conjecture that the confinement-deconfinement phase transition takes place at precisely the value where the dilute instanton calculation breaks down. At large the weakly interacting phase (CS) is realized. Instantons are well localized configurations with a typical size . Color in CS phase is not confined. At low the strong interacting regime is realized and color is confined. Instantons are not well localized configurations, but rather are represented by instanton quarks which can propagate far away from each other. The value of the critical chemical potential as a function of temperature, is given by saturating the inequality (6).

Few remarks are in order. First, we can estimate the critical not only at , but also at as long as the temperature is relatively small such that our approach is justified. Indeed, in the weak coupling regime the dependence of the instanton density is determined by a simple insertion into the expression for the density (II). The temperature dependence also enters the expression for . As long as does not vanish and we are in CS phase, our calculations (II) are justified, and the critical can be estimated as a function of at relatively small as shown in FIG.1.

2.3 | 1.4 | 3.5 | |

2.6 | 1.5 | 3.5 |

We should emphasize that in our picture the nature of the phase transition is universal and it is not sensitive to the specific values of and , in spite of the fact that the ground state of the superconducting phase is very sensitive to the values of , and quark mass (CFL, 2SC, crystalline or even more complicated phases).

## V Numerical Results.

The critical chemical potential as a function of temperature is implicitly given by . We can calculate from (II). We are however limited to temperatures where Cooper pairing takes place, i.e. for pr . We have determined the critical chemical potential in different cases at nonzero baryon or isospin chemical potential. We find that the value of the critical chemical potential at are given by (we use which is numerically close to for ).

As an example, we explicitly show the results as a function of temperature for at nonzero in FIG. 1, where direct lattice calculation are possible. We notice that with our conventions the transition from the normal phase to pion condensation happens at .

As expected (see Table 1), for given the critical value for decreases when increases. This is due to the fact that an extra fermion degree of freedom suppresses the instantons, such that the instanton density becomes smaller. As direct consequence of that suppression the critical value at the point when the instanton dilute gas approximation breaks down is smaller for than for .

As a final remark: while we expect that the instanton density (II) suffers from large uncertainties at , the numerical results for are not very sensitive to these uncertainties due to the extraction of the large power from the instanton density, .

## Vi Instanton Quarks as Monopoles. Confinement.

Having formulated our conjecture and the results which follow from it, the question about the relation between the standard ’t Hooft and Mandelstam Hooft picture of the confinement and our proposal (when confinement is due to the instanton-quarks) can be formulated. The key point of the ’t Hooft - Mandelstam approach is the assumption that dynamical monopoles in QCD exist and Bose condense. The goal of this section is to argue that the instanton-quarks carry the magnetic charges. Therefore, in principle, they may play the role of the dynamical monopoles which are the key players in the ’t Hooft and Mandelstam Hooft framework. In this case both pictures could be the two sides of the same coin.

Expression (III) clearly shows that the statistical ensemble of particles interact according to the Coulomb law. An immediate suspicion following from this observation is that these particles carry a magnetic and/or electric charge, since charges of that type interact precisely in the above manner. This suspicion will be corroborated in a moment. The charges were originally introduced in a very formal manner so that the QCD effective low energy Lagrangian (8) can be written in the dual CG form (III). In the previous sections we presented arguments that the particles carry fractional topological charges and can be identified with instanton quarks. Now we shall argue that these particles also carry the magnetic charges.

As a short detour, let us remind few important results regarding the Georgi-Glashow model in the weak coupling regime, with a -term when the scalar have a large VEV. The monopole solution can be constructed explicitly and the well-known Witten’s effect Witten1 , where the monopole acquires an electrical charge, takes place. Let denote the generator of large gauge transformations corresponding to rotations in the subgroup of picked out by the gauge field, i.e. rotations in about the axis . Rotations by an angle of about this axis must yield the identity for arbitrary configurations, which implies Witten1 that the magnetic monopoles carry an electric charge proportional to . Indeed,

(12) |

where,

are the magnetic and electric charge operators respectively, expressed in terms of the original fields, and is the vacuum expectation value at infinity. The combination in Eq. (12) is an integer and determines the magnetic charge of the configuration. As usual, it is assumed that (12) remains correct in the strong coupling regime when is not large and/or in the more radical case when is not present in the original formulation. Indeed, as explained in Hooft1 the existence of is not essential and some effective fields may play its role. One finds that monopoles do exist and the Witten effect expressed by formula (12) remains unaltered even when monopoles appear as singularities in the course of the gauge fixing procedure as described in Hooft1 .

Restricting attention to terms which are proportional to the
-parameter, a comparison between the CG representation,
Eq.(III), and Eq.(12) will now be carried out. From
the CG representation,
Eq.(III) the relevant term is the total charge, , of the
configuration, while in Eq.(12) the relevant factor is the total
magnetic charge for each time slice. The following
identification is then made^{3}^{3}3 Of course we assume here that a configuration is
static, or slowly depending on time. Therefore,
the identification (13) should be considered as a relation if the instanton quarks
were treated as classical sources. It is definitely not the case for the dynamical system
under study.
Nevertheless, relation (13) serves as a good argument suggesting that instanton quarks
carry the magnetic charges. The crucial questions are: can these monopoles
propagate far away from each other? do these monopoles
condense?,

(13) |

From these simple observations one can immediately deduce that our fractional magnetic charges cannot be related to any semi-classical solutions, which can carry only integer charges; rather, configurations with fractional magnetic charges should have pure quantum origin.

One should notice here that the connection between monopoles and instantons on the classical level is not a very new idea nahm . Indeed, for example, quite recently, such a relation was established for the periodic instantons (also called calorons) defined on vanbaal , see also Diakonov and khoze where monopoles and instantons are intimately related objects in semiclassical construction.

Furthermore, a similar relation was seen in the study of Abelian projection for instantons Polikarpov ; Brower , albeit at the classical level. In particular in ref. Brower it was demonstrated that the instanton’s topological charge, , is given in terms of the monopole charge forming the loop as follows . This formula is very similar to our relation (13), where the total topological charge, , for a configuration containing a number of particles, described by the system (III) was identified with the total magnetic charge for each Euclidean time slice for the same configuration. Further to this point, lattice simulations do not contradict this picture where large instantons induce the magnetic monopole loops forming large clusters, see e.g.Diakonov and references therein.

We conclude this section with few following remarks.

1). The relation between topological charge in 4d and magnetic charge
in 3d is understood only on the level of classical equations of motion, nahm -Brower .
However, this knowledge does not provide us with answers on the crucial questions
such as: “ what is dynamical properties of these monopoles?”, “do they condense or, rather,
they propagate only for short distances for a short period of time?”

2). A similar to eq.(13) identification
could be made for a different system with large (7)
when only small size instantons are present. However, in this case
it is quite obvious that
the description in terms of the monopole loops makes no sense because the typical size of
the loops is very small, of order , and the magnetic charge is obviously
screened on large distances. Therefore, monopole charge of constituents
play no role for such ensembles.

3). In contrast with the small instantons, the constituents of
large size instantons (instanton quarks with charge
) may propagate far away from each other. In this case
the description in terms of the instanton quarks which carry the monopole charges
could be appropriate. For such configurations the magnetic charges
of the instanton quarks should manifest themselves in some way.
In particular, if the magnetic
charges Bose-condense, this indicates the onset of quark
confinement. To investigate the possibility for such a condensation
an expression for the magnetic charge creation operator, , must
be found and its VEV (magnetization) calculated.
Such a program is very
ambitious, and obviously beyond the scope of the present work.

## Vii Concluding Comments

### vii.1 Main Results

The main leitmotiv of this talk is based on the conjecture that the confinement-deconfinement phase transition at nonzero chemical potential and small temperature is driven by instantons. The instantons qualitatively change the shapes at the transition: they small well-localized objects at large ; they become arbitrary large, strongly overlapped configurations at small in which case description in terms of the instanton quarks become appropriate. Let us emphasize again: the instanton quarks are point like defects which have pure quantum origin and can not be described as semiclassical configurations. They are characterized by translational collective variables, such that units of the topological charge are represented by coherent superposition of instanton quarks (per unit charge) to make together collective variables. This number precisely matches the number of the instanton parameters with topological charge . While the instanton quarks can be arbitrary far away from each other, they keep the information about their origin; they are correlated. Therefore, instanton quarks form not a random, but rather, the coherent large size configurations.

Furthermore we make a quantitative prediction for the critical value of the chemical potential where this transition between two descriptions takes place: at . This prediction can be readily tested on the lattice at nonzero isospin chemical potential.

### vii.2 Future Directions

There are well established lattice method which allow to
introduce isospin chemical potential into the system, see
e.g. kogut .
Independently, there are well- established lattice methods which allow to measure
the topological charge density distribution, see e.g.
Horvath ; Gattringer ; Horvath1 .
We claim that the topological charge density distribution measured as
a function of
will experience sharp changes at the same critical value
where the phase transition (or rapid crossover) occurs.
Indeed, the changes in the topological charge
density distribution are expected due to the fundamental differences in
dependence in two different regimes. We identify these changes
with confinement-deconfinement transition based on the arguments presented above.
We strongly advocate the lattice community to perform such an analysis to see
whether corresponding “ accidental coincidence” indeed takes place.
Such an analysis would provide an unique opportunity to study a transition
from “Higgs -like” gauge theory to “Non -Higgs” gauge theory by varying
the external parameter which plays the role of the vacuum expectation value
of a Higgs field^{4}^{4}4 “Higgs -like”
gauge theories characterized by some finite expectation value of the Higgs field,
when topological defects have finite size, dependence is trivial, ,
and weak coupling regime is realized. This is in contrast with “Non -Higgs” gauge theories,
like QCD at zero temperature and chemical potential when no fundamental scalar fields
exist, dependence appears in form of
and weak coupling regime can not be achieved in description of the large distance physics..
In such an analysis one could explicitly study what is happening with finite size
instantons (which are under complete theoretical control
at large ) when transition from weak coupling regime to strong coupling regime
occurs.

### vii.3 Relation to Other Studies

Here we would like to make few comments on relation to other works.

i). As we already mentioned, at the intuitive level there seems to be a close relation between instanton quarks and the “periodic instanton” vanbaal ; Diakonov ; Gattringer . Indeed, in these papers it has been shown that the large size instantons and monopoles are intimately connected and instantons have the internal structure resembling the instanton-quarks. Also, it has been shown that the constituents carry the magnetic charges. More than that, it has been also argued that large size instantons likely were missing in the lattice simulations, which is consistent with the picture advocated in the present work. Unfortunately, one should not expect to be able to account for large instantons using semiclassical technique to bring this intuitive correspondence onto the quantitative level. However, such a mapping may help us to understand the relation between pictures advocated by ’t Hooft and Mandelstam Hooft on one hand and picture where instanton-quarks are the key players, on the other hand.

ii). There seems to be another close relation (albeit at the intuitive level) between the instanton quarks and configurations with center vortices and nexuses with fractional fluxes , see recent papers on the subject and earlier references thereinEngelhardt . In particular, the total topological charge for entire configuration in both cases is always integer. Locally, however, essentially independent units carry fractional charges . While the geometrical and topological properties are very similar in both cases, there is, however, a fundamental difference between the two: center vortices/nexuses are classical configurations, while the instanton quarks (and everything which accompanying them) have pure quantum origin. This remark is also applied to the “periodic instanton” mentioned above. This difference, in particular, manifests itself for the gauge group different from . In this case the fractional topological charge carried by instanton quarks is , see Section III. At the same time, in general, is not related to the center of the group playing a crucial role in construction of center vorticesEngelhardt .

iii). Using the overlap formalism for chiral fermions Narayanan:1993ss , it has been demonstrated Edwards:1998dj that there is a strong evidence for there existence of gauge field configurations with fractional topological charge for gauge theory.

iv). There is an interesting recent development in lattice computations which in principle would allow to study the topological charge fluctuations in QCD vacuum without any assumptions or guidance based on some specific models for QCD vacuum configurationsHorvath1 . Our remark here that the picture based on the instanton quarks advocated here is consistent with these recent lattice resultsHorvath1 . Indeed, the most profound finding of ref.Horvath1 is demonstration that the topological density distribution in QCD has “ inherently global ” structure. It is definitely consistent with our picture when the point like instanton quarks can be far away from each other, but still keep the correlation at arbitrary large distances.

Another interesting observation by ref.Horvath1 can be explained as follows. If 4D structures of finite size (such as instantons with finite size ) dominate the continuum limit, than these coherent regions of size should exhibit scaling behavior when the lattice spacing is changed. This feature has not been observed in ref.Horvath1 . Therefore, it has been suggested that, in physical units, the corresponding 4D structures should shrink to mere points in the continuum limit. Such an observation is certainly not in contradiction with our picture where instanton quarks are indeed, the effective 4D point like constituents classified by translational zero modes.

As we discussed earlier in the text, the instanton quarks in static limit carry the magnetic charges. At the same time, the magnetic charge of the entire large-size instanton (with all its constituents with fractional charges ) must be zero. Therefore instanton quarks are attached to each other by magnetic strings such that total magnetic flux of whole system is zero. While the fluxes are , they can be probed by quarks in fundamental representation. This picture, again, is consistent with feature of the “sceleton ” (minimal hard-core substructure exhibiting the global behavior) from ref.Horvath1 which is viewed as a network of world lines for point-like objects.

Finally, the dual picture of our CG representation (describing the instanton quarks) is nothing but the effective chiral lagrangian for Goldstone fields, see eq.(8). This “ obvious” connection between confinement and chiral symmetry breaking phenomenon in our framework is consistent with speculation of ref.Horvath1 that the corresponding long distance correlations might be associated with long range propagation of Goldstone fields.

It is too early to say whether ref.Horvath1 finds precisely the features we have been advocating to exist for quite a whileJZ , but the results of ref.Horvath1 look very exciting and promising to us.

v). There seems to be that instanton quarks have been identified in the brane construction in SUSY gauge theories as -brane Brodie:1998bv . While these objects were called as “ fractional instantons” or “merons” in ref. Brodie:1998bv , they obviously have all features of the instanton quarks described above. In particular, the objects from ref. Brodie:1998bv are point like configurations classified by four translational collective variables, precisely as discussed above. It also has been argued Brodie:1998bv that they condense in SYM which leads to the confinement in the theory. At the same time, it has been argued that the same fractional constituents ( -branes) do not play any role in Seiber-Witten model Brodie:1998bv , see alsoBuchel:2001nw . This is perfectly consistent with our proposal that the instanton quarks are not important in “Higgs- like” gauge theories, but play a crucial role in “Non-Higgs” gauge theories. More than that, we conjecture that the instanton quarks is the driving force for the phase transition separating these two fundamentally different types of gauge theories. We suggest to use chemical potential as a parameter which allows us to interpolate between these two types of behavior.

vi). As the final remark: the parameter played a key role in all discussions presented above. However, the region of where transition is expected to occur (see Table 1) is not very sensitive to value of . Indeed, the dependence in physical observable comes with extra suppression which is very small factor. This is exactly the reason why all results for are quoted for . This is definitely not the case when transition from normal to superfluid phase is considered as a function of baryon chemical potential at , or as a function of isotopical chemical potential at . In these cases the transitions are happening at where very nontrivial dependence on is expectedMetlitski:2005db .

## Acknowledgements

Author thanks Pierre van Baal for numerous, very insightful and never ending discussions on the subject. Author also thanks all collaborators of the papers TZ -JZ which constitute the main bulk of this talk. Author also thanks Rajamani Narayanan for the correspondence regarding papers Narayanan:1993ss and Edwards:1998dj , Alex Buchel for correspondence regarding the papers Brodie:1998bv and Buchel:2001nw , and Michael Creutz for correspondence regarding the papers Metlitski:2005db . I am also thankful to the organizers of the Light Cone Meeting, Cairns, Australia, 2005, for inviting me to speak on this subject. The work was supported, in part, by the Natural Sciences and Engineering Research Council of Canada.

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