Large-N volume independence in conformal and confining gauge theories
PPreprint typeset in JHEP style - HYPER VERSION
SLAC-PUB-14160
Large- N volume independence in conformal andconfining gauge theories Mithat ¨Unsal ∗ and Laurence G. Yaffe † SLAC and Physics Department, Stanford University, Stanford, CA 94305 Department of Physics, University of Washington, Seattle, WA 98195–1560
Abstract:
Consequences of large N volume independence are examined in conformal and confining gaugetheories. In the large N limit, gauge theories compactified on R d − k × ( S ) k are independent of the S radii,provided the theory has unbroken center symmetry. In particular, this implies that a large N gauge theorywhich, on R d , flows to an IR fixed point, retains the infinite correlation length and other scale invariantproperties of the decompactified theory even when compactified on R d − k × ( S ) k . In other words, finitevolume effects are 1 /N suppressed. In lattice formulations of vector-like theories, this implies that numericalstudies to determine the boundary between confined and conformal phases may be performed on one-sitelattice models. In N = 4 supersymmetric Yang-Mills theory, the center symmetry realization is a matter ofchoice: the theory on R − k × ( S ) k has a moduli space which contains points with all possible realizations ofcenter symmetry. Large N QCD with massive adjoint fermions and one or two compactified dimensions has arich phase structure with an infinite number of phase transitions coalescing in the zero radius limit.
Keywords: /N expansion, lattice QCD, nonperturbative effects. ∗ [email protected] † [email protected] a r X i v : . [ h e p - t h ] J un ontents
1. Introduction 12. Compactification and center symmetry 33. Conformal theories and conformal windows 54. N = 4 SYM 75. Massive QCD(adj) 96. Multiple compactified dimensions 157. Prospects 20
1. Introduction
Wide classes of large N gauge theories, when studied on toroidal compactifications of R d , have proper-ties which are independent of the compactification radii. For brevity, we will refer to independence oncompactification radii as volume independence . Examples (and counterexamples) of large N volumeindependence have been discussed since the 1980’s [1, 2, 3, 4, 5], but there has been a recent resurgenceof interest in the subject.In SU ( N ) gauge theories on R d − k × ( S ) k , with only adjoint representation matter fields, large N volume independence holds provided two symmetry realization conditions are satisfied as N → ∞ [6]: • Translation symmetry is not spontaneously broken. (1a) • The ( Z N ) k center symmetry is not spontaneously broken. (1b) Our discussion applies to compactifications on R d − k × ( S ) k where k , the number of compactified dimensions, mayrange from 1 to d . If k < d , so that some dimensions remain uncompactified, then it is an abuse of language to referto independence on the compactification radii as volume independence. We hope that our use of this name, whichoriginated in discussions of compactifications of R d to ( S ) d with cubic symmetry, does not cause confusion. In latticegauge theories, large N equivalence between the decompactified theory and a single site model is sometimes referred toas “large N reduction” or “Eguchi-Kawai (EK) reduction”. Center symmetry transformations are gauge transformations which are periodic only up to an element of the center ofthe gauge group. For SU ( N ) theories on R d − k × ( S ) k , the group of such transformations, modulo gauge transformationscontinuously connected to the identity, is ( Z N ) k . Center transformations associated with a particular toroidal cyclemultiply Wilson loops by a phase factor z n where z ∈ Z N and n is the winding number of the loop around the cycle.Hence, topologically non-trivial Wilson loops serve as order parameters for center symmetry. Z N center transformationsare symmetries of SU ( N ) gauge theories provided all matter fields are in representations (such as the adjoint) withvanishing N -ality. To simplify the presentation, we limit our discussion to this class of theories. However, it should benoted that there are additional large N equivalences which relate SO ( N ), Sp ( N ) and SU ( N ) gauge theories, and whichrelate theories with matter fields in rank-2 symmetric, antisymmetric and bifundamental representations to theorieswith adjoint representation matter [7, 8, 9, 10]. – 1 –olume independence applies to the leading large N behavior of expectation values and connectedcorrelators of topologically trivial Wilson loops and similar single-trace observables. See Ref. [6] formore detail.In SU ( N ) Yang-Mills theory, large N volume independence holds as long as all compactificationradii are larger than a critical radius L c ∼ Λ − [11]. Volume independence fails below this criticalradius due to a center-symmetry breaking phase transition. In the case of a single compactified di-mension, this is the usual confinement/deconfinement thermal phase transition. Two approaches formodifying a lattice gauge theory in order to suppress this center-symmetry breaking phase transition,and restore the validity of volume independence down to arbitrarily small radius (or down to a singlesite in a lattice regulated theory), are known. One may add double trace deformations to the actioninvolving absolute squares of Wilson loops wrapping the compactified directions [17]. Such deforma-tions can suppress spontaneous breaking of center symmetry while leaving the large N dynamics inthe center-symmetric sector of the theory completely unaffected. Alternatively, one can add one ormore massless adjoint representation fermions, with periodic (not antiperiodic) boundary conditions[6]. The fermions modify the effective potential for topologically non-trivial Wilson loops in a mannerwhich prevents the breaking of center symmetry even for arbitrarily small compactification radii.In this work we discuss the physical basis for large N volume independence and examine its impli-cations in both confining and conformal gauge theories. We highlight the presence of nonuniformitieswhen the gauge group rank N → ∞ and compactification size L →
0. For simplicity of presentation,much of our discussion will focus on the case of a single compactified dimension. If center symmetryis unbroken on R d − × S , we show that the long distance physics for finite values of N is sensitive to L only via the combination N L . Specific consequences we discuss include the following:1. For asymptotically free theories with a strong scale Λ, the compactified theory differs negligiblyfrom the decompactified theory provided
N L Λ (cid:29)
1. Conversely, a dimensionally reduced longdistance effective theory characterizes the long distance dynamics only when
N L Λ (cid:28) R d , have scale-invariant long distance dynamics, compactification on R d − × S leads to a correlation length of order N L ; a factor of N times longer than would benaively expected. Maximally supersymmetric Yang-Mills theory ( N = 4 SYM) is a special casewhere the correlation length of the compactified theory remains infinite and all realizations ofcenter symmetry coexist.3. For QCD with n f massless adjoint fermions, the lower boundary of the “conformal window” ( i.e. ,the minimal number of flavors n ∗ f for which the theory flows to a non-trivial IR fixed point) may,in the large N limit, be determined by studying the compactified theory with arbitrarily smallradius.4. For QCD with massive adjoint fermions, the symmetry realization and phase structure of thetheory, compactified on R × S , sensitively depends on the value of N Lm .In a final section, we discuss the situation with multiple compactified dimensions. There are severalnew issues in this case, but most of the R d − × S analysis generalizes in a straightforward fashion. Earlier schemes known as quenched Eguchi-Kawai reduction [3] and twisted Eguchi-Kawai [4, 5] have recently beenshown to fail due to nonperturbative effects [12, 13, 14, 15]. However, see Ref. [16] for a recent proposed fix for twistedreduction. – 2 – ( )/( ) π L
4 / N → ∞ π NL π L
4 / π L
4 / π NL π L
2 / π L
2 / π L
2 / (a) Center−broken (c) Center−symmetric (b) Center−symmetric Figure 1:
Dependence of the Kaluza-Klein spectrum on the realization of center symmetry. (a) Maximallybroken center symmetry with holonomy Ω = 1 gives the usual 2 π/L level spacing. (b) Unbroken centersymmetry produces a finer 2 π/ ( NL ) level spacing. (c) Sending N → ∞ with unbroken center symmetry leadsto continuous spectra.
2. Compactification and center symmetry
Consider an asymptotically free SU ( N ) gauge theory on R × S . Let Λ denote the strong scale of thetheory, and let Ω denote the holonomy of the gauge field around the compactified direction. [In otherwords, Ω( x ) is the path-ordered exponential of the line integral of the gauge field around the S atspatial position x .] We assume that the theory has a Z N center symmetry, so our discussion appliesto QCD with adjoint representation fermions [denoted QCD(adj)], provided the number of flavors n f is below the asymptotic freedom limit, n f < n AFf ≡ .
5. A center symmetry transformation multipliesthe holonomy by a phase factor, Ω → z Ω with z ∈ Z N . Hence, Wilson line expectations (cid:104) tr Ω k (cid:105) , forany non-zero k mod N , are order parameters for the realization of center symmetry.Compactification produces Kaluza-Klein (KK) towers — discrete frequency spectra of field modes.The form of the KK spectra is critically dependent on the realization of center symmetry, as illustratedin Figure 1. Fig. 1a shows the “standard” form of Kaluza-Klein towers, in which allowed momentaare spaced at integer (or half-integer) multiples of 2 π/L . For adjoint representation fields, each levelhas an O ( N ) degeneracy. This is the situation when the holonomy Ω = 1 and the center symmetryis completely broken. For physics sensitive to energies small compared to the inverse compactificationradius, E (cid:28) L , (2.1) The results of this discussion also apply to QCD-like theories with fermions in symmetric or antisymmetric rank-two tensor representations, or ordinary QCD with n f fundamental representation fermions provided n f is held fixed as N → ∞ . As noted in Ref. [6] and discussed more fully in Ref. [18], an “emergent” Z N center symmetry appears in thelarge N limit of theories with symmetric or antisymmetric tensor representation matter fields. For these theories, theaddition of explicit double-trace center-symmetry stabilizing terms is needed to prevent center symmetry breaking atsmall L [17]. – 3 –r length scales large compared to L , the non-zero frequency components of all fields may be integratedout, leading to a dimensionally reduced 3 d effective theory describing long distance dynamics [19, 20,21]. In contrast, when the center symmetry is unbroken the eigenvalues of Ω are (on average) evenlydistributed around the unit circle. This follows from the vanishing of traces of Ω k for all non-zero k mod N . Such a non-degenerate eigenvalue distribution of Ω produces a finer KK spectrum withspacing of 2 π/ ( N L ) and O ( N ) degeneracies, as illustrated in Fig. 1b. To see this, recall that a non-trivial holonomy shifts the phase acquired by an excitation propagating around the S , or equivalentlyshifts the frequency moding of field components. When the eigenvalues of Ω are non-degenerateand uniformly distributed, every field breaks up into N pieces with different offsets in the frequencyquantization. The range of energies for which a dimensionally reduced effective description is valid isnow E (cid:28) N L . (2.2)The resulting long distance effective theory, relevant only for distances large compared to
N L , cor-responds perturbatively to a SU ( N ) → U (1) N Higgsing of the theory. The holonomy Ω acts as aneffective 3 d adjoint representation Higgs field which gives masses in the [2 π/N L, π/L ) range to all off-diagonal field components. In addition to perturbative fluctuations, there are also non-perturbativetopological defects, both self-dual monopoles and non-self-dual magnetic bions (or monopole-anti-monopole bound states) of various charges [22]. A semiclassical analysis (generalizing Polyakov’sclassic treatment [23] to gauge theory on R × S ) shows that these topological defects, even whenarbitrarily dilute, can generate a mass gap and area law behavior for spatial Wilson loops, as discussedin detail for QCD(adj) in Ref. [22].A key point is that integrating out the non-zero frequency modes perturbatively, and analyzingthe monopole/bion dynamics using semiclassical methods, is only valid when the theory is weaklycoupled on the scale of N L . In other words, the dimensionally reduced description of the long distancedynamics, with semi-classical Abelian confinement, is only valid when
N L Λ (cid:28) N → ∞ , with unbroken center symmetry, the frequency spacing in Kaluza-Klein towers ap-proaches zero and the spectrum approaches the continuous frequency spectrum of the decompactifiedtheory, as illustrated in Fig. 1c. The domain of validity (2.2) of the 3 d long distance descriptionshrinks to zero energy (and diverging distances). Observables probing any fixed energy scale becomeunable to resolve the vanishing discreteness in the frequency spectrum. Compactification effects are1 /N suppressed and the leading large N behavior of expectations, or connected correlators, of singletrace, topologically trivial observables is volume independent. For dynamics on the scale Λ, the effectof compactification is negligible when N L Λ (cid:29)
1. In this regime, there is no weakly coupled descriptionof the long distance physics.Recognition of the connection between unbroken center symmetry and a 1 /N suppressed spacingin the KK spectrum does not constitute a proof of large N volume independence. For that, one mustcompare the large N loop equations in lattice regularized theories [8], or the N = ∞ classical dynamicsgenerated by appropriate coherent states [24]. But consideration of the KK spectrum does provide asimple physical understanding of the origin of large N volume independence, and clarifies the relevantscales which control the approach to the N = ∞ limit.– 4 –o reiterate, for theories on R × S , when the center symmetry is not spontaneously brokenthe physically relevant length scale appearing in finite volume effects is not L , but rather N L . Forconfining theories with a strong scale Λ, there are two distinct characteristic regimes:
N L Λ (cid:28) , semi-classical, Abelian confinement = ⇒ volume dependence; (2.3) N L Λ (cid:29) , non-Abelian confinement = ⇒ volume independence. (2.4)
3. Conformal theories and conformal windows
Interest in possible extensions to the standard model involving strongly coupled, nearly conformalsectors [25, 26, 27, 28, 29, 30, 31] has stimulated substantial efforts to determine “conformal windows”in QCD-like gauge theories with varying fermion content [32, 33, 34, 35, 36, 37, 38, 39, 40, 41].Numerical simulations must, of course, work with UV and IR regulated theories, and finite volumetoroidal compactifications are virtually always used as the IR regular of choice. As with all numericalsimulations of lattice gauge theories, efforts to find the boundary between confining and conformalbehavior must adequately control multiple sources of systematic error: finite volume effects, non-zerolattice spacing artifacts, and extrapolations to the chiral limit.For theories with infinite correlation length (when defined on R d ), compactification of a spatialdimension introduces a new length scale, the compactification size L . This modifies the behavior offluctuations with wavelengths comparable or larger than L and typically produces a finite correlationlength of order L . But for theories satisfying large N volume independence, for the reasons discussedin the previous section, the characteristic size of a compactification-induced correlation length is notthe physical size L , but instead equals L times a positive power of N . This means that finite volumeeffects are 1 /N suppressed as long as the center symmetry is unbroken. Even for simulations at modestvalues of N , this additional suppression of finite volume effects decreases the lattice size required toachieve acceptably small systematic errors. For larger values of N , it implies that very small latticeswill be sufficient to extract infinite volume observables. The basic point we wish to stress is that large N volume independence is not restricted to confining phases; it is equally applicable to conformalphases provided the symmetry realization conditions (1) are satisfied.As an example, consider QCD(adj) with massless fermions, and one dimension periodically com-pactified, as a function of the number of fermions n f and compactification size L . Figure 2 illustratesthe situation. In the decompactified limit, L = ∞ , one expects a confining phase with sponta-neously broken chiral symmetry for sufficiently small n f , and a conformal phase ( i.e. , a phase in whichthe theory flows to a non-trivial IR fixed point) with unbroken chiral symmetry in some window n ∗ f ≤ n f < n AFf , where n AFf is the asymptotic freedom limit. Consequently, as L decreases frominfinity, there should be both chirally symmetric and chirally asymmetric phases extending into the( L, n f )-plane phase diagram. This figures treats the number of fermion flavors n f as a continuous variable. In the Euclidean theory, one can definenon-integer n f by taking the fermion determinant, which is real and positive in QCD(adj), to a fractional power. More precisely, there is a continuous non-Abelian chiral symmetry when n f >
1, plus a flavor-independent discretechiral symmetry. Our discussion focuses on the continuous chiral symmetry. Unlike the non-Abelian chiral symmetry,in the compactified theory the discrete chiral symmetry can be spontaneously broken even at weak coupling [42]. Anunconventional order parameter for the discrete chiral symmetry is a topological disorder (or monopole) operator. – 5 – /( Λ) N n f AF n f R χ SB χ SB N < ∞ (a) L confined Confined CFTconfined n f * N n f AF n f * n f R χ SB χ SB (b) N = ∞ L confined Confined CFTconfined
Figure 2:
Contrasting finite N and infinite N phase diagrams of massless QCD(adj) on R × S , as afunction of the compactification size L and the number of fermion flavors n f . In the decompactified limit,one expects a confining phase with spontaneously broken chiral symmetry for sufficiently small n f , and aconformal phase with unbroken chiral symmetry in the window n ∗ f ≤ n f < n AFf (with n AFf the asymptoticfreedom limit). For finite N (left), as one decreases L the chiral transition line must bend and approach anintercept at an “unconventional” scale of 1 / ( N Λ). To the right of this line is a phase with unbroken continuouschiral symmetry and finite, compactification-induced correlation length, smoothly connecting the analyticallytractable NL Λ (cid:28) L = ∞ boundary. At N = ∞ (right), the theoryexhibits volume independence in both the chirally broken and chirally symmetric phases. The phase transitionline extends straight down from L = ∞ and n f = n ∗ f . This implies that numerical studies on very small latticescan be used to determine the conformal window boundary n ∗ f . For finite N and sufficiently small L [small compared to 1 / ( N Λ)], one can reliably analyze thetheory using perturbative and semiclassical methods [22], as mentioned earlier. One finds unbrokencontinuous chiral symmetry, broken discrete chiral symmetry, a non-zero mass gap, and area-lawbehavior for large topologically trivial spatial Wilson loops.The simplest, most plausible, scenario is that the chirally symmetric phase at small L (and any n f ) smoothly connects to the chirally symmetric conformal phase at L = ∞ and n f ≥ n ∗ f . Thechiral transition line separating these two phases must bend as L decreases, as shown, and approachan intercept at an “unconventional” scale of 1 / ( N Λ) [43]. This is the scale below which the longdistance semiclassical analysis is valid. To the left of this line, one has a typical confining phase withspontaneously broken continuous chiral symmetry. The mass gap (inverse correlation length) vanishesdue to the presence of Goldstone bosons. The spatial string tension (or area law coefficient for largespatial Wilson loops) will have finite, non-vanishing limits as L → ∞ . To the right of this line one hasa phase with unbroken continuous chiral symmetry and finite, compactification-induced correlationlength which diverges as L → ∞ . Similarly, this phase will have a non-zero spatial string tension forfinite L , which vanishes as L → ∞ .Massless adjoint fermions, with periodic boundary conditions on R × S , prevent the spontaneousbreaking of center symmetry [44, 6]. Consequently, at N = ∞ (illustrated in Fig. 2b), both the chirally– 6 –roken and chirally symmetric phases will exhibit volume independence. The chirally asymmetricphase will have a finite correlation length and non-zero spatial string tension, both independent of L . The chirally symmetric phase will have an infinite correlation length, irrespective of L . In thislong-distance conformal phase, large Wilson loop expectation values will show Coulombic behavior.For rectangular R × T loops,lim T →∞ − T ln (cid:10) W ( R × T ) (cid:11) ∼ σ R , confined phase; − g ∗ /R , IR conformal phase , (3.1)with σ and g ∗ having large N limits which are independent of the compactification size L . Thephase transition line must extend straight down from L = ∞ and n f = n ∗ f . The value of n ∗ f inthe decompactified theory can be extracted from numerical studies on lattices with arbitrarily small L . In effect, large N volume independence allows one to trade extrapolations in lattice volume forextrapolations in the number of colors, N . N = 4 SYM When a Z N center-symmetric theory is compactified on R d − × S , the dynamics of the theory maygenerate an effective potential for the Wilson line holonomy which causes its eigenvalues to attract,as in pure Yang-Mills theory, leading to spontaneous breaking of center symmetry. Alternatively,repulsive interactions between eigenvalues may be generated, as in massless QCD(adj) (with periodicfermion boundary conditions), or engineered (by the addition of explicit stabilizing terms), therebypreventing center symmetry breaking. But there is a third possibility, which is realized by maximallysupersymmetric Yang-Mills ( N = 4 SYM) theory: a strictly vanishing effective potential for the Wilsonline holonomy.Consider N = 4 SYM theory with gauge group SU ( N ) on R . The theory possess an SU (4) R R -symmetry group, and contains adjoint representation scalars Φ [ IJ ] and fermions λ I in the and of SU (4) R , respectively. The renormalization group beta function vanishes identically, so the gaugecoupling g is a scale independent physical parameter of the theory. There is a continuous modulispace of vacua, M R = R N / S N , (4.1)corresponding to mutually commuting scalar field expectation values with completely arbitrary valuesfor their eigenvalues (modulo Weyl group permutations). The physical theory at the origin of modulispace is an interacting non-Abelian CFT. Generic points in moduli space correspond to Higgsing ofthe SU ( N ) gauge group down to a maximal Abelian U (1) N − subgroup.When compactified on R × S with periodic spin connection ( i.e. , periodic boundary conditionsfor fermions), the theory has a Z N center symmetry and remains supersymmetric. The behavior ofthe compactified theory was previously discussed by Seiberg [45]. He argued that the theory at theorigin of the moduli space (4.1) flows to three-dimensional N = 8 superconformal SYM theory at lowenergies, E (cid:28) g ≡ g /L , with an emergent SO (8) R symmetry. The compactified theory is manifestlynot volume independent: results depend on the compactification scale L and long distance propertieson R × S differ from the uncompactified theory on R .– 7 –owever, the compactified theory on R × S has a larger moduli space than does the uncom-pactified theory. Due to the maximal N = 4 supersymmetry, no superpotential for the Wilson lineholonomy, either perturbative or non-perturbative, is generated. Consequently, compactification addsa new branch to the moduli space, with the holonomy Ω behaving as an adjoint Higgs field. Themoduli space of N = 4 SYM on R × S is M R × S = [ R N × ( (cid:101) S ) N ] / S N , (4.2)where (cid:101) S is the dual circle on which eigenvalues of the Wilson line holonomy Ω reside. In thiscompactified theory, the realization of the Z N center symmetry is entirely a matter of choice. Genericpoints in the moduli space (4.2) will involve a set of eigenvalues { e iθ a } , a = 1 , · · · , N , of Ω which arenot invariant under any Z N transformation (other than the identity), and hence completely break thecenter symmetry. But since the choice of eigenvalues for Ω is arbitrary, the moduli space also includespoints where the set of eigenvalues is completely Z N symmetric, as well as points where the set ofeigenvalues is invariant only under some subgroup of Z N (if N is composite).Prior discussion [45] of compactification in N = 4 SYM has focused on the case where Ω = 1. Withall eigenvalues of Ω clustered at a single point, volume independence fails, as noted above. Here, weinstead wish to examine the situation at center-symmetric points in moduli space where eigenvaluesare evenly distributed around the unit circle and Ω = diag (cid:16) , e πi/N , e πi/N , · · · , e πi ( N − /N (cid:17) . (4.3)For simplicity, focus on the case where scalar field expectations are small compared to 1 /L . For finite N , at such center symmetric points in the moduli space, the dynamics Abelianizes at large distancesand reduces to an effective U (1) N − gauge theory. This will be a valid description for distances largecompared to the inverse of the (lightest) W -boson mass, m W = 2 πLN . (4.4)Off-diagonal components of all fields acquire masses of order m W or greater, due to their coupling tothe component of the gauge field in the compactified direction. The surviving components are thosealigned along the Cartan subalgebra of SU ( N ). The effective theory describing physics at energies wellbelow m W is just a free U (1) N − Abelian gauge theory with neutral massless fermions and scalars, L = 1 g N − (cid:88) a =1 (cid:34) ( F aij ) + ( ∂ i A a ) +
12 4 (cid:88)
I 5. Massive QCD(adj) In SU ( N ) Yang-Mills theory, compactified on R × S with compactification size L (cid:46) / Λ, the one-loop contribution to the Wilson line effective potential produces eigenvalue attraction and spontaneousbreaking of center symmetry. This symmetry realization is stable if very massive adjoint represen-tation fermions are added to the theory, since their effects are negligible when the fermion mass m is large compared to 1 /L . But when massless adjoint representation fermions are present, with pe-riodic boundary conditions, their contribution to the Wilson line effective potential favors eigenvalue In the strong coupling domain where λ ≡ g N (cid:29) 1, an AdS/CFT analysis [46] shows that the criterion for thehigh-energy non-Abelian CFT regime becomes E (cid:29) λ/ ( LN ). – 9 –epulsion and overcomes the attractive pure gauge contribution, leading to unbroken center symmetry.Consequently, it is inevitable that QCD(adj) on R × S , with periodic spin connection, exhibits oneor more phase transitions as the fermion mass m is varied for fixed small compactification radius L .In asymptotically free QCD(adj), one would naively expect the behavior of the massive theory toclosely resemble the massless limit when the mass m is small compared to 1 /L . As we next discuss,this expectation is too simplistic. Quite a rich phase diagram emerges as the fermion mass and thecompactification radius are varied. When N L Λ (cid:28) 1, non-zero frequency Kaluza-Klein modes are weakly coupled and may be in-tegrated out perturbatively. With n f adjoint fermions having a common mass m , the resultingone-loop potential may be conveniently expressed in the form V [Ω] = 2 π L ∞ (cid:88) n =1 (cid:2) − n f ( nLm ) K ( nLm ) (cid:3) | tr Ω n | n . (5.1)Here K ( z ) is the modified Bessel function of the second kind, with asymptotic behavior K ( z ) ∼ (cid:40) z − O ( z ) , z (cid:28) (cid:112) π z e − z , z (cid:29) . (5.2)As the mass m → ∞ , the fermions decouple and the effective potential (5.1) reduces to the puregauge result, V YM [Ω] = − π L (cid:80) ∞ n =1 | tr Ω n | /n , up to exponentially small O ( e − Lm ) corrections. Inthe opposite limit of massless fermions, the coefficient of the | tr Ω n | in the series (5.1) reduces to[ − n f ] /n , previously found in [6], and V [Ω] → − ( n f − V YM [Ω].The n ’th term in the series (5.1) is evidently an effective mass term for the winding number n Wilson line tr(Ω n ). For arbitrary SU ( N ) matrices, tr(Ω n ) is reducible to lower order traces when | n | > N/ 2, but traces with windings up to | n | = (cid:98) N/ (cid:99) may be regarded as independent. If theeffective masses m n ≡ π n (cid:2) − n f z n K ( z n ) (cid:3) , z n ≡ nLm , (5.3)are positive for all n ≤ (cid:98) N/ (cid:99) , then the minimum of the one-loop potential lies at the center symmetricpoint where trΩ n = 0 for all non-zero n mod N . If some of these effective masses are negative, then thecorresponding Wilson lines will develop non-zero expectation values, implying spontaneous breakingof center symmetry. Since z K ( z ) decreases monotonically for z > 0, if the first mass m is negativethen so are all higher masses. In this case, the Z N center symmetry is completely broken. If N is large(and composite), there can be a plethora of intermediate phases where the center symmetry partiallybreaks to different discrete subgroups. The effective mass squared (5.3) vanishes when z n exceeds athreshold z ∗ whose value depends on n f , z ∗ ( n f ) = { . , . , . , . } , (5.4) Related analysis of QCD(adj) on R × S has been performed in Refs. [47, 48, 49]. Numerical simulations on a finitelattice mimicking this geometry are reported in Ref. [50]. When Ω ∼ L (cid:28) 1. But, as discussed in section 2, when the centersymmetry is unbroken the KK-mode frequency spacing is smaller by a factor of 1 /N , so the weak coupling conditionapplies to the length scale NL instead of L . – 10 –or n f = 2 , , , Consequently, the Wilson line with winding number n ≤ (cid:98) N/ (cid:99) becomes unstablewhen the compactification size L exceeds L n ≡ z ∗ nm , (5.5)Since L (cid:98) N/ (cid:99) < · · · < L < L , if L lies in the interval [ L k , L k − ] then m k < m k − > 0, implyingthat Wilson lines wrapping fewer than k times are stable, while loops with k up to (cid:98) N/ (cid:99) windingsare unstable. When L < L (cid:98) N/ (cid:99) , all independent Wilson lines are stabilized, and center symmetry isunbroken.When n f > 1, for any finite mass m , as one increases L from zero the initial phase is fully Z N symmetric. For sufficiently small L a perturbative analysis is reliable for any value of m . Onemay regard this regime as one in which the SU ( N ) gauge group is Higgsed down to the maximalAbelian subgroup U (1) N − . When L exceeds L (cid:98) N/ (cid:99) , the highest (independent) winding modebecomes unstable and develops a non-zero expectation value — provided the weak coupling condition N L Λ (cid:28) L ∼ L (cid:98) N/ (cid:99) = O (1 /N m ), this condition impliesthat the fermion must be heavy, m (cid:29) Λ. As L continues to increase (with m (cid:29) Λ), successively lowerwinding modes become unstable and will develop expectation values each time L passes a threshold L k . All modes are locally unstable when L exceeds L , so the center symmetry will be completelybroken when L > L (and N L Λ (cid:28) L , large- m corner of the ( L, m )-plane phase diagram, when n f > 1. The behavior in other regions of the phase diagram, especially the small mass regime, dependson the number of fermion flavors. We will discuss separately three cases: multiple flavors below theconformal window, 2 ≤ n f < n ∗ f ; a single flavor, n f = 1; and multiple flavors within the conformalwindow, n ∗ f ≤ n f < n AFf . Multiple flavors: 2 ≤ n f < n ∗ f With fewer than n ∗ f flavors, the decompactified theory on R , for any value of fermion mass, has atypical confining phase with a strong scale of Λ. Consequently, for L large compared to 1 / Λ the centersymmetry will be unbroken. This absence of any center symmetry breaking for L (cid:29) / Λ should holdfor all values of m .Heavy fermions, with m (cid:29) Λ, make a negligible contribution to dynamics on the scale of Λ, andso the difference between imposing periodic and antiperiodic boundary conditions on such fermionsis also negligible. Consequently, for large m there will be a conventional confining/deconfining phasetransition at L = L c = O (1 / Λ), where L c = 1 /T c is the inverse transition temperature in pure Yang-Mills theory. Across this transition, the center symmetry changes from completely broken ( L < L c ) The single flavor case n f = 1 is discussed separately below. In QCD(adj), asymptotic freedom is lost at n AFf = 5 . At large m and small L , confinement is the result of monopoles carrying a net magnetic charge and topologicalcharge 1 /N . At m = 0 and small L , confinement results from magnetic “bions,” magnetically charged, topologicallyneutral combinations of monopoles and antimonopoles of differing types, which become bound due to fermion zero modeexchange [22]. Turning on a non-zero fermion mass lifts the fermion zero modes and allows the bions to unbind, smoothlyconverting the small m bion-induced confinement into monopole-induced confinement at large m . The slopes of the phase transition lines (5.5) for small L agree with Ref. [51], which studied the same class of theorieson S × S . Our values (5.5) agree with the numerical values on the mL axis of Fig. 4b of Ref. [51]. – 11 – Z Z m L c m* Center broken Center symmetric ∞ L (a) SU(2) ∞ Z m L c m* Z Z Z Z Center broken ∞ L ∞ Center symmetric (b) SU(6) Figure 3: Center symmetry realization of QCD(adj) on R × S with 2 ≤ n f < n ∗ f and periodic spinconnection, as a function of compactification size L and inverse fermion mass 1 /m . Illustrated are the cases N = 2 (left) and N = 6 (right). The left hand axis where m = ∞ corresponds to pure Yang-Mills theory,where a “confining/deconfining” phase transition occurs at L c ∼ Λ − . Phases are labeled according to the unbroken subgroup of Z N center symmetry; the “ Z ” region is the phase with totally broken center symmetry.The slopes of the transition line(s) approaching the origin are perturbatively computable. Within the centersymmetric phase, the limit of vanishing size L (bottom axis) is expected to be continuously connected to thelarge L , large m domain (upper left corner) which corresponds to decompactified Yang-Mills theory. The doton the right-hand boundary indicates the chiral symmetry transition point of the massless theory, expected tooccur at an O (1 / ( N Λ)) value of L . to fully restored ( L > L c ). The completely broken phase just below L c presumably connects directlyto the completely broken phase at L > L identified in the small L analysis.As noted above, the conventional understanding of confinement in QCD-like theories (on R )implies that a center-symmetric phase will be present at vanishingly small m and sufficiently large L . The perturbative analysis valid when Λ (cid:28) / ( N L ) shows that a center-symmetric phase is alsopresent when the fermion mass is sufficiently small, m (cid:46) / ( N L ). The most plausible scenario is thatthese center symmetric regions are part of a single connected center symmetric phase which exists forall L when the fermions are sufficiently light, m < m ∗ = O (Λ). The resulting phase diagram, as afunction L and 1 /m , is illustrated in Fig. 3 for two representative values of N . Single flavor: n f = 1 Single flavor QCD(adj), in the massless limit, is N = 1 supersymmetric Yang-Mill theory. Turning on asmall but non-vanishing mass corresponds to a soft breaking of supersymmetry. Exact supersymmetryimplies that the one loop potential vanishes at m = 0, as one may easily confirm after substituting n f = 1 into the result (5.1) and sending m → 0. For small but non-zero fermion mass, the sub-leading The chiral limit of QCD(adj) with periodic boundary conditions on S × S was studied in Ref. [52], where itwas argued that there is no center symmetry changing phase transition, consistent with expectations in the partiallydecompactified R × S limit. – 12 – c Z Z m n = 1 f ∞ YM SYM Center broken Center symmetric (a) SU(2), ∞ L L c m Z Center broken Z Center symmetric Z Center symmetric n f ∞ (b) SU(2), ∞ L f n * ≤ < n f AF Figure 4: Center symmetry realization of QCD(adj) on R × S with periodic spin connection, as a functionof compactification size L and inverse fermion mass 1 /m , for the case of a single flavor (left), or multiple flavorswithin the conformal window (right), n ∗ f ≤ n f < n AFf . For n f = 1, the right-hand m = 0 axis corresponds to N = 1 SYM, which does not break center symmetry for any L . With non-zero fermion mass m , a single adjointfermion is insufficient to stabilize the center symmetry at arbitrarily small L . For n ∗ f ≤ n f < n AFf (right), thephase boundaries separating center symmetry broken and unbroken phases are expected to extend to L = ∞ as m → 0, with no continuous connection between the center-symmetric phases at large and small L . term in the small z asymptotic behavior (5.2) contributes and the n f = 1 effective potential becomes V [Ω] = − m π L ∞ (cid:88) n =1 n | tr Ω n | + O ( m ) . (5.6)Within the domain of validity of the perturbative analysis, this shows that Wilson lines with all windingnumbers are unstable when the fermion mass is non-zero, despite the periodic boundary condition forfermions. Consequently, the n f = 1 theory at any non-zero mass m and sufficiently small L will havecompletely broken center symmetry.When m = 0 there is no perturbative contribution (at any order) to the Wilson line effectivepotential, but there is a non-perturbatively induced effective potential which ensures unbroken centersymmetry in the supersymmetric theory on R × S [44] . For small but non-zero m , and small L ,there will be competition between the one-loop O ( m ) soft supersymmetry breaking potential and thenon-perturbatively induced superpotential, leading to non-uniformity in the m → L → L = m = 0 corner of the phase diagram, as illustrated for an SU (2) theory on the left side of Fig. 4.Unlike the previous multi-flavor case, for larger values of N there is no reason to expect the phasediagram to contain any region with partially broken center symmetry. Multiple flavors: n ∗ f ≤ n f < n AFf For this range of flavors, the chiral limit of the theory is in the conformal window, with long distancebehavior (on R ) described by a non-trivial renormalization group (RG) fixed point. When the theory– 13 –s compactified on R × S and a non-zero fermion mass is introduced the scale invariant long distancebehavior is cut-off, either by m or 1 /L . The resulting behavior in the ( L, m ) phase diagram will bequalitatively identical to the case 2 ≤ n f < n ∗ f discussed above — except in the corner where m → L → ∞ . Specifically, nothing changes in the small L perturbative analysis. For all finite valuesof m , a center symmetric phase will be present for sufficiently small compactification size. One ormore transition lines (depending on the value of N ) emerge from the L = 1 /m = 0 corner of thephase diagram with slopes given by Eq. (5.5). In the large mass, large L portion of the phase diagram, m (cid:29) Λ (cid:38) /L , the theory reduces to pure Yang-Mills theory and exhibits a confinement/deconfinementtransition at L = L c = O (1 / Λ).In the massless limit, on R , the gauge coupling g ( µ ) ceases to run logarithmically below the scaleΛ and asymptotes to a non-zero fixed-point value g ∗ . Ref. [53] argues that the fixed point coupling willbe relatively weak, and that the massless theory on R × S will have unbroken center symmetry forall L . Turning on a small fermion mass m (cid:28) Λ will have negligible effect on the RG flow for µ (cid:29) m ,but will eliminate the fermion contribution to the RG evolution at scales small compared to m . Sincethe massless fermion contribution to the beta function was essential for producing a fixed point, theeffective gauge coupling will only remain (nearly) constant for scales between Λ and m . Below thescale m the coupling will resume its increase, growing just as it does in pure Yang-Mills theory andeventually becoming strong and driving confinement at a scale Λ YM ∼ m e − π / ( b g ∗ ) .When the mass-deformed theory is compactified on R × S , the confining phase will be stable, andcenter symmetry unbroken, provided 1 /L (cid:28) Λ YM . If the prediction of Ref. [53] of a rather small valueof the fixed point coupling is correct, then Λ YM will be substantially smaller than m (even though thereis no arbitrarily large parametric separation). Assuming so, a conventional confining/deconfining phasetransition, just as in pure Yang-Mills theory, must occur when L decreases to L c = O (1 / Λ YM ), sincethe fermions have negligible effect on physics at scales well below m . When the compactification sizedecreases further to O (1 /m ) or smaller, the fermions can influence the resulting symmetry realization.When the fermions (with non-thermal periodic boundary conditions) are sufficiently light comparedto 1 /L they will stabilize the center symmetric phase, just as in the massless limit.The simplest possibility for the resulting phase diagram is sketched on the right hand side ofFig. 4, for the case of N = 2. The novel feature is the “sliver” of totally broken center symmetryphase extending all the way to L = ∞ and m = 0, surrounded on either side by a phase with unbrokencenter symmetry. For N > 3, there will also be phases with partially broken center symmetry emergingfrom L = 1 /m = 0 and lying in between the totally broken and small- L symmetric phases, just asin Fig. 3(b). We expect these partially broken phases to terminate at a non-zero O (Λ) value of m ,but cannot exclude the logical possibility that they also persist in slivers extending to the upper right m = 1 /L = 0 corner. Large N limit of massive QCD(adj) As N → ∞ , Z N center symmetry transformations become a dense set within the continuous U (1)group. At N = ∞ , any discrete cyclic group Z p may be regarded as a subgroup of the center symmetry.For n f > 1, when the compactification size L lies in the interval [ L p , L p − ], then the effective potential(5.1) is minimized when the eigenvalues of Ω form p clumps equi-spaced around the unit circle, sothat tr Ω p (cid:54) = 0 while all traces of Ω to lower powers vanish. At these minima, the center symmetry is– 14 –roken to Z p . For fixed m (cid:29) Λ, as L decreases from O (1 / Λ) to zero, passing through each L p ∝ /p ,there is an infinite sequence of phase transitions with an accumulation point at L = 0:Compactification size : L > L > L > · · · > L p − > L p > · · · Residual center symmetry : Z | Z | Z | · · · | Z p | · · · (5.7)For very large but finite values of N , not all of these intervals will correspond to distinct symmetryrealizations. Which transitions are associated with changes in symmetry realization will depend onthe prime factorization of N . But within each interval [ L p , L p − ], the eigenvalues of Ω will form p clumps — some having (cid:98) N/p (cid:99) eigenvalues and others having (cid:100) N/p (cid:101) . In other words, the finite N eigenvalue distribution will approximate p equi-distributed clumps as closely as possible. The Wilsonline expectation value (cid:104) N tr Ω k (cid:105) will be O (1) when the winding number k is an integer multiple of p ,and O (1 /N ) when the winding k is not a multiple of p .As discussed above, for arbitrarily large fermion mass, an unbroken center symmetry phase willexist both at large compactification size, L (cid:38) O (1 / Λ), and sufficiently small size, L < L (cid:98) N/ (cid:99) when n f > 1. Consequently, one might hope that large N volume independence would relate massiveQCD(adj) on R to the theory in the small L center symmetric domain. Since properties of QCD(adj)with m (cid:29) Λ differ negligibly from pure Yang-Mills theory, this suggests that massive QCD(adj) withsufficiently small L and large m could serve as a reduced model exactly reproducing properties of large- N Yang-Mills theory. This, however, is not the case on R × S . The problem is that the condition forunbroken center symmetry in the reduced theory, L < L N/ = 2 z ∗ / ( N m ), combined with the strongcoupling condition (2.4) for large N volume independence, N L Λ (cid:29) 1, together imply that m (cid:28) Λ. Inother words, when taking N → ∞ for fixed mass m (cid:29) Λ, staying within the center symmetric small- L phase forces one to send L to zero, L (cid:46) / ( N m ), and this makes it impossible to remain within the N L Λ (cid:29) N volume independence. This means that massive QCD(adj) onsmall S × R . cannot be used as a large N reduced model for (continuum) Yang-Mills theory on R .Nevertheless, provided the phase diagram sketched in Fig. 3 is qualitatively correct as N → ∞ ,large N volume independence is valid in massive QCD(adj) with 2 ≤ n f < n ∗ f , for all L , when thefermion mass is below the O (Λ) lower limit m ∗ of any broken center symmetry phase. And likewisefor n ∗ f ≤ n f < n AFf and m < O (Λ), large N volume independence will relate the center-symmetricphases at large and small L , despite the presence of an intervening broken symmetry phase. 6. Multiple compactified dimensions We now turn to a consideration of QCD(adj) on R × ( S ) . The extension to compactifications onhigher dimensional torii will be discussed below.Let Ω and Ω denote the holonomy of the gauge field around the two independent cycles of the2-torus T = ( S ) . The set of vacua of the classical gauge theory is the space of flat connections, F µν = 0, which implies commuting covariantly constant holonomies,[Ω , Ω ] = 0 . (6.1)Hence, for vacuum configurations the two holonomies can be simultaneously diagonalized by a suitablegauge transformation. In other words, the classical vacua may be parametrized by two sets of N – 15 –igenvalues, Ω = diag (cid:0) e iα , e iα , · · · , e iα N (cid:1) , (6.2a)Ω = diag (cid:0) e iβ , e iβ , · · · , e iβ N (cid:1) , (6.2b)up to a global gauge transformation (which simultaneously conjugates Ω and Ω ). Each diagonalizedholonomy Ω i may be regarded as taking values in the maximal torus of SU ( N ), which we label as T N . For generic configurations with distinct eigenvalues, the subgroup of global gauge transforma-tions which preserve the diagonalized form (6.2) is the Weyl group W of SU ( N ), whose elements simultaneously permute the eigenvalues of the two holonomies, ( α i , β i ) → ( α σ ( i ) , β σ ( i ) ), where σ ∈ S N (with S N the N -element permutation group). Consequently, the classical moduli space of the theorycan be described as M = ( T N ) / S N . (6.3)Another useful way to describe M is as follows. Let (cid:101) T denote the torus dual to the base space2-torus. If (cid:126)L ≡ ( L , L ) are the periods of T , then the dual torus (cid:101) T has periods (cid:101) L = 2 π/L and (cid:101) L = 2 π/L . Each pair of simultaneous eigenvalues ( e iα , e iβ ) may be written as ( e ik L , e ik L )where (cid:126)k ≡ ( k , k ) ∈ (cid:101) T . Consequently, the moduli space M may also be viewed as the configurationspace of N identical “eigenparticles” which are placed on the dual torus (cid:101) T . (Clearly, this descriptiongeneralizes immediately to compactifications on higher dimensional torii.)Integrating out fluctuations will produce an effective potential which depends on the commutingholonomies Ω i . In QCD(adj) with n f fermions (having a common mass m and periodic spin connection)the one-loop result may be expressed as a sum of pairwise interactions between eigenparticles, V [Ω , Ω ] = N (cid:88) i,j =1 v [ (cid:126)α i − (cid:126)α j ] , (6.4)with v [∆ (cid:126)α ] ≡ (cid:88) (cid:126)n ∈ Z (cid:48) m (cid:126)n cos[ (cid:126)n · ∆ (cid:126)α ] , (6.5)and m (cid:126)n ≡ π | (cid:126)Ln | (cid:104) − n f ( m | (cid:126)Ln | ) K ( m | (cid:126)Ln | ) (cid:105) . (6.6)Here (cid:126)α i ≡ ( α i , β i ), (cid:126)n = ( n , n ), (cid:126)Ln ≡ ( L n , L n ), and the prime on the sum indicates omission ofthe term with (cid:126)n = 0. Alternatively, one may express this potential as a sum of squares of Wilson lines, V [Ω , Ω ] = (cid:88) (cid:126)n ∈ Z (cid:48) m (cid:126)n (cid:12)(cid:12) tr(Ω n Ω n ) (cid:12)(cid:12) . (6.7)Unbroken ( Z N ) center symmetry implies vanishing of the traces tr(Ω n Ω n ) for all winding numbers n and n which are not multiples of N . The fermion contribution (proportional to n f ) to the effectivemass squared m (cid:126)n is positive. One may see from the pairwise potential (6.5) that this means fermionsgenerate repulsive interactions between every pair of eigenparticles. The equivalent form (6.7) showsthat the positive fermion contribution to m (cid:126)n works to stabilize the center symmetric configuration withvanishing traces. In contrast, the ( − 1) pure gauge contribution to m (cid:126)n produces attractive interactions– 16 –etween eigenparticles which work to destabilize center symmetry. This exactly parallels the situationon R × S , except that eigenvalue pairs on (cid:101) T take the place of single eigenvalues on S .Naturally, the competition between fermion and gauge field contributions leads to different typesof minima depending on the values of the periods (cid:126)L , the fermion mass m , and the number of flavors n f . For sufficiently large fermion mass (within the small L domain of validity of the perturbativeanalysis), the gauge field contribution will dominate, causing all N eigenparticles to coalesce andproducing complete spontaneous breaking of the ( Z N ) center symmetry. For sufficiently light andnumerous fermions, the repulsive fermion-induced interaction between eigenparticles will dominate.This will cause the eigenparticles to disperse throughout the dual torus. Finding the arrangementwhich minimizes the effective potential is analogous to a sphere-packing problem; the precise resultwill depend sensitively on N and the aspect ratio L /L . When, for example, L = L ≡ L and N isa perfect square, the minimum of V [Ω , Ω ] for sufficiently light mass and n f > N eigenparticles into a √ N ×√ N square array. Such a configuration producesvanishing order parameters, tr (Ω n Ω n ) = 0, for windings n and n which are non-zero modulo √ N ,corresponding to spontaneous breaking of the ( Z N ) center symmetry down to a ( Z √ N ) subgroup.For these configurations, the evaluation of the effective potential is reliable provided √ N L Λ (cid:28) 1; thisis the condition that the coupling be weak on the O [1 / ( √ N L )] scale of the lightest off-diagonal degreesof freedom ( i.e. , charged W ’s) which were integrated out to obtain the effective potential.More generally, for O (1) values of L /L and large values of N , the minimum of the effectivepotential will correspond to configurations where the eigenparticles are distributed with approximatelyuniform density on the dual torus. The typical nearest-neighbor separation between eigenparticles onthe dual torus will be ∼ π/ √ N L , where L is the geometric mean of L and L , and the appropriateweak coupling condition will again be √ N L Λ (cid:28) L (cid:29) L , the eigenparticles will form a one-dimensional array.When, for example, L = pL for some large integer p and M ≡ N/p is an integer factor of N , then (cid:126)α j = (2 πj/N, πj/M ) is a minimum energy configuration. This configuration is invariant (up to Weylpermutations) under a Z N subgroup obliquely embedded within the Z N × Z N center symmetry groupand corresponds to a partial breaking of the Z N × Z N symmetry down to a single Z N .In contrast to the situation on R × S , no configuration of eigenparticles on the dual toruscan be invariant under the entire ( Z N ) center symmetry. The unbroken subgroup of the centersymmetry will be some Z p × Z q with pq ≤ N . Consequently, there will be (at least) O ( N ) gauge-inequivalent degenerate minima, related by center transformations. But before one can concludethat this truly implies spontaneous breaking of center symmetry, the effects of fluctuations must beconsidered. On R × T , fluctuations of eigenvalues away from the minimum energy configurationsbehave like two-dimensional scalar fields. These fluctuations, for n f > 1, acquire a non-zero massof order √ λ/L from the one-loop effective potential, with λ ≡ g N the ’t Hooft coupling. When L is small and the perturbative evaluation of the effective potential is reliable, the probability of suchfluctuations overcoming the action barriers separating different minima is negligible. Consequently, More explicitly, this means Ω ∝ C √ N ⊗ √ N and Ω ∝ √ N ⊗ C √ N , where 1 √ N is a √ N -dimensional identity matrixand C √ N = diag (1 , ω, ω , · · · , ω √ N − ), with ω ≡ exp(2 πi/ √ N ) a Z √ N generator. However, we have not checked tosee if, for example, a bcc lattice is preferred over a simple cubic lattice. For suitable values of N the minimum energyconfiguation of eigenvalues will be some regular lattice; exactly which lattice is inessential to our discussion. – 17 –n R × T , multiple light adjoint representation fermions cannot prevent at least partial breaking ofcenter symmetry in the limit of small compactification radii. A rich pattern of phase boundaries willemerge from the L = L = 1 /m = 0 corner of the phase diagram, just as for the R × S case, when n f > n f = 1, or single flavor QCD(adj), is a special case. With a non-zero mass m , thecenter-symmetry stabilizing fermion contribution can never dominate the destabilizing gauge fieldcontribution in the coefficients (6.6). Consequently, as the compactification sizes L , L → m > 0, the center symmetry will be completely broken, just like the situation on R × S .In the massless limit, the n f = 1 theory becomes N = 1 supersymmetric Yang-Mills theory. Centersymmetry is surely unbroken when the compactification size is sufficiently large. In supersymmetrictheories with supersymmetry preserving boundary conditions, it is believed that there are no phasetransitions as a function of volume. The singularities in the holomorphic coupling are of (real) co-dimension two [54, 55], so even if a singularity was encountered as the compactification size decreased,it is possible to analytically continue around it. Hence, center symmetry in N = 1 SYM (compactifiedon R × T ) should be unbroken for any value of L and L .It is instructive to see how this conclusion can emerge from a small L analysis. The perturbativeeffective potential vanishes identically in this supersymmetric theory but, as noted above, a non-perturbative superpotential is generated on R × S [44], and hence necessarily also on R × T . (Thisfollows from considering the regime L (cid:28) L , where physics must approach the R × S case.) Theparametric size of the resulting bosonic potential is Λ L where L = min( L , L ). This potentiallifts the classical moduli space (6.3) but, for Λ L (cid:28) 1, it is a very weak “pinning” potential forthe eigenparticles and generates an effective mass µ for fluctuations of eigenvalues which is tiny, µ ∼ O ( √ λ Λ L ). The resulting fluctuations, on scales which are large compared to L but smallcompared to 1 /µ , have amplitudes large enough to wash out the distinction between neighboringdegenerate minima related by center transformations. Hence, for N = 1 SYM, a classical analysis ofthe Wilson line effective potential is invalid; the effects of nearly massless two-dimensional eigenvaluefluctuations cannot be neglected. Integrating out the effects of these fluctuations on length scalesbetween L and µ − will modify the effective potential relevant for even longer scales, leading tomerging of the multiple minima and restoration of the full center symmetry.Returning to the case of n f > 1, we noted above that for arbitrarily small L there will be at leastpartial breaking of the center symmetry. The implications of this for large N volume independence is,perhaps surprisingly, negligible. With multiple fermion flavors and sufficiently small compactificationsize, all coefficients m (cid:126)n in the Wilson line effective potential (6.7) with up to N windings are positive,favoring unbroken center symmetry. But no configuration of eigenvalues, for finite N , can force all orderparameters tr(Ω n Ω n ) to vanish (for windings n and n non-zero modulo N ). Nevertheless when, for Consider, for example, the specific case of N = M and L = L = L where the minimum energy configuration ofeigenparticles should be a square lattice. Eigenvalue fluctuations behave like massless 2 d fields, with (cid:104) δα ( x ) δα (0) (cid:105) ∼ ( λ/ πN ) ln 1 / ( µ x ), for | x | (cid:46) /µ . Examining, e.g. , the order parameter O ≡ N tr(Ω M ), the difference in (cid:104)O(cid:105) between neighboring minima is | e πi/M − | = O (1 /M ). If O d denotes O averaged over a region of size d , one finds (cid:104) ( δ O d ) (cid:105) ∼ N [ λ π ln µ d ] . So the rms fluctuation in O d is O (1 /M ) or larger when d ≤ ¯ d ∼ µ − e − π/λ . Inserting µ − ∼ ( L/ √ λ )(Λ L ) − ∼ ( L/ √ λ ) e π /λ shows that ¯ d ∼ ( L/ √ λ ) e (8 π − π ) /λ , where λ = λ (1 /L ). So for small L , thereis a parametrically large hierarchy, L (cid:28) ¯ d (cid:28) /µ . – 18 –xample, N = M and L = L , the unbroken Z √ N × Z √ N subgroup of center symmetry still forcesall loops with winding numbers less than √ N to vanish. As N → ∞ , this (plus unbroken translationsymmetry) is sufficient to guarantee that the loop equations of the compactified and decompactifiedtheories coincide. For other values of N or L /L , the minimum energy configurations of eigenparticleson the dual torus (cid:101) T will lead to non-zero values of Wilson lines even when the winding numbers aresmall. But repulsion of eigenparticles on the dual torus will cause the order parameters N tr(Ω n Ω n )to be O (1 /N ) — not O (1) — provided n , n (cid:28) N . This means that the large N limit of theexpectation value of any topologically non-trivial Wilson loop vanishes, showing that the full centersymmetry is effectively restored as N = ∞ . Consequently, large N volume independence will relate n f ≥ R × T with sufficiently small compactification size to the decompactified theoryon R .Most aspects of the above discussion of QCD(adj) on R × T generalize immediately to thecase of R × T or T . The only difference in expressions (6.4)–(6.7) is that (cid:126)α and (cid:126)n change fromtwo-component to three- or four-component vectors. However, fluctuations of eigenvalues becomeprogressively more important as the number of uncompactified dimensions decreases. For finite valuesof N , the discrete center symmetry cannot break spontaneously when the theory is compactified on R × T or T . However, as is well known, spontaneous symmetry breaking can occur at N = ∞ even in finite volume theories. Consequently, one must carefully examine the effects of eigenvaluefluctuations as N → ∞ . When repulsive interactions dominate and eigenparticles are distributedthroughout the dual torus, one may show from the expression (6.7) that the potential barrier whichseparates degenerate minima related by a center transformation is O ( N ), and does not grow as N becomes large. On T , where there is no infinite volume of uncompactified dimensions, this impliesthat fluctuations sampling all center-symmetry related minima will have non-vanishing probabilitiesas N → ∞ , thereby ensuring unbroken center symmetry at N = ∞ . The situation is different whenattractive interactions between eigenparticles dominate and the eigenparticles clump. In this case,the potential barrier between different minima is O ( N ), and the probability of symmetry-restoringfluctuations vanishes at N = ∞ .On R × T , the relevant fluctuations are tunneling transitions which are localized in the un-compactified dimension, but the basic conclusion in the same: when the repulsive fermion-inducedinteraction between eigenparticles dominates the attractive gauge field contribution, fluctuations sam-pling all center-symmetry related configurations should retain non-zero probabilities as N → ∞ , sothat center symmetry remains unbroken at N = ∞ .The basic interaction between eigenparticles switches between attractive and repulsive at an O (1)value of mL (for n f > who investigated a single-site model of QCD(adj) and found that the center symmetry is intact for a There is one noteworthy change for T compactifications: the sum over (cid:126)n ∈ Z which now appears in the coefficients(6.5) is only conditionally convergent for ∆ (cid:126)α (cid:54) = 0, and has a logarithmic divergence at ∆ (cid:126)α = 0. This reflects the factthat coinciding eigenparticle positions on the dual torus represent special configurations where some of the off-diagonalfluctuations which were integrated out to produce the effective potential (6.4) become massless. On T with coincidingeigenvalues, these are off-diagonal zero-modes for whom quartic interactions remain relevant. Inappropriately applyinga Gaussian approximation to these zero modes leads to the unphysical logarithmic singularity in the effective potential. For related work, also see Ref. [57, 58]. – 19 –ide range of fermion mass, up to an O (1) value of ma (with a the lattice spacing), for reasonablylarge values of N and values of the bare ’t Hooft coupling in the range typically used in lattice QCDstudies. For compactifications on R × T or T the bottom line, once again, is that large N volumeindependence will be valid when the fermions are sufficiently light and numerous. But, due to enhancedfluctuations in lower dimensions, the upper limit on the range of fermion masses for which unbrokencenter symmetry persists down to L → 0, at infinite N , should be the larger of ( O (1 /L ), O (Λ)) andnot just O (Λ) as in the earlier cases of R × T and R × S . See Refs. [59] for more discussion of thisissue. 7. Prospects Volume independence is an exact property of certain large- N gauge theories. Although the idea is old,it was widely believed that, for four dimensional gauge theories, only a partial reduction was possibledown to a minimal compactification size L = L c ∼ / Λ, and it has often been asserted that the smallvolume theory is weakly coupled whenever L Λ (cid:28) 1. This proves not to be the case. The understandingthat emerges from valid examples of volume independence (as L → 0) leads to a modified picture.When center symmetry is unbroken, a weak coupling description is only possible when L Λ is smallcompared to a positive power of 1 /N . It is possible for a QCD-like large N gauge theory, formulatedin a box much smaller than the inverse strong scale, to reproduce infinite volume results.There is an ongoing effort in the lattice gauge theory community to determine the lower boundaryof the conformal window for various QCD-like theories. One of the technical issues in all latticegauge theory simulations is controlling finite-volume effects and performing reliable infinite volumeextrapolations. The 1 /N suppression of finite-volume effects in large- N center symmetric theoriesallows one to trade a large volume extrapolation for a large N extrapolation, and should be helpfulfor studies of conformal windows in large N theories.In a lattice formulation of gauge theories, large- N volume independence implies a non-perturbativeequivalence between four-dimensional field theories and zero dimensional matrix models or one dimen-sional quantum mechanics of large- N matrices [6, 17, 56]. 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