Phases of one dimensional large N gauge theory in a 1/D expansion
aa r X i v : . [ h e p - t h ] F e b TIFR/TH/09-14
Phases of one dimensional large N gauge theoryin a /D expansion Gautam Mandal, Manavendra Mahato and Takeshi Morita
Department of Theoretical Physics,Tata Institute of Fundamental Research,Mumbai 400 005,
INDIA email: mandal, manav, [email protected]
Draft date: October 19, 2018
Abstract
We consider large N Yang Mills theory with D adjoint scalar fields in d dimensions for d = 0 or 1. We show the existence of a non-trivial saddle pointof the functional integral at large D which is characterized by a mass gap forthe adjoint scalars. We integrate out the adjoint scalars in a 1 /D expansionaround the saddle point. In case of one dimension which is regarded as acircle, this procedure leads to an effective action for the Wilson line. We findan analogue of the confinement/deconfinement transition which consists ofa second order phase transition from a uniform to a non-uniform eigenvaluedistribution of the Wilson line, closely followed by a Gross-Witten-Wadiatransition where a gap develops in the eigenvalue distribution. The phasetransition can be regarded as a continuation of a Gregory-Laflamme transi-tion. Our methods involve large values of the dimensionless ’tHooft coupling.The analysis in this paper is quantitatively supported by earlier numericalwork for D = 9. ontents d = 1 model: preliminaries 43 The d = 0 model 8 D limit . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Large D, N limit . . . . . . . . . . . . . . . . . . . . . . . . . 103.3 The saddle point . . . . . . . . . . . . . . . . . . . . . . . . . 11 d = 1 model: result 11 S ( △ , { u n } ) in leading large D . . . . . . . . . 124.2 1 /D correction to the effective action . . . . . . . . . . . . . . 184.3 1 /D expansion vs. numerical calculation . . . . . . . . . . . . 21 M ab,cd A.1 Algebraic properties of M ab,cd . . . . . . . . . . . . . . . . . . 27A.2 Results involving M ab,cd . . . . . . . . . . . . . . . . . . . . . 30 B Details of the d = 0 model 32 B.1 Calculation of S : the 1 /D correction . . . . . . . . . . . . . . 33 C Evaluation of a toy integral using a complex saddle point 35D The Y -Propagator for d = 1
36E All loop corrections up to /D in the d = 1 model 37 E.1 Two-loop correction . . . . . . . . . . . . . . . . . . . . . . . . 39E.2 Three-loop correction . . . . . . . . . . . . . . . . . . . . . . . 40E.3 ( n + 1)-loop correction to effective potential . . . . . . . . . . 411 Introduction and Summary
Matrix models in low dimensions (especially 0 and 1) have served as usefultools in many contexts. These include (i) c ≤ N reduced models andtheir variants [2, 3], (iii) BFSS matrix theory, which corresponds to DLCQof M-theory [4], (iv) IKKT matrix model of type IIB string theory [5, 6], (v)D0 brane black holes [7, 8, 9, 10, 11, 12, 13, 14], (vi) KK reduction of 4-dimensional N = 4 SYM on S [15, 16, 17, 18], (vii) The BMN matrix model[19], (viii) the matrix model of unstable D0 branes [20, 21], (ix) D braneson tori [22, 23, 24, 25], etc. In many cases, one can regard these modelsas dimensional reduction of large N Yang-Mills theories. In models arisingfrom D branes, the YM theories are typically supersymmetric; however, insome situations the theory is effectively described by the bosonic sector. Theobvious advantage of such a description, when it is possible, is that it is easilyamenable to numerical calculations, and in some fortunate circumstances, tosome powerful exact methods.In this paper, we consider the following system for d = 0 , S = 1 g Z d d x Tr D X I =1 D µ Y I D µ Y I − X I,J
14 [ Y I , Y J ] ! , (1.1)where D µ refers to the covariant derivative ∂ µ − i [ A µ , . ]. A µ , Y I are SU ( N )matrices. For d = 0 there is no gauge field and the first term is absent. For d = 1, there is a single gauge field A which is non-dynamical.The action (1.1) can be regarded as a dimensional reduction of D + d dimensional bosonic YM theory to d dimensions. The specific physicalcontext we have in mind in the present paper is related to case (ix) above,which is discussed in [22, 23, 24, 25] and reviewed below in Section 5.In the d = 1 case, we consider the dimension as a circle, of circumference β , and study the partition function and other quantities as a function of β .It was conjectured in [22, 23, 24, 25], on the basis of numerical investigations,that the d = 1 system exhibits a phase transition which is analogous to theconfinement/deconfinement transition of N = 4 SYM on S . The phasetransition was argued to be the weak coupling continuation of the blackhole/black string transition of the d = 2 model. One of the motivationsfor the present work was to understand the nature of the phase transition2nalytically.Our results are briefly as follows:1. In the limit of large D , the model (1.1) has a non-trivial saddle pointcharacterized by a non-zero value of h Tr Y I Y I i . The large D scaling isdefined by keeping a modified ’tHooft coupling ˜ λ = λD fixed.2. In this limit, fluctuations around the above saddle point are suppressedby powers of 1/ D . This enables us to develop a systematic expansion,in 1 /D , of the partition function and other quantities as a function ofthe radius of the circle.3. In the d = 0 case, the exact partition function is calculable up to 1 /D at finite N (first shown in [6]), whereas in the d = 1 case our resultsare obtained in the leading large N limit.4. Since the condensate provides a dynamically generated mass to theadjoint scalars Y I , it is possible to explicitly integrate them out. In the d = 1 case, this allows us to compute an effective action S eff ( W ) forthe Wilson line W = P exp[ i Z β dtA ] (1.2)in a 1 /D expansion.5. The S eff computed in this fashion provides the first analytic evidence,in a 1 /D expansion, for the phase transitions mentioned above. It con-firms the appearance of a double transition : (i) a second order phasetransition characterized by the onset of non-uniformity in the eigen-value distribution of W , followed by (ii) a third order Gross-Witten-Wadia (GWW) transition signalling the appearance of a gapped phase[26, 27, 28]. The appearance of a double transition is supported bythe numerical works in [24, 25] . The phase transition temperatures T c and T c , computed up to 1 /D , show excellent agreement with theirresults in the D = 9 case, as shown in the following table (see Section4.3 for more details and other comparisons): We thank S. Minwalla for a discussion on this point. We differ from [24, 25], though, regarding the order of the two transitions. See Section4.3 for details. c T c Our result 0.895 0.917Numerical result 0.8761 0.9056. Our methods involve large values of the dimensionless ’tHooft coupling.The large D technique used in this paper for d = 0 , d = 0 context. Our results for d = 0 are in complete agreementwith those of [6], though our method is slightly different (see Section 3 fordetails) in a way that enables a natural generalization to higher dimensions.A large D expansion has also been used in certain lattice theories [29]. Ourmethods also have some overlap with that of [8, 9] where a series of self-consistent equations (gap equations) are introduced to determine variouscondensates and a GWW type transition is suggested.The paper is organized as follows. In Section 2 we set up the main cal-culational method for the d = 1 model. The method consists of introducingan SO ( D )-invariant dynamical field to get rid of the commutator-squaredinteraction. This leads to an action quadratic in the adjoint scalars with adynamical mass term, which allows us to integrate them out. We work outthe example of the d = 0 model in Section 3 to test the method. We showthe existence of a non-trivial saddle point in which a mass is generated forthe adjoints. We explain the nature of the large D limit and compute thepartition function at finite N in a 1 /D expansion. In Section 4 we comeback to the d = 1 model to derive the effective action for the Wilson line ina 1 /D expansion. We show the existence of a second order phase transitionfollowed by a GWW transition. In Section 5 we provide a D brane realizationof our model, following earlier work [22, 23, 24, 25] which we briefly review.In Section 6 we conclude with a discussion.This paper arose as part of a larger project of exploring dynamical blackhole/black string transitions in terms of a dynamical large N transition in aunitary matrix model [30]. d = 1 model: preliminaries We will set up the d = 1 model in this section, test the formalism with thesimpler case of d = 0 in Section 3, and continue on to Section 4 to solve the4 = 1 model. A D brane realization of the d = 0 , Y I in (1.1) to gY I , so thatthe action becomes S = Z β dt Tr D X I =1 (cid:0) D Y I (cid:1) − X I,J g Y I , Y J ][ Y I , Y J ] ! . (2.1)We have assumed the theory to be on a circle, of circumference β , which canbe interpreted either as a Euclidean time direction or as a spatial circle. Thecovariant derivative is defined by D Y I = ∂ t Y I − i [ A , Y I ].The large N limit is defined by keeping the ’tHooft coupling λ = g N fixed, as N → ∞ . It is convenient to define the following related dimension-less quantities β eff = βλ / = β ( g N ) / , (2.2) λ eff = λβ = β . (2.3)For later convenience it is useful here to summarise various alternative def-initions of coupling constant which we will use in this paper in differentcontexts: λ ’tHooft coupling g N ˜ g ’tHooft like coupling at large D and finite N g D ˜ λ ’tHooft like coupling at large D and large N λDλ eff dimensionless ’tHooft coupling λβ The last line refers only to the d = 1 case.We would like to explore the partition function Z = Z D A D Y I e − S , (2.4)as a function of β eff . We will also be interested in the effective action S eff ( A ),defined by exp[ − S eff ( A )] = Z D Y I e − S . (2.5)It will turn out that S eff depends only on the gauge-invariant content of thegauge potential, namely on the eigenvalues of the Wilson line W (1.2).5he first step in solving the model consists of making the action (2.1)quadratic in the Y I by introducing an auxiliary field B ab . Let us write Y I = P N − a =1 Y Ia λ a , where λ a are the generators of SU ( N ). This leads to anexpression − Tr[ Y I , Y J ][ Y I , Y J ] = ( Y Ia Y Ib ) M ab,cd ( Y Jc Y Jd ) , (2.6)which is written in terms of SO ( D )-invariant Y -bilinears, where M ab,cd = − n Tr[ λ a , λ c ][ λ b , λ d ] + ( a ↔ b ) + ( c ↔ d ) + ( a ↔ b, c ↔ d ) o . (2.7)Properties of the matrix M are discussed in detail in Appendix A. Using thefact that M is invertible , we can write Z = N Z D B D A D Y I e − S ( B,A ,Y ) ,S ( B, A , Y ) = Z β dt (cid:20) (cid:0) D Y Ia (cid:1) − i B ab Y Ia Y Ib + 14 g B ab M − ab,cd B cd (cid:21) , (2.8)with the following classical equation of motion for B ab :1 g M − ab,cd B cd = iY Ia Y Ib . (2.9)In the above 1 / N ≡ R D B exp[ − R dt B ab M − ab,cd B cd / (4 g )], which we willignore in the rest of the paper since it involves only a numerical factor. Sincethe action (2.8) is quadratic in Y I , we can formally integrate them out, to It might be puzzling, at first sight, that M does not have a zero mode since (2.6)clearly vanishes for special configurations of the ‘ Y I ’s, e.g. when all ‘ Y I ’s commute. Theresolution is that M has both positive and negative eigenvalues (see (A.13)) and thereforehas light-like vectors which, however, do not correspond to zero eigenvalues. We thankToby Wiseman for a useful discussion on this point. The indefinite signature of M (A.13) involves some subtlety in choosing the contourof the functional integral in (2.8). E.g. we need to choose real contours for componentsof B ab along the positive eigenspace of M and purely imaginary contours otherwise. Thecorrect choice ensures finiteness of the normalization constant N . Z = Z D B D A e − S eff ( B,A ) ,S eff ( B, A ) = Z β dt g B ab M − ab,cd B cd + D (cid:0) ( D ) ab + iB ab (cid:1) . (2.10)For large D which scales as 1 /g , we may expect the one-loop determinantto be comparable with the classical term in (2.10) and hence modify the naiveclassical solution B ab = 0. We will assume, and shortly justify, that (2.10)admits a gauge-invariant time-independent solution, of the form (see, e.g.Eqns. (3.15) and (4.8)) B ab = i △ δ ab . (2.11)The appearance of ‘ i ’ is due to the fact, as we will see later, that the solutioncorresponds to a saddle point in the complex plane. The condensate in termsof the original physical variables Y I , however, turns out to be real: h Tr Y I Y I i = N g △ , (2.12)where we have used (2.9), and also M − ab,cd δ cd = N δ ab , which is derived inEqn. (A.12).To proceed, we write the B ab field as the sum of a constant trace pieceand the rest, as B ab ( t ) = B δ ab + gb ab ( t ) . (2.13)where b ab ( t ) satisfies R dt b aa ( t ) = 0. We will show below that B has a saddlepoint value of the form B = i △ , consistent with (2.11). Substituting (2.13) In much of this paper, we will treat D as an arbitrarily specifiable parameter, exceptin the section dealing with comparison with D branes, where we put D = 9. The precisescaling at large D is defined in Eqns. (3.6) and (3.12). See also comments at the end ofSection 4.1. Although the large N scaling is not apparent in (2.10), the action as well as themeasure admits a topological expansion in powers of 1 /N for fixed λ , as is apparent fromexpressions such as (3.13).
7n the action in (2.8) we get Z = Z dB D A D b ab D Y I e − S , S = S + S q + S int , (2.14)where , S = βN B g ,S q = Z β dt (cid:18) b ab M − ab,cd b cd + 12 (cid:0) D Y Ia (cid:1) − i B Y I a (cid:19) ,S int = − Z β dt (cid:18) ig b ab Y Ia Y Ib (cid:19) . (2.15)Before proceeding to solve this model, let us discuss the d = 0 matrix modelas a partial test of our formalism. d = 0 model The d = 0 model is defined by a partition function Z = Z dY I exp[ − S ] , S = − g X I,J [ Y I , Y J ] . (3.1)Here Y I are SU ( N ) hermitian matrices which are normalized the same wayas in (2.1). A D brane interpretation of this model is discussed in Section 5. Note the appearance of an effective mass term for the adjoint scalars Y Ia in the saddlepoint B = i △ . In [23] such a mass term is added by hand to integrate out the Y I ; herethe mass term is dynamically generated. Strictly speaking, the last term in S q is a cubic term and should be regarded as aninteraction. However, when we consider B as an external parameter (unintegrated) wecan regard this term as quadratic. Most of the results in this section are in [6] who have discussed this model earlier in thecontext of the IKKT matrix model. Our method, however, is slightly different, especiallyin the way we distinguish between the diagonal and the off-diagonal fluctuations of theauxiliary field B ab (see, e.g. (2.13) and (B.2)) which allows for a natural generalization tothe d = 1 model. d = 1, we can rewrite (3.1) as Z = Z dB db ab dY I exp[ − S ] , (3.2) S = S + S q + S int ,S = N B g , S q = 14 b ab M − ab,cd b cd − i B Y I a , S int = − ig b ab Y Ia Y Ib . (3.3)Integrating out the Y I gives us the d = 0 analogue of (2.10) where B ab issplit into B and b ab : Z = Z dB db ab e − S eff ( B ,b ab ) ,S eff ( B , b ab ) = N B g + 14 b ab M − ab,cd b cd + D iB δ ab + igb ab ) . (3.4) D limit Let us make a formal Taylor expansion of S eff in powers of b ab : S eff ( B , b ab ) = S (0) ( B ) + 12 S (2) ab ; cd ( B ) b ab b cd + S int eff ,S (0) ( B ) = N B g + D iB δ ab ) ,S (2) ab ; cd ( B ) = 12 M − ab,cd + g D ∂ log det (cid:0) iB δ rs + i ¯ b rs (cid:1) ∂ ¯ b ab ∂ ¯ b cd (cid:12)(cid:12)(cid:12) ¯ b rs =0 ,S int eff = O ( b ) , (3.5)where in defining S (2) we have used ¯ b ab ≡ gb ab in order to exhibit the g dependence explicitly. There is no linear term in the above Taylor expansionsince b aa = 0. Let us define a large D limit by keeping˜ g ≡ g D, (3.6)fixed. It is easy to see that S (0) ( B ) = O ( D ) , S (2) ab ; cd ( B ) = O (1) , S int eff = O (1 /D ) . (3.7)9et us now do the integral in (3.4) over b ab , to give Z = Z dB exp[ −S ( B )] , (3.8) S ( B ) D = S ( B , ˜ g ) + 1 D S (˜ g ) + O (cid:18) D (cid:19) , (3.9)where S ( B , ˜ g ) = 1 D S (0) ( B ) = N B g + 12 log det ( iB δ ab ) , S ( B , ˜ g ) = 12 log det S (2) ab ; cd . (3.10)The first ‘log det’ is essentially a 1-loop integral over Y I , while the second‘log det’ is a 1-loop integral over b ab (see Figure 6(a) and (b)). In AppendixB we will present an explicit computation of these quantities. We find (see(B.7)) S ( B ) = N B g + ( N − (cid:18) − B ˜ g N (cid:19) , S ( B ) = N −
12 log (cid:18) − ˜ g NB (cid:19) + N ( N + 1)( N − (cid:18) − g B (cid:19) + N ( N − N + 3)8 log (cid:18) g B (cid:19) . (3.11)In the d = 1 case, an explicit finite N , large D result such as Eqn. (3.11)is difficult to obtain, but we will derive an analogue of Eqn. (3.14) below.Furthermore, as remarked at the end of Section 4.1, for d = 1 we will nottake the strict D = ∞ limit since criticality involves 1 /D effects. D, N limit
It is easy to see that both S and S admit a ’tHooft-like expansion in which˜ λ = λD = g N D = ˜ g N, (3.12)is kept fixed. We obtain an expansion of the sort S DN = (cid:18) S , + 1 N S , + · · · (cid:19) + 1 D (cid:18) S , + 1 N S , + · · · (cid:19) + · · · . (3.13)10n the diagrammatic evaluation described in the Appendices, such an expan-sion indeed corresponds to a topological expansion. Explicitly, from (3.11),we get S DN = B λ + 14 log − B ˜ λ + 1 D ˜ λB − ˜ λB ! + 12 log − ˜ λB ! + O (cid:0) /D (cid:1) + O (cid:0) /N (cid:1) . (3.14) We are left with evaluating (3.8). Because of the appearance of an overallfactor of N in (3.14), we can evaluate (3.8) using a saddle point method(see a similar calculation presented in Appendix C in a toy example). Thesaddle point value is given by B = i △ , △ = 2˜ λ (cid:18) D (cid:19) + O (1 /D ) . (3.15)The same result is also derived in [6] in a slightly different manner. [6]also performed a numerical analysis which agrees with the above analyticalcalculation and also with the numerical calculations of [22, 23].Using the above saddle point, we get the free energy, F = − log ZDN = −
14 + log 24 + 1 D (cid:18) −
58 + 12 log 32 (cid:19) + O (cid:18) D (cid:19) . (3.16) d = 1 model: result After gaining some experience with the d = 0 matrix model, we now returnto the more involved case, the d = 1 model. We start with (2.15), and as withEqns. (3.8) and (B.2), we first integrate out the Y I and the b ab to obtain Z = Z dB D A e −S ( B ,A ) , (4.1)where e −S ( B ,A ) = Z D b ab D Y I e − S , (4.2)11ith S defined as in (2.15). Different from the previous section, we considerlarge N case only in this section.It is convenient to parametrize A by choosing a gauge in which A istime-independent and is also diagonal: A ij = α i δ ij . The gauge-invariantcontent of A is then given by the moments u n = 1 N Tr W n = 1 N N X i =1 e inβα i , (4.3)where W is the Wilson loop operator, defined in (1.2). The above gaugefixing gives rise to a Faddeev Popov Jacobian (See [16]) D A = Y i dα i e − S F P , S
F P = N X n n | u n | . (4.4)It is convenient to parametrize B = i △ , since the saddle point valuewill be real in terms of △ as in the d = 0 case (3.15). From now on, we willdenote S ( B , A ) as S ( △ , { u n } ).Note that there is a Jacobian involved in changing from the integrationmeasure over the eigenvalues α i to the integration measure over u n , ¯ u n ; how-ever, it is O ( N ) and is hence subleading compared to the classical actionwhich is O ( N ) [31]. Since in this section we will be concerned with theleading term in the 1 /N expansion, we will ignore this Jacobian. S ( △ , { u n } ) in leading large D As in Section 3, we can ignore the interaction S int in the large D limit. Hencethe leading result of the effective action is obtained by integrating out the Y I from S q in (2.15).We can integrate out Y I by using the propagator studied in Appendix D(following [16]) and obtain D (cid:0) det (cid:0) − D + △ (cid:1)(cid:1) = DN β △ − D ∞ X n =1 x n n | u n | . (4.5)Here x = e − β △ and we have ignored 1 /N terms and irrelevant constant terms.We also ignored a temperature dependent divergent term.12ombining the above equation with S from (2.15), and adding the con-tribution from (4.4), we get S ( △ , { u n } ) DN = − β △ λ + β △ ∞ X n =1 (cid:18) /D − x n n (cid:19) | u n | . (4.6)The 1 /D term comes from (4.4). The reason it is kept here is that near thecritical temperature this term will turn out to be more significant than other O (1 /D ) terms from S int which we will encounter in the next subsection.Our task is to evaluate (4.1), with S ( B , A ) = S ( △ , { u n } ) given above.It is useful to first perform the integral over B , using a saddle point methodsimilar to Section 3. In other words, for fixed external u n , let us now solvethe saddle point equation − △ λ + 12 + ∞ X n =1 e − nβ △ | u n | = 0 . (4.7)It is difficult to solve this equation for △ ( { u n } ) exactly. However, for small u n , it is possible to solve it in a power series in the u n . This leads to thefollowing saddle point solution △ ( { u n } ) = ˜ λ / ∞ X n =1 ¯ x n | u n | ! + · · · , (4.8)where ¯ x = exp[ − β ˜ λ / ] . (4.9)Substituting (4.8) in (4.6), we get a Landau-Ginzburg type effective actionfor the u n : S ( { u n } ) DN = 38 β ˜ λ / + a | u | + b | u | + ∞ X n =2 a n | u n | + · · · ,a n = 1 n (1 /D − ¯ x n ) ,b = 13 β ˜ λ / ¯ x , (4.10)13 S /DN | u | / T < T c T = T c T c < T < T c T = T c T > T c Figure 1: Phase transitions: S vs | u | (see (4.10), (4.26)). As T crosses T c , u becomes tachyonic and there is a second order phase transition whichsignals an onset of non-uniformity in the eigenvalue distribution ρ ( α ). At T = T c , characterized by a potential minimum at | u | = 1 /
2, a gap developsin the eigenvalue distribution, signalling a GWW transition.where the · · · involve other u n terms for n >
1, which are ignored for reasonsstated below.Let us analyze the phase structure of the theory by using (4.10) (seeFigure 1). Our analysis will be similar to [16]. Note that for ¯ x < /D all‘ a n ’s are positive. This implies that { u n = 0 ∀ n = 1 , , ... } is a minimum ofthe potential. Recall that u = 0 = Tr W corresponds to an analog of the confinementphase in gauge theory. The vanishing of all u n also has a familiar interpre-tation. Let us define an eigenvalue density of the Wilson line (1.2) by ρ ( α ) = 1 N N X n =1 δ ( α − α i ) . (4.11) We will show below that inclusion of higher loop terms does not change the natureof phase transitions, although it changes the critical temperature and numerical values ofvarious thermodynamical quantities. The issue of whether it is a local or a global minimum is more subtle, and depends ondetails of higher order terms in (4.10). We will argue below that in a 1 /D expansion thehigher order terms can be ignored and u n = 0 is a global minimum.
14n terms of this, the u n are given by u n = Z πβ dαρ ( α ) e − inβα . (4.12)The vanishing of all u n therefore corresponds to a uniform eigenvalue distri-bution.Let us now consider the effect of increasing the temperature, or equiva-lently increasing ¯ x . As ¯ x crosses 1 /D , i.e. T crosses a critical temperaturegiven by T c = ˜ λ / log D = λ / D / log D , (4.13)the sign of a in (4.10) flips, while the coefficient of | u | and the coefficientsof all | u n | , n > | u n | ’s remain zero for n > whereas | u | assumes a small non-zero value h| u |i = r − a b = s (3 D log D ) δT λ / − √ D (log D ) / (cid:18) δT ˜ λ / (cid:19) / + · · · , (4.14)at T = T c + δT for small and positive δT . The order of the phase transition can be determined by studying the freeenergy. It is easy to show that, for small δT , the Landau Ginzburg freeenergy is of the form S / ( DN ) = constant + constant( δT ) Θ( δT ) + O (cid:18) D , N (cid:19) . (4.15)The second derivative of the above function with respect to δT is discon-tinuous at δT = 0. Thus we have a second order phase transition. Weshould remark that the transition is characteristic of the large N limit andis expected to be smoothened at finite N , as has been argued in [16]. Because of couplings such as u − u discussed in Section 4.2, higher u n ’s pick up somenon-zero values at higher orders in 1 /D . However, these can be ignored in the presentdiscussion. δT here is assumed to be small enough such that h| u |i satisfies the bound | u | ≤ /
15s we increase the temperature further, we encounter another phase tran-sition. To understand this phase transition, we first note that when u n = 0for all n > x does not cross 1 / √ D ),the eigenvalue density can be represented as ρ ( α ) = β π (1 + 2 | u | cos( βα )) . As | u | increases from small values to 1 /
2, the eigenvalue density vanishes at βα = π . In the present case, the saddle point value h| u |i , (4.14), reachesthe value 1 / T equals a critical temperature T c = T c + ˜ λ / D log D . (4.16)As T crosses T c we have a GWW type phase transition [26, 27] from a gaplesseigenvalue distribution to a gapped one. The nature of this transition hasbeen discussed in detail in [16, 17], where a Landau-Ginzburg potential ofthe form (4.10) was assumed, with vanishing u n , n >
1. Our analysis abovesupports this assumption, hence we can use their analysis. Following Eqn.(6.18) in [16], we find that for temperatures just above T c , | u | grows as | u | = 12 + log D D s D ˜ λ / ( T − T c ) − + O (cid:18) D (cid:19) . (4.17)By comparing this equation with the form of | u | below T c (4.14), we findthat the second derivative of | u | (or equivalently the third derivative of theLandau Ginzburg free energy) with respect to temperature is discontinuousat T c , although the first derivative or the value of | u | is continuous. To beprecise, we find | u | = 14 + 32 D log D T − T c ˜ λ / − D (log D ) (cid:18) T − T c ˜ λ / (cid:19) + · · · , T < ∼ T c | u | = 14 + 32 D log D T − T c ˜ λ / − D log D (cid:18) T − T c ˜ λ / (cid:19) + · · · , T > ∼ T c (4.18) By a suitable shift of the origin of the angle α to absorb the phase of u . The rate of change of | u | below T c is given by the expansion parameter (log D ) | T − T c | / ˜ λ / while above T c it is given by the expansion parameter D | T − T c | / ˜ λ / . Hence | u | changes at a much faster rate above T c . c T | u | / T c Figure 2: Phase transitions: | u | vs T . As T crosses T c , | u | starts growingfrom zero and equals 1 / T = T c . The transition at T c is a third orderGWW type transition. Because of the sharp change in the second derivativeof | u | across T c within a very small range of temperature (see Eqn. (4.18)and also footnote 15), it almost appears like a discontinuity in the slope ofthe plot. However, when we zoom into a small temperature interval around T c , the ∂ | u | /∂T is seen to be continuous, as is analytically evident from(4.18). Although we have plotted | u | vs T here, to facilitate comparisonwith [24], a plot of | u | vs T shows exactly similar features.where we have ignored corrections of O (1 /D ) and terms proportional to( T − T c ) and higher. Thus, the phase transition at T = T c is third order,as in the original GWW transition. (See Figure. 2.)Beyond T c it is in principle possible to have further phase transitions tomultiple-gap phases. In case of unitary matrix models with general singletrace actions of the form P n c n u n +c . c . it was shown in [32, 33] that additionalgaps can open up in the eigenvalue distribution as the temperature is varied.We will not attempt to study this issue in this paper, since the analysis inthe relevant ranges of temperatures is complicated. The numerical analysisof [24], appears to suggest, however, that the only phase transitions in themodel are the two already discussed above. High temperature:
Once the temperature becomes very high, β eff ≪ u n = 1. Thesaddle point equation (4.7) then becomes − △ λ + 12 + e − β △ − e − β △ = 0 . (4.19)17e can approximately solve it for small β as, △ = λβ ! / . (4.20)This result is consistent with the d = 0 condensate in (3.15) by identifying˜ λ | d =0 as ˜ λ/β .We end this subsection with a few comments: (a) Small | u | approximation: Since the second phase transition hap-pens when | u | reaches 1/2, one may worry about the validity of small u n approximation in (4.8). There is no problem, however, since terms involv-ing u n in the saddle point equation (4.7) are also suppressed by x n ∼ /D n around the critical temperature. Therefore, even if | u | is not small, ouranalysis is valid. (b) Large D limit vs /D expansion: It is clear from the phase transi-tion temperatures (4.13) and (4.16) that the phase transitions disappear inthe large D limit. Hence we should simply regard D as large but not takethe strict D → ∞ limit if we want to explore criticality. /D correction to the effective action In the previous section, we have considered the effective theory (4.6) includingthe 1 /D term (4.4) from the gauge fixing and discussed the phase transition.However, in addition to this 1 /D term, other 1 /D corrections can arise from S int in (2.15). It corresponds to S , in (3.13) in the 0 dimensional model.Hence we have to evaluate them and show that these corrections are sub-dominant around the critical point.The terms we should look for at this order, in so far as the issue of phasetransitions is concerned, are as follows. Besides the explicit corrections to | u | and | u | in (4.10), the corrections to the gauge-field independent termsare also relevant, since they contribute to the saddle point equation (4.7).Interaction terms like u u − also affect the Landau-Ginzburg type potential(4.10) by generating an effective | u | term. However as we show below18qn. (E.12), the corrections to the coefficient of | u | from these interactionsare order 1 /D and we ignore them here . Thus the relevant terms in theeffective action are S ( △ , { u n } ) / ( DN ) = C + C | u | + C | u | + · · · , (4.21)and we can explicitly calculate them, C = − β △ λ + β △ β △ D λ △ ! − − ˜ λ △ ! − ˜ λ △ ! , (4.22) C = (cid:18) D − x (cid:19) + β △ D " ˜ λ △ ! λ △ ! − + ˜ λ △ ˜ λ △ − ˜ λ △ ! − ˜ λ △ ! x + O ( x ) , (4.23) C = β △ D − ˜ λ △ ! λ △ ! − − ˜ λ △ ! x + β △ D (2 + β △ ) − (cid:16) ˜ λ △ (cid:17) (cid:16) ˜ λ △ (cid:17) − ˜ λ △ ! x + O ( x ) . (4.24)Here we have omitted higher x terms, since we are interested in a range oftemperatures below or around the critical temperature x ∼ /D details ofthe derivation are shown in Appendix E. Although the cubic interaction u u − merely renormalizes the coefficient of the | u | term in the Landau-Ginzburg potential (4.10), one needs to be careful about integratingout the u consistent with the positivity constraint of ρ ( α ). Note that the small x expansion is valid for low temperatures. It means that ouranalysis works well for large effective coupling λ eff (2.3) (as long as it does not scale with D ). This assumption is in particular valid around the phase transitions.
19s in (4.8), we solve the saddle point equation for △ in powers of u , toobtain: △ ˜ λ / = 1 + 1 D √ − ! + 23 ¯ x | u | + · · · . (4.25)Here the | u | and higher order terms do not affect (4.26) and are dropped.Substituting this in (4.21), we get S / ( DN ) = β ˜ λ / ǫ + a ′ | u | + b ′ | u | + · · · , (4.26)with ǫ = 38 + 1 D − √ ! , (4.27) a ′ = 1 D − ¯ x − ˜ λ / βD − √ ! ¯ x, (4.28) b ′ = ˜ λ / β x − ˜ λ / βD ˜ λ / β √ − ! + 391 √ − ! ¯ x . (4.29)It is obvious that these equations constitute O (1 /D ) fractional correctionsto various quantities appearing in (4.10).As argued in the previous subsection, the phase transition temperature T c is characterized by vanishing of a ′ and T c is given by b ′ = − a ′ . Thisgives us the following corrected values of the transition temperatures: β c ˜ λ / = log D D − √ !! . (4.30) β c ˜ λ / − β c ˜ λ / = log DD " −
16 + 1 D − √ ! log D − √ ! . (4.31)Although the analysis in this subsection leads to subleading corrections tothe phase transition temperatures, it is easy to see that the nature of thephase transitions derived in the previous subsection remains unaltered.20 .3 /D expansion vs. numerical calculation In this section, we evaluate the critical temperatures and a few other quanti-ties, using results in the previous subsections in the large D expansion, andcompare them with the numerical results for D = 9, which were studied in[22, 23, 24, 25].The d = 1 model was numerically analyzed in [22, 23] (see Section 5 forconnection to D branes) where it was suggested that the system undergoes aweakly first order Gross-Witten-Wadia type transition (characterized by thedevelopment of a gap in the eigenvalue distribution ρ ( α )). A more detailednumerical study [24] subsequently claimed that in stead of a single first ordertransition, it consists of two higher order phase transitions: (a) from uniformto non-uniform ρ ( α ), followed closely by (b) a GWW type transition in whicha gap appears. This is in agreement with the picture of the two transitions wederived in the previous subsections (see Figure 3). Let us compare betweenour results and those of [24] in some detail.We first compare the two critical temperatures derived from the numericalanalysis in [24] with our large D expansion. In order to do it, we note thatthe dimensionless temperature defined in [22, 23, 24, 25] and (2.2) is T eff ≡ β eff = 1 λ / β = D / ˜ λ / β . (4.32)In the units λ = g N = 1, used by [24], T eff is simply T . By employing thesame unit, we obtain the critical temperatures as in Table 1. The leadingorder results in the 1 /D expansion are from Eqns. (4.13) and (4.16), and thenext order is from Eqns. (4.30) and (4.31).Similarly, we can also compare the value of the condensate △ and thefree energy in the confinement phase, which are given by R ≡ g N h Tr Y I Y I i| β →∞ = 12 △ | β →∞ , F ≡ − βN log Z | β →∞ . (4.33)Those can be derived from (4.8) and (4.10) in the leading order, and (4.25)and (4.26) in the next order. The results are also summarized in table 1 andour results agree with the numerical result remarkably well . Note that we call the critical temperature from uniform to non-uniform distributionas T c and the next GWW type as T c . However, in [24], they used the opposite notation. R and F in (4.33) are defined as r and ǫ in [24]. c T c R F Numerical result 0.8761 0.905 2.291 6.695Leading large D result 0.947 0.964 2.16 7.02Large D including 1 /D effect 0.895 0.917 2.28 6.72Table 1: Comparison with numerical results derived in [24] for D = 9 andour large D analysis. Here we have used λ = g N = 1 units. We list thecritical temperature T c and T c , and the condensation and the free energy atthe confinement phase defined in (4.33). The first line is the numerical result.The values in the second line are the leading large D results. The third line isthe result including the first 1 /D correction. The fractional differences fromthe numerical results can be checked to be order 1 /D in the second line and1 /D in the third line, as expected.We note here that although we find excellent quantitative agreement with[24], the more qualitative inferences in [24] regarding the order of the phasetransitions are different from ours. The phase transitions at T c and T c areclaimed in [24] to be of 3rd order and 2nd order, respectively, while in ouranalysis they are the other way around. We believe that this difference maybe due to the fact that in numerical work it is not easy to ascertain the orderof a transition except when it is a strong first order transition. The transitionat T c in our analysis is described by a classic Landau Ginzburg potentialwhich describes a second order transition. For a LG potential involving only u to describe a third order transition as suggested in [24], we need a u termwhich is disallowed by the symmetries of the theory. Likewise, a second ordertransition at T c is inferred in [24] by noting a jump in ∂ | u | /∂T . We find, onthe other hand, that there is a sharp, but continuous change in this quantity(see Figure 2 and Eqn. (4.18)). We note that our analysis, of course, ignorescorrections of order 1 /D ; however, we do not expect any qualitative changesin the above conclusions for large values of D such as D = 9 . J. Nishimura informed us that he agrees with the conclusion obtained in this paperand that the numerical data in [24] around T c are also consistent with a second orderphase transition. He also mentioned that fitting the data with that assumption leads toa slight increase in their estimate on T c , which further improves the agreement with ourvalue of T c . The estimate of T c in [24], on the other hand, does not depend on theassumed order of transition in their analysis. Thus, the agreement with our value of T c is not affected. We thank J. Nishimura for providing us with the results of this reanalysis. /D expansion for small D : Ref. [25] also numerically analysed thetransition for D = 2 and 3, and found two transitions as in the D = 9 casein [24]. In our study too, we have two transitions for D = 2 and 3, since b ′ in (4.29) is positive even for these values of D . We summarise these resultsas follows: T c ( D = 2) T c ( D = 2) T c ( D = 3) T c ( D = 3)Our result 1.4 1.6 1.1 1.2Numerical result 1.12 1.3 0.93 1.1The critical temperatures in their numerical results are close to our results . Our phase transition temperatures also show agreement with the criticaltemperature numerically evaluated in Ref. [23] for D = 4.These results suggest that our analysis seems to work even for such smallvalues of D . However, we do not have a detailed understanding of such anunexpected agreement. The d = 1 model (2.1) appears in various contexts, as mentioned in theIntroduction. The context closest to the contents of this paper is that of[22, 23, 24], which we briefly review in this section.Let us consider thermal D0 branes in R × S . The distribution of thebranes is dynamically determined and for a certain parameter region, thegeometry becomes a black string winding around the S . If we increase theradius of the S beyond a critical radius, the Gregory-Laflamme instabilitymode [34] appears and a black hole solution, localized on the S , is favoured.It is argued in [22] (see also [35]) that this black string/black hole transitionis first order.Through gauge/gravity duality [7], we expect this transition to be repro-duced by a d = 1 thermal SYM with a compact adjoint scalar at strong’tHooft coupling. By using a T-duality [36], this model can be mapped to a Their interpretation of the order of the phase transitions is the same as in [24], andis different from ours. The explanation of this discrepancy is similar to the D = 9 case asmentioned above. niformnon-uniformgapped2d SYM 1d YM0d YM t ′ λ ′ λ ′ = t ′ λ ′ = 1 /t ′
1d SYM
Figure 3: Phase diagram of the d = 2 SYM (5.1). Below λ ′ = t ′ , thetemporal KK modes can be ignored and below λ ′ = 1 /t ′ , the spatial KKmodes can be ignored. Thus the effective d = 1 (bosonic) YM descriptionis valid on the right of the curve λ ′ = t ′ . The overlap of this region withthe region below λ ′ = 1 /t ′ additionally admits an effective d = 0 description.The two phase transition lines below λ ′ = t ′ are given by λ ′ t ′ = 1 /T c and λ ′ t ′ = 1 /T c , where T c ,c are given in (4.30) and (4.31). A similar phasestructure was earlier inferred in [24] on the basis of numerical analysis.2d SYM on T S = 1 g Z L dx Z β dt Tr F µν + 12 X I =1 D µ Y I D µ Y I − X I,J [ Y I , Y J ] ! + fermions . (5.1)This theory is characterized by two independent dimensionless constants: (a) λ ′ = λ L where λ = g N is the ’tHooft coupling, and (b) t ′ = L/β , thedimensionless temperature. However, the analysis of this theory at strong’tHooft coupling is difficult. Instead of investigating the above transition atstrong coupling, the gauge theory allows us to study a continuation of thephase transition to weak ’tHooft coupling.It has been argued in [22, 23] that in the range of temperatures givenby λ ′ / < t ′ < /λ ′ all fermionic modes as well as both the spatial andtemporal KK modes can be ignored. The theory is then governed by just thezero modes which describe the d = 0 model studied in Section 3. (See Figure3). As λ ′ t ′ grows to order unity, the spatial KK modes cannot be ignored24ny more, though the temporal KK modes can still be ignored. In fact, inthe range of temperatures and coupling t ′ > λ ′ , the theory (5.1) reduces tothe d = 1 model (2.1) with the spatial circle of length L identified with β of (2.1). The d = 1 ’tHooft coupling λ is identified with λ /β ′ so that λ eff appearing in (2.3) is identified as λ eff = λ L /β = λ ′ t ′ . (5.2)Note that the transitions in the d = 1 model, which we studied in Section4, happen around λ eff = 1 /T ≈ . d = 2 model (5.1) if t ′ > λ eff . Thuswe can reliably expect these transitions to be the continuation of the blackstring/black hole transition to weak ’tHooft coupling .As has been suggested first in [37], the eigenvalue distribution of theWilson loop (4.11) is related to the geometry of the D0 branes. A uniform(non-uniform) gapless eigenvalue distribution corresponds to a uniform (non-uniform) black string winding around the S , whereas a gapped distributioncorresponds to a black hole localized on the S . Now, we found in Section 4that the uniform eigenvalue distribution is favoured at low temperature anda gapped distribution is favoured at high temperature while a non-uniformdistribution exists between those two phases. Since the temperature in the d = 1 model is mapped to the radius of the original S , there should be aphase transition from a black string to a black hole as the radius of the circleis increased, which indeed is the case. The fact that our transition consists oftwo closely spaced transitions disagrees, however, with the single first ordertransition in the gravity description. A plausible resolution is as follows. It iseasy to see that if b in (4.10) is negative when a vanishes, there is only one,first order, phase transition instead of the two transitions [16]. Therefore thegravity picture can be reconciled with the gauge theory calculation if the signof b in (4.10) flips at some higher value of coupling in the two dimensionalmodel. At such a value the two phase transition lines found at weak couplingwill merge and yield a single first order transition line (see Figure 3). The “weak” coupling here refers to the d = 2 coupling λ ′ which satisfies λ ′ < t ′ (seeFigure 3). We should remark that the analysis in this paper is valid even for large valuesof the d = 1 ’tHooft coupling λ eff as we explained in footnote 17. The equivalence withthe d = 2 model, however, is valid only for temperatures t ′ > λ eff . Conclusion
In this paper we have developed a technique of solving matrix models ( d =0 ,
1) which are dimensional reductions of D + d dimensional bosonic YMtheory to d dimensions. The technique involves working in a 1 /D expan-sion, which allows us to analytically compute free energies and other ther-modynamic quantities. In the d = 1 case our results show that the systemundergoes a double phase transition: a second order phase transition whichsignals onset of a non-uniformity of the eigenvalue distribution ρ ( α ) of theWilson line, followed by a third order GWW phase transition which signalsdevelopment of a gap in ρ ( α ). Following the arguments in [22, 23, 24, 25], weinterpreted this double transition as a continuation of the Gregory-Laflamme(black string/black hole) phase transition to weak coupling. Our results agreewith the numerical results of [24], and offers an analytic resolution of the issueof the order of the phase transitions.The large D technique developed in this paper is in principle applicableto a variety of bosonic matrix models involving commutator-squared inter-actions. The applicability of our techniques would be greatly enhanced if weare able to extend them to higher dimensional models d > D µ + iB ) with dynamical gaugefields without making a coupling constant expansion. One possibility is tofind regions of parameter space in which an effective d = 1 description arises(as in the d = 2 toroidal model described in Section 5) and work around thatlimit. The supersymmetric extension of the large D methods appears morechallenging, even qualitatively, since the number of bosons and fermions growat different rates as D grows large. We hope to come back to some of theseissues in a future publication.In this paper we have been concerned with thermodynamic properties ofthe matrix models. Another possible application of our methods could be toaddress dynamical questions. Indeed, one of the motivations of this paperwas to apply these techniques to derive an effective action for gauge fields inthe time-dependent context and to study dynamical phase transitions usingthis effective action. Work in this direction is in progress [30].26 cknowledgement We would like to thank Spenta Wadia for collaboration in an ongoing work ondynamical black hole/black string transitions [30] which inspired the presentpaper, and for sharing numerous insights. We would like to thank AdelAwad, Avinash Dhar, Sumit Das, Ian Ellwood, Barak Kol, Oleg Lunin, SamirMathur, Jun Nishimura, Toby Wiseman and especially Shiraz Minwalla foruseful discussions. We would like to thank Jun Nishimura for sharing with usthe result a reanalysis of [24] which further improves the agreement with ouranalysis. T.M. would like to thank the theory group at KEK for their kindhospitality, where part of this work was done. G.M. would like to thank theorganizers of the Benasque conference on Gravity (July 2009), the organizersof the QTS6 meeting in Lexington, the University of Kentucky, Lexingtonand the School of Natural Sciences, IAS, Princeton for hospitality duringpart of this project.
A Some results involving M ab,cd In this section, we will calculate several quantities involving M ab,cd , for exam-ple, Tr M n , M − , the eigenvalues of M etc. which are important in solvingboth the d = 0 and the d = 1 models. We begin by first investigatingalgebraic properties of M ab,cd . A.1 Algebraic properties of M ab,cd As in (2.7), M ab,cd is defined as M ab,cd = − n Tr[ λ a , λ c ][ λ b , λ d ] + ( a ↔ b ) + ( c ↔ d ) + ( a ↔ b, c ↔ d ) o . (A.1) M ab,cd has four adjoint indices and is symmetric under the interchanges a ↔ b and c ↔ d . Hence, in the SU ( N ) case, we can regard M ab,cd as an N ( N − / × N ( N − / ab and cd as two single indices.Equivalently, we can regard M as an operator acting on the N ( N − / V B (whose elements can be regarded as B ab ) labeledby a symmetric pair of adjoint indices ab , on to the same vector space ( i.e M is an endomorphism of the vector space V B ).27 (cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:32)(cid:3)(cid:3)(cid:20)(cid:3)(cid:14)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:14)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:14)(cid:3)(cid:3) N -1 N -2 N -1 N -1 N -1 N -1 Figure 4: Irreducible decomposition of the N ( N − / V B labelled by a symmetric pair of adjoint indices ab , regarded as asymmetric product of ‘adjoint’ × ‘adjoint’. The dimensions of the representa-tions in the RHS are 1, N − N ( N + 1)( N − / N ( N − N + 3) / K , K , K and K . Note that SU (2) and SU (3) are exceptional.In SU (2) the second and the third representations are absent and in SU (3)the third one is absent [38].To proceed, let us decompose this N ( N − / V B into irreducible representations of SU ( N ). The decomposition is shownin Figure 4 and we obtain four irreducible representations. Let us define avector in this space as B ab , then the first in Figure 4 is the ‘trace’ part B aa and the latter three constitute irreducible decomposition of the symmetric‘traceless’ vector. Correspondingly we can define four projection operators K iab,cd ( i = 0 , , ,
3) acting on this vector space such that K iab,cd B cd belongsto the i -th irreducible representation, B ab = ( K B ) ab + ( K B ) ab + ( K B ) ab + ( K B ) ab . We will show that the endomorphism M ab,cd can also be decomposed followingthe above equation and hence can be represented as a linear combination ofthe K i (see (A.10)).It is possible to construct the K i explicitly. Let us define the following Here ‘trace’ of a vector B ab means the sum of the two adjoint indices. e.g. B aa . F ab,cd = 14 (cid:2) Tr (cid:0) λ a λ b λ c λ d (cid:1) + ( a ↔ b ) + ( c ↔ d ) + ( a ↔ b, c ↔ d ) (cid:3) ,G ab,cd = 12 (cid:2) Tr (cid:0) λ a λ c λ b λ d (cid:1) + ( a ↔ b ) (cid:3) ,H ab,cd = δ ab δ cd ,I ab,cd = 12 ( δ ac δ bd + δ ad δ bc ) . (A.2)Those matrices satisfy the following relations, F = 12 (cid:18) N (cid:19) H + N (cid:18) − N (cid:19) F, F G = − N F + 1 N H,F H = N (1 − N ) H, G = I + 1 N H − N F, GH = − N H,H = ( N − H, F I = F, GI = G, HI = H , I = I (A.3)where the product AB is defined by ( AB ) ab,ef = A ab,cd B cd,ef . Because of thecyclic property of the trace, this product satisfies AB = BA . Note that I ab,cd plays the role of the identity matrix. In order to derive these relations, wehave employed the identity N − X a =1 λ aij λ akl = δ il δ jk − N δ ij δ kl , (A.4)where i, j are fundamental indices.By using these matrices, the four projection operators are represented as: K ≡ N − H, K ≡ NN − (cid:18) F − N H (cid:19) ,K ≡ (cid:18) I − G − N − (cid:18) F − N H (cid:19) − N ( N − H (cid:19) ,K ≡ (cid:18) I + G − N + 2 (cid:18) F − N H (cid:19) − N ( N + 1) H (cid:19) . (A.5)We can show that these matrices satisfy, K i K j = δ ij K i , (A.6)29nd I ab,cd = K ab,cd + K ab,cd + K ab,cd + K ab,cd . (A.7)Thus the K i are indeed projection operators. Actually, we can find the cor-respondence between K i defined in (A.5) and the irreducible representationsin Figure 4. For example, K ab,cd acts on a vector B cd as K ab,cd B cd = 1 N − δ ab B cc . (A.8)Thus K maps the vector to the ‘trace’ B cc . Therefore K is the projectionoperator corresponding to in the RHS of Figure 4. Similarly we can findthe correspondence for other K i .In addition to such explicit identifications, we can calculate the traces of K i as K ab,ab = 1 , K ab,ab = N − ,K ab,ab = N ( N + 1)( N − , K ab,ab = N ( N − N + 3)4 . (A.9)The values of the traces are equivalent to the dimensions of the irreduciblerepresentations shown in Figure 4. This is another evidence for the corre-spondence between K i and those representations.By using (A.1), (A.2) and (A.5), M ab,cd can be described as M ab,cd = 2( F ab,cd − G ab,cd )= 2 N K ab,cd + N K ab,cd + 2 K ab,cd − K ab,cd . (A.10) A.2 Results involving M ab,cd In this subsection we use the tools developed above to calculate several quan-tities associated with M ab,cd necessary in the study of the matrix models inthis paper. Here ‘trace’ of an endomorphism matrix A ab,cd is defined as Tr A ≡ A ab,ab he inverse M − : Equation (2.8) assumes the existence of M − . From(A.7) and (A.10), we explicitly obtain M − ab,cd = 12 N K ab,cd + 1 N K ab,cd + 12 K ab,cd − K ab,cd . (A.11)By using this, we can calculate M − ab,cc , which is necessary to derive (2.12)and (2.15). First we can show K iab,cc = 0 for i = 0, since the irreduciblerepresentations in Figure 4 are ‘traceless’ except . Then we obtain M − ab,cc = 12 N K ab,cc = 12 N δ ab . (A.12) Eigenvalue of M ab,cd : We now derive the eigenvector and eigenvalue of M ab,cd . By using (A.10), we obtain the eigenvector as M ab,cd { ( K B ) cd + ( K B ) cd + ( K B ) cd + ( K B ) cd } =2 N ( K B ) ab + N ( K B ) ab + 2( K B ) ab − K B ) ab . (A.13)where B ab is a general vector. Note that the eigenvalue of ( K B ) ab is nega-tive, which makes the action (2.8) not positive definite. As we have remarkedin footnote 4, this can be dealt with by making appropriate choices for inte-gration contours for different elements of B ab . Calculation of Tr M n : We now calculate Tr M n = M nab,ab , which appears inthe loop calculation of the matrix model. By using Eqns. (A.6) and (A.10),we obtainTr M n = (2 N ) n Tr K + N n Tr K + 2 n Tr K + ( − n Tr K . (A.14)In the large N limit, we obtain by using (A.9)Tr M = − N , Tr M = 3 N , Tr M n = N n +2 ( n ≥ . (A.15)It is also possible to calculate the effective action for finite N in the d = 0model. To do this, note that in the loop diagrams including b ab , the vector b ab satisfies the ‘traceless’ condition b aa = 0 as in (2.13). This condition changesthe propagator for b ab with M replaced by M ′ : M ′ ab,cd = N K ab,cd + 2 K ab,cd − K ab,cd . (A.16)31igure 5: Feynman rules for the matrix model (3.3).Hence the b ab -loops actually involve Tr M ′ n which are given byTr M ′ n = N n ( N −
1) + 2 n N ( N + 1)( N − − n N ( N − N + 3)4 . (A.17)Note that it gives the same value to (A.15) under the large N limit and thetraceless condition indeed affects only finite N correction. B Details of the d = 0 model In this Appendix, we will present details pertaining to the d = 0 model ofSection 3. Specifically, we will derive (3.11).We start with (3.3). The Feynman rules of (3.3) are shown in Figure 5.The propagators are given by h Y Ia Y Jb i = iB δ ab δ IJ , h b ab b cd i = 2 M ′ ab,cd . (B.1)The matrix M ′ ab,cd is defined in (A.16), which is the propagator for b ab satisfy-ing the traceless condition b aa = 0. Note that M ′ ab,cd is obtained by removing K from M ab,cd , where K is the projection operator corresponding to thetrace part b aa (See eq.(A.8)). As remarked below (A.17), the difference be-tween M and M ′ appears only at subleading orders in 1 /N .The effective action S ( B ) in (3.8) is formally given byexp[ −S ( B )] = Z dY I dbe − S − S q ∞ X n =1 ( − S int ) n n ! ! = e − S ( − iB ) − D ( N − ∞ X n =1 ( − ) n n ! h S nint i ! . (B.2)32igure 6: Some examples of vacuum diagrams of (B.2). Figure (a) is theleading order in the D expansion ( O ( D )). (b) is next order ( O (1)) and (c)is O (1 /D ) . The diagrams in Figure 8 also contribute to O (1).where we have dropped some irrelevant normalization factor. Thus, S ( B )is given by a sum of all connected vacuum diagrams represented by theabove equation, with external B . The term in the above equation obtainedby putting S int = 0 corresponds to S (0) ( B ) in (3.5). Diagrammatically itcorresponds to the Y I -loop in Figure 6(a) plus terms independent of Y and b . Since each Y -loop gives rise to a factor D , this term is of order O ( D ),consistent with the arguments in Section 3.1.Therefore, we obtain S in (3.10) as, S /D = − N △ g + N −
14 log (cid:18) △ ˜ g (cid:19) , (B.3)where ˜ g = g D , as defined in (3.6). Here we have used the notation B = i △ , (B.4)in anticipation of the fact that the saddle point value (3.15) will be given interms of real △ . B.1 Calculation of S : the /D correction Now we calculate S in (3.10).We can obtain S directly by using the full propagator for h bb i , as inSection 3.1. However we calculate it diagrammatically here , since this The connection with Section 3.1 can be made by the formal Schwinger-Dyson sum S (2) = M + M GM + M GM GM + · · · where S (2) is the quantity appearing in (3.5), while G = G (2) appears in (B.5). h bb i propagators. The correction to h bb i is order ˜ g . The 1PI propagator comes from this diagram only at this orderin the 1 /D expansion.Figure 8: The O (1) corrections to the effective action in the large D expan-sion.derivation will be more convenient. It is easy to show that at leading orderin 1 /D , the relevant correction to the 1PI h bb i propagator comes entirelyfrom the one-loop diagram of Figure 7. Hence only diagrams described inFigure 8 contribute to the effective action. Note that each diagram includesplanar and non-planar structures in the N counting. Higher loop terms arecharacterized by higher powers of the dimensionless quantity ˜ λ/ △ . However,as evident from (3.15), this quantity is order 1. Hence we must sum over allloops, which we describe below.In each diagram in Figure 8, the two Y I s in a vertex b ab Y Ia Y Ib are con-tracted with two Y J s in a different vertex b cd Y Jc Y Jd . Therefore it is convenient34o define a composite propagator G (2) ab,cd ≡ X I,J (cid:0) h Y Ia Y Jc ih Y Ib Y Jd i + h Y Ia Y Jd ih Y Ib Y Jc i (cid:1) = 2 D △ I ab,cd , (B.5)where I ab,cd is defined in (A.2) and it satisfies M ′ ab,cd I cd,ef = M ′ ab,ef . Thiscomposite propagator corresponds to one double line loop in Figure 7 and 8.By using this propagator, we can calculate the ( n + 1)-th loop correction to N S as − n )! h S nint i c = − n )! (cid:18) − ig (cid:19) n h b a b Y I a Y I b · · · b a n b n Y I n a n Y I n b n i c = − ( − ) n n (cid:18) g D △ (cid:19) n M ′ a b ,a b I a b ,a b M ′ a b ,a b · · · I a n b n ,a b = − ( − ) n n (cid:18) g D △ (cid:19) n Tr M ′ n . (B.6)Here h· · · i c denotes the connected diagram. Tr ′ M n has been calculated in(A.17). Now we can sum over n and obtain the effective action, S /D = − N △ g + ( N − (cid:18) △ ˜ g N (cid:19) + N − D log (cid:18) g N △ (cid:19) + N ( N + 1)( N − D log (cid:18) g △ (cid:19) + N ( N − N + 3)8 D log (cid:18) − g △ (cid:19) + O (cid:18) D (cid:19) . (B.7) C Evaluation of a toy integral using a com-plex saddle point
In Section 3, we evaluated the partition function (3.8) by a saddle pointmethod. A similar calculation was done in Section 4. In this Appendix, weillustrate the procedure by considering a toy example.Let us consider the integral I = Z ∞−∞ dye − ay − by ( a, b > . (C.1) See footnote 24 for another motivation for defining this propagator. b is small we can expand in powers of b and obtain, I = r πa − b √ π a / + · · · . (C.2)Now we try to solve this integral by using the auxiliary variable. We canrewrite the integral as1 √ πb Z ∞−∞ dydx exp (cid:18) − ay − x b + 2 ixy (cid:19) = 1 √ b Z ∞−∞ dx exp (cid:18) − x b −
12 log( a − ix ) (cid:19) . (C.3)Let us try to evaluate this integral by using the saddle point method, in thelimit b →
0. The exponent has two saddle points x = − ia/ ± i √ a + 4 b/ b = 0, the extremum is at x = 0, we should choose thesaddle point corresponding to the “+” sign. We get I = r πa − b √ π a / + · · · , (C.4)reproducing the earlier expression. D The Y -Propagator for d = 1 In this section, we derive the Y I propagator in the d = 1 model. The kineticterm of Y I in (2.15) can be written as, Z β dt Tr 12 (cid:2) Y I (cid:0) − D + △ (cid:1) Y I (cid:3) = β X i,j,n Y Inij (cid:18) π n β − πn ( α j − α i ) β + ( α j − α i ) + △ (cid:19) Y I − nji , (D.1)where we have used the constant diagonal gauge A ij = α i δ ij and we haveexpanded Y I ( t ) = P n Y In e πin/β . We have also used the notation B = i △ as in Section 4. Hence the propagator for each mode is given by h Y Inij Y Jmkl i = 1 β π n β − πn ( α j − α i ) β + ( α j − α i ) + △ δ il δ jk δ IJ δ n + m, . (D.2)36hen the propagator for Y I ( t ) becomes h Y Iij ( t ) Y Jkl (0) i = X n β e πinβ t π n β − πn ( α j − α i ) β + ( α j − α i ) + △ δ il δ jk δ IJ = X n − i π △ " e πinβ t n − β ( α j − α i )2 π − i β △ π − e πinβ t n − β ( α j − α i )2 π + i β △ π δ il δ jk δ IJ = e i ( α j − α i ) || t || △ " e −△|| t || − e iβ ( α j − α i ) e − β △ − e △|| t || − e iβ ( α j − α i ) e β △ δ il δ jk δ IJ . (D.3)Here || t || denotes || t + nβ || = t for 0 ≤ t < β . In order to derive it, we haveused the formulae , ∞ X n = −∞ sin( a − n ) xa − n = π, ∞ X n = −∞ cos( a − n ) xa − n = π cot( πa ) . (D.4)We can write the expression (D.3) further as h Y Iij ( t ) Y Jkl (0) i =12 △ " e ( i ( α j − α i ) −△ ) || t || ∞ X n =0 x n u jn u i − n + e ( − i ( α j − α i ) −△ )( β −|| t || ) ∞ X n =0 x n u j − n u in δ il δ jk δ IJ , (D.5)where x = e − β △ and u in = e iβnα i which satisfies P Ni =1 u in = N u n . E All loop corrections up to /D in the d = 1 model In this appendix, we will show the derivation of the effective action of the d = 1 model including the leading 1 /D correction (4.21). This correctioncorresponds to S , in (3.13) in the 0 dimensional model. Even in d = 1model (2.15), the same diagrams as in the 0 dimensional model (Figure 8) Eqns. (D.4), (E.25) and (E.33) are shown in [39]. X j,p,I,J h Y Iij ( t ) Y Jpq ( t ′ ) ih Y Ijk ( t ) Y Jlp ( t ′ ) i ≡ DN X n G (2) n,ik e i πnβ ( t − t ′ ) δ iq δ kl . (E.1)Note that in this propagator we have only taken into account contractionswhich corresponds to planar diagrams. It turns out that the effective actionobtained in this way corresponds to the leading term in a 1 /N expansion. Wewill make a brief remark about non-planar terms at the end of this Appendix.We can calculate this composite propagator by using (D.5) and a formulafor Fourier integrals involving || t || ,1 β Z β Z β dtdt ′ e s || t − t ′ || e − πinβ t e − πimβ t ′ = δ n + m, e sβ − sβ − πin . (E.2)Then the composite propagator can be obtained as G (2) n,ik = 18 △ (cid:0) P − n,ik S − ik + P + n,ik S + ik + Q n,ik S Q,ik (cid:1) , (E.3)where the n -independent quantities are given by S − ik = 1 + ∞ X m =1 x m ( u i − m u m + u km u − m ) , (E.4) S + ik = ( S − ik ) ∗ = 1 + ∞ X m =1 x m ( u im u − m + u k − m u m ) , (E.5) S Q,ik = x ∞ X l,m =0 x l + m [ u l + m +1 u i − l u k − m ( u i − − u k − ) + u − ( l + m +1) u il u km ( u k − u i )] , (E.6)and the n -dependent quantities are given by P − n,ik = 1 πi − iβ ( α k − α i ) − △ β πi − n , P + n,ik = 1 πi iβ ( α k − α i )+2 △ β πi − n ,Q n,ik = 1 πi iβ ( α k − α i )2 πi − n . (E.7)38y using the composite propagator (E.1), we can calculate the loop cor-rection to the effective action as we studied in Appendix B. The ( n + 1)-loopcorrection in Figure 8 is given by − d n ( − ) n n (cid:0) βg DN (cid:1) n ∞ X m = −∞ N X i,j =1 (cid:16) G (2) m,ij (cid:17) n , (E.8)where d n is a factor derived from the number of the planar diagrams and wecan fix it by using (A.15), d = − , d = 3 , d n = 1 ( n ≥ . It is difficult to evaluate (E.8) in general. However, we are interested in thetheory around the critical temperature ¯ x ∼ /D where ¯ x is given by (4.9).Hence we can expand the effective potential with respect to x and the lowestorder of x is enough to evaluate the 1 /D correction. Especially, only thecoefficient of x | u | and x | u | and the gauge-field independent terms willgive us the relevant information of the dynamics around the critical points. E.1 Two-loop correction
The two-loop correction to the effective action corresponds to the n = 1 termin (E.8) and is given by − βg DN N X i,j =1 ∞ X m = −∞ G (2) m,ij . (E.9)We sum over the Fourier mode first. ∞ X m = −∞ P − m,ij = 1 + 2 ∞ X m =1 x m u i − m u jm , ∞ X m = −∞ P + m,ij = 1 + 2 ∞ X m =1 x m u im u j − m , ∞ X m = −∞ Q m,ij = u i − + u j − u i − − u j − , (E.10)39here we have used (D.4). After summing over i, j , we obtain the correctionto the two-loop effective action as, S two − loop = − N β △ ˜ λ △ ! ∞ X m =1 ( x m + 2 x m ) | u m | ! + S int , (E.11) S int = − N β △ ˜ λ △ ! (cid:0) x ( u u − + u − u ) + O ( x ) (cid:1) . (E.12)Here we have omitted higher x terms in the interaction, which are irrelevantaround the critical points, as argued before.We notice that this correction includes a cubic interaction x u u − . Sincethe effective action (4.6) has the term | u | / | u | term can be induced through this interaction after integrating out u .However, the coefficient of the | u | terms obtained this way will be O ( x )and we can ignore it compared to the | u | potential in (4.26). Generally, wecan show that the lowest order coefficient of the cubic interaction from thehigher loops is also x and we will ignore them here. E.2 Three-loop correction
We evaluate the three-loop correction (the n = 2 term in (E.8)) − (cid:0) βg DN (cid:1) ∞ X m = −∞ N X i,j =1 (cid:16) G (2) m,ij (cid:17) . (E.13)In order to calculate the sum of the Fourier mode m , we derive the productof P ± and Q as P − m,ij P + m,ij = 12 △ β (cid:0) P − m,ij + P + m,ij (cid:1) ,P − m,ij Q m,ij = 1 △ β (cid:0) P − m,ij + Q m,ij (cid:1) , P + m,ij Q m,ij = 1 △ β (cid:0) − P + m,ij + Q m,ij (cid:1) . (E.14)We also calculate the sum of squares of these quantities by using the formula ∞ X m = −∞ (cid:18) a − m (cid:19) = − ∂∂a ∞ X m = −∞ a − m = π sin πa . (E.15)40his leads to ∞ X m = −∞ ( P − m,ij ) = 4 x u i − u j (1 − u i − u j x ) , ∞ X m = −∞ (cid:0) P + m,ij (cid:1) = 4 x u i u j − (1 − u i u j − x ) , ∞ X m = −∞ ( Q m,ij ) = 4 u i − u j (1 − u i − u j ) . (E.16)Now we can sum over i, j and obtain the leading order of the corrections as S three − loop = N " − β ˜ λ △ + − β ˜ λ △ x + O ( x ) ! | u | + − β ˜ λ △ x (cid:18)
52 + β △ (cid:19) + O ( x ) ! | u | + · · · . (E.17) E.3 ( n + 1) -loop correction to effective potential Up to three loops, the leading order of the coefficient of | u | is x and | u | is x in the x expansion. We can find that these are true even in an arbitraryloop. Thus it is enough to fix the coefficient of these terms in each loop. Inorder to evaluate the ( n + 1)-loop, we have to calculate ∞ X m = −∞ N X i,j =1 (cid:16) G (2) m,ij (cid:17) n = (cid:18) △ (cid:19) ∞ X m = −∞ N X i,j =1 (cid:0) P − m,ij S − ij + P + m,ij S + ij + Q m,ij S Q,ij (cid:1) n . (E.18)in (E.8). However we can reduce this equation. Since S Q,ij is an order x quantity as in (E.6), we can ignore S kQ ( k ≥
3) terms here. Then we shouldevaluate only (cid:0) P − S − + P + S + (cid:1) n + nQS Q (cid:0) P − S − + P + S + (cid:1) n − + n ( n − QS Q ) (cid:0) P − S − + P + S + (cid:1) n − , (E.19)where we have omitted the indices.First we calculate the first term in (E.19). It is convenient to define a k,l = ( P + ) k ( P − ) l so that the equation becomes, (cid:0) P − S − + P + S + (cid:1) n = n X k =0 (cid:18) nk (cid:19) a n − k,k S n − k + S k − . (E.20)41hrough the relation (E.14), a k,l satisfy a k,l = 12 △ β ( a k,l − + a k − ,l ) . (E.21)Here we can approximate a k, = a ,k = 0 if k ≥
2. This is because wecan show that they are order x quantities and the lowest order terms onlycontribute to x | u | by using a similar logic as in (E.15) and (E.16). Then,through (E.21), we can obtain a k,l = 1(2 △ β ) k + l − k X k l − =1 k l − X k l − =1 · · · k X k =1 k X k =1 ( a , + a , )= 1(2 △ β ) k + l − ( k + l − k − l − a , + a , ) . (E.22)Since a , = P + m,ij and a , = P − m,ij , we can sum over the i, j and m indices in(E.20) as ∞ X m = −∞ N X i,j =1 (cid:0) P + m,ij + P − m,ij (cid:1) (cid:0) S + ij (cid:1) n − k (cid:0) S − ij (cid:1) k =2 N (cid:0) nx | u | + n ( n − x | u | + · · · (cid:1) , (E.23)where · · · denotes the irrelevant higher order terms. Then (E.20) becomes2 n N ( △ β ) n − (2 n − n − (cid:0) nx | u | + n ( n − x | u | (cid:1) + · · · , (E.24)where we have used n X k =0 (cid:18) nk (cid:19)(cid:18) n − k − (cid:19) = 2 n − (2 n − n − . (E.25)Next we evaluate QS Q (cid:0) P − S − + P + S + (cid:1) n − , (E.26)in (E.19). First we can show that ∞ X m = −∞ N X i,j =1 (cid:0) P ± m,ij (cid:1) (cid:0) S + ij (cid:1) l (cid:0) S − ij (cid:1) k S Q,ij , ∞ X m = −∞ N X i,j =1 Q m,ij S Q,ij (cid:0) P − m,ij S − ij + P + m,ij S + ij (cid:1) n − = 1( △ β ) n − ∞ X m = −∞ N X i,j =1 Q m,ij S Q,ij (cid:0) S − ij + S + ij (cid:1) n − + · · · = 4 N n − ( △ β ) n − (cid:0) x | u | + ( n − x | u | (cid:1) + · · · . (E.27)Finally we evaluate ( QS Q ) (cid:0) P − S − + P + S + (cid:1) n − . (E.28)Here we can show that ∞ X m = −∞ N X i,j =1 (cid:0) P ± m,ij (cid:1) (cid:0) S + ij (cid:1) k (cid:0) S − ij (cid:1) l S Q,ij , does not contribute to the relevant potential. Thus we obtain ∞ X m = −∞ N X i,j =1 ( Q m,ij S Q,ij ) (cid:0) P − m,ij S − ij + P + m,ij S + ij (cid:1) n − = 2 n − ( △ β ) n − ∞ X m = −∞ N X i,j =1 ( Q m,ij S Q,ij ) + · · · = 2 n − ( △ β ) n − (cid:0) N x | u | (cid:1) + · · · . (E.29)Let us summarize all relevant terms of the ( n + 1)-loop effective action.The gauge-field independent term becomes − ( − ) n d n N β △ ˜ λ △ ! n (2 n − n )!! . (E.30)Note that this result is exact. The leading | u | potential in the x expansionis given by − ( − ) n d n N β △ ˜ λ △ ! n (cid:18) (2 n − n − (cid:19) x | u | + O ( x ) . (E.31)43he leading | u | potential is − ( − ) n d n N β △ n − ˜ λ △ ! n (cid:18) (2 n − n − △ β (cid:19) x | u | + O ( x ) . (E.32)We can sum over n by using the following formula: ∞ X n =1 ( − ) n − (2 n − n )!! x n = √ x − , (E.33)and its derivative with respect to x . With this, we finally obtain the effectiveaction (4.21).Note that it is possible to extend the calculation in this section to finite N case as we have done in the d = 0 model. The finite N result for the gauge-field independent constant term is simply obtained by replacing d n N n +2 in(E.30) with Tr M ′ n in (A.17). The terms including the gauge potential aremore complicated and we have to modify the composite propagator (E.1). References [1] E. Brezin and S. R. Wadia, “The Large N expansion in quantum fieldtheory and statistical physics: From spin systems to two-dimensionalgravity,”