Burgers' equation in 2D SU(N) YM

It is shown that the logarithmic derivative of the characteristic polynomial of a Wilson loop in two dimensional pure Yang Mills theory with gauge group SU(N) exactly satisfies Burgers' equation, with viscosity given by 1/(2N). The Wilson loop does not intersect itself and Euclidean space-time is assumed flat and infinite. This result provides a precise framework in 2D YM for recent observations of Blaizot and Nowak and was inspired by their work.

the average characteristic polynomial, and depending also on the initial condition.
In addition to making the insight of [10] particularly transparent, I hope that this result would also aid future efforts to exploit large N universality in dimensions higher than two for obtaining analytical quantitative estimates of the ratio between a scale describing perturbative phenomena and the scale of confinement. This was the original motivation for seeking to establish numerically large N phase transitions in Wilson loops [1].

Characteristic polynomial.
An N × N simple unitary Wilson loop matrix W , defined on a curve that does not self intersect, with τ denoting the dimensionless area in units of the 't Hooft gauge coupling, has the following probability distribution: The sum is over all irreducible representations R with character χ R (W ) and second order Casimir C 2 (R). dW is the Haar measure. Normalization conventions are standard [2] and τ ≥ 0. We introduce the average characteristic polynomial One can think about Q N (z, τ ) as the generating function for the χ R (W ) with totally antisymmetric R. Simple manipulations [2] produce an integral representation: It is more convenient to study where, for the time being, y is kept real. q N (y, t) is even in y and this is the main reason for extracting the exponential factor from Q N . Changing the integration variable u to x = y + τ (u + 1/2) gives:

Main result.
It is now a trivial matter to observe that with initial condition lim τ →0 [q N (y, τ )] = (2 cosh(y/2)) N The behavior at y → ±∞ prevents solving (6) by Fourier decomposition and any associated general conclusions about boundedness as τ → +∞. The initial condition is a consequence of for any W 0 ∈ SU (N ). This equation can be also directly derived from the polynomial formula of Q N , without going to the integral representation. This heat equation is related to Burgers' equation (for example, see [12], problem 12(a), p. 214) by Burgers' equation and the initial condition are At N = ∞, N drops out of the equation giving the inviscid limit: 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) The solution is given implicitly by This equation is known to produce a shock at a time τ * > 0 which is the first time at which multiple solutions become available. τ * is the smallest positive value satisfying 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 3 + cy 5 + ....
This implies that y * = 0 and therefore A shock will form if a < 0. In the case of N = ∞ 2D YM we have Therefore, the critical area corresponds to the well known critical value [4,6]. Universality can be invoked now in a sense that applies to the nonlinear equation producing a generic shock [13,14]. This means taking the simplest polynomial h(y) capable of producing shocks: h(y) = ay + by 3 with a < 0, 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 of the structure of the evolving Riemann surface y(φ, τ ) parameterized by τ ≥ 0. One can also take τ into the complex plane.

Large N = small viscosity.
Making the viscosity nonzero is a singular perturbation which eliminates the shock and has the same effect as making N finite. Large N universality will hold in the vicinity of the critical area and corresponds to universal behavior in the vicinity of the would-be shock for small viscosities, which is the simplest dissipative 1 regularization of the shock. The important new insight is that the large N transition is equivalent to a movable singularity, determined by the initial condition, rather than by the evolution rule. 2 Thus, the simplest initial condition producing a shock will also lead to a universal small viscosity smoothing of the shock.
Running the derivation backwards, with the minimal initial condition 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 higher dimensions than two in [1,2]. The critical exponents µ = 1/2, 3/4 associated with the scalings N µ that need to be taken [2] are identical to those found in defining the small viscosity 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].

Higher critical points.
We have become accustomed to expect higher critical points, of reduced degrees of stability, to accompany a basic large N critical point. Looking at (15) it seems plausible that setting b = 0 and making c > 0 would produce a critical point of one degree of stability less. Obviously, if this is true, a whole hierarchy will be generated, by initial conditions of the form ay + by 2m−1 with integer m ≥ 2. If one is not worried about the convergence of the associated universal integrals and one is also willing to give up the y → −y parity symmetry, also higher critical points with half integer m could be studied, at least as formal originators of asymptotic series. It would be intriguing if parent models existed with physical symmetries that selected one of these higher critical points. More work on this is left for the future.

Product representation.
It certainly is true that One may view the z i (τ ) as certain averages of the eigenvalues of W , but not as usually defined: In [2] it was proved that |z i (τ )| = 1 for all i = 1, ..., N (see [17] for a plot); this indicates that the z i (τ ) are to be viewed as an approximations to theẑ av i (τ ). By applying large N factorization N times, one can argue, at least away from large N critical points, that identically orderedẑ av i (τ )'s and z i (τ )'s are equal to each other. I suspect that this stays true also in the double scaling limit. If this suspicion is validated, we shall obtain a new method 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 inserting the product (21) into the heat equation (6) and applying (4), standard manipulations of the kind employed in the study of Calogero systems producė Hereż j = dz j (τ )/dτ and 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 → 1/z j and z j → z * j , as expected from the structure of the polynomial. In addition, again as expected, the product of all zeros is constant in τ . Moreover, the equations of motion (23) imply d|z j (τ )| 2 /dτ = 0, j = 1, .., N .
The map z = −e y creates an infinite number of copies of the zeros z j , which are all on the imaginary axis. We choose one specific y j for each z j , j = 1, .., N . The equation of motion for the y j 's is: The universal description changes the equation obeyed by the y j 's. However, as pointed out in [16] on the basis of an old theorem [18], the y j (τ ) still stay on the imaginary axis for all τ . In the universal case periodicity under y j (τ ) → y j (τ ) + 2m j πi, m j ∈ Z is lost, since the initial condition on the y i 's no longer is periodic. Thus, one needs to use the y j variables to make the connection between the exact equations of motion and the universal ones. I leave a more detailed study of the universal limit of the eigenvalue motion to the future.
The regularization of the shock provides a smooth connection between small and large loops. In two dimensions Burgers' equation provides an exact renormalization group type of equation allowing the evaluation of φ N (y, τ ) when τ → ∞, given φ N (y, τ ) in the limit τ → 0. The approach to the limit τ → ∞ gives the dimensionless string tension 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 + (−1) N . This simply says that at infinite τ all W m terms, for any m > 0, can be replaced by zero. Using (4), we conclude that the large τ behavior is given by: We now wish to recover the ensuing φ N (y, τ = ∞) from Burgers' equation. The route is again in reverse of our derivation: First go to the heat equation, then get the integral representation in order to incorporate the initial condition. Finally, in order to get the asymptotic behavior for large τ , change variables in the integral representation, arriving at: For large τ , one of the two exponents making up each hyperbolic function dominates, depending on the sign of u: Here, ε(u) is the sign function. The above equation implies that This is the expected result. At infinite N , the hyperbolic tangent becomes a sign function. In an electrostatic picture it is obvious that the above result holds if the poles of φ(y, τ ) are uniformly spaced 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 = −1 as y goes through zero along the real axis [2]. That the solution has this limiting behavior is essential for confinement, which would be indicated by the leading correction to the above result being exponentially small in τ .
Note that τ was taken to infinity at finite N ; the final result admits a subsequent infinite 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 implicit equation defining the solution for τ < 4. With the wrong initial conditions this jump might not grow to the full size required for consistency with confinement; thus, the transition in itself is insufficient to guarantee confinement. If we want to add the input that 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 would hope that the analogy to the three dimensional incompressible Navier-Stokes equation does 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 Nif 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 equation holding in two dimensional Yang Mills theory with gauge group SU (N ) defined on the infinite Euclidean plane. A surprising simplicity in the area dependence of the average characteristic polynomial of simple Wilson loops was found. Nevertheless, the essential feature of the existence of a large N phase transition is captured by this observable. In this respect the average characteristic polynomial of the Wilson loop is superior to traces of the Wilson loop in some fixed representation. As explained in [2] this observable has other advantages, in dimensions three and four. The simple and exact finite N relation to Burgers' equation presented above seems to provide opportunity for progress in different directions, as emphasized in the course of this paper. The secondary objective of the paper was to present enough observations to convince the reader that there are many interesting issues left to explore. Last, but not 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 → 4 − , a jump in certain particularly sensitive quantities in 4D YM [19].
It should also be mentioned that workers in lattice field theory [20] have shown numerically that in four dimensions the trace 2 cos θ of a Wilson loop for SU (2) seems to evolve with the area as if θ were diffusing on the SU (2) group manifold where the eigenvalues of W are e ±iθ . For N = 2 there is no essential distinction between the characteristic polynomial and any other gauge invariant observable related to the matrix W .

Added note.
Blaizot and Nowak [21] have independently identified the viscosity as 1 2N .