Phase structure of twisted Eguchi-Kawai model
Tatsuo Azeyanagi, Masanori Hanada, Tomoyoshi Hirata, Tomomi Ishikawa
aa r X i v : . [ h e p - l a t ] D ec Preprint typeset in JHEP style - HYPER VERSION
KUNS-2068RIKEN-TH 97RBRC-682WIS/19/07-Nov-DPP
Phase structure of twisted Eguchi-Kawai model
Tatsuo Azeyanagi a , Masanori Hanada b,c , Tomoyoshi Hirata a and Tomomi Ishikawa d a Department of Physics, Kyoto University,Kyoto 606-8502, Japan b Theoretical Physics Laboratory, RIKEN Nishina Center,Wako, Saitama 351-0198, Japan c Department of Particle Physics, Weizmann Institute of Science,Rehovot 76100, Israel d RIKEN BNL Research Center, Brookhaven National Laboratory,Upton, New York 11973, USAE-mail: [email protected], [email protected],[email protected], [email protected]
Abstract:
Twisted Eguchi-Kawai model is a useful tool for studying the large- N gaugetheory. It can also provide a nonperturbative formulation of the gauge theory on noncom-mutative spaces. Recently it was found that the Z N symmetry in this model, which iscrucial for the above applications, can break spontaneously in the intermediate couplingregion. In this article, we study the phase structure of this model using the Monte-Carlosimulation. In particular, we elaborately investigate the symmetry breaking point from theweak coupling side. The simulation results show that we cannot take a continuum limitfor this model. Keywords:
Matrix Models, Lattice Gauge Field Theories, Non-Commutative Geometry. ontents
1. Introduction 12. Twisted Eguchi-Kawai model 3 Z N symmetry 52.4 Limiting procedure 6 Z N symmetry breaking in the TEK model 6
4. Discussions 10 Z N symmetry breaking point 104.2 Continuum and large- N limit 12
5. Conclusions 13A. Double scaling limit as the noncommutative Yang-Mills theory 14
1. Introduction
The large- N gauge theories provide fruitful features to both phenomenology and string the-ory. They are simplified in the large- N limit while preserving essential features of QCD [1].Additionally, dimensional reductions of ten-dimensional N = 1 super Yang-Mills theory(matrix model) are expected to provide nonperturbative formulations of superstring theory[2, 3, 4, 5], and can also be regarded as effective actions of D-branes [6]. Furthermore, theirtwisted reduced versions, which we study in this article, can provide a nonperturbativeformulation of the gauge theories on noncommutative spaces (NCYM) [7, 8]. In order tostudy the nonperturbative nature of these theories, numerical simulations using lattice reg-ularizations are quite efficient. (Non-lattice simulations are also applicable for the reducedmodels. See references [9, 10] for the recent progress.)In the large- N limit there is an equivalence between the gauge theory and its zero-dimensional reduction, which is known as Eguchi-Kawai equivalence [11]. Here, we considerthe SU ( N ) gauge theory (YM) on D -dimensional periodic lattice with the Wilson’s pla-quette action S W = − βN X x X µ = ν Tr U µ ( x ) U ν ( x + ˆ µ ) U † µ ( x + ˆ ν ) U † ν ( x ) , (1.1)– 1 –here U µ ( x ) ( µ = 1 , ..., D ) ∈ SU ( N ) are link variables and β is the inverse of the bare’t Hooft coupling. In the large- N limit the space-time degrees of freedom can be neglected,and then this theory can be equivalent to a model defined on a single hyper-cube, S EK = − βN X µ = ν Tr U µ U ν U † µ U † ν , (1.2)which is called the Eguchi-Kawai model (EK model). The equality was shown by observingthat the Schwinger-Dyson equations for Wilson loops (loop equations) in both theories arethe same. In the EK model the loop equations can naively have open Wilson line terms,which do not exist in the original gauge theory side due to the gauge invariance. Thereforewe need to assume that the global Z DN symmetry U µ → e iθ µ U µ , (1.3)which eliminates the non-zero expectation value of the open Wilson lines, is not brokenspontaneously. However, soon after the discovery of the equivalence, it was found that the Z DN symmetry is actually broken for D > Z DN sym-metry is ensured in the weak coupling limit. It is not obvious whether the symmetry isbroken or not in the intermediate coupling region. There is no guarantee for not violatingthe symmetry. Numerical simulations in the 1980s, however, suggested that the Z DN sym-metry is not broken throughout the whole coupling region. Then we have believed thatthe TEK model actually describes the large- N limit of the gauge theory.Recently some indication about the Z DN symmetry breaking was surprisingly reportedin several context around the TEK model [16, 17][18, 19]. The most relevant discussionfor the present article was done by Teper and Vairinhos in [17] . They showed that the Z DN symmetry is really broken in the intermediate coupling region by the Monte-Carlosimulation for the D = 4 TEK model with the standard twist. Our work in this articleis along this line and we mainly concentrate on investigating locations of the symmetrybreaking from the weak coupling side in ( β, N ) plane. By the Monte-Carlo simulation weclarify the linear behavior of critical lattice coupling β Lc ∼ L , (1.4)where β Lc represents critical lattice coupling from the weak coupling side and L is the latticesize we have considered. This result means that the continuum limit of the planar gauge In [18, 19] a similar model with two commutative and two noncommutative dimensions were studiedin the context of NCYM. In this case the instability of Z N preserving vacuum was observed even in aperturbative calculation. This instability arises due to UV/IR mixing. – 2 – heory cannot be described by the TEK model from the argument of the scaling behavioraround the weak coupling limit. This discussion can be also applied to the NCYM case.This article is organized as follows. In the next section we review the TEK modelbriefly and fix our setup. In section 3 we show the numerical results for the Z DN symmetrybreaking of the TEK model and find the scaling behavior (1.4). In section 4 we givethe validation for the numerical result, and also discuss whether the TEK model has acontinuum limit or not.
2. Twisted Eguchi-Kawai model
In this study, we treat the D = 4 case. The TEK model [15] is a matrix model defined bythe partition function Z T EK = Z Y µ =1 dU µ exp( − S T EK ) , (2.1)with the action S T EK = − βN X µ = ν Z µν Tr U µ U ν U † µ U † ν , (2.2)where U µ and dU µ ( µ = 1 , , ,
4) are link variables and Haar measure. The phase factors Z µν are Z µν = exp (2 πin µν /N ) , n µν = − n νµ ∈ Z N . (2.3)The Wilson loop operator also contains the phase Z ( C ) as W T EK ( C ) ≡ Z ( C ) h ˆ W ( C ) i , (2.4)where ˆ W ( C ) is the trace of the product of link variables along a contour C and Z ( C ) isthe product of Z µν ’s which correspond to the plaquettes in a surface whose boundary is C . This model is obtained by dimensional reduction of the Wilson’s lattice gauge theorywith the twisted boundary condition. With these definitions, the loop equations in theTEK model take the same form as those in the ordinary lattice gauge theory if the Z N symmetry, which we discuss in section 2.3, is not broken. In the weak coupling limit, the path-integral is dominated by the configuration which givesthe minimum to the action. This configuration U (0) µ = Γ µ satisfies the ’t Hooft algebraΓ µ Γ ν = Z νµ Γ ν Γ µ , (2.5)and is called “twist-eater”. The most popular twist might be the minimal symmetric twist(standard twist) n µν = L L L − L L L − L − L L − L − L − L , N = L . (2.6)– 3 –his twist represents L lattice. In order to construct the classical solution for this twist,it is convenient to use the SL (4 , Z ) transformation for the coordinates on T [20]. Usingthe SL (4 , Z ) transformation we can always rewrite the n µν in the skew-diagonal form n µν −→ n ′ µν = V T n µν V = L − L L − L , (2.7)where V is a SL (4 , Z ) transformation matrix. This form makes the construction of thetwist-eater easy. Here we define L × L “shift” matrix ˆ S L and “clock” matrix ˆ C L byˆ S L = · · · · · · · · · , ˆ C L = O e πi/L e πi · /L . . . O e πi ( L − /L , (2.8)which satisfy the little ’t Hooft algebraˆ C L ˆ S L = e − πi/L ˆ S L ˆ C L . (2.9)Using these matrices, the twist-eater configuration for the skew-diagonal form (2.7) is easilyconstructed as Γ = ˆ C L ⊗ L , Γ = ˆ S L ⊗ L , Γ = L ⊗ ˆ C L , Γ = L ⊗ ˆ S L . (2.10)From (2.7) we can also construct the twist-eater configuration for the minimal symmetrictwist (2.6) as Γ = ˆ C L ⊗ L , Γ = ˆ S L ˆ C L ⊗ ˆ C L , Γ = ˆ S L ˆ C L ⊗ ˆ S L , Γ = ˆ S L ⊗ L . (2.11)Although these forms are different only by the coordinate transformation, they can givedifferent results except the weak coupling limit as seen in next section.Another kind of the twist we consider in this article is n µν = mL − mL mL − mL , N = mL (2.12)with classical solution Γ = ˆ C L ⊗ L ⊗ m , Γ = ˆ S L ⊗ L ⊗ m , Γ = L ⊗ ˆ C L ⊗ m , Γ = L ⊗ ˆ S L ⊗ m . (2.13)– 4 –hile we write the twist using the skew-diagonal form here, we can always rewrite it inthe symmetric form by the SL (4 , Z ) transformation. We call this twist “generic twist” inthis article, and the minimal twists (2.6) and (2.7) are particular cases ( m = 1) of thegeneric twist. As is well known, the TEK model can describe the NCYM theory [7, 8].Expanding the matrix model around noncommutative tori background, we can obtainnoncommutative U ( m ) Yang-Mills theory on fuzzy tori. (Note that this interpretation ispossible even at finite- N .) Because fuzzy torus can be used as a regularization of fuzzy R , it is naively possible to give a nonperturbative formulation of the NCYM on fuzzy R by taking a suitable large- N limit in the TEK model. (See appendix A for details.)However, we will see later it is not the case because of the Z N symmetry breaking. In theNCYM interpretation the shift and clock matrices can be regarded as matrix realizationof a fuzzy torus. From this point of view, twist prescription (2.12) provides YM theorieson m -coincident four-dimensional fuzzy tori. Z N symmetry The Z N symmetry plays a crucial role in the Eguchi-Kawai equivalence. Generally, the YMtheory with a periodic boundary condition has a critical size. If we shrink the volume of thesystem beyond the critical size, we encounter the center symmetry breaking, which is justthe same as the finite temperature system. In the EK model, which is a single hyper-cubicmodel, the critical size corresponds to β c ∼ .
19 in the lattice coupling. In the region lessthan the β c – the strong coupling region – the center symmetry Z N is maintained. Onthe other hand, in the region larger than β c – the weak coupling region – the symmetry isspontaneously broken, and then the EK equivalence does not hold.The TEK model avoids this problem by imposing the twisted boundary condition onthe system instead of the periodic one. In the weak coupling limit the path integral isdominated by the vacuum configuration, which is twist-eater configurations, as we alreadymentioned. These configurations are invariant under global Z L transformation U µ → e iθ µ U µ , e iθ µ ∈ Z L , (2.14)which is regarded as the U (1) symmetry in the large- N limit. As a result, W T EK ( C ) iszero if C is an open contour in the weak coupling limit.A key point is that the solution for this problem is obvious only at the classical level.That is to say, there is no guarantee to maintain the Z L symmetry if we take into account thequantum fluctuation. Going away from the weak coupling limit, the configurations fluctuatearound the twist-eater. The situation can be displayed in the eigenvalue distribution ofthe link variables. In the weak coupling limit the N eigenvalues distribute regularly anduniformly on the unit circle in the complex plane, and then they are Z L symmetric. Ifwe decrease β , the eigenvalues begin to fluctuate around the location of the twist-eater.If the fluctuation is not too large, the Z L symmetric distribution is maintained. However,large fluctuation can make the uniform distribution shrink to a point, which corresponds to U µ = N configuration. In the strong coupling region the distribution is randomly uniform,and then the symmetry is restored. – 5 –lthough there is no guarantee to maintain the Z N symmetry in the intermediatecoupling region, the 1980s numerical simulations suggested that the symmetry was unbro-ken. And this caused us to believe that the EK equivalence in the TEK model does holdthroughout the whole coupling region. As is well known, the scaling of the YM lattice theory behaves as β ∼ log a − aroundthe weak coupling limit, where a is the lattice spacing, and which is obtained by one-loopperturbative calculation of the renormalization group equation. If we wish to constructthe TEK model which corresponds to the YM theory by the EK equivalence, the scalingof the TEK model should obey that of the YM theory. In the TEK model, the lattice size L relates to N . (For the twist we consider in this article, the relation is N = mL .) Then,the YM system with fixed physical size l = aL can be obtained by the scaling β ∼ log a − ∼ log N. (2.15)In order to obtain the large- N limit with infinite volume, we should increase β slower thanthe scaling (2.15). If it is not the case, the system shrinks to a point.In the case of the NCYM, the scaling near the weak coupling limit is essentially sameas the YM theory, that is, β ∼ log a − . (See appendix A.) But if we wish to make theTEK model corresponding to the NCYM, there is a constraint a L = al = f ixed , whichmeans that we take a scheme in which the noncommutative parameter θ is fixed. Then,both the continuum limit and the infinite volume limit are simultaneously taken (doublescaling limit). Regardless of difference of the constraint, the scaling for the NCYM weshould take is the same as that of the ordinary YM (2.15) by the nature of the logarithmscaling. Z N symmetry breaking in the TEK model As mentioned in the previous section, the Z N symmetry breaking had not been observedin the older numerical simulation. However, there are several recent reports which indicatethe symmetry breaking [16, 17, 19]. In [17], the symmetry breaking in the D = 4 SU ( N )TEK model was studied in the case of the standard twist up to N = 144 = 12 . Theauthors of [17] performed the Monte-Carlo simulation starting both from a randomizedconfiguration (“hot start”) and from the twist-eater solution (“cold start”). In both casesthe Z N symmetry begins to break at N ≥
100 = 10 . At N = 144 the symmetry breakingand restoration patterns they observed are Z N β Hc −−→ Z N −→ Z N −→ Z N −→ Z N ( N = 144, standard, hot start) , Z N ←− Z N ←− Z N ←− Z N ←− Z N β Lc ←−− Z N ( N = 144, standard, cold start) , (3.1)where β Hc and β Lc are the first breaking point for the hot start and that for cold start,respectively. Note that although there is recovery of the symmetry for the cold start, thesymmetry remains broken for the hot start.– 6 –n this section we show the results of the numerical simulation for this symmetrybreaking phenomena. In order to argue about the possibility of the continuum and large- N limiting procedure for this model, we mainly focus on the first breaking point for thecold start β Lc , which depends on N . In our simulation we use the pseudo-heatbath algorithm. The algorithm is based on [21],and in each sweep over-relaxation is performed five times after multiplying SU (2) matrices.The number of sweeps is O (1000) for each β . We scan the symmetry breaking on theresolution of ∆ β = 0 . ± . Z N symmetry is a first-order transition. As an order parameter for detecting the Z N breakdown, we measure the Polyakov lines P µ ≡ (cid:28)(cid:12)(cid:12)(cid:12)(cid:12) N Tr U µ (cid:12)(cid:12)(cid:12)(cid:12)(cid:29) . (3.2) First of all we treat the minimal symmetric twist (2.6). This twist is the most standardand is also used in the paper [17]. In our study we only investigate the first Z N symmetrybreaking point from weak coupling limit, that is, β Lc for this twist. (For more detailedinformation about the symmetry breaking phenomena, see [17].) The obtained results arein table 1 and plotted in figure 1. The symmetry breaking points and patterns ( Z N β Lc −−→ Z N for N = 100; Z N β Lc −−→ Z N for N > N = 144.In this work we explore the simulation for larger N . From figure 1 we can find clear lineardependence of β Lc on N (= L ) for N & N ≥
169 data is β Lc ∼ . N + 0 . . (3.3)A theoretical argument for this linear behavior is discussed in section 4. Minimal skew-diagonal twist
Twists can be always transformed into the skew-diagonal form by SL (4 , Z ) transformationas we mentioned in section 2.2. As it were, the minimal symmetric twist (2.6) is equivalentto the minimal skew-diagonal twist (2.7) in the weak coupling limit. However, both formscan represent different features by taking into account the quantum fluctuation. Actually,the Z N symmetry is already broken at N = 25. This fact enables us to observe the N -dependence of the critical points easily. Not only is the symmetry breaking point differentfrom the symmetric form, so is the breaking and restoration pattern. Figure 2 shows theexpectation value of the plaquette (top) and the Polyakov lines (besides the top) versus Strictly speaking, the symmetry preserved in the weak coupling region is not Z N but Z L . However, Z L is sufficient for the Eguchi-Kawai equivalence so we do not dare to distinguish them in this article. – 7 – L β Lc N L β Lc
100 10 0 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . Table 1:
Critical lattice coupling from the weak coupling side β Lc for the minimal symmetric twist. )0.30.350.40.450.50.550.60.65 β c L Figure 1:
Plot of β Lc versus N for the minimal symmetric twist. Fit line is equation (3.3), whichis obtained using N ≥
169 data. β for the cold start at N = 100. For N ≥
100 we find the Z N symmetry breaking andrestoration pattern: Z N ← Z N ← Z N ← Z N β Lc ←−− Z N ( N = 100, minimal skew-diagonal, cold start) , (3.4)which represents a difference from the symmetric form case (3.1). The first breakingpattern Z N β Lc −−→ Z N is, however, the same as that in the symmetric twist. (We notethat for N ≤
81 the first breaking pattern is Z N β Lc −−→ Z N , which resembles the pattern Z N β Lc −−→ Z N at N = 100 for the symmetric form [17].)Table 2 shows the first breaking points for the cold start β Lc and for the hot start β Hc .These data are plotted in figure 3 for β Lc and figure 4 for β Hc . Again, we find clear lineardependence on N for β Lc , as we found for the symmetric form. Additionally, we also findclear dependence on 1 /N for β Hc . The fitted results are β Lc ∼ . N + 0 . , (3.5) β Hc ∼ . N + 0 . , (3.6)– 8 – < p l aque tt e > P P P β P s t r o n g c o u p l i n g e x p a n s i o n w ea k c oup li ng e x pan s i on Z N4 Z N0 Z N2 Z N3 Z N4 Figure 2:
Expectation value of the plaquette (top) and the Polyakov line (besides the top) versusthe lattice coupling β for N = 100 with the minimal skew-diagonal twist (cold start). As β isdecreased, the Z N symmetry is broken and restored as Z N ← Z N ← Z N ← Z N β Lc ←−− Z N . N L β Hc β Lc N L β Hc β Lc . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . Table 2:
Critical lattice coupling from the weak coupling side β Lc and from strong coupling side β Hc for the minimal skew-diagonal twist ( m = 1). where we used only N ≥
64 data for β Lc , whereas all data are used for β Hc . As N isincreased the β Hc approaches a point 0 . Z N β Hc −−→ Z N takesplace in the original EK model. These results suggest that the quantum fluctuation is solarge that the Z N symmetry is broken in exactly the same region as that in the original EKmodel. The lines for transitions β Lc and β Hc seem to intersect around the bulk transitionpoint β Bc ∼ .
35, which corresponds to N ∼
20 for the twist considered here. For smallervalues than N ∼
20, we did not observe a signal of breakdown of the Z N symmetry. Generic skew-diagonal twist
Here, we show the numerical result of the generic twist (2.12). For this twist we use theskew-diagonal form because the Z N symmetry breaking occurs at smaller N than that inthe symmetric form, which makes our investigation much easier.– 9 –
50 100 150N ( =L )0.30.40.50.60.70.8 β c L Figure 3:
Plot of β Lc versus N for the minimalskew-diagonal twist. The fit line is equation(3.5), which is obtained using N ≥
64 data. )0.150.20.250.3 β c H Figure 4:
Plot of β Hc versus 1 /N for the mini-mal skew-diagonal twist. The fit line is equation(3.6), which is obtained using N ≥
25 data. Ex-trapolation to 1 /N = 0 gives β Hc → .
18, whichis close to the critical point in the original EKmodel, β = 0 . We measure β Lc for this twist up to m = 4. Table 3 shows the β Lc for m = 2 , , m = 1 is presented in table 2. These data are plotted in figure 5. From this figurewe can find that the β Lc for each L are reduced as we increase m , and the dependenceis linear in 1 /m . The data at 1 /m = 0 in this plot are linearly extrapolated values. Aninteresting point is the behavior for the case L = 5. While the Z N symmetry breakingis observed for m = 1 , m = 4 because the β Lc reaches the bulktransition point β Bc ∼ .
35 by increasing m . Figure 6 represents the same data in figure 5,but the horizon axis is L . As we have seen in the m = 1 case, the data for L ≥ L for each m . From these figures, we find that the datafor L ≥ β Lc ∼ . L + 0 . m + 0 . . (3.7)
4. Discussions
In this section we discuss the numerical results obtained in the previous section and thevalidity of taking the large- N and continuum limit for this model. Z N symmetry breaking point In the previous section we showed our numerical results. In particular, we elaboratelyinvestigated β Lc , the first Z N breaking point from the cold start. From our investigation,– 10 – = 2 m = 3 m = 4 L N β Lc N β Lc N β Lc . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . Table 3: β Lc for the generic skew-diagonal twist ( m = 2 , , m = 1. β c L L=10L= 9L= 8L= 7L= 6L= 5
Figure 5: β Lc versus 1 /m for L = 5 , · · · , β Lc for m = ∞ is evaluated by extrapolatingthese data with straight line. β c L m=1m=2m=3m=4m= ∞ Figure 6:
Scaling of β Lc for m = 1 , , β Lc for m = ∞ , which isobtained by an extrapolation shown in figure 5. we found the clear linear behavior like (3.3), (3.5) and (3.7). These behaviors can beobtained through the following consideration. Energy difference between twist-eater Γ µ and identity N configurations We simply assume that the Z N breaking is a transition from twist-eater phase U µ = Γ µ to identity configuration phase U µ = N . For plainness, we consider Z N β Lc −−→ Z N typebreaking here. Of course we can treat Z N β Lc −−→ Z N β Lc −−→ Z N β Lc −−→ Z N β Lc −−→ Z N (cascade) typebreaking at a β Lc , but the obtained behavior is not different from the former type. Firstly,we focus on the classical energy difference between these configurations. The energy can– 11 –e easily calculated from the action (2.2) as∆ S = S T EK [ U µ = N ] − S T EK [ U µ = Γ µ ]= βN X µ = ν (cid:26) − cos (cid:18) πn µν N (cid:19)(cid:27) ≃ π β X µ = ν n µν . (4.1)For the generic twist, it becomes∆ S = ( π βm L (symmetric form) , π βm L (skew-diagonal form) . (4.2)Note that the symmetric form is roughly three times more stable than the skew-diagonalform if both twists have equal quantum fluctuations. This is the reason that the Z N symmetry breaking for the skew-diagonal form can occur at quite smaller N than that forthe symmetric form, as is observed in our simulation. Quantum fluctuations and symmetry breaking
Going away from the weak coupling limit, the system has quantum fluctuations. Herewe naively expect that the Z N symmetry is broken if the fluctuation around twist-eaterconfiguration exceeds the energy difference ∆ S . Because the system describes O ( N )interacting gluons, it is natural to assume that their quantum fluctuations provide O ( N )value to the effective action. For the generic twist the quantum fluctuation is O ( m L ).Combined with the fact (4.2), we can estimate the critical point β Lc as β Lc ∼ L , (4.3)which is consistent with the numerical results (3.3), (3.5) and (3.7). In addition we canexplain the difference of the coefficient of L in (3.3), (3.5) and (3.7) between the symmetricand the skew-diagonal form, which is roughly three times different, by the factor in (4.2).Although the above crude estimation reproduces the linear L behavior of β Lc , wecannot explain the dependence on m . To catch the behavior completely, we need to makethe discussion more sophisticated. However, we do not pursue this issue here because the m dependence can be negligible at the larger N .This argument can be applied for other twist prescriptions like taking the twist phaseas exp( iπ ( L + 1) /L ), which is usually used for describing noncommutative spaces. (Seeappendix A.) N limit We have shown that the linear L dependence of the critical point β Lc could be explainedby the theoretical discussion in this section. While our simulation is restricted in the small N region, we confirm that the behavior must continue to N = ∞ by combining with thediscussion. Then the EK equivalence is valid only in the region β > β Lc ∼ N even in theweak coupling limit and the large- N limit. As we mentioned in the section 2.4, both theordinary YM with fixed physical volume and the NCYM theory with fixed noncommutative– 12 –arameter have essentially logarithm scaling (2.15) near the weak coupling limit. Then,because β Lc grows faster than the logarithm, the EK equivalence does not hold in thecontinuum limit.
5. Conclusions
In order to study the nonperturbative nature of the large- N gauge theory by lattice simu-lations, the large- N reduction is very useful property for saving the computational effort.In this paper, we studied the phase structure of the TEK model, which has been a majorway to realize the large- N reduction. Contrary to the naive hope in old days, at leastin ordinary twist prescriptions as investigated in this paper, the Z N symmetry is brokeneven in the weak coupling region and hence a continuum limit as the planar gauge theorycannot be described by the TEK model. For the NCYM, the situation is the same. We canalso consider a lot of variation for the twist prescription and the combination of reducedand non-reduced dimension. For example, in [18, 19], four-dimensional model with twocommutative and two noncommutative directions was studied using two-dimensional lat-tice action. However, the Z N symmetry is broken also in this model, and hence we cannottake a naive continuum limit.Another way for the reduction is the QEK model, in which the eigenvalues of thelink variables are quenched. The QEK model might have no problem in principle, but itscomputational cost is larger than that of the TEK model. Although the TEK and QEKmodel are reduced models to a single hyper-cube, recent studies deviate from them. Thecontemporary method might be the partial reduction [22]. This work showed that thelarge- N reduction is valid above some critical physical size l c . This means that for a latticesize L the reduction holds below some lattice coupling β ( L ). In order to take continuumlimits we should avoid the bulk transition point β Bc , causing the condition β Bc < β ( L ) tobe necessary. That is, there is a lower limitation for the lattice size L c for the continuumreduction. In addition, the twist prescription is also applicable to the partial reduction [23].Due to the twisted boundary condition, the lower limitation L c can be reduced. Therefore,combination of the twist prescription and the partial reduction would be quite efficient inthe current situation.Note also that NCYM on fuzzy R could be realized by using TEK with quotientconditions [8] which give a periodic condition to eigenvalues and hence quantum fluctuationis suppressed. Further study in this direction would be important. Acknowledgments
The numerical computations in this work were in part carried out at the Yukawa In-stitute Computer Facility. The authors would like to thank Sinya Aoki, Hikaru Kawai,Jun Nishimura, Masanori Okawa, Yoshiaki Susaki, Hiroshi Suzuki and Shinichiro Yamatofor stimulating discussions and comments. M. H. was supported by Special PostdoctoralResearchers Program at RIKEN. T. H. would like to thank the Japan Society for thePromotion of Science for financial support.– 13 – . Double scaling limit as the noncommutative Yang-Mills theory
The TEK model can be used to formulate gauge theories on noncommutative spaces non-perturbatively [7, 8, 24]. In this appendix, we give a review for the construction of theNCYM from the TEK model [7], a discussion for the scaling and some supplemental com-ments for our analysis.By taking U µ = e iaA µ , where a corresponds to the lattice spacing, and expanding theaction of the TEK model (2.2), we have its continuum version S T EK,continuum = − g X µ = ν Tr ([ A µ , A ν ] − iθ µν ) (A.1)up to higher order terms in a , where θ µν = 2 πn µν N a , g = a βN. (A.2)Then, by expanding the action around a classical solution of (A.1) A (0) µ = ˆ p µ , [ˆ p µ , ˆ p ν ] = iθ µν , (A.3)we obtain the U (1) NCYM on fuzzy R as follows. Let us define the “noncommutativecoordinate” ˆ x µ = (cid:0) θ − (cid:1) µν ˆ p ν . Then we have[ˆ x µ , ˆ x ν ] = − i ( θ − ) µν . (A.4)This commutation relation is the same as that of coordinate on fuzzy R with noncommu-tativity parameter θ , and hence functions of ˆ x can be mapped to functions on fuzzy R .More precisely, we have the following mapping rule: f (ˆ x ) = P k ˜ f ( k ) e ik ˆ x ↔ f ( x ) = P k ˜ f ( k ) e ikx ,f (ˆ x ) g (ˆ x ) ↔ f ( x ) ⋆ g ( x ) ,i [ˆ p µ , · ] ↔ ∂ µ , Tr ↔ √ det θ π R d x, (A.5)where ⋆ represents the noncommutative star product, f ( x ) ⋆ g ( x ) = f ( x ) exp (cid:18) − i ← ∂ µ ( θ − ) µν → ∂ ν (cid:19) g ( x ) , (A.6)and we obtain U (1) NCYM action S U (1) NC = − g NC Z d x F µν ⋆ F µν (A.7)with coupling constant g NC = 4 π g / √ det θ. (A.8)In the same way, by expanding the action (A.1) around A (0) µ = ˆ p µ ⊗ m , U ( m ) NCYM canbe obtained. From (2.13), it is apparent that the generic twist gives the U ( m ) NCYM.– 14 –ntuitively, the vacuum configuration (2.13) describes m -coincident fuzzy tori and fuzzy R is realized as a tangent space.In order to keep the noncommutative scale θ finite, we should take the double scalinglimit with a − ∼ √ L ∼ N / . (A.9)One-loop beta function for U ( m ) NCYM is given by [26] β − loop ( g NC ) = − π ) mg NC + O ( g NC ) . (A.10)Therefore, the ’t Hooft coupling β scales as β ∼ g NC ∼ log N. (A.11)Then, the scaling we should take for the NCYM is just the same as that for the ordinaryYM, and Z DN symmetry is broken in the scaling limit. Therefore, fuzzy torus crunches toa point and hence the fuzzy R cannot be realized .Of course, we can also use other twist prescriptions. In order to make the periodicityof the discretized fuzzy torus correct, we usually take the twist as exp( iπ ( L + 1) /L ) [8].Regardless of the difference of the twist, the conclusion might not be altered. Here werepeat the discussion in section 4.1. In this case, the Z N is likely to break down to Z . Thedifference between potentials in twist-eater and Z -preserving configurations is∆ S ∼ βN n − cos (cid:16) πL (cid:17)o ∼ βm L , (A.12)which is the same order as (4.1). Then the behavior of the critical point β Lc (4.3) is notchanged. References [1] G. ’t Hooft, “A PLANAR DIAGRAM THEORY FOR STRONG INTERACTIONS,”
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