Planar limit of orientifold field theories and emergent center symmetry

Adi Armoni,1 Mikhail Shifman,2 and Mithat Unsal3,4

1Department of Physics, Swansea University, Singleton Park, Swansea, SA2 8PP, United Kingdom2William I. Fine Theoretical Physics Institute, University of Minnesota, Minneapolis, Minnesota 55455, USA

3SLAC, Stanford University, Menlo Park, California 94025, USA4Physics Department, Stanford University, Stanford, California, 94305, USA

(Received 10 December 2007; published 7 February 2008)

We consider orientifold field theories [i.e., SU�N� Yang-Mills theories with fermions in the two-indexsymmetric or antisymmetric representations] on R3 � S1 where the compact dimension can be eithertemporal or spatial. These theories are planar equivalent to supersymmetric Yang-Mills theory. The latterhas ZN center symmetry. The famous Polyakov criterion establishing confinement-deconfinement phasetransition as that from ZN symmetric to ZN broken phase applies. At the Lagrangian level the orientifoldtheories have at most a Z2 center. We discuss how the full ZN center symmetry dynamically emerges in theorientifold theories in the limit N ! 1. In the confining phase the manifestation of this enhancement isthe existence of stable k strings in the large-N limit of the orientifold theories. These strings are identicalto those of supersymmetric Yang-Mills theories. We argue that critical temperatures (and other features) ofthe confinement-deconfinement phase transition are the same in the orientifold daughters and theirsupersymmetric parent up to 1=N corrections. We also discuss the Abelian and non-Abelian confiningregimes of four-dimensional QCD-like theories.

DOI: 10.1103/PhysRevD.77.045012 PACS numbers: 11.15.Pg, 11.10.Wx

I. INTRODUCTION

In this paper we consider dynamical aspects of four-dimensional orientifold field theories compactified onR3 � S1. The compactified dimension S1 is either temporalor spatial. Whether we deal with thermal or spatial for-mulation of the problem depends on the spin connection offermions along the compact direction. In the latter case wearrive at a zero-temperature field theory where phase tran-sitions (if any) are induced quantum mechanically. Ineither case, if the radius of S1 is sufficiently large we returnto four-dimensional theory, R3 � S1 ! R4.

By orientifold field theories we mean SU�N� Yang-Millstheories with Dirac fermions in two-index representationsof SU�N�—symmetric or antisymmetric.1 Our startingpoint is the large-N equivalence between these theoriesand N � 1 super-Yang-Mills (SYM) theory [1–5] (seealso Ref. [6] for the relation with string theory). Sincesupersymmetric gauge theories are better understood thannonsupersymmetric, we hope to learn more about non-supersymmetric daughters from planar equivalence. Thisexpectation comes true: the center symmetry of SYMtheory turns out to be an emergent symmetry of the ori-entifold daughters in the large-N limit. This fact was firstnoted in [7] while the first mention of the problem of centersymmetry on both sides of planar equivalence can be foundin [8]. Here we investigate the reasons that lead to theemergence of the center symmetry and its implications, asthey manifest themselves at small and large values of theS1 radius.

For SU(3) the orienti-AS theory reduces to one-flavorQCD. If the large-N limit is applicable to N � 3, at leastsemiquantitatively, we can copy SYM theory data tostrongly coupled one-flavor QCD. A concrete example isthe temperature independence observed in [9]. It wasshown that certain observables of SYM theory are tem-perature independent at large N and so is the charge-conjugation-even subset of these observables in thelarge-N orientifold field theory. It implies a very weak(suppressed by 1=N) temperature dependence of certainwell-defined observables in the confining phase of QCD.This analytical result is supported by lattice simulations[10–12]. For a recent review, see [13]. The planar equiva-lence between SYM theory and oreinti-AS is valid in anyphase which does not break the charge conjugation sym-metry (C invariance). This implies coinciding Polyakovloop expectation values in the low-temperature confinedand high-temperature deconfined phases and the equalityof the confinement-deconfinement transition temperaturein orienti and SYM theories in the large-N limit. Otherfeatures of the phase transition are predicted to coincidetoo.

In the spatial compactification of SYM theory, the centersymmetry is unbroken at any radius. For orienti-AS, it isdynamically broken, along with C and CPT, at small radii,and restored at a critical radius of the order of ��1 [14,15].These zero-temperature, quantum phase transitions areobserved in recent lattice simulations by two independentgroups [16,17]. The unbroken center symmetry in the smallS1 regime of the vectorlike gauge theories, unlike thedynamically broken center symmetry, leads to Higgsingof the theory. The long-distance dynamics of such QCD-like theories are intimately connected to the Polyakov

1They will be referred to as orienti-S and orienti-AS,respectively.

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model [18]. We discuss both the strong-coupling and weak-coupling confinement regimes. At weak coupling we getPolyakov (Abelian) confinement which is analytically trac-table. The region of validity of the Abelian confinement inQCD-like theories is a vanishingly small window whichdiminishes with increasing N. The fact that the Abelianconfinement regime is vanishingly small is a consequenceof volume independence. We discuss this issue in somedetail.

Summarizing, our findings are as follows:Orientifold field theories on R3 � S1 exhibit a number of

a priori unexpected features. These theories are planarequivalent to supersymmetric Yang-Mills theory. The latterhas ZN center symmetry. The famous Polyakov criterionestablishing confinement-deconfinement phase transitionas that from ZN symmetric to ZN broken phase applies.At the Lagrangian level the orientifold theories have atmost a Z2 center. The full ZN center symmetry dynamicallyemerges in the orientifold theories in the limit N ! 1. Inthe confining phase the manifestation of this enhancementis the existence of stable k strings in the large-N limit of theorientifold theories. These strings are identical to those ofsupersymmetric Yang-Mills theories. The critical tempera-tures of the confinement-deconfinement phase transitionsare the same in the orientifold daughters and their super-symmetric parent up to 1=N corrections. Depending on thesize of S1 one can identify the Abelian and non-Abelianconfining regimes of four-dimensional QCD-like theories.

The organization of the paper is as follows. In Sec. II weoutline basic facts on planar equivalence, define thePolyakov line, and spatial Wilson lines. Section III isdevoted to the center symmetry in SYM theory, QCDwith fundamental fermions and orienti-S/AS. It gives analternative derivation of the approximate, exact, and emer-gent center symmetries. In Sec. IV we discuss the strong-coupling manifestation of the emergent center symmetry:existence and stability of k strings. In Sec. V we discussstrong versus weak coupling regimes. Section VI is de-voted to Polyakov’s mechanism of Abelian confinement. Inparticular, we address the issue of how it can be general-ized to theories with adjoint and two-index fermions. Weconfront the thermal confinement-deconfinement phasetransitions in SYM and orienti theories in Sec. VII. Herewe prove the equality of the critical temperatures at N !1. Finally, Sec. VIII briefly summarizes our results. One-loop potentials are derived in the Appendix.

II. PLANAR EQUIVALENCE AND POLYAKOVLINE

Planar equivalence is equivalence in the large-N limit ofdistinct QCD-like theories in their common sectors. Agreat deal of attention has been given to the equivalencebetween supersymmetric gluodynamics and its orientifoldand Z2 orbifold daughters. The Lagrangian of the parentsupersymmetric theory is

L � �1

4g2P

Ga��Ga

�� �i

g2P

�a�D� _���a _�; (1)

where �a� is the gluino (Weyl) field in the adjoint repre-sentation of SU�N�, and g2

P stands for the coupling constantin the parent theory. The orientifold daughter is obtainedby replacing �a� by the Dirac spinor in the two-index(symmetric or antisymmetric) representation (to be re-ferred to as orienti-S or orienti-AS). The gauge couplingstays intact. To obtain the Z2 orbifold daughter we mustpass to the gauge group SU�N=2� � SU�N=2�, replace �a�

by a bifundamental Dirac spinor, and rescale the gaugecoupling, g2

D � 2g2P [19–23]. We will focus on orienti-AS.

Consideration of orienti-S runs in parallel, with the sameconclusions.

Planar equivalence between the parent and daughtertheories can be applied in arbitrary geometry, in particular,on S1 � R3, S1 � S3, and R4. The equivalence implies thatcorrelation functions of the daughter theory are equal to thecorrelation functions of the parent theory at N ! 1.Hence, in those cases where the underlying ‘‘microscopic’’symmetries of the planar-equivalent partners do not coin-cide, the theory with a lower microscopic symmetry willreflect the symmetries of the parent theory, which arenaively absent in the daughter (or vice versa) [4]. Themost profound effect of such a symmetry mismatch—and enhancement—occurs when one dimension is com-pactified onto a circle, and the symmetry under considera-tion is the center symmetry. The corresponding orderparameter is the Polyakov line which we define below.

Assume that one dimension is compactified (it may beeither time or a spatial dimension). For definiteness, wewill assume z to be compactified. The Polyakov line(sometimes called the Polyakov loop) is defined as apath-ordered holonomy of the Wilson line in the compac-tified dimension,

U � P exp�iZ L

0azdz

�� VUVy; (2)

where L is the size of the compact dimension while V is amatrix diagonalizing U. Moreover,

U � diagfv1; v2; . . . ; vNg � eiaL; (3)

where

a �X

Cartan gen

acTc � diagfa1; a2; . . . ; aNg;

XNk�1

ak � 0:

(4)

It is obvious that

ai � �i lnvi mod 2�: (5)

The planar equivalence implies definite relations amongthe expectation values of the Polyakov loops in SU�N�SYM and orienti theories—two gauge theories with dis-tinct center symmetries at the Lagrangian level. In the next

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section, we will discuss the vacuum structure of thesetheories from the center symmetry viewpoint.

III. CENTER SYMMETRY (EXACT ANDAPPROXIMATE)

In SYM theory all dynamical fields—gluons and glui-nos—are in the adjoint representation of SU�N�. Thismeans that the gauge group is

G � SU�N�=ZN (6)

rather than SU�N�. This fact manifests itself as a ZNsymmetry on the elementary cell of fa1; a2; . . . ; aNg.Under SU�N� transformations from ZN

U ! e2�ik=NU; k � 0; 1; . . . ; N � 1: (7)

The ZN symmetry, usually referred to as the center sym-metry, may or may not be spontaneously broken. There is afamous Polyakov criterion regarding confinement/decon-finement transition in SU�N� Yang-Mills theories. If oneconsiders the Polyakov line along the compactified direc-tion, and its expectation value hTrUi does not vanish, thecenter symmetry is broken implying deconfinement. Onthe other hand, if hTrUi � 0 the center symmetry is un-broken implying confinement.2

Introducing fundamental dynamical fermions removesthe center symmetry (see Ref. [24]). However, one can stillmake sense out of the center symmetry as an approximatesymmetry. The simplest way to study the impact of thefundamental fermions is to integrate them out implying thefollowing (formal) result:

logdet�i��DF� �m� �

Xn2Z

XCn

��Cn�TrU�Cn� (8)

where the superscript F stands for fundamental, ��Cn� arecoefficients scaling with N as O�N0�. The integer n is thewinding number of the loop C along the S1 circle which isvalued in the first homotopy group of S1,

�1�S1� � Z:

A small mass m is inserted as an infrared regulator. Notethat log of the determinant in (8), the fermionic contribu-tion to the action expressed in terms of the gluonic observ-ables, scales as N1.

The n � 0 sector has net winding number zero. Hence,the corresponding term in (8) is neutral under the center

symmetry transformations. However, for instance, the n �1 sector operators are Polyakov loops charged under thecenter group transformations. Thus, the fermion contribu-tion (8) to the action is explicitly noninvariant with respectto the center symmetry.

A typical term in the sum from the winding class ntransforms as

TrU�Cn� ! hn TrU�Cn�; (9)

where h 2 ZN . Despite this fact, the center symmetry is anapproximate symmetry, since the contribution of the fun-damental fermions (8) is suppressed as 1=N relative to thepure glue sector whose action sales as N2. At N � 1, thefundamental fermions are completely quenched and thecenter symmetry becomes exact. The connected correlatorsor expectation values of the gluonic observables—includ-ing the Polyakov loop correlators—are the same as in pureYang-Mills theory.3

For orienti-AS the dynamical AS fermions are not sup-pressed in the large-N limit. The above rationale applicableto fundamental fermions no longer holds. However, planarequivalence will lead us to the same conclusion—emer-gence of an approximate center symmetry at large N.

The Lagrangian of the orientifold theories has the form

L � �1

4g2 Ga��G

a�� �

i

g2 �ijD� _�

� _�ij; (10)

where ij is the Dirac spinor in the two-index antisym-metric or symmetric representation. Obviously, there is noZN symmetry at the Lagrangian level. The center symme-try is Z2 for even N and none for odd N. Indeed, the two-index fermion field, unlike that of gluino, does not stayintact under the action of center elements. The action of acenter group element on an adjoint fermion is trivial, �!h�hy � �. The action on an AS fermion is ! h h �h2 . Thus, for N even (odd), h � 1 (h � �1) are thecenter group elements which leave the AS fermion invari-ant, in accordance with at most a Z2 center symmetry fororienti-AS theory.

As was mentioned, the antisymmetric fermions are notsuppressed in the large-N limit. Integrating out the two-index antisymmetric fermion yields

logdet�i��DAS� �m� � N2

Xn2Z

XCn

��Cn�2

��Tr

NU�Cn�

�2

�1

NTr

NU2�Cn�

�: (11)

In the large-N limit we can ignore the single-trace terms

2In QCD-like theories with fermions, the boundary conditionson fermions—antiperiodic versus periodic—(to be denoted asS) determine interpretation of the center symmetry. If thefermions obey S�, the partition function has a thermal inter-pretation; a change in the (temporal) center symmetry realizationmust be interpreted in terms of the jump in the free energy of thesystem. If the fermions obey S�, then the partition function is ofthe twisted type, Tr��1�Fe��H, and realization of the spatialcenter symmetry has interpretation in terms of the jump in thevacuum energy.

3The above suppression is analogous to the isotopic symmetryin QCD. Since the two lightest flavors are very light compared tothe strong scale, mu;d=�� 1, QCD possesses an approximateSU(2) invariance despite the fact that the up and down quarkmasses differ significantly. The appropriate parameter mu;d=�plays the same role as 1=N.

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since they are suppressed by 1=N compared to the O�N2�double-trace term. The single-trace term contributionscales as that of the fundamental fermions, and is quenchedin the same fashion.

A typical double-trace term �TrU�Cn��2 is O�N2� and isa part of the leading large-N dynamics. Thus, the impact ofthe two-index antisymmetric fermions on dynamics is asimportant as that of the glue sector of the theory. Thedouble-trace term is explicitly noninvariant under the ZNcenter transformations.

We see that in orientifold theories the center symmetryimplementation is much less trivial than in theories withfundamental quarks. As was argued in [7], the centersymmetry emerges dynamically in the planar limit N !1. Here we will carry out a thorough consideration andpresent independent albeit related arguments.

The action of the pure glue sector is local and manifestlyinvariant under the ZN center. Integrating out fermionsinduces a nonlocal sum (11) over gluonic observables.This sum includes both topologically trivial loops withno net winding around the compact direction (the n � 0term) and nontrivial loops with nonvanishing windingnumbers. The topologically trivial loops are singlet underthe ZN center symmetry by construction, while the loopswith nonvanishing windings are noninvariant.

Let us inspect the N dependence more carefully. If weexpand the fermion action in the given gluon backgroundwe get

�exp

��N2

Xn�0

XCn

��Cn�2

�Tr

NU�Cn�

�2��; (12)

where h� � �imeans averaging with the exponent combiningthe gluon Lagrangian with the zero winding number term.This weight function is obviously center-symmetric. If h isan element of the SU�N� center, a typical term in the sum(12) transforms as

�Tr

NU�Cn�

Tr

NU�Cn�

�! h2n

�Tr

NU�Cn�

Tr

NU�Cn�

�

� h2n��

Tr

NU�Cn�

��Tr

NU�Cn�

�

�

�Tr

NU�Cn�

Tr

NU�Cn�

�con

�;

(13)

where we picked up a quadratic term as an example. Theconnected term in the expression above is suppressedrelative to the leading factorized part by 1=N2, as followsfrom the standardN counting, and can be neglected at largeN. As for the factorized part, planar equivalence impliesthat all expectation values of multiwinding Polyakov loopsare suppressed in the large N limit by 1=N,

�1

NTrU�Cn�

�SYM� 0;

�Tr

NU�Cn�

�AS� O

�1

N

�! 0; n 2 Z� f0g;

(14)

where the first relation follows from unbroken center sym-metry in the SYM theory and the latter is a result of planarequivalence (in the C-unbroken, confining phase of orienti-AS).

Consequently, the noninvariance of the expectationvalue of the action under a global center transformation is

h�Si � hS�hn TrU�Cn�� � S�TrU�Cn��i � O�

1

N

�hSi;

(15)

which implies, in turn, dynamical emergence of centersymmetry in orientifold theories in the large-N limit. Letus emphasize again that the fermion part of the Lagrangianwhich explicitly breaks the ZN symmetry is not subleadingin large N. However, the effect of the ZN breaking onphysical observables is suppressed at N ! 1.

This remarkable phenomenon is a natural (and straight-forward) consequence of the large-N equivalence betweenN � 1 SYM theory and orienti-AS. Despite the fact thatthe center symmetry in the orienti-AS Lagrangian is atmost Z2, in the N � 1 limit all observables behave as ifthey are under the protection of the ZN center symmetry.(This point is also emphasized in [25].) We will refer to thisemergent symmetry of the orienti-AS vacuum as the cus-todial symmetry. The custodial symmetry becomes exact inthe N � 1 limit, and is approximate at large N.

The immediate implication of this discussion is as fol-lows: when we integrate out fermions in the N � 1 limit,the dynamical pattern in orienti-AS/S in the confined phasesimplifies. The sum over all homotopy classes in (11)reduces to a single term—the one over the loops withthe vanishing winding number,

Pn2Z��Cn� ! ��C0�.

Thus,

logdet�i��DAS� �m� � N

2XC0

��C0�

2

�Tr

NU�C0�

�2: (16)

Consequently, the action and other observables of the N �1 orienti-AS/S are indistinguishable from the ‘‘reduced’’theory with action

Sreduced � SYM �XC0

��C0�

2

�Tr

NU�C0�

�2: (17)

Clearly, the ZN center is a manifest symmetry of thereduced theory. Our derivation also provides a direct deri-vation of the temperature independence [9] of the orienti-AS theory in the confining low-temperature phase. See alsoSec. VII.

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The ZN-symmetric vacuum structure at low tempera-tures, in the C-unbroken phase, can also be phrased interms of the eigenvalue distribution of the Polyakov loop.We want to argue that the vacuum of the orienti-AS theoryis invariant under the custodial ZN center symmetry (whichis the symmetry of the supersymmetric parent). Let ��denote the distribution of the eigenvalues of the Polyakovloop in orienti-AS in the confined phase. Let us decompose�� into its Fourier modes,

�� �1

2�

Xk2Z

eikk:

All moments (other than that with k � 0) are restricted tobe O�1=N� due to planar equivalence in the low-temperature phase,

k��

Tr

NUk��

1

N

XNi�1

eiaik!Zd��eik�

1

N; k� 0:

(18)

Consequently, in the N � 1 limit, the eigenvalue distri-bution of the orientifold theory is flat,

�� �1

2�: (19)

This implies, in particular, that under the action of a centergroup element, U ! U expf2�ikN g, the eigenvalue distribu-tion

�� ! ��

2�kN

�� �� (20)

remains invariant.In line with our conclusion are recent calculations on

S3 � S1 [25,26] (based on techniques developed in[27,28]) showing that the vacuum of the large-N orienti-fold theory in the confining phase is characterized by thedistribution (19), i.e., supports the ZN center symmetryrather than the naively expected Z2. The authors ofRefs. [25,29] also reached the conclusion that perturbativetransitions which take place on S3 � S1 capture the natureof the nonperturbative transition taking place in the semi-decompactification limit of R3 � S1, as probed in latticesimulations.

IV. MANIFESTATION OF ZN AT STRONGCOUPLING

If the circumference of S1 is large enough, L > L , weare in the non-Abelian confinement regime both in SYMand orientifold theories.4 The signature of the ZN center inSYM theory is the existence of the k strings. The tensionsand thicknesses of the k strings are class functions of the

center group ZN . The planar equivalence implies that theorienti-AS theory must have the very same k strings despitethe presence of the dynamical fermions charged under thecenter group. Below we discuss how this arises.

The simplest SYM string is a (chromoelectric) flux tubethat connects heavy (probe) color sources in the funda-mental representation. Usually it is referred to as thefundamental string. The flux tubes attached to colorsources in higher representations of SU�N� are known ask strings, where k denotes the n-ality of the color repre-sentation under consideration. The n-ality of the represen-tation with ‘ upper and m lower indices (i.e., ‘fundamental and m antifundamental) is defined as

k � j‘�mj: (21)

It is clear that for stable strings the maximal value of k ��N=2� where � � denotes the integer part. The stability ofthese �N=2� varieties of strings is a question of energetics.For Tk � kT1, which is the observation in lattice studiesand certain supersymmetric theories, all k strings are stable[30,31].

On the other hand, in the orientifold-AS theory at finiteN (with N even) the only stable k string is the fundamentalone. If N is odd, there are no stable k strings at all, as inQCD with fundamental matter [32]. This is due to the factthat probe charges with even n-alities can be completelyscreened by two-index antisymmetric quarks, while thosewith odd n-alities can be screened down up to a singlefundamental index (if N is even) or completely (if N isodd). A similar screening takes place in the orientifold-Stheory with the quark �ij� replaced by fijg.

However, at N ! 1 the breaking amplitude of a colorsinglet into two is 1=N suppressed. Note that the samebreaking amplitude in large-N QCD with fundamentalfermions dies off as 1=

Np

. Consequently, in the limit N !1 all k strings become stable against breaking, and iden-tical to those of the SYM theory. The k-string tension, inleading order in N, is Tk � kT1, where T1 is the tension ofthe fundamental string, and is marginally stable.

LetWk�C� denote a large Wilson loop in a representationwith n-ality k, with C being the boundary of a surface �.The expectation values of such Wilson operators in orienti-AS with odd N is given by the formula

hWk�C�iAS � e�TkA��� �1

Ne��kP�C�; (22)

where Tk is the string tension, A��� denotes the area of thesurface spanned by the loop C, and P�C� its perimeter (seeSec. 10 of [33]). This formula captures two asymptoticregimes and exhibits noncommutativity of the long-distance and large-N limits, a general and quite obviousfeature,

limR!1

limN!1

�loghWk�C�iAS

R

�� lim

N!1limR!1

�loghWk�C�iAS

R

�:

(23)

4The value of L will be discussed in Sec. V. Since neitherparent nor daughter theories have massless states in the limitL! 1, where we recover R4 geometry, the limit must besmooth.

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At any finite N, at asymptotically large distances, theperimeter law dominates, and the potential asymptotes toa constant, i.e.,

limR!1

V�R� � const:

In this regime, infinite separation of probe color chargescosts only a finite energy. However, if we take the large-Nlimit first, linear confinement holds at large distances, i.e.,

limR!1

V�R� � TkR:

Planar equivalence, which implies taking the limit N ! 1first, guarantees equality of the string tensions in N � 1SYM theory and orienti-AS,

TSYMk � TAS

k ; for all k; N � 1: (24)

We stress again that, unlike fundamental fermions whichare quenched in the large-N limit, the two-index antisym-metric (symmetric) fermions are not suppressed. The emer-gence of �N=2� varieties of chromoelectric flux tubes isassociated with the suppression of quantum fluctuations inthe large-N limit. This is a nontrivial dynamical effect.

V. STRONG VERSUS WEAK COUPLING

SYM theory is planar equivalent to the orientifold theo-ries provided C (charge conjugation) invariance is notspontaneously broken [14,15]. So far we discussed thelarge-L limit, i.e., L� ��1 where � is the dynamicalscale parameter. In this limit both SYM theory and itsnonsupersymmetric daughters are expected to confinemuch in the same way as pure Yang-Mills theory on R4.We will refer to this regime as non-Abelian confinement.

On the other hand, if

L� ��1; (25)

the gauge coupling is small at the compactification scale.SYM theory with periodic spin connection (which pre-serves supersymmetry) undergoes gauge symmetry break-ing at the high scale �L�1. The running law of the four-dimensional gauge coupling is changed at the scale wherethe gauge coupling is still small; i.e., we are in the weak-coupling Higgs regime. In further descent of the scale thecorresponding evolution of the coupling constant is deter-mined by a three-dimensional theory.

In the fully Higgsed regime ai � aj for all i; j �1; . . . ; N. If one chooses a generic set of all different ai’s,SYM theory is maximally Higgsed; more exactly, SU�N�gauge symmetry is broken down to the maximal Abeliansubgroup U�1�N�1. The gauge fields from the Cartan sub-algebra (as well as the fermions) remain massless in per-turbation theory5 (they will be referred to as ‘‘photons’’),while all other gauge fields acquire masses (they will be

referred to ‘‘W bosons’’). For generic sets of ai there is noregular pattern in the W boson masses. However, if theHiggsed theory is described by ZN-symmetric expectationvalues of the diagonal elements vk,

vk � e2�ik=N; k � 1; . . . ; N (26)

(or permutations), see Fig. 1, the pattern of the W bosonmasses is regular. N lightest W bosons [corresponding tosimple roots and affine root of SU�N�] are degenerate andhave masses 2�

LN , while all others can be obtained as k 2�LN

where k is an integer. Thus, there are �N2 gauge bosonswhose mass scales as 1=L and �N gauge bosons whosemass scales as 1=�LN�.

In the Higgs regime one can consider two distinct sub-regimes. If

L & L �1

�N; (27)

there exists a clear-cut separation between the scale of thelightest W bosons 2�

LN and nonperturbatively induced pho-ton masses� expf� 8�2

Ng2g, where g is the gauge coupling in

the four-dimensional Lagrangian (1) or (10). How thenonperturbative photon mass is generated and why it leadsto linear Abelian confinement is explained in Sec. VI. Inthis regime, the vast majority of W bosons acquire massesthat scale as N and, therefore, decouple in the large-Nlimit. Thus, below the scale L�1

we deal with three-dimensional Abelian theory at weak coupling. In thistheory Abelian confinement sets in due to a generalizationof the Polyakov mechanism. If the vacuum field is chosenaccording to (26) the Polyakov order parameter vanishes.We are in the ZN-symmetric regime, much in the same wayas in non-Abelian confinement at L>��1. Note thatEq. (26) implies that

FIG. 1. ZN symmetric vacuum fields vk. For definitions seeEq. (3).

5The nonperturbative mechanism which generates mass gap� expf� 8�2

Ng2g is discussed in Sec. VI.

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fakLg � �2��N=2�

N;�

2���N=2� � 1�

N; . . . ;

2��N=2�

N;

(28)

where �A� stands for the integer part of A.On the other hand, if we increase L, starting from L and

eventually approaching ��1, the masses of ‘‘typical’’ Wbosons become of order � (the number of such typical Wbosons�N2), while the masses of ‘‘light’’ W bosons scaleas �=N (the number of light W bosons �N1). In this casethe effective low-energy description of the theory at N !1 must include light W’s along with the exponentiallylight photons. The expectation value of the Polyakov loopstill vanishes.

This suggests that the L dependence (in one-flavor theo-ries) of all physical quantities is smooth across the board:from L & L to L� ��1. Rather than a phase transitionthere may be a crossover. Theoretical considerations at themoment do not allow us to prove or disprove the aboveconjecture.

The general features of the L behavior are expected to besimilar to those of the � evolution of the Seiberg-Wittensolution of N � 2 Yang-Mills theory [34]. (Here � is theN � 2 breaking parameter in Ref. [34].) If j�j is small,the Seiberg-Witten solution applies exhibiting Abelianconfinement. As j�j evolves to larger values, eventuallybecoming � �, non-Abelian confinement is expected toset in. Since � is a holomorphic parameter one does notexpect to have a phase transition on the way; the � evolu-tion is expected to be smooth.

The nature of the transition between Abelian and non-Abelian confinement is a good question for lattice studies.In terms of the expectation value of the Polyakov loop,hTrUi � 0 in all regimes. However, at small L, this is dueto the fact that the eigenvalues of the Polyakov line are, atzeroth order, frozen at the roots of unity as in Fig. 1(Higgsing), and fluctuations are negligibly small. At largeL, these eigenvalues are strongly coupled and randomizedover the �0; 2�� interval. Consequently, there is no gaugesymmetry breaking (non-Abelian confinement).

The set of fields in Eq. (28) is automatically invariantunder the C transformation. Therefore, planar equivalencebetween SYM and orientifold theories must hold both inthe Higgs regime and at strong coupling where C invari-ance was argued to hold too [7,15].

We would like to stress that the set of fields in Eq. (28)should be considered, for the time-being, as a fixed back-ground field configuration. Generally speaking, it does notminimize the energy functional. For instance, in the ther-mal compactification this field configuration realizes themaximum of the effective potential, rather than the mini-mum. To get the set of fields Eq. (28) as a vacuum con-figuration (i.e., that minimizing the effective potential) wehave to change the pattern of compactification (e.g., S3 �S1) or introduce a deformation of the theory through addi-tion of source terms or both. What is important for us here

is that this is doable and it is perfectly reasonable toquantize the theory in the ZN invariant background (28)which realizes maximal Higgsing.

Then, it is instructive to illustrate how planar equiva-lence between SYM and orientifold theories works in thisregime by examining the one-loop example. Given thebackground fields (4) the effective potential for the SYMtheory is

Veff �1

24�2

� XNi;j�1

�ai � aj�2�2�� �ai � aj��2 �8

15�4N

� 2XNi<j�1

�ai � aj�2�2�� �ai � aj��2�; (29)

where everything is measured in the units of L. The effec-tive potential for the orientifold-AS theory (see Appendix)

Veff �1

24�2

� XNi;j�1

�ai � aj�2�2�� �ai � aj��

2 �8

15�4N

� 2XNi<j�1

�ai � aj�2�2�� �ai � aj��

2

�: (30)

For the ZN symmetric background (4) the expressions (29)and (30) are identical up to terms suppressed by powers of1=N.

VI. POLYAKOV’S 3D CONFINEMENT

Long ago Polyakov considered three-dimensional SU(2)Georgi-Glashow model (a Yang-Mills adjoint Higgs sys-tem) in the Higgs regime [18]. In this regime SU(2) isbroken down to U(1), so that at low energies the theoryreduces to compact electrodynamics. The dual photon is ascalar field � of the phase type (i.e., it is defined on theinterval �0; 2��)

F�� �g2

3

4�"���@

��; (31)

where g23 is the three-dimensional gauge coupling with

mass dimension �g23� � �1. In perturbation theory the

dual photon � is massless. However, it acquires a massdue to instantons (technically, the latter are identical to the’t Hooft–Polyakov monopoles, after the substitution of onespatial dimension by imaginary time). In the vacuum of thetheory, one deals with a gas of instantons interactingaccording to the Coulomb law. The dual photon mass isdue to the Debye screening. In fact, it is determined by theone-instanton vertex,

m� �m5=2W g�3

3 e�S0=2; (32)

where S0 is the one-instanton action,

S0 � 4�mW

g23

; (33)

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mW is theW boson mass. As a result, the low-energy theoryis described by a three-dimensional sine-Gordon model,

L � �g2

3

32�2 �@���2 � c1m

5Wg�43 e�S0 cos�; (34)

where c1 is an undetermined prefactor. This model sup-ports a domain line6 (with� field vortices at the end points)which in 1� 2 dimensions must be interpreted as a string.Since the � field dualizes three-dimensional photon, the �field vortices in fact represent electric probe charges in theoriginal formulation, connected by the electric flux tubeswhich look like domain lines in the dual formulation.

As is well known [35], addition of one (or more) Diracfermions in the adjoint representation eliminates the aboveAbelian confinement in the Polyakov model. This is due tothe fact that the instanton monopole acquires fermion zeromodes due to the Callias index theorem [36,37]. Instantonmonopoles, instead of generating mass for the � fieldthrough the potential ei� � e�i�, produce a vertex withcompulsory fermion zero modes attached to it. For one-flavor theory, this is given by

e�S0�ei� � e�i� � � �: (35)

The three-dimensional microscopic theory with an adjointfermion possess a nonanomalous U(1) fermion numbersymmetry. This U(1) invariance is manifest in Eq. (35); itintertwines the fermion global rotation with a continuousshift symmetry for the dual photon,

! ei� ; �! �� 2�: (36)

The continuous shift symmetry (unlike a discrete one)prohibits any mass term (or potential) for the � field. Aswas shown in [35], the U(1) fermion number is sponta-neously broken, and the dual photon � is the associatedGoldstone particle. Thus, the � field remains masslessnonperturbatively, and linear confinement does not occur.This is because domain lines become infinitely thick (infi-nite range) in the absence of the dual photon mass. Onedoes need a nonvanishing dual photon mass to make thedomain line thickness of the order of m�1

� . Only then, atdistances � m�1

� linear confinement will set in.As was noted in [38], this obstacle is circumvented if we

consider a three-dimensional model obtained as a low-energy reduction of the four-dimensional model compac-tified on S1 � R3. The adjoint Weyl fermion in four di-mensions becomes an adjoint Dirac fermion in threedimensions. In this case, there is no U(1) fermion numbersymmetry. There is an anomalous U�1�A; because of theanomaly only a discrete subgroup of U�1�A is a validsymmetry. For SYM, the anomaly free subgroup is Z2N;A.As stated earlier, the discrete shift symmetry does notprohibit a mass term for the dual photon; hence, it mustbe generated [38]. Whatever nonperturbative object is

responsible for the dual photon mass, it should have nofermionic zero mode ‘‘attached.’’ Otherwise, it will gen-erate vertices as in (35) which do not result in the bosonicpotential.

The microscopic origin of the mass term can be traced tothe compactness of the adjoint Higgs field whenever weconsider a theory on S1 � R3. This is the feature which isabsent in the Polyakov model [18] and its naive fermionicextension [35]. When the adjoint Higgs field is compact asin Fig. 1, in additional to N � 1 ’t Hooft–Polyakov mono-pole instantons [whose existence is tied up to �1�S1� � 0]there is one extra, which can be referred to as the Kaluza-Klein (KK) monopole instanton.7 Each of these monopolescarries two zero modes; hence, they cannot contribute tothe bosonic potential. The bound state of the ’t Hooft–Polyakov monopole instanton with magnetic charge�i andantimonopole with charge ��i�1 has no fermion zeromode: in the sense of topological charge, it is indistin-guishable from the perturbative vacuum. Hence, such abound state can contribute to the bosonic potential. If wenormalize the magnetic and topological charges of themonopoles as

�ZF;ZF ~F

��

��i;

1

N

�; (37)

where �i stand for roots of the affine Lie algebra, then thefollowing bound states are relevant:

��i;

1

N

��

���i�1;�

1

N

�� ��i ��i�1; 0�: (38)

This pair is stable, as was shown in Ref. [38], where it isreferred to as a magnetic bion. Thus, we can borrowPolyakov’s discussion of magnetic monopoles and applydirectly to these objects. The magnetic bions will induce amass term for the dual photons via the Debye screening,the essence Polyakov’s mechanism.

In the SU�N� gauge theory on R3 � S1, which isHiggsed, SU�N� ! U�1�N�1, the bosonic part of the effec-tive low-energy Lagrangian is generated by the pairs (38),and hence the potential is proportional to e�2S0 , rather than

6Similar to the axion domain wall.

7The eigenvalues shown in Fig. 1 may be viewed as EuclideanD2 branes. N split branes support a spontaneously broken U�1�Ngauge theory, whose U(1) center of mass decouples, and theresulting theory is U�1�N�1. The N � 1 Bogomolny-Prasad-Sommerfield monopoles may be viewed as the Euclidean D0branes connecting eigenvalues �a1 ! a2�; �a2 !a3�; . . . ; �aN�1 ! aN�. Clearly, we can also have a monopolewhich connects �aN ! a1� which owes its existence to theperiodicity of the adjoint Higgs field, or equivalently, to thefact that the underlying theory is on S1 � R3. Usually it is calledthe KK monopole. The Euclidean D0 branes with the oppositeorientation, connecting �aj aj�1�, j � 1; . . . ; :N are the anti-monopoles. This viewpoint makes manifest the fact that the KKand ’t Hooft–Polyakov monopoles are all on the same footing.The magnetic and topological charges of the monopoles con-necting �aj $ aj�1� is��j;

1N� where the direction of the arrow

is correlated with the sign of the charges.

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e�S0 in the Polyakov problem. If we introduce an �N �1�-component vector �,

� � ��1; . . . ; �N�1�; (39)

representing N � 1 dual photons of the U�1�N�1 theory,and �i (i � 1; . . . ; N) are the simple and affine roots of theSU�N� Lie algebra, the bosonic part of the effectiveLagrangian can be written as

L��1; . . . ; �N�1� �g2

3

32�2 �@���2

� cm5Wg�43 e�2S0

�XNi�1

cos��i ��i�1��; (40)

where c is an undetermined coefficient and g3 is the three-dimensional coupling constant,

g23 � g2L�1: (41)

We remind the reader that �i (i � 1; . . . ; N � 1) representthe magnetic charges of �N � 1� types of the ’t Hooft–Polyakov monopoles while the affine root

� N � �XN�1

i�1

�i (42)

is the magnetic charge of the KK monopole. Note that theconfigurations that contribute to the effective Lagrangianhave magnetic charges�i � �i�1 and vertices ei��i��i�1�� ,corresponding to a product of a monopole vertex ei��i withcharge �i, and antimonopole vertex e�i�i�1� with charge��i�1. We used Eq. (28) to guarantee that the vacuumconfiguration is ZN symmetric; hence the actions (fugac-ities) e�2S0 are all equal.

Equation (40) implies that nonvanishing masses aregenerated for all �, proportional to e�S0 , albeit muchsmaller than the masses in the Polyakov model in whichthey are �e�S0=2. There are N � 1 distinct U(1)’s in thismodel, corresponding to N � 1 distinct electric charges.These are the electric charges qi of all color components ofa probe nondynamical quark Qi (i � 1; . . . ; N) in the fun-damental representation of SU�N�.8 Correspondingly, thereare N � 1 types of Abelian strings (domain lines). Theirtensions are equal to each other and proportional to e�S0 .Linear confinement develops at distances larger than eS0 .

Needless to say, the physical spectrum in the Higgs/Abelian confinement regime is much richer than that inthe non-Abelian confinement regime. If in the latter caseonly color singlets act as asymptotic states, in the Abelianconfinement regime all systems that have vanishing N � 1electric charges have finite mass and represent asymptoticstates.

VII. THERMAL COMPACTIFICATION

As was already mentioned, the requirement for planarequivalence to hold is to have a vacuum with an unbrokenC conjugation symmetry. In the non-Abelian confinementregime C parity is unbroken. In the Higgs regime thisdepends on the choice of vk’s. The set (26) automaticallyguarantees C parity. Now we abandon this choice and willfocus on the case ak � 0; � relevant to the thermal com-pactification. The vacuum field ak � 0 or ak � � for all kis not ZN symmetric while, as we already know, the non-Abelian confinement regime is ZN symmetric. Hence, weexpect a phase transition at a deconfinement temperatureT .

When the temperature is high, namely T > T (thedeconfining phase), the Polyakov loop expectation valuein orienti-AS/S does not vanish; the minimal energy statesare C preserving vacua ak � 0 or ak � � for all k. If wechoose the same minima of the effective potential in thehigh-temperature phase of the SYM theory, planar equiva-lence will hold. Namely, the SYM theory and orienti-AS/Swill have the same Green functions, condensates, spectra,etc. in the common sector.

Since all the eigenvalues of the Polyakov line coincide inthe high-temperature phase, say at ak � 0 for all k, thehigh-temperature theory is not Higgsed. The would-be Wbosons remain massless. Its dynamics is that of the non-Abelian theory, albeit three dimensional rather than fourdimensional. It is believed that the phase transition at T separates the ZN symmetric phase from that with brokenZN in pure Yang-Mills or SYM theory. The orientifoldtheories seemed puzzling from this standpoint [8]. Nowwe understand that the phase transition at T in orienti-AS/S is quite similar: it separates the high-temperature phasewith no ZN center symmetry (at best, it is Z2 for even Nwhich is spontaneously broken) from the low-temperaturephase with emergent unbroken ZN.

The physical observables in the common sector of SYMtheory and orienti-AS/S must coincide throughout thewhole phase diagram, since charge conjugation symmetryis unbroken in neither phase of the theory. This implies, inparticular, the phase transition temperatures must coincide,

TSYM � Torienti-AS=S

; N !1: (43)

The same applies to the Polyakov order parameter. At N �1,

�1

NTrU

�SYM;orienti-AS=S

�

�0; T < T 1; T > T ;

(44)

where the sign double-valuedness corresponds to two pos-sible vacuum fields, ak � 0,8 k or ak � �,8 k. (In SYMtheory these two minima correspond to the ones invariantunder the naive C, where the comparison is straightfor-ward.) In other words, the notions of high and low tem-peratures, defined relatively to the deconfinementtransition temperature, must coincide in the two theories.8qi are subject to the condition

PNi�1 qi � 0.

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Otherwise we would get an inconsistency among the com-mon sector observables. This gives us a nontrivial dynami-cal result, at finite N,

TSYM � Torienti-AS=S

�1�O

�1

N

��: (45)

This prediction of planar equivalence should be easilytestable on lattices.

VIII. CONCLUSIONS

The strong coupling dynamics of nonsupersymmetricvectorlike gauge theories, despite many efforts over theyears, remains elusive. Currently, the nonperturbativelarge-N equivalences provide deep hints into the structureof the vectorlike gauge theories. One of the most profoundexamples of the large-N equivalences is that betweenN � 1 SYM theory and its orientifold daughters. Theparent and daughter theories are clearly distinct, withdifferent fundamental symmetries and dynamics. At theLagrangian level, SYM theory is supersymmetric and hasZN center symmetry, while orienti theories are nonsuper-symmetric and have at most a Z2 center.

However, the large-N equivalence tells us that thesetheories become indistinguishable in their neutral sectorin the large-N limit. More specifically, it tells us that thephysical Hilbert spaces of these two theories in the C-evensubsectors coincide, their confinement-deconfinementtemperatures are identical, and the k-string tensions mustmatch.

Apparently, all these observables are associated withcertain symmetries which are explicit in the parent theory,but not in the daughter ones. In the large-N limit thecorrelators of the daughter theories carry benchmarks ofthe custodial symmetries of its parent. We refer to suchsymmetries, which are absent at the Lagrangian level butappear dynamically in the neutral correlators, as emergentsymmetries. In orienti-AS, the ZN center symmetry andsupersymmetry (protecting the degeneracy of the bosonicspectrum) are emergent symmetries in this sense.

At the level of the Schwinger-Dyson equations (or loopequations), the equivalence is a consequence of the quan-tum fluctuation suppression in the large-N limit. The (sup-pressed) fluctuations are well aware of distinctions in theparent/daughter theories, which have no place in the lead-ing large-N dynamics. This is true for orienti-AS/S, as wellas for orbifold SU�N� � SU�N� Yang-Mills theory withbifundamental quark (assuming unbroken Z2 interchangesymmetry). This is also valid for one-flavor QCD withorthogonal and symplectic gauge groups with AS/S repre-sentation fermions [39]. In our opinion, attempts to under-stand such universal behavior (which is a naturalconsequence of planar equivalence) may provide insightsinto the strongly coupled regimes of QCD, and otherstrongly coupled systems. This question is left for futurework.

ACKNOWLEDGMENTS

We are grateful to Larry Yaffe for discussions. A. A. issupported by PPARC. The work of M. S. is supported inpart by DOE Grant No. DE-FG02-94ER408. The work ofM. U. is supported by the U.S. Department of Energy GrantNo. DE-AC02-76SF00515.

APPENDIX: ONE-LOOP POTENTIALS

The one-loop effective potentials for Polyakov lines inthe case of SYM theory and orienti-AS are

VSYMeff �U� �

2

�2L4

X1n�1

1

n4 ���1� an�jTrUnj2�VAeff�U�

�2

�2L4

X1n�1

1

n4

��jTrUnj2

� an

��TrUn�2 � TrU2n

2� c:c:

��; (A1)

where the first terms are due to gauge boson (and ghosts),and the second term is due to fermions endowed with spinstructure

an ����1�n for S�

1 for S�:(A2)

In the large N limit, the single-trace term can be neglectedsince it is suppressed by O�1=N� relative to the double-trace terms. In terms of the simultaneous eigenstates of theC and center symmetry,

Tr �n � TrUn Tr�U �n; CTr� � Tr�;

(A3)

the potential takes the form

VASeff ���;��� �

2

�2L4

X1n�1

1

4n4 ���1� an�jTr�n�j

2

� ��1� an�jTr�n�j

2�: (A4)

The form of the one-loop potential in QCD(AS) is notsurprising. The first class of terms jTr�n

�j2 is the images

of the jTrUj2. The second category is the square of thetwisted (non-neutral) C-odd operators, which are not theimage of any operator in the orienti partner. They do,however, get induced via a one-loop Coleman-Weinberganalysis. Whether or not the twisted operator Tr�� ac-quires a vacuum expectation value and induces the sponta-neous breaking of charge conjugation is correlated with thespin structure S of fermions along the S1 circle. Inthermal case (S�), despite the presence of such operators,C is unbroken at high temperature. In spatial compactifi-cation (S�), C is broken at small S1. Regardless of spinstructure, C is preserved at large radius (either temporal orspatial) continuously connected to R4.

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