JHEP07(2001)019 July 17, 2001 branes: D0 Accepted: bosons of the Georgi-Glashow ± W March 29, 2001, ab [email protected] confining strings , Received: 2+1 and Martin Schvellinger We study the behaviour of Polyakov’s confining string in the Confinement, Duality in Gauge Field Theories, Bosonic Strings, a [email protected] [email protected] HYPER VERSION Theory Division, CERN CH-1211, Geneva 23, SwitzerlandDepartment and of Mathematics andPlymouth Statistics, PL4 University 8AA, of Uk Plymouth E-mail: Theoretical Physics, Department of Physics,1 University Keble of Road, Oxford Oxford,Center OX1 3NP, for Theoretical Uk Physics, Massachusetts77 Institute Massachusetts of Avenue, Technology CambridgeE-mail: MA 02139, Usa b a Abstract: Georgi-Glashow model in threetion dimensions described near in confining-deconfining [33]. phaseof In transi- the the string confining language, string the into transition D0 mechanism branes is the (charged decay Alex Kovner Ian I. Kogan Hagedorn transition, vortices and lessons from model). In the world-sheetperature picture are the represented world-lines as of world-sheetcorresponds heavy vortices D0 of to branes a the at certain condensation finitebe” type, tem- and Hagedorn of the transition these transition in vortices.corresponds the to confining the We string monopole also (which binding is transition showfact in not that that the realized the field the in theoretical decay “would language. ouris into model) The understood D0 branes as occurs thefinite at consequence mass lower of of than the the the D0 large Hagedorn branes. thickness temperature of theKeywords: confining string and D-branes. JHEP07(2001)019 (1.1) bm e a m 1 ∼ ) m ( ρ bosons — the D0 branes of confining string 10 ± W A lot of effort has been invested into the study of the Hagedorn transition in the 3.1 Deconfinement in3.2 the Georgi-Glashow model The monopole3.3 binding as Vortices the on3.4 Hagedorn the transition world-sheet — D0 open branes strings destroy and the charged string particles 17 16 14 20 2.1 The2.2 string action Thestringissoft...andthereforerigid!2.3 The 7 6 led to the famouscontext Hagedorn of transition [1]. statistical bootstrapindicates This type that models of they [1, spectrum are 2,mental first composed string 3] arose theories of and, in we more the a find for fundamental the search hadrons, constituents for same such [4]. hints kind behaviour tolies of In the spectrum beyond funda- existence (see the of for Hagedornhigh-temperature ‘string example temperature? phase constituents’ [5]), is which and Is of revealsit great the this true interest. fundamental temperature degrees that What limiting of fordifferent or freedom? classes all is may And types there have is totally a of different string high-temperature theories behaviour? therecritical (super)strings. is There the is same(but enormous by universal literature no on physics means this or all) subject, can some be references found, for example in [6]. For weakly coupled critical 4. Discussion 22 3. How does the string melt? 13 1. Introduction What happens to stringsstring at theory. very In highthe the temperatures exponential early is growth days in one of the of one-particle dual the density resonance Big of models states questions and of hadronic bootstrap Contents 1. Introduction2. Confining strings in the Georgi-Glashow model 4 1 JHEP07(2001)019 [21]. Above /N 1 ) gauge theory) [19] N and thus what is their role at (for SU( 3 N Z 2 = 4 supersymmetry admits BPS solutions that do not N limit we hope it is a weakly coupled string with coupling of order 1 2 N Since non abelian confining gauge theories are strongly linked to strings, we may Although much have been understood, many aspects of hot string theory remain In some cases extra symmetry can protect the theory from tachyonic instabilities. For example M,F,S theories, Matrix Models, etc. give us a lot of new exciting hints, but we still do not have In the recent paper [13] it was shown that in non-commutative open string theory which is 2 3 1 high temperature. Recently itdescription was of suggested the in Hagedorn thebranes transition, framework [16] of that (for the the references Matrix fundamentalreferences on Model string therein). thermodynamics decays Thus in into it Matrix D0 freedom could String in be theory the that see hot D0 phase. [17] branes and are the fundamental degreeshope of to glean somethe insight study about of the thefundamental theory high deconfined of temperature strong phase interactions, behaviour it ofphase was of suggested gauge transition that strings [18] theories. there during is from a whichelectric When deconfining the representation is symmetry QCD broken. of emerged In therestoration as the centre dual of of the magnetic the the picture true gauge this global corresponds group magnetic to in the symmetry the Hagedorn temperature the vorticesphase. populate From the the world-sheet target andities space we point for have of non a view new zeromodes). we winding are modes talking in about the tachyonic instabil- imaginary time direction (thermal winding — for more details seeconfining [20]. phase The of lore QCD is shouldlarge that be these describable two by transitions some are string the theory. same. At least The in the (super)strings the HagedornThouless transition (BKT) can [7, 8] be transitionsation” described on of a as the world-sheet Berezinsky-Kosterlitz- world [9]–[12].in sheet It some vortices. is cases due may It to actually the has be ”conden- been first also order suggested [11] that (see the also transition discussion in [12]). it was shown in [14] that the theory with a solid theoretical concept. decoupled from gravity, insecond order. no more than five space-time dimensions the Hagedorn transition is suffer from Hagedorn-type instabilities.happen. However for generic string theory these instabilities must mysterious. For example, it isof thought using that canonical for Gibbs critical ensemblemicro-canonical (super)strings may the ensembles be whole are not idea found justified,notion since to of the be temperature canonical inequivalent and andof [3, the canonical gravity 15]. ensemble (see Besides iscritical [11] strings. the not and whole In well references the definedcertainty therein) more what in general the which fundamental case the degrees necessarily of presence of interacting exists freedom strings are in we do theories not of know with Above the deconfining transitionmust the mean string that description the does string not itself make undergoes sense. some This kind of a phase transition. This JHEP07(2001)019 (1) one can O 3 while in confining phase it must be 2 N limit suggests that the effective string at short distances has an infinite N All things considered, it would be very interesting to have a model where we know Such a model in fact does exist. It is the confining Georgi-Glashow model in Recently the deconfining transition in this theory has been studied in detail The outline of this paper is the following. In section 2 we discuss the properties Still, it is likely that the hot phases of gauge theories and string theories have argue [26] that smooth world-sheets must disappear. both, the mechanism of confinement andside, the and string the description detailed at picture lowphase energies of on on deconfinement one the phase other transitioncan side. and about the the A deconfining transition model in like string this theory would in be a2 + useful controlled 1 setting. to dimensions. learnlinearly as It much confining was as due shown we 1996 to by Polyakov the Polyakov proposed long screening the agoon description in the [27, of the so-called the 28] confining plasma strings confining thatfor [29]. of phase this compact Induced monopole-instantons. of theory QED string this action [29, is theory In canexpansion 30] based be describing and explicitly rigid it string derived [31]. was found For more that details this see [32] nonlocal andin action references [33]. therein. has derivative Theshown critical that temperature transition for itself themodel. is transition in has The the been universality discussionof calculated class in the and of present [33] it the paper was was twopicture dimensional is and however Ising to to purely discuss relate field various the aspects theoretical. analysis of of the The transitionof [33] purpose from the to the confining Polyakov’s string confining string perspective. Kalb-Ramond string in field the [34] (like Georgi-Glashow inconfigurations model. [29]) and we Rather the use than summation the introducing over direct the closed correspondence string between field world-sheets. The integration common physics. Forfree energy example, is using proportional the to fact that in the deconfining phase the number of world-sheet fields [23]. Whetheris deconfinement related transition of to gauge the theories between Hagedorn the transition two, of using strings AdS/CFTalso is correspondence earlier also was discussed paper not recently quite [25])deconfinement in clear. where transition [24] takes (see The it place relation wasat before found weak the that coupling Hagedorn (the in transition. physical the limit The strong for situation coupling gauge theories) regime is the not known. picture is verygrey attractive. cat in If a thisquarks dark and is room, gluons the since and the case, weproperties. know high we the temperature The are fundamental phase lagrangian problem definitelydealing is which with is described not determines (recent their in that results looking based terms weto for on of be still AdS/CFT a useful duality have in [22] this no mayof respect) turn QCD idea for out confinement. the what eventually simple It string reason: maythe in analysis theory so fact of far we be the there quite high-temperature are isin an no partition the unusual consistent function large theory string of theory. a For chromoelectric example, flux tube JHEP07(2001)019 (2.1) in the , h 2 ) 2 v − h † h bosons) appear as D0 ± (2tr 4 λ W + 2 ) h µ D tr( 3 dx 4 Z ]+ µν F µν F tr[ 3 dx Z 2 g 1 2 − ]= ,h µ In section 3 we discuss the confining string at finite temperature. First, we show Finally in section 4 we conclude with discussion of our results. Consider the 2+1 dimensional SU(2) gauge theory with a scalar field A [ S adjoint representation. The action of the theory is branes in the string description. that the Hagedorn temperaturetemperature corresponds at in which the the fieldature monopole-instantons theoretical arises become language naturally bind to in in the the charged pairs. Georgi-Glashow particles model, This at if temper- finite onewhich temperature. neglects are the We effects also usually of showcorrespond discussed the that to in the the world-sheet the trajectories vortices compact context of euclidean the of time. endpoints the of TheGeorgi-Glashow an Hagedorn actual model open transition, mechanism though string is of physically propagating quite theance different. in deconfining of the The the transition transition plasma in is of the due chargeda to particles, trajectory the or of appear- in a the D0 stringalso brane language corresponds — bounds to D0 a branes. a world-sheetof Since of vortex a an on somewhat open different the string, kind, stringthan such since world-sheet. a Neuman the trajectory string boundary These coordinates conditions. vorticesdestroyed satisfy by are the At appearance Dirichlet however of rather high these temperaturebroken vortices, into the or short in string segments the which world-sheet target connectit is space the can picture D0 be the branes. presented string in Thisis is the mechanism, driven although string by physics language, on isthe short essentially distance string, field scales where theoretical — the in shorterness origin. stringy than of degrees the It of the physical thickness freedom confining of transition are string precedes absent. and the relative Due Hagedorn lightness to transition, of the rendering the large the thick- D0 latter branes, irrelevant. the deconfining 2. Confining strings in the Georgi-GlashowIn this model section we giveof the its derivation of striking Polyakov’s physical confining properties.Kalb-Ramond string fields. and Note discuss that some in the present derivation we do not use over the field degrees ofwill freedom be leads referred to then henceforth directlyis as to the that the confining it string string. action. makes Anstring, explicit This advantage namely of string an this that derivation important thethe and fluctuations inverse quite of thickness unusual of this the property stringof string) of with do the the large not so cost confining momenta called energy.the (larger zigzag This heavy than charged is symmetry particles a introduced of dynamical the extension by Georgi-Glashow Polyakov model [29]. ( We also show that JHEP07(2001)019 , a a πn τ 0 =0 a ζ (2.4) (2.3) (2.2) h i φ 2 =2 φ , respec- = 2 v h 2 )], with g ], ν H W = M M ,A ( µ . 2 W A W 787 and approaches ) M . 2 φ g +[ πM µ 4 . We will be working in A and cos − ν . Thus, the string tension ab 2 ∂ δ − C − λv exp[ become linearly confined with ν (1 , , 2 φ A 2 ± / 2 } 7 )=2 . We use the notations of refer- µ g W g =2 b π ζ 2 ∂ m γ W M 2 τ 2 2 0 π H a 16 g M = ζ σS τ 16 . M ( + γ µν = tr {− = F φ M ζ , µ 2 2 γ a 5 g τ M φ∂ a µ µ ∝ =exp . The short distance physics will be important ∂ A i σ i W 2 ) 2 . In this regime the SU(2) gauge group is broken 2 π C M 2 = g ( g 32 µ ) takes values between 1 and 1 W A H h W 3 (see [33] and references therein). M M v ( dx H W M M Z ]= ] and the normalisation φ [ is the classical ground state. Therefore the classical equations of ,h S µ n A are separated by a domain wall. The action density per unit area of such +[ is the minimal area subtended by the contour h π µ gauge bosons are heavy with masses m ∂ S ± =2 = φ The effective action eq. (2.2) describes only the dynamics of the photon field and W h µ and a domain wall is easily estimated from the action eq. (2.2) as where As discussed in detailthe in path integral [27, with 28] actionresult the eq. (2.2) fundamental induces Wilson such loop a domain when wall inserted solution into with the tively. Furthermore, perturbatively theone theory charged looks matter. like At electrodynamics largeportant, distances with so the that spin non-perturbative the effects photon acquires become mass very and im- the the weakly coupled regime, The monopole induced photon mass is ence [33], where the monopole fugacity is numerical constant while is precisely equal tonothing the but wall tension the of world-sheet of the the domain confining wall. QCD The string. domain wall in fact is D motion have wall-like solutions — where the two regions of space, say with for us to some extentthis in question the later, discussion but of for the nowIt let phase exemplifies us transition. in discuss a We the will effective veryFirst, come lagrangian simple note eq. back manner that (2.2) to the as the it mechanismfor theory is. of has any confinement more integer in than this one theory. vacuum state. In particular a non-perturbatively small stringto tension. Polyakov [28] These are due non-perturbative to effectshave the Coulomb according contributions of interactions monopole and instantons. formclidean The a monopoles path plasma in integral. thephoton sense The of mass. “Debye the 3-dimensional screening” Thelagrangian Eu- in written low this in energy terms plasma physics of provides the of for dual the a photon theory field finite is described by the effective where we use matrix notations to U(1) by theton large associated expectation with value thethe of unbroken the group Higgs is field. massless, but Perturbatively the the pho- Higgs field as well as unity for large values of is valid at energies below the scale JHEP07(2001)019 η η η is . ˆ φ (2.7) (2.5) (2.6) ) ,then ˆ φ m in order S cos π − is necessarily (1 φ 2 2 g π 2 γ into the continuous 16 M only takes values be- φ ˆ φ )+ have to be solved in the must jump by 2 ˆ φ πη when approaching 6= 0, the field ˆ φ π ) +2 x , ( ˆ ) φ η ( x µ ( (in this case a plane) is depicted on satisfies normal classical equations. ∂ ) ˆ πη S φ <π πη ) = 0. Thus, the points in space where x ( )+2 φ 6 +2 ˆ φ x = 1, the field ( ˆ φ ˆ ( φ =0,itraisesto η µ ˆ π< φ ∂ 2 − )= andkeepinginmindthat 2 π x η ( g to the kinetic term, since otherwise the action is UV 32 φ can be integrated out in the steepest descent approx- η . Then, the qualitative structure of the solution for on which 3 ˆ vanishes, the field η φ η S dx Z − is continuous but bounded within the vicinity of one “vacuum” . Let us assume for this solution that the surfaces along which exp( ˆ φ π is integer valued. Clearly, whenever η on the other side of the surface and again approaches zero far from the η D ˆ π φ and − D 1. In the context of the effective lagrangian eq. (2.2) this is the “dilute gas” π − approaches its vacuum value − Z , ˆ φ 1 = The expression for the expectation value of the Wilson loop is the starting point We now substitute the decomposition eq. (2.5) into eq. (2.2) and integrate over , Z . The partition function is ˆ approximation. However as aphysically matter allowed of values of fact, as we will see below, this exhausts all clear. In the bulk, where to cancel the contribution of divergent. Thus the solutionfield is that of a “broken wall” — far from the surface the does not vanish are fewto 0 and far between. Also we will limit the possible values of When crossing a surface presence of the externaltween source imation. This means that the equations of motion for At weak coupling the field and the discrete parts surface. The profile offigure the 1. solution for smooth jumps to where the field and the field for Polyakov’s derivation of theever, action follow for a the slightly confining different stringto path which in understand in [29]. some our physical We view will, properties makes how- it of more the straightforward confining2.1 string. The string action Since the string world-sheet is thein domain wall in the the effective partition action,position we function will of integrate all the degrees domain of wall. freedom To do apart this from let those us that split the mark field the φ not in the vicinity of the trivial vacuum does not vanish mark inadjacent a vacua. very real sense the location of a domain wall between two JHEP07(2001)019 (2.8) . However, for our 1 field as a function of and the two halfs of − γ ˆ φ m M S , are the coordinates in the µ α µ is different from zero, is clearly X in this situation. Now, however ∂ξ ∂X ˆ φ ˆ φ µ α φ and ∂ξ ∂X ξ S 2 d 7 Z . Thus, when two surfaces come within the solves classical equations of motion, clearly π 1 σ ˆ φ − γ = M −π σS with respect to each other. The sole purpose of this π ]= η [ S φ is precisely the profile of the domain wall we discussed above. The ˆ φ Schematic representation of the solution profile of the are the coordinates on the surface ξ to the kinetic term in the action. Thus the action of our solution is precisely Of course the confining strings in the Georgi-Glashow model are not free. The Since outside the surface the field η only difference is that this wall is broken along the surface the profile of where distance of this ordercalculated by they just start finding the towe classical want interact. solution to for discuss In one principle, particular property this Thestringissoft...andthereforerigid! of2.2 interaction the can confining strings be - theirThe rigidity. Polyakov’s action wethe have derived long in wavelength approximation theis to previous only the subsection valid actual is for action ofphoton of mass. course string the only world-sheets Expansion confining in whichit powers string. are will of give smooth It derivatives corrections can on to be the the in action scale principle on performed of scales and comparable the to inverse 3-dimensional space-time. Thiswhich is classically is precisely equivalent the to Polyakov’s the action well Nambu-Gotos of stringthickness a action. of free the region string, inof figure order 1 of in which the the inverse field photon mass the solution are displaced by 2 Figure 1: the coordinate perpendicular to the domain wall. discontinuity, as noted above isof to cancel the “would be” UV divergent contribution JHEP07(2001)019 but 1 − γ M = 1 on an η have with this ac- field with a “rough” 1 ˆ φ − γ M , as in figure 2. d The solid line represents the string. The dashed lines mean d φ weak 1, fluctuations on the scale of the . Thus, all string configurations which γ γ Figure 3: actual the contours of the effective thick string. /M 2 dM 8 π g = 2 γ −π σ/M φ of the old surface. However, Schematic representation of the solution profile of the d To calculate the action we have now We believe that this independence of the action on short wave length fluctua- with the new boundary condition — the ˆ add to it some wiggles on a much shorter distance scale string tension are not penalised at all. tions is a dynamical manifestation of the so called zigzag symmetry introduced by curacy the same energy!.it tolerates The any string numberfigure is of 3. therefore wiggles extremely In on soft,the particular, short in square since distance of the the the scale sense string photon without that tension mass, cost for in our energy, string is much greater than φ surface of the discontinuity is wiggly.new This boundary condition changesfile the of pro- the classical solution onlythickness within the since the action ofspace the close to classical the solutionthat discontinuity, of is the the in action old solution of no with the way the new accuracy dominated solution by will the be region the of same as to solve the classical equation for the field domain wall profile. confining string strange thingsis happen quite in peculiar, the sincechanges the ultraviolet. of action the Physically of world-sheetour the the on derivation situation string short of has distance the absolutely scales. string no action. sensitivity This to should Suppose the be that rather obvious from than taking Figure 2: differ from each other only on small resolution scales absolutely smooth surface, we make the surface look the same on the scale JHEP07(2001)019 (2.9) ,wemust itself does σ O be suppressed. 1 − γ M x, ~ b D ··· + 2 sD + σ in the string path integral. If 9 pair. Therefore, we believe that most of the string x ~ O a − D W does depend on them, then its average vanishes ab − O gg + √ W 4 ξ 2 d Z = S . This means that our string is intrinsically “thick”. If we still want 1 − γ M . 2 γ The confining string is therefore very different from a weakly interacting string In a somewhat perverse way, this softness of the “mathematical” string leads to The derivative expansion for the confining string action has been discussed in Thus, for all practical purposes, we should just exclude the high momentum Indeed, the states with high angular momentum may very well be absent too, due to the fact 4 Thus, such a string theoryσ/M must have a curvature term with the coefficient of order to describe this situation inmake terms the of string a rigid thin so string that with any the bend string on tension the scale smaller than Polyakov [29]. Indeed,itself, Polyakov notes physically that nothing if hasaction. a happened This situation segment and is of sowiggle an a extreme with such example string infinite a of goes momentum. a zigzagmomentum back wiggle What should modes, we on happens have not but just physically, costenergy. also discussed is — any that finite a not but only large infinite momentum string modesusually considered do in not the cost modes string with theory. momentum In of the orderwhich weakly of is interacting the of string square the momentum root sametum of order. the modes string In do tension the not carry confiningconfining carry energy string string energy on to at the contain all. otherwith states hand, Thus, low with these we angular large momen- do momentum). spatial not momentum expect (heavy the string spectrum states of the thatalongstringcandecayintoa rigidity of the physicalprevious string. discussion, The point the is highfreedom. the momentum The following. action modes As of area any it in string calculation is of configuration a obvious some does from sense physical not the quantity “gauge” depend on degrees them. of Consider not depend on thefactors string out. high If momentum on modes, the the other integration hand over these modes the literature (see [32]) and references therein). It has in general the form states will not appearin in [29]. the spectrum of the Georgi-Glashow model contrary to belief expressed since their fluctuation inanalogous the path to integral the is situation completelyinvariant, and in random. for gauge This calculation theories, isfactors of absolutely where out, those all independently the of observables integration the must over observable the be under gauge gauge consideration. modesstring always modes fromexample the when calculating consideration the altogether entropythe of (“gauge our states string fix” which we them are shouldscale only different of to take order on into zero). account the course For grained level with the course graining JHEP07(2001)019 than (2.12) (2.10) (2.11) is negative, in its frame- s ± ). The rigidity is W 2 parametrically ,ξ 1 ξ x ~ ( does not take values in b x ~ φ D . One can define a vortex 2 π D . 2 γ } ,where σ χ M iχ { k symmetry represented in the effective + exp 2 σ χ 2 Z / 1 10 x ~ =2 a 2 2 φ D π g 8 ab branes of confining string gg = √ D0 ξ 2 V on the embedded world-sheet can proceed. d x ~ ± b Z W x∂ ~ = a ∂ S = bosons. It is therefore important to understand how they fit into are the determinant and the covariant derivative with respect to ab ± g a bosons — the W symmetry. This symmetry in the confining phase is spontaneously D ± 2 Z W . This in principle does not preclude us from discussing ) is equivalent to a large and positive curvature term and γ a coefficient of order one. Hence in this way the fact that the confining W g M k M on the scale of their Compton wave length, but this is of no importance to us in As mentioned in the introduction, the Georgi-Glashow model possesses the global The Polyakov effective lagrangian eq. (2.2) is only valid at energies much lower ≤ ± , but is rather a phase. More precisely, the field D work, since being charged particlesponent they associated have a with long them, range-low andthe momentum this framework field “Coulomb” of com- field eq. is (2.2).W in We principle will describable not in be able to describe the internal structure of lagrangian eq. (2.2)? The answer is, that in fact the field any case. There isbefore however the one discussion important of element that we have to add to eq. (2.2) magnetic the “natural” string scale given by the string tension. 2.3 The with string lacks reallarge stringy rigidity high term in momentum theis degrees effective very of description. large, freedom We and emphasise is translates again effective that itself on the the into rigidity distance scales much larger the string picture. than broken in the vacuum [19, 20]. How is this controlled by the second term.and In the the system derivative is expansion stabilised thecumulative by effect stiffness the of higher (the order infinite terms. number( of) Our the argument higher shows order that terms the at short distances induced metric where R As discussed in [33] thedriven deconfining by phase the transition in the Georgi-Glashow model is is the phase fieldoperator [20] which takes values between 0 and 2 JHEP07(2001)019 . C V χ which (2.14) (2.17) (2.15) (2.16) (2.13) .This ,which χ L L W . when going )) + W M x , and the unit π ( M µ C ) boson is a state πj of length − 2 C W − . ( , φ ) + µ 2 ) . 2 ∂ S . ∗ W 2 ) ∈ V ))( V L x x ( + λ ( . W µ ∂ 2 δ ) .The ∗ ) M V πj χ x V V ( ( 2 λ ν − 2 µ 2 γ ∂ ∂ − n ∗ 4 ) M . φ V µνλ φ ( µ V )+ ν − ( ∂ 1 ∂ ( ∗ . The action also has to be augmented by a cos S 11 W 2 V χ 2 π µνλ 1 µ ∈ − M →− g 4 i 32 x g (1 V ( V∂ x = δ 2 3 µ 2 3 ) g µ ∂ d π x 2 γ dx J ( is 1 µ 16 3 g π Z M µ n Z j dx = + − bosons are represented in this effective theory. According )= Z does not vanish is illustrated on Fig.4. δS x that would mean that it has to change by a 4 ± ( boson in the string language, let us consider a path integral = µ is a phase, it is obvious that the effective theory eq. (2.13) µ exp j φ j W S χ W Dφ are the normal vectors to these surfaces. The shape of the surfaces symmetry acts as are two surfaces which both terminate on the curve 2 2 2 Z n Z S . By core energy we mean the energy concentrated at distances of order ]= W and . The relevant partition function is and C 1 [ M 1 C n Z S . This higher derivative term is χ Since the field To represent a around with positive (negative) unit winding of allows topologically nontrivial configurations with non-vanishing winding of of the UV cutoff offield our effective theory which is not contained in the low momentum we should insert a sourceIn in terms the of path the integral field which forces a winding on the field in the presence of one such particle. To create a winding state with world-line Thus, charged states carry unit winding number of vectors In terms of this operator the lagrangian eq. (2.2) is to [19, 20], the electric current is identified with the topological current in eq. (2.13), higher derivative term in orderequal that to the winding configuration has the “core” energy on which the current where is precisely the action of the world-line of a massive particle of mass The density of thisis action singular does — not that vanish it only the at points the of points winding. where For the a phase closed of curve carries the winding, the contribution of this extra term to the action is is precisely how the Here the “external current” The magnetic JHEP07(2001)019 6 . 2 S is the boson, C and + and again 1 W π S by 2 1 1 . The integration boson. 2 S is non-vanishing.The S S 2 W µ S j and therefore and π 1 we see that the presence of S η and ˆ φ is arbitrary as long as they both terminate 2 µ changes by 4 2 the result will not depend on exact position n S 1 µ η φ into n φ and 12 1 to jump across the surface S along which the current φ 2 S the field which is the world-line of the C C and 1 the two surfaces introduced in eq. (2.17) specify the positions of by one on the two surfaces 5 one has two extra string world-sheets along S η η η in order to cancel the, otherwise UV divergent contribution forces the field boson. 2 µ j S + W is performed in exactly the same way as before. The only difference C η across The two surfaces . However at fixed π 2 S at fixed ˆ and φ . Thus when going around . In particular they could coincide, which is the form used in [33]. Here we prefer to use a As before, splitting the field variable The insertion of µ We note that the position of the surfaces We again stress that after the integration over 1 is large on all scales relevant to zero temperature physics. The contribution of the just shifts the variable C j 5 6 S µ jump by 2 over now is that for any given surfaces come together at the line in the string representation isbrane, given which by serves the as sum a overare source surfaces thought for in of two the as extra presence infinitelythis string of world-sheets. heavy. respect. a D0 The Usually They situation D0 areW branes in not the infinitely GG heavy, model but is very heavy very indeed similar since in the mass of We thus see that the field theoretical path integral in the presence of a Figure 4: of of the two extra world-sheets. j world-line of the on more general form with non coincident surfaces for reasons which will become clear immediately. JHEP07(2001)019 , is } L η + parti- W − M =2ona W {− η and + with W η The Ising domain cre- − cles. For static particles theration configu- propagates in thetion, time perpendicular direc- tothe the page. plane of Figure 5: ated by thethe two positions strings of connecting in V 13 1. present. =1and − − i is negative, see figure 5. V .Inthein- W terminate on ,andonthe h V − 2 + V S W W and + and 1 magnetic symmetry, turning it into a local rather than a W S 2 Z , we see that in between V . It may be easier to visualise 2 S and 1 1. Thus having two domain walls is S − = We close this section by noting, that the reason we have two string world-sheets Note, that in physical terms there are only Incidentally, going back to our definition i V terminating on a D0 brane isrepresentation that of the field SU(2). theory We in can questionmodel. has imagine only adding fields We heavy in would adjoint fundamentalterminating particles then on to have a the also D0and brane. allowed would configurations These have a of D0 different branes one massIn would and the string however therefore field be world-sheet a theoretic different physically terms, weightthe different in presence properties the of path of the integral. fundamental the charges changes drastically closed surface isallowed physically to equivalent take to are limited vacuum. to 0,1 Therefore and the values that the two world-sheets the value of while outside them itcreated is positive. domain Thus of we the have second vacuum of onesideontheworld-lineof In this case the surfaces the situation with both other side on the world-line of frared therefore our stringsIsing are nothing domain but walls, the andcreates an the Ising pair domain. of D0-branes two distinct vacua in the model They are thus boundaries ofthe a second closed domain vacuum of of the field and vanishes for infinitelythe large situation system. changes dramatically As at wesystem finite will temperature, has see where finite in one extent. the dimension next of the section however, 3. How does the stringNow melt? that we have anstring understanding of we the can basicquestion physical move we properties on want of to the to address confining the is what analysis is the of mechanism the of the deconfining melting phase of the transition. string in The of the vortex field D0 brane to the partition function is suppressed by a very small factor exp between global symmetry [35]. the same as havingconfiguration a is wall equivalent to and the an vacuum. anti-wall, and A if configuration they of coincide spatially such a h JHEP07(2001)019 14 at large distances. However, at finite temperature the /r In this section we will first explain how this mechanism is related to the well un- Although such a mechanism of transition is a logical possibility, it turns out that The first one is the binding of magnetic monopoles at high temperature. This interaction in the infrared becomes logarithmic. The reason is that the finite temper- the deconfined phase. Inin this the framework respect of the thetemperature basic string is mechanism theory reached that is due the has toIt Hagedorn been excitations has transition, discussed of also where the an the highvortices limiting interpretation energy on as spectrum the the of string BKT the world-sheet. string. transition due toderstood the physics condensation of of deconfinement the intransition the corresponds GG to model. “confinement” Ittemperature. or turns out binding Then that of the we magneticstood Hagedorn show monopoles as that the at the world-line high of of vortex vortices the on on end the the point world-sheet” world-sheetand of is there should the an statement is open be that no string. under- contains once linear the Thus arbitrarily potential monopoles the long are “condensation between strings. bound,objects, charges they any If appear longer, open in thestrings strings the thermal are exist thermal ensemble allowed, ensemble in asinated (world-sheet the is vortices). by the theory closed case If as strings in onlymap dynamical of the closed into world-sheet GG arbitrary vortices model, length.slightly in the different. the World thermal The string lines presence ensemble description, ofof of is arbitrarily although charges, dom- charged long or these strings particles “plasma would vortices phase” mean also are deconfinement also of these vortices. the actual mechanism inthe the thickness Georgi-Glashow of model the physical is string,temperature”. completely the transition It different. happens occurs long Due because before the to large the density enough, “Hagedorn of charged so particles that (D0 branes) thethe becomes distance string. between them So becomeslength the smaller of the than thermal strings the ensemble is widthpicture shorter is of becomes than dominated their useless. thickness. by The At theoryD0-branes. configurations this really point in becomes This of the destruction which course plasma thestring of the of string point the weakly of interacting string view. “froma In within” a string is sense theoretical it quite nature. is peculiarstring of thickness, from distinctly It where the field occurs theoretical as duestring nature, we modes rather to have than exist physics seen at on the these the string momenta. scale language3.1 is smaller inadequate, Deconfinement than in since the the no Georgi-GlashowWe model start with a briefare two recap mechanism of for the the main deconfining points transition of that [33]. onemechanism can Within was contemplate. the first GG suggestedclidean model in path there integral [36]. formalism the monopoles The“potential” form point Coulomb decreasing gas here as with the is 1 interaction the following. In the eu- JHEP07(2001)019 , . β the /T . (3.1) 2 NC =1 /g T 2 β T ,alreadyat 2 γ M ∝ . The universality C model. which is in the uni- M T ρ π 2 XY / 2 the mean distance between g therefore interact via a two π = 4 β / 2 T g = . C 2 T π , the field strength far from the core is . If this where the mechanism of the g 2 π 8 = π/g 15 / ’s. Thus the system again resembles a two 2 the monopole gas becomes two dimensional. g MB W γ T = M ∝ NC T T ’s is very small, so that the transition does not occur since the charges W , and thus the strength of the infrared logarithmic interaction is i 2 pairs bound by strings, but the ensemble rather looks like a neutral plasma. x x g T ± in the ensemble becomes equal to the thickness of the confining string. At this The other candidate mechanism is the appearance of the charged plasma of Two dimensional Coulomb gas is known to undergo the BKT phase transition. The truth however lies somewhere in between [33]. It turns out that at . If one neglects the monopole effects and considers a “non-compact” theory, the = W ± ± i ˜ versality class of 2D XY model. W and thus clearly the fieldTwo profile monopoles is separated two dimensional by on a distance distance scales larger larger than than dimensional rather than a three dimensionalat Coulomb large potential, distances. which is Since logarithmic the density of the monopoles is tiny extremely low temperatures Below this temperature the photonit should should be be massive, while masslessThus above since this one the temperature may cosine expect term the in deconfining the transition lagrangian at eq. (2.2) is irrelevant. point it does not makeof sense any more to think of the thermal ensemble as dilute gas W density of are bound by the linear potential. However at potential between charged particlescertain, is logarithmic. albeit very At finite smalldimensional temperature density Coulomb one of gas. has a phase This to gas a undergoes plasma a phase BKT transition at from a confined The strength of theof logarithmic the interaction monopole is far easilyThe enough calculated. field from the The strength core magnetic shouldSince spreads flux have the evenly total only in flux components theF of parallel compact the direction. to monopole the is spatial 2 directions. transition, the universality class would be again that of 2D class of the transition is that of the 2D Ising model corresponding to the restoration Due to the peculiar propertyinteraction that grows for with the temperature, monopoles themonopole the high binding strength temperature of phase. phase the correspondsare logarithmic The bound to transition is the temperature above which the monopoles ature path integral is formulated withtime periodic direction. boundary conditions The inin field the Euclidean this lines direction. are thereforebend The prevented close from to magnetic the crossing fieldflux boundary the lines is and boundary emanating go squeezed parallel into from to two the it. dimensions. monopole So, The have effectively length the to of whole magnetic the time direction is Indeed it is shown rigorously in [33] that the transition occurs at JHEP07(2001)019 L (3.6) (3.5) (3.4) (3.3) (3.2) In this simple is the number 7 z ,where L a scales exponentially with z L . . Let us calculate the free energy σL . } L bound states. . Such a prefactor is not essential for our √ L − σ αL { xT √ W - 1 x σL . − σL , + 16 = = √ W σL x H E )=exp T L ( ]= L N [ F is determined by the physical thickness of the string. α symmetry. The transition can be pictured as condensation of . The only natural lattice spacing for such a discretization is the 2 z Z ln a = α . Then clearly the number of possible states is a a number of order one. The free energy then is x We disregard in this argument the fact that the string has to close on itself. This extra condition 7 thickness of the string.as For an the almost scale free associated string with the the thickness string is tension. naturally Thus the the same entropy is the free energy becomespear negative, in the which thermal means ensemble in thatbecomes a strings completely a unsuppressed of “soup” way. arbitrary of Thependent” length thermal arbitrarily vacuum string long ap- tension strings. vanishes Thus, and it effectively is the not “temperature possible de- to talk about strings anymore At the temperature with The dimensional constant Imagine that the stringspacing can take only positions allowed on a lattice with the lattice The number of states for a closed string of the length 3.2 The monopole binding asLet the us Hagedorn now consider transition theogy phase with transition the from discussion thecharged in string particles. the perspective. From previous the First, string subsection,contributions in point anal- let of of us the view completely this heavyclosed disregard means strings. D0 the that Naively we branes, one neglectture, and expects possible beyond in thus which such the are a string can entirely theoryis not existence of within exist. of order the In Hagedorn of an tempera- theoryConsider almost the free a of string string closed this scale. temperature string One of can a visualise given fixed this length phenomenon in simple terms. situation of the magnetic of order one, equal to the number of nearest neighbours on the lattice. Ising domains which fill the interior of would lead to aargument, prefactor and we with therefore power do dependence not on worry about it. of such a string. The energy of the string is JHEP07(2001)019 (3.7) (3.8) . . 2 2 σ g g ∝ ∝ γ σ α 17 M = ∝ H T α The noteworthy feature of this formula is, that the Hagedorn temperature of the Since we have completely neglected the possibility of appearance of D0 branes The situation is very similar to the BKT phase transition, where the free energy In the string partition function language this is just a restatement of the well The same physical effect is there for the confining GG string. There is one dif- We thus have for the confining GG string This is indeed theprevious correct subsection. magnitude of the critical temperature as discussedconfining in string the is muchbecause higher the entropy than of thedue the string to thick the scale. string fact is thatto This high much the momentum smaller is modes entropy than easy of at that thethe all, to of confining string as understand string the to discussed do free a not in string much contribute the higher previous temperature section. for entropy(charged Thus, effects particles), one the to transition needs become we to have important. of been heat the discussing is monopole the binding string transition representation the in the monopoles Georgi Glashow bind, model. thestring At tension photon the point disappears of where and thepicture the this GG is thickness model just of the becomes the duallong statement string massless that strings. the diverges. and ensemble The is In thus BKT dominated the nature by the the Hagedorn of infinitely both transitions3.3 underlines Vortices this on point. the world-sheet —Sometimes the open Hagedorn strings transition is and discussedsheet. charged in What particles terms is of the the vortices physical on nature the world- of these objects? in the hotstrings phase. see [37] For and more references details therein. about thisof “random a vortex walk” becomes descriptionthe negative of vacuum at in the hot the critical hot temperature, phase. and the vorticesknown populate fact that thea partition torus function which diverges windswinding around in in the the the compact sector Euclidean Euclidean withenergy time time in the the direction. physically sector topology corresponds with Fixing of one tothe the closed string string. the unit is The calculation the integration cause of over of all the possible the divergence lengths free of of the partitionference, function at which high however temperature. turns outThe to thickness be of crucial the for GG theto nature string the of is inverse the not photon phase given mass, transition. by and the thus string tension. Rather it is equal JHEP07(2001)019 b are locations of b and L a s C a represents the compact dimension. b R L s 18 a A R . Since this is true for any contour of arbitrarily β string which winds around the compact direction. Figure 6 L s open the vortices on the world-sheet are bound in pairs. The corresponding varies from 0 to H ab 0 B x The open string world-sheet conformally transformed into a closed string world- R T