aa r X i v : . [ h e p - t h ] J un Burgers’ equation in 2D SU(N) YM.
H. Neuberger
Department of Physics and Astronomy, Rutgers UniversityPiscataway, NJ 08855, U.S.A
October 23, 2018
Abstract
It is shown that the logarithmic derivative of the characteristic polynomial of a Wilsonloop in two dimensional pure Yang Mills theory with gauge group SU(N) exactly satisfiesBurgers’ equation, with viscosity given by 1/(2N). The Wilson loop does not intersectitself and Euclidean space-time is assumed flat and infinite. This result provides a preciseframework in 2D YM for recent observations of Blaizot and Nowak and was inspired bytheir work. ontents N = small viscosity. 55 Higher critical points. 66 Product representation. 67 Large τ behavior. 78 Discussion. 89 Added note. 9 Recent numerical work provides evidence that Wilson loops in SU ( N ) gauge theory intwo, three and four dimensions exhibit an infinite N phase transition as they are dilatedfrom a small size to a large one; in the course of this dilation the eigenvalue distributionof the untraced Wilson loop unitary matrix expands from a small arc on the unit circleto encompassing the entire unit circle [1, 2]. An analogous effect takes place in the twodimensional principal chiral model for SU ( N ) [3].The universality class of this transition is that of a random multiplicative ensembleof unitary matrices. The transition was discovered by Durhuus and Olesen [4] (DO)when they solved the Makeenko-Migdal [5] loop equations in two dimensional planarQCD. The associated multiplicative random matrix ensemble [6] can be axiomatizedin the language of noncommutative probability [7]. It provides a generalization of thefamiliar law of large numbers. The essential feature making a difference is that onecase is commutative and the other not. Various recent insights into the DO transition[8, 9, 10] point to possibly deeper interpretations of the transition.In this note, motivated by a recent paper by Blaizot and Nowak [10], I present anexact map from the average characteristic polynomial associated with a Wilson loop toBurgers’ equation. This extends to finite N the original work of DO at N = ∞ , wherethe inviscid Burgers’ equation plays a central role. The main observation is that all finite N effects are exactly represented by reinstating a finite viscosity in Burgers’ equation,given by N . Positive N gives positive viscosity, so the equation knows at least that N should not be negative. I suspect that integral N ’s are identified as special by a Mittag-Leffler [11] representation of the solution, stemming from a product representation of he average characteristic polynomial, and depending also on the initial condition.In addition to making the insight of [10] particularly transparent, I hope that thisresult would also aid future efforts to exploit large N universality in dimensions higherthan two for obtaining analytical quantitative estimates of the ratio between a scaledescribing perturbative phenomena and the scale of confinement. This was the originalmotivation for seeking to establish numerically large N phase transitions in Wilsonloops [1]. An N × N simple unitary Wilson loop matrix W , defined on a curve that does not selfintersect, with τ denoting the dimensionless area in units of the ’t Hooft gauge coupling,has the following probability distribution: P N ( W, τ ) dW = X R d R χ r ( W ) e − τC ( R ) dW (1)The sum is over all irreducible representations R with character χ R ( W ) and second orderCasimir C ( R ). dW is the Haar measure. Normalization conventions are standard [2]and τ ≥
0. We introduce the average characteristic polynomial Q N ( z, τ ) = h det( z − W ) i P N ( τ ) (2)One can think about Q N ( z, τ ) as the generating function for the h χ R ( W ) i with totallyantisymmetric R . Simple manipulations [2] produce an integral representation: Q N ( z, τ ) = s N τ π Z ∞−∞ due − N τu h z − e − τ ( u +1 / i N (3)It is more convenient to study q N ( y, τ ) = ( − N e − Ny e Nτ Q N ( − e y , τ ) (4)where, for the time being, y is kept real. q N ( y, t ) is even in y and this is the main reasonfor extracting the exponential factor from Q N . Changing the integration variable u to x = y + τ ( u + 1 /
2) gives: q N ( y, τ ) = s N πτ Z ∞−∞ dxe − N τ ( y − x ) e N log(2 cosh( x/ (5) It is now a trivial matter to observe that ∂q N ∂τ = 12 N ∂ q N ∂y (6) ith initial condition lim τ → [ q N ( y, τ )] = (2 cosh( y/ N (7)The behavior at y → ±∞ prevents solving (6) by Fourier decomposition and any asso-ciated general conclusions about boundedness as τ → + ∞ . The initial condition is aconsequence of P N ( W,
0) = δ ( W, ) with Z dW δ ( W, W ) f ( W ) = f ( W ) (8)for any W ∈ SU ( N ). This equation can be also directly derived from the polynomialformula of Q N , without going to the integral representation. This heat equation isrelated to Burgers’ equation (for example, see [12], problem 12(a), p. 214) by φ N ( y, τ ) = − N ∂ log q N ( y, τ ) ∂y (9)Burgers’ equation and the initial condition are ∂φ N ∂τ + φ N ∂φ N ∂y = 12 N ∂ φ N ∂y , φ N ( y,
0) = −
12 tanh y N = ∞ , N drops out of the equation giving the inviscid limit: ∂φ∂τ + φ ∂φ∂y = 0 (11)The initial condition is N independent so we can drop the N subscript on φ at N = ∞ .So long as φ is uniquely defined, this is the point-wise N = ∞ limit of φ N .The equation can be solved by the method of characteristics (for example, see [12],p. 16.) for an arbitrary initial condition φ ( y,
0) = h ( y ) (12)The solution is given implicitly by φ ( y, τ ) = h ( y − τ φ ( y, τ )) (13)This equation is known to produce a shock at a time τ ∗ > τ ∗ is the smallest positive value satisfying τ ∗ = − dh/dy )( y ∗ ) with ( d h/dy )( y ∗ ) = 0 (14)We are interested only in solutions odd in y ; hence, assuming h ( y ) to be smooth near y = 0 we expand: h ( y ) = ay + by + cy + .... (15)This implies that y ∗ = 0 and therefore τ ∗ = − a (16) shock will form if a <
0. In the case of N = ∞
2D YM we have h ( y ) = −
12 tanh y − y/ y / − .... (17)Therefore, the critical area corresponds to τ ∗ = 4 , (18)the well known critical value [4, 6].Universality can be invoked now in a sense that applies to the nonlinear equationproducing a generic shock [13, 14]. This means taking the simplest polynomial h ( y )capable of producing shocks: h ( y ) = ay + by (19)with a < , b >
0. The y location of the shock is at the origin, y = y ∗ = 0. Extending h and y to the complex plane provides a geometric view of this universality in terms ofthe structure of the evolving Riemann surface y ( φ, τ ) parameterized by τ ≥
0. One canalso take τ into the complex plane. N = small viscosity. Making the viscosity nonzero is a singular perturbation which eliminates the shock andhas the same effect as making N finite. Large N universality will hold in the vicinity ofthe critical area and corresponds to universal behavior in the vicinity of the would-beshock for small viscosities, which is the simplest dissipative regularization of the shock.The important new insight is that the large N transition is equivalent to a mov-able singularity, determined by the initial condition, rather than by the evolution rule. Thus, the simplest initial condition producing a shock will also lead to a universal smallviscosity smoothing of the shock.Running the derivation backwards, with the minimal initial condition h ( y ) = − y/ y /
48 (20)produces an integral representation on which a double scaling limit can be taken directly,exactly reproducing the limit used in matching to the large N transitions in higherdimensions than two in [1, 2]. The critical exponents µ = 1 / , / N µ that need to be taken [2] are identical to those found in defining the smallviscosity limit [16]. The associated integral, studied in detail in [2] (see [17] for a plot),is related to Pearcey’s integral by a contour change, as indicated in [10].The particular initial condition (19) has been analyzed in great detail in [16]. The shock can be regulated also dispersively, in which case we could use a third derivative on the righthand side of the inviscid Burgers’ equation, producing the KdV equation. If there were a symmetry restrictingto a Hamiltonian partial differential equations, this might have been the equation defining the universalityclass. Something similar happens in the context of models consisting of one or several large matrices, wherePainlev´e equations enter (see for example [5] and [15]). Higher critical points.
We have become accustomed to expect higher critical points, of reduced degrees ofstability, to accompany a basic large N critical point. Looking at (15) it seems plausiblethat setting b = 0 and making c > ay + by m − with integer m ≥
2. If one is not worried aboutthe convergence of the associated universal integrals and one is also willing to give upthe y → − y parity symmetry, also higher critical points with half integer m could bestudied, at least as formal originators of asymptotic series.It would be intriguing if parent models existed with physical symmetries that selectedone of these higher critical points. More work on this is left for the future. It certainly is true that Q N ( z, τ ) = h det( z − W ) i P N ( τ ) = N Y ( z − z i ( τ )) (21)One may view the z i ( τ ) as certain averages of the eigenvalues of W , but not as usuallydefined: det( z − W ) = N Y ( z − ˆ z i ( W )) , ˆ z avi ( τ ) = h ˆ z i ( W ) i P N ( τ ) (22)In [2] it was proved that | z i ( τ ) | = 1 for all i = 1 , ..., N (see [17] for a plot); this indicatesthat the z i ( τ ) are to be viewed as an approximations to the ˆ z avi ( τ ). By applying large N factorization N times, one can argue, at least away from large N critical points, thatidentically ordered ˆ z avi ( τ )’s and z i ( τ )’s are equal to each other. I suspect that this staystrue also in the double scaling limit. If this suspicion is validated, we shall obtain a newmethod to identify, using numerical simulations, the location and nature of the large N transition in dimensions higher than two.It is therefore interesting to derive evolution equations for the z i ( τ ). After insertingthe product (21) into the heat equation (6) and applying (4), standard manipulations ofthe kind employed in the study of Calogero systems produce˙ z j z j = 12 N X k ′ z k + z j z k − z j , for j = 1 , .., N (23)Here ˙ z j = dz j ( τ ) /dτ and P k ′ means that the k = j term is dropped from the sum,where the index k runs from 1 to N . This equation is form invariant under z j → /z j and z j → z ∗ j , as expected from the structure of the polynomial. In addition, again asexpected, the product of all zeros is constant in τ . Moreover, the equations of motion(23) imply d | z j ( τ ) | /dτ = 0 , j = 1 , .., N . he initial condition is z j (0) = 1 , j = 1 , .., N and is degenerate. However, at any τ > τ , we have z j ( ∞ ) = e πi ( j +1 / − N/ /N .The map z = − e y creates an infinite number of copies of the zeros z j , which are allon the imaginary axis. We choose one specific y j for each z j , j = 1 , .., N . The equationof motion for the y j ’s is:˙ y j = 12 N X k ′ coth y k − y j N X k ′ X n ∈ Z y k − y j + 2 nπi (24)The universal description changes the equation obeyed by the y j ’s. However, as pointedout in [16] on the basis of an old theorem [18], the y j ( τ ) still stay on the imaginaryaxis for all τ . In the universal case periodicity under y j ( τ ) → y j ( τ ) + 2 m j πi, m j ∈ Z is lost, since the initial condition on the y i ’s no longer is periodic. Thus, one needs touse the y j variables to make the connection between the exact equations of motion andthe universal ones. I leave a more detailed study of the universal limit of the eigenvaluemotion to the future. τ behavior. The regularization of the shock provides a smooth connection between small and largeloops. In two dimensions Burgers’ equation provides an exact renormalization grouptype of equation allowing the evaluation of φ N ( y, τ ) when τ → ∞ , given φ N ( y, τ ) inthe limit τ →
0. The approach to the limit τ → ∞ gives the dimensionless stringtension associated with the dimensionless area τ . Here we only show how the correct φ N ( y, τ = ∞ ) is obtained. It is clear that Q N ( z, τ = ∞ ) = z N + ( − N . This simplysays that at infinite τ all h W m i terms, for any m >
0, can be replaced by zero.Using (4), we conclude that the large τ behavior is given by:lim τ →∞ (cid:16) e − Nτ q N ( y, τ ) (cid:17) = 2 cosh N y φ N ( y, τ = ∞ ) from Burgers’ equation. Theroute is again in reverse of our derivation: First go to the heat equation, then get theintegral representation in order to incorporate the initial condition. Finally, in order toget the asymptotic behavior for large τ , change variables in the integral representation,arriving at: N ∂ y log q N ( y, τ ) = R due − Nu sinh(( u √ τ + y ) / u √ τ + y ) / N − R due − Nu (2 cosh(( u √ τ + y ) / N (26)For large τ , one of the two exponents making up each hyperbolic function dominates,depending on the sign of u : lim τ →∞ (cid:18) N ∂ y log q N ( y, τ ) (cid:19) = τ →∞ R due − Nu ε ( u ) e N [ ε ( u )( u √ τ + y ) / R due − Nu e N [ ε ( u )( u √ τ + y ) / (27)Here, ε ( u ) is the sign function. The above equation implies that φ N ( y, ∞ ) = lim τ →∞ (cid:18) − N ∂ y log q N ( y, τ ) (cid:19) = −
12 tanh
N y N , the hyperbolic tangent becomes a sign function. In an electrostaticpicture it is obvious that the above result holds if the poles of φ ( y, τ ) are uniformlyspaced and dense on the circle | e y | = 1: Viewing the poles as charges, the jump ε ( y )comes from crossing the line charge at z = − y goes through zero along the realaxis [2]. That the solution has this limiting behavior is essential for confinement, whichwould be indicated by the leading correction to the above result being exponentiallysmall in τ .Note that τ was taken to infinity at finite N ; the final result admits a subsequentinfinite N limit. Had we taken N → ∞ first, we could have interpreted the shock,appearing first at τ = 4, as a jump between two extremal solutions of the implicitequation defining the solution for τ <
4. With the wrong initial conditions this jumpmight not grow to the full size required for consistency with confinement; thus, thetransition in itself is insufficient to guarantee confinement. If we want to add the inputthat there is confinement we need to put a constraint on the initial condition.Regarding [10], following [22], I opt not to address here the question how Burgers’equation relates to turbulence. As a start, I refer the reader to [23]. In general, one wouldhope that the analogy to the three dimensional incompressible Navier-Stokes equationdoes not hold too literally. Large N would map to large Reynolds numbers, while small N to small Reynolds numbers; however, I am hoping that matters simplify at large N –if they do not, one would be better off concentrating on N = 3.Again, I leave details for further work. The primary objective of this paper was the derivation of (10) as an exact equationholding in two dimensional Yang Mills theory with gauge group SU ( N ) defined on theinfinite Euclidean plane. A surprising simplicity in the area dependence of the averagecharacteristic polynomial of simple Wilson loops was found. Nevertheless, the essentialfeature of the existence of a large N phase transition is captured by this observable. Inthis respect the average characteristic polynomial of the Wilson loop is superior to tracesof the Wilson loop in some fixed representation. As explained in [2] this observable hasother advantages, in dimensions three and four.The simple and exact finite N relation to Burgers’ equation presented above seemsto provide opportunity for progress in different directions, as emphasized in the courseof this paper. The secondary objective of the paper was to present enough observations o convince the reader that there are many interesting issues left to explore. Last, butnot least, the insights of Blaizot and Nowak [10] deserve further study.The shock at τ = 4 is reminiscent of the possibility that instantons at infinite N might herald, as τ → − , a jump in certain particularly sensitive quantities in 4D YM[19].It should also be mentioned that workers in lattice field theory [20] have shownnumerically that in four dimensions the trace 2 cos θ of a Wilson loop for SU (2) seemsto evolve with the area as if θ were diffusing on the SU (2) group manifold where theeigenvalues of W are e ± iθ . For N = 2 there is no essential distinction between thecharacteristic polynomial and any other gauge invariant observable related to the matrix W . Blaizot and Nowak [21] have independently identified the viscosity as N . Acknowledgments.
I acknowledge partial support by the DOE under grant number DE-FG02-01ER41165at Rutgers University and by the SAS of Rutgers University. I note with regret that myresearch has for a long time been deliberately obstructed by my high energy colleaguesat Rutgers. I gratefully acknowledge an email from the authors of [10] on Feb. 27, 2008,drawing my attention to their work and a further email commenting on the presentmanuscript and identifying seminars where they presented their work in progress. Anongoing collaboration on related topics with R. Narayanan, as well as comments on thismanuscript, are also gratefully acknowledged. I also wish to thank J. Feinberg for somecomments on the manuscript.
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