D-dependence of the gap between the critical temperatures in the one-dimensional gauge theories

We investigate the temperature dependence (gap) between the uniform/non-uniform and the non-uniform/gapped transitions in the large-N one-dimensional bosonic gauge theories (1D models) with D matrix scalar fields on a one-dimensional circled space. We use the equations of its critical temperatures given in the 1/D expansion in arXiv:0910.4526. Those transitions are related with the Gregory-Laflamme (GL) instabilities in the gravities and the Rayleigh-Plateau (RP) instabilities in the fluid dynamics, and qualitative similarities between those are expected. We find that the tendency in the dimensional dependence of the gap is opposite from those in the gravities and fluid dynamics. This would be interesting as a counterexample of the gauge/gravity and gauge/fluid correspondences.


Introduction
The model we consider in this paper is the one-dimensional large-N gauge theories given by the BFSS matrix model [2] with general D. The BFSS matrix model has appeared in the evolutions of the superstring theory. Let us overview it.
In the superstring theory there are five theories defined in the ten-dimensional space-time. The low energy sectors of those are the five types in the D=10 supergravity. [3] has proposed that superstring theory emerges from the S 1 -compactification of the eleven-dimensional membrane theory [4], and the strings in the ten-dimensional space-time would be curled up membranes, where the membrane theory can be obtained as the classical solution in the D=11 supergravity [5]. [6] has proposed the relation R = g s l s for the S 1 -compactification (R is its radius), and shown that the D=11 supergravity emerges as the low energy sector of the strongly coupled type IIA superstring theory. This has been reached by focusing on the mass spectra between BPS black hole solutions in the type IIA supergravity and the KK modes in the S 1 -compactified D=11 supergravity theory. [7] has first referred to this theory as M-theory.
The objects to play the critical role in the M-theory are zero-dimensional BPS states, but since there are also specially p-dimensional BPS states in the type IIA supergravity, it is also needed to show the correspondences of those with objects in the D=11 supergravity theory. [8] has discovered Dp-branes, then [9] has shown that the membranes not winding on the S 1 -compactified space in the eleven dimensional space-time are the D2-branes in the type IIA superstring theory.
Based on these, [9] has proposed that the membranes are composed of a large number of D0-branes. The model for an infinite number of D0-brane's dynamics had been obtained as a quantum mechanics given by SU(N) matrix model (matrix theory) [10,11], but it had been considered to be valid only at the low energy level due to the spikes inducing instabilities. However, [2] has proposed that the descriptions by the matrix theory are the ones at the infinite momentum frame (IMF) in the eleven dimensional space-time, which is valid at any energy scales. By this, we have reached the microscopic descriptions of the M-theory in the IMF using the matrix theory (BFSS matrix model). N is taken to infinities in the IMF, but [12,13,14] have proposed that finite N are possible by changing the compactified direction to the light-cones.
One of the interesting interpretation of the bosonic BFSS (bBFSS) matrix model would be the low-energy dynamics of bosonic D0-branes on R D × S 1 (L ′ ) , where D = 9. According to [1,15,16], a way to reach this interpretation is considering a D = 2 SYM on R D−1 × S 1 (L) × S 1 (β) first. This corresponds to the low-energy D1-brane system at finite temperature T = β −1 , where D1-branes wind around the L-direction overlapping. Then, we perform a T-duality to the L-direction. As a result, L exchanges to L ′ = 2πα ′ /L, and D1-branes exchange to D0-branes. We also take the high temperature limit. As a result, the dependence on the β-direction decouples, which leads R D−1 × S 1 (β→0) to R D (More specifically, see Sec.2.2). Fermions are also decouple. Finally, we can reach the bBFSS matrix model above. The eigenvalues of Wilson line wrapping around the L ′ -direction represent the position of D0-branes in the L ′ -direction.
As such, BFSS matrix models have originally come up from contexts of the Mtheory, however that with general D (1D gauge theories) also play the role of the effective microscopic description of the low energy dynamics of D0 branes. Exploiting that, we can try to obtain understanding for the D-brane systems and black objects.
The 1/D expansion has been performed in a 1D bosonic gauge theory on a onedimensional circled space [1] † . It is very important because it is the method regardless of the coupling constants; It is not the expansion with regard to the coupling constants but around large D. Actually, [1] has succeeded in obtaining the results for not only the critical temperatures but also the transition-orders in the model above. This is very wonderful, because normally we need non-perturbative analysis to obtain transitionorders. Since the 1/D expansion takes the similar fashion with usual perturbative expansions, the analysis of the transition-orders has been possible for the first time.
The phase transitions occurring in the 1D bosonic gauge theories are two: 1) the uniform/non-uniform transition and 2) uniform/gapped transition.
The critical temperatures obtained by the 1/D expansion agree with the results of Monte Carlo (MC) simulations very well, however the transition-orders are obtained differently among [1], [16] and [40]: As temperature is risen, it is concluded that 1). in [1], the second-order transition first, then the third-order transition occur, 2). in [16], the third-order transition first, then the second-order transition occur, 3). in [40], only the first-order transition occurs until D = 20, expecting that the transition switches to the situation in [1] at some large D, (see [33] in [40]). The scenario in [40] would be right based on the following three notion: 1) The 1/D expansion should be right at large D, 2) since there is the noise of the finite-N in the method of [16], remaining some ambiguity in the conclusion (see [32] in [40]), and 3) the numerical data in [40] are clear. In those studies, the D-dependence of the temperature difference (gap) between the critical temperatures has not been investigated (see [32] in [40]). Whether its behavior agrees or not with that in the gravity and fluid sides is one of the interesting points in terms of gauge/gravity and gauge/fluid correspondences. Here, let us turn to the critical phenomena in the gravity and fluid sides.
The critical phenomena in the gravity and fluid sides are Gregory-Laflamme (GL) instabilities [42,43] and Rayleigh-Plateau (RP) instabilities. GL instabilities have deep relations with the uniform/non-uniform and non-uniform/gapped transitions [44]. RP instabilities show similar behaviors with GL instabilities. Therefore RP instabilities show similarities with the gauge theories. [45,46,47,48] and [49,50,51] address those issues from the gravity side and the fluid side, respectively.
Among those studies, we would like to focus on the results in [48] and [51] on how the transition-orders vary depending on the number of transverse space dimensions. ( [53,54,55,56,57,51] are studies related with this issue.) According to [48], 1). One first-order transition occurs in d = · · · , 9, 10, 11, 2). A first-order transition, then a higher-order transition occur in d = 12, 13, 3). A second-order transition, then a higher-order transition occur in d = 14, 15, · · · (d is the number of space dimensions in the transverse directions in a D = d + 1 S 1 -compactified spaces).
Regarding the results in [51], we would like to refer readers to Table.1 in [51]; As the point in [51], only one first-order transition occurs at not-large D, while second-order and some transitions occur separately in succession at large D.
As such, we would like to address the gap in the large-N 1D bosonic gauge theories on a S 1 -circled space with D matrix scalar fields.
The point we finally argue in this study is that the gap does not narrow even if D becomes smaller. On the other hand, as D becomes larger, the gap narrows. Those means that the two transitions keep on occurring separately at small D, while the two transitions merge and occur as a single transition effectively at large D. These tendencies are opposite of the gravity and fluid sides above.
Of course there is no guarantee that the correspondences with the gravities and fluids are always held; Actually the interpretation of our gauge theory as the gravity is valid only at D = 9. However, since some qualitative similarities are expected, the point we argue in this study is interesting as a specific counterexample of that.
As the organization of this paper, in Sec.2, the model is given. In Sec.3-6 are the review for the 1/D expansion, obtaining the equations of the critical temperatures.
In Sec.7, we show the D-dependence of the gaps, then based on that, we argue that the correspondences with gravity and fluid are not always held. In Sec.8, we argue this in the Z m symmetric solutions.

Our model
We begin with the one-dimensional SU(N) bosonic Yang-Mills gauge theory given by the bosonic BFSS type matrix model (1D model): where A 0 and Y I are the N × N bosonic Hermitian matrices, and t is the Euclidean time which can be related with the temperature T as . A 0 and Y I obey the boundary conditions: Y I (t) = Y I (t + β) and A 0 (t) = A 0 (t + β). D is a parameter. Performing a rescaling: Y I → g Y I , we rewrite the one above into We omit to write the summations for I in what follows. We take g 2 N to a constant: g 2 N ≡ λ while taking large N as the large-N limit ‡ . We can see [λ] = M 3 . Hence we define a dimensionless parameter λ eff = λβ 3 .

Possible λ and β for the description by our model
Our model (1) with D = 9 can be obtained from the high temperature limit and the T-duality of the SU(N) N = 8 SYM on a circle with a period L at finite temperature T 2 = β −1 2 : [Y I , Y J ] 2 + fermions, (2) where µ, ν take two values t, x, L is common to the L in the description of Sec.1, and fermions are anti-periodic in the t-circle. We refer to (2) as 2D SYM in what follows. The 2D SYM is characterized with the two dimensionless parameters: where λ 2 ≡ g 2 2 N is the 't Hooft coupling in the 2D SYM. The high temperature limit is taken, which leads to decoupling of the t-dependence. As a result the 8 change to 9. Fermions also decouple. We also take the T-duality § . ‡ We can change the overall factor g 2 arbitrarily as g 2 → κg 2 by the rescalings: (Y I , A 0 ) → (κ −1/3 Y I , κ 1/3 A 0 ) and (t, β) → κ 1/3 (t, β). We can change g 2 arbitrarily by this rescaling without changing physics as long as λ eff is fixed.
§ One reason to perform the T-daul is to look at the regions other than λ ′ ≫ 1. In such a parameter regions the winding modes and the α ′ -corrections become effective, which break the fact that D1-branes are solutions at the supergravity level. However we can keep those as a solution at the supergravity level by performing the T-dual [15].
It is considered as the effective theory for the D0-branes in the S 1 -compactified D = 9 space-time at finite temperature, where the x-cycle plays the role of the finite temperature after the T-duality. We denote the period of the S 1 direction as L ′ . We have noted the relation between L ′ and L in Sec.1. D0-branes are assumed to be distributed on a same S 1 -circle.
When λ ′ is large, the dynamics on both the x-cycle and the β 2 -cycle becomes active. However, even if λ ′ is large, if β 2 is small, the final contributions of the dynamics from the t-cycle can be ignorable since the space itself is small. Likewise, even if λ ′ is large, if L ′ is some small values, the final contribution from the x-cycle can be ignorable. These can be written in the qualitative manner as [15]: The boundary of λ ′1/3 < t ′ is plotted in Fig.1.
In particular, when we realize the following situation: by taking the high-temperature limit, the 2D SYM reduces to our 1D model (1). At this time, the parameters in the 2D SYM and our model (1) are linked as Using these we can rewrite the condition (4) as where λ eff is given under (1). Therefore, when the condition (6) is held, we can consider our 1D model (1) instead of the 2D SYM.
Let us mention the conclusion in this section. Since the high temperature limit is taken, t ′ goes to ∞. At this time, we can assign any finite values to β and λ without breaking (6) by exploiting the rescaling in the footnote under (1). Therefore, practically we can always include the uniform/non-uniform and the non-uniform/gapped transitions in the parameter region where the description by our 1D model (1) is possible. Figure 1: Phase structure in 2D SYM (2). As going to the right side, it becomes more higher temperature region, on the other hand, as going up, it becomes more strongly coupled region. The "1st" in the upper-left region can be known from the GL instability in the gravity side. The bottom-right region separated by the fine dotted line is the region effectively described by our 1D model (1). "uniform", "non-uniform" and "gapped" represent the phases.

Preliminary for the analysis of the effective action
From this Section to Sec.5, we review obtaining the effective action in [1], and in Sec.6, we review obtaining the equations of the critical behaviors in [1].
where t a are the generators of SU(N) with the orthogonal condition: tr(t a t b ) = δ ab , and Y i a are coefficients.
(1) can be written as Here, when we introduce B ab , some factor the distribution function accompanies, but we ignore it as it is just a numerical factor [1]. We can see that B ab plays the role of the squared masses for Y I a . We can confirm the equality between (1) and (8) using M −1 ab,cd δ cd = 1 2N δ ab . Integrating out Y I , we can write the action as In the one above, it is known that B ab will get some value for the large D [1]. If we write it as B ab = i∆ 2 0 δ ab , ∆ 2 0 will turn out to be real and play the role of squared mass, which guarantees that we are on a stable vacuum.
We consider B ab with quantum fluctuations as Replacing B ab in (8) with thisB ab we can obtain The SU(N) gauge symmetry exists in our model at each t ∈ [0, β]. We can separate off the volume factor for the gauge transformation by inserting the unity (56) as whereλ ≡ λD, and u n = 1 Here, we are now taking the static diagonal gauge (A 0 ) ij = α i δ ij /β, (i, j = 1, · · · , N). u n are the Wilson lines twining around the t-direction n times. Let us look at the terms in (13). The second term will be turned out to be indispensable, because it plays the critical role in the determination of the sign of the |u 1 | 2 's coefficient in the effective action (32). Thus let us include it. Therefore, we have to take into account the 1/D correction to 1/D order.
The term of the summation in (12) is the interaction term. We comment on the contribution from this term in Appendix.B. The θ-integral gives just a gauge volume, which we disregard.

One-loop integral of Y I
Taking SU(3) to make our calculation process concrete, we write down the expression for the part to become the one-loop integration of Y I , explicitly. Then deducing the expression for arbitrary N, we perform the one-loop order path-integral.
We start with where t a are 1/2 of Gell-Mann matrices, and A θ 0 a and Y a are some constants as the components of the vector. Since we take the time-independent diagonal gauge, we can take the components A θ 0 a freely as long as this gauge is kept with the traceless condition. Therefore, we have taken A θ 0 a in the (15), where We proceed our calculation by performing the plane-wave expansion:

Expression of action
We can write our action as where Z = DY exp(−S). We now compute the expressions of the kinetic and mass terms.
We obtain the expression of the kinetic term, In the one above, we have used Kronecker-delta function, 1 β t = δ mn ¶ , and k −n = −k n and α ij = −α ji . We have written the components relevant to the trace at the last. We can obtain the expression of (16) in the same way, which agrees to (16).
We next obtain the expression of the mass term, which is written as where Y ij in the third line are given in (14). The third line appears to depend on N, but in forth and fifth lines, we can deduce the expression at arbitrary N. ¶ Kronecker-delta function in the non-compactified space is 1 2π 2π 0 dx e i(m−n)x = δ mn . In the calculation (21), the transitions from the second to the third lines and from the third to the fourth lines might be difficult to understand instantly, so we have written those explicitly.
From (20) and (21), we can now write the action as We are omitting the index "I" in Y I in the description above.

Degree of freedom to be integrated
We confirm the degree of freedom with regard to Y to be integrated. To this purpose, let us write the plane-wave expanded scalar matrix field Y and its Hermitian conjugate in a rough manner as where the characters used above, a, b, · · · , j, are the ones used only in this subsection. From the condition: Y = Y † , we can obtain the following condition: c −n = f +n , d −n = −g +n , f −n = c +n , g −n = −d +n , for the non-diagonal elements a −n = a +n , b −n = −b +n , h −n = h +n , j −n = −j +n , for the diagonal elements Plugging these into the Y in (23), it can be written as We can see that the degrees of freedom to be integrated are the parts corresponding to the following ones: • For all the diagonal elements: -Real-part: a n (n = 0, 1, 2, · · · ), h n (n = 0, 1, 2, · · · ) -Imaginary-part: b n (n = 1, 2, · · · ), j n (n = 1, 2, · · · ) • For one-side of the non-diagonal elements: -Real-part: c n (n = −2, −1, 0, 1, 2, · · · ) -Imaginary-part: d n (n = −2, −1, 0, 1, 2, · · · ) Therefore the integral measure except for the factors is given as

Path-integral
We can see from (24) that there is the relation: Y n ij = Y −n ji * . Exploiting this, we can decompose the description of the action (22) into each component as In the one above, we have written the expression at general N based on (22). We perform the path-integrals of Y in (27). We show its calculation process in Appendix.C. As a result we get the following result: Adding the FP term obtained in Appendix.A.3 and the corrections arisen from the interaction term to 1/D order (we quote from (4.21) in [1]), where x ≡ e −β∆ and y ≡λ 4∆ 3 .
S int represents the corrections from the interaction term and "· · · " represnets negligible corrections. All the 1/D order terms except for "1/D" in c 2 are the terms from S int . 1/N corrections from S int do not appear in our analysis, because it starts from 1/N 2 in S int as written under (E.33) and (A.17) in [1].

Evaluation of ∆ at the saddle-point
We fix ∆ to the saddle-point by taking its variation in (30) instead of performing the path-integral. Note that this is the saddle-point method, so it can work at the large-N. It turns out that we cannot obtain the ∆ exactly. However we can solve in the power series of |u 1 | 2 . We finally obtain the approximated solution to the |u 1 | 2 order in the 1/D expansion as We can see that the 1/D part is consistent with (4.25) in [1]. "· · · " represents some corrections which will be ignoreble when |u 1 | is small.
Plugging (31) into the effective action (30), we can obtain the following Ginzburg-Landau (GL) type effective action: * * The one above is consistent with (4.26) in [1]. * * The term 1/D in |u 1 | 2 in (32) comes from the gauge-fixing. Other terms come from the integrals for Y and b, roughly saying. We can see that the uniform/non-uniform transition in our model are determined by which one is larger.

Equations of the critical temperatures
Let us obtain the equations of the critical phenomena based on GL action (32). We can see that the coefficient of |u 1 | 2 is positive for β ≫ 1, which means that |u 1 | = 0 and the confinement (uniform) phase is realized. However when the temperature is risen, the sign of the coefficient of |u 1 | 2 will flip to negative at some temperature. As a result, |u 1 | gets some finite value and the phase switches to the deconfinement (non-uniform). We can get the critical temperature T 1 = β −1 1 for this from the condition c ′ 2 | β=β 1 = 0. In actual calculation, we obtain where we have put β 1 as ln D λ 1/3 (1 + α D D ) and obtained with regard to α D . Its result is α D = 203 160 − √ 5 3 . Finally, T 1 is obtained as The one above is consistent with (4.30) in [1].
It is known in [44] that the eigenvalue density function is given as ρ(α) = β 2π 1 + 2|u 1 | cos(βα) . Therefore, a gap arises in the eigenvalue distributions when |u 1 | reaches 1/2. According to [44], the third-order phase transition occurs at that time. We obtain the critical temperature for this by solving with regard to δβ in δS GL ∆ at s.p.
Expanding the one above regarding δβ to the first-order, then solving it, we finally obtain ( 168491 153600 = 505473 460800 does not agree with [1], but this would be just a typo in [1]).) Denoting the critical temperature for this as T 2 , its result is where , then have expanded with regard to 1/D.
Finally, we can check the transition-order of the uniform/non-uniform. However, since it is not important in the issue we treat in this study, we perform it in Appendix.D.
7 D-dependence of the gap between T 1,2 In this section, we check the D-dependence of the gap between the critical temperatures associated with the uniform/non-uniform and non-uniform/gapped transitions.
In Fig.2, we represent T 1,2 in (33) and (37) against D, where we treatλ = Dλ in those expressions, given under (13), by plugging unit in λ and having D appear explicitly.
We can see that even if D becomes smaller, the gap between T 1,2 does not close, while as D grows, the gap narrows. Those means that the two transitions do not merge at small D, while merge and become a single transition effectively at large D.
Since higher-order corrections of the 1/D expansion become effective when D is small, what we mentioned above concerning small D might be an error of that. However, we can see in the Table in the last of Sec.4 in [1] that the results of the 1/D expansion is not incorrect so much from the numerical results of the Monte Carlo simulation (MC simulation) at D = 2, and as can be seen there the numerical difference between T 1,2 is 1.3 − 1.12 = 0.18. This numerical value can be considered as the sign of the existence of the gap. Therefore, the gap keeps appearing at small D even in the MC simulation. Therefore, we can consider that the tendency we have found above is right even at small D.
These tendencies are completely opposite from the tendency of GL and RP instabilities, where we have summarized those tendencies in Sec.1.
From those results, we can conclude that the gauge/gravity and gauge/fluid correspondences are not always held. Of course there is no guarantee that the correspondences with the gravities and fluids are always held; Actually the interpretation of our gauge theory as the gravity is valid only at D = 9. However qualitative similarities can be expected. Therefore, the point we argue is interesting as a specific counterexample.    Figure 2: D-dependence of the gap between T 1,2 against D; the red and blue points represent T 1 and T 2 , respectively. We can see that the gap does not narrow even for small D, while gets smaller as D gets larger, which are opposite tendency from GL instabilities in gravities and RP instabilities in fluid dynamics.

D-dependence of the gap in the Z m symmetric solutions
In this section, we generalize the D-dependence of the gap between the uniform/nonuniform and non-uniform/gapped transitions into the critical temperatures of the Z m symmetric solutions.
First, let us define the Z m symmetric solutions. Since we are now taking the static diagonal gauge, we can write the gauge field as Then considering a set {N 1 , N 2 , · · · , N m }, where N k ∈ Z with m k=1 N k = N, let us consider the following gauge field's configuration: We can consider 2πl/m as the mean position of α i belonging in N l .
We can see that this configuration is Z m symmetric if α (l) j are expanding evenly around 2πl/m. This (39) is the definition for the Z m symmetric solutions. We will refer to the Z m symmetric solutions as "Z m -solution" in what follows. We can understand that (39) can be the solutions in what follows.
What we have treated so far can be considered as the case with m = 1, and what we will perform in this section is the generalization of the D-dependence of the gap between the uniform/non-uniform and non-uniform/gapped transitions in Sec.7 into the framework of the Z m symmetric solutions.
Here, if α (l) j belonging to 2πl/m for some l are completely separated from α (l ′ ) j belonging to 2πl ′ /m for any l ′ except for l and forming a mob, we refer to those configurations as "multi-cut Z m -solution".
In what follows we assume N l ∼ O(N) (which leads to m ≪ N) and N 1 = N 2 = · · · = N m . In addition, normally α (l) j ≪ 1 might be assumed, however since in this section we consider the transitions between the uniform phase and the Z m -solutions, we assume that α (l) j are expanding widely in such a way that α (l) j and α (l ′ ) j belonging to the mobs next to each other merge and form a uniform state, or are at the moment to start to separate and form the Z m -solutions. We will not consider the situations with α (l) For the Z m -solutions, we can see Therefore, in the situation with a Z m -solution, we can write the effective action (30) in the following form: where S (m) eff. means the effective action for a Z m -solution. In the one above, there is no 1/D corrections as long as we consider up to the 1/D order. This is because it turns out that all the 1/D order corrections are below 1/D 2 order for the Z m -solutions with m ≥ 2, Let us explain the one above more. Considering x given in (29), x p (p ≥ 2) always accompany to the terms concerning |u p | 2 (we can know this in the appendix in [1]), and we can see that in the higher temperature regions where the Z m -solutions with m ≥ 2 begin to appear as a saddle-point solution (We mention the reason of this in what follows), x behaves as x ∼ 1 D exp 1 1+λ −1/3 ln D∆T ∼ 1/D for D ≫ 1 and ∆T ≫ 1. We have obtained this "x ∼ 1/D" by writing the higher temperatures T high temp. and ∆ as T high temp. = T 1 + ∆T ∼λ 1/3 / ln D + ∆T and ∆ ∼λ 1/3 , where we have taken the leadings of those.
Considering (30) with removing the 1/D corrections arisen from the S int and ignoring the overall factor 1/m, we can see that (41) can match with such a (30) only by identifying β → mβ and |u 1 | → |u km |. Therefore we can write the effective action for a Z m -solution by referring (32) in the case of Z 1 solution as S (m) eff.
Note that the contribution with k = 1 in (41) are dominant in the one above corresponding to the fact that the contribution with n = 1 in (28) is dominant in (32).
Since the effective action (42) is (32) in which just the temperature is exchanged as β → mβ, we can get the critical temperatures T where T

Conclusion
Let us summarize the result we have newly provided in this study, which is the totally opposite tendency of the D-dependence of the gap between the two critical temperatures compared with the gaps in GL and RP instabilities the gravity and fluid sides. We have plotted the D-dependence of the gap in Fig.2.
Of course there is no guarantee that the correspondences with the gravities and fluids are always held; Actually the interpretation of our gauge theory as the gravity is valid only at D = 9. However, since some qualitative similarities are expected, the point we argue in this study is interesting as a specific counterexample of gauge/gravity and gauge/fluid correspondences.

A Our Faddeev-Popov (FP) term
Let us begin with a general formula of the delta-function. We consider some function f (x) (f (x 0 ) = 0) expanded around x = x 0 in a delta-function as At this time, the following formula is held: f ′ (x) x=x 0 corresponds to the FP determinant.
Here, let us mention that we represent the unitary matrices as U = exp(igǫ), where ǫ ≡ N 2 −1 a=1 θ a t a (t a are the generators of SU(N) Lie algebra and θ a are these coefficients) in what follows.
From now on, we consider the one-dimensional system with SU(N) gauge freedom such as our model. Gauge transformations act on gauge fields A 0 (t) as A θ , and the θ in the shoulder of A 0 (t) means that A 0 (t) got a gauge transformation for θ from the configuration of A 0 (t).
We pick up the time-independent configuration: in the path-integral for SU(N) transformation. Even if we remove the t-dependence from the gauge matrix field A 0 , there still remains the t-independent SU(N) gauge freedom in the A 0 . We fix it by the diagonalization gauge: A η 0 0 = diag(α 1 , · · · , α N ). In what follows, we first obtain the FP terms arisen from the time-independent gauge and the diagonal gauge individually. Then we obtain the FP term as a whole by summing the FP terms in the each gauge fixing.
A.1 FP term from the gauge-fixing, ∂ 0 A 0 = 0 In order to compose the unity ((45) in the case) for the time-independent gauge (46), we consider the deviations arisen by the gauge transformation from the configuration satisfying (46) as where the δ in the l.h.s. means the gauge transformation, and (D 0 ) ij = ∂ t I ij + iα ij (α ij ≡ α i − α j and i, j = 1, · · · , N). Here we shall note that the analysis in what follows will be performed in the situation that the configuration of the gauge matrix field on which the gauge transformations act is the time-independent and diagonal one. Therefore, the unity for (46) can be written as where Θ θ ij ≡ ∂ 0 (A θ 0 ) ij , and the integral is for the SU(N) gauge transformation space. "∂ 2 0 " and "∂ 0 D θ 0 " are just formal expressions for here only. We omit to write β t=0 in the second line and from now on.
We here evaluate the FP determinant part in (48).
where we have performed the plane-wave expansion without the zero-mode [29]. Note α i in α ij in (49) are the elements in the diagonalized gauge matrix field. We then calculate a part of (49). Here, where we have assumed that N is even numbers. Therefore, A.2 FP term from gauge-fixing, A 0ij = α i δ ij We fix the remaining t-independent SU(N) gauge freedom by the diagonal gauge: Since gauge transformations from A η 0 0 can be written as the configuration deviated from the diagonalized one simultaneously can be written as This can be seen from the case of SU (3), (16). The unity for the diagonalized constraint is therefore given as

A.3 Total FP term
We can now obtain the unity when we impose the t-independent diagonalized constraints by combining (48) and (55) as Note that β t=0 attaching to the whole is omitted in the expression above.
B Derivation of (E.8) in [1] From (12), we write the contribution arisen from the interaction term as where (57) is a summation of the (n + 1)-loops diagrams (n = 1, 2, · · · , ∞) in Fig.3. We can see (2n−1)!! (n−1)! 2 n−1 2 2n (2n)! 2 2n comes out from Wick contractions. We mention origins of each factor: a). "(2n)! 2 2n " in the denominator comes from the denominator in (57). b). "(2n − 1)!!" is the number of pairs made by combining the interaction terms bY Y two by two (each pair forms a 1PI diagram made of one loop of Y with two lines of b). c). "(n − 1)!" is the number of patterns to combine those 1PI diagrams to form a one big loop as the n-loops of Fig.3. d). "2 n−1 " comes from the option to put each 1PI diagram upward or downward. e). "2 2n " comes from the two parttens in combining Y 's in the two interaction terms bY Y each to form one 1PI diagram.
Since there is no explanation between (E.1) and (E.8) in [1], let us consider how to derive the (E.8). Upon evaluating (57), we consider the contribution of the 2-loops as the most simple example.
Focusing on a part with n = 1 in (57), where b a 1 b 1 b a 2 b 2 = M a 1 b 1 ,a 2 b 2 , and we have used Y I a = 2 tr(t a Y I ) with Y I = N 2 a=1 Y I a t a . As the invariance: M a 1 b 1 ,a 2 b 2 = M a 1 b 1 ,b 2 a 2 , (58) can be written as (58) = 4g 2 M a 1 b 1 ,a 2 b 2 t a 1 ij t b 1 kl t a 2 qp t b 2 nm dt 2 dt 1 Y I 1 ji (t 2 )Y I 2 pq (t 1 ) Y I 1 lk (t 2 )Y I 2 mn (t 1 ) .
We take the leading contribution in the large-N (this would be the point). In this case, the contribution in the case that each indeces for the inner and outer lines in the Y's loop becomes respectively same will be picked up. Therefore, Using the composite propagator given in (E.1) of [1], (60) can be written as It would be difficult to evaluate this. However based on the following two points: 1). The result should become (E.8) when D = 0 including coefficients except for β, 2). standing up behavior of Wilson line |u 1 | just above T 1 , (35), we would be able to analogize how (60) will be written finally as (60) = − β d 1 g 2 DN 2 n q,m G n,qm .
Performing the one above by rising n in the (n + 1)-loops, we can educe the contributions at the (n + 1)-loops as where d 1 = −1, d 2 = 3, and d n = 1 for n = 3, 4, · · · . C Calculation process from (27) to (28) We show the calculation process from (27) to (28). In the ones above, we have used the following relations: D Transition-order of the uniform/non-uniform transition In this appendix we check the transition-order associated with the uniform/non-uniform transition at T 1 .