### 2+1 D Yang-Mills at Large-N

A while back, I pointed to an announcement by Leigh, Minic and Yelnikov, proposing a solution to 2+1 D Yang-Mills at large-$N$. Well, the long version of their paper has appeared.

The main previously-missing detail which is filled in here (perhaps making sense of my previous discussion) is how exactly they propose to regularize the Karabali-Nair Hamiltonian (see Appendix A of their paper) $\mathcal{H}= \frac{g_{\text{YM}}^2 N}{2\pi}\int J^a \frac{\delta}{\delta J^a} +\doubleintegral \Omega^{a b}(z,w) \frac{\delta}{\delta J^a(z)}\frac{\delta}{\delta J^b(w)} +\frac{1}{2 g_{\text{YM}}^2}\int \overline{\partial}J^a\overline{\partial}J^a$ where $\Omega^{a b}(z,w) = \frac{N}{\pi} D^{a b}_w \mathcal{G}(w-z)$ Here $D = \partial - \frac{\pi}{N} [J, \cdot]$ and $\mathcal{G}(w-z)$ is the ordinary Green’s function, satisfying $\overline{\partial}_z\mathcal{G} = \delta^{(2)}(z)$.

They take as an ansatz that the ground state wave functional takes the form
$\Psi_0(J)= \exp\left(-\frac{\pi}{2N m^2}\int\tr \overline{\partial}J\, K\left(\frac{\Delta}{m^2}\right)\, \overline{\partial}J\right)$
where $m= \frac{g_{\text{YM}}^2 N}{2\pi}$ and $\Delta = \{D,\overline{\partial}\}/2$, for *some* kernel $K\left(\frac{\Delta}{m^2}\right)$. Formally, they then expand
$K(L) = \sum_{n=0}^\infty c_n L^n$
and find that the Schrœdinger equation, $\mathcal{H}\Psi_0=0$ is equivalent to a Riccati equation,
$-\frac{1}{2L}\frac{d}{d L} (L^2 K) +L K^2 + 1 =0$
which is solved^{1} by a ratio of Bessel functions.
$K(L) = \frac{1}{\sqrt{L}}\frac{J_2(4\sqrt{L})}{J_1(4\sqrt{L})}$
(There’s a constant of integration that is fixed by demanding that $\Psi_0$ be normalizable. This choice also, as they argue, yields the correct asymptotically-free UV behaviour and IR behaviour corresponding to confinement and a mass-gap.)

In addition to the predictions for the spectrum of spin-$0$ glueballs that I discussed previously, they also produce predictions for the spectrum of higher-spin glueball states.

^{1} The trick is to write
$K(L)= -\frac{4}{f(x)}\frac{d}{d x} (f(x)/x)$
with $x=4\sqrt{L}$, which converts the (nonlinear) Riccati equation into the (linear) Bessel equation,
$f'' +\frac{1}{x} f' +\left(1-\frac{2}{x^2}\right) f =0$

## Re: 2+1 D Yang-Mills at Large-N

Its certainly an interesting paper. There are certain parts that get a little hazy to me, for instance, its not clear to me why they restrict to quadratic order in solving the Schroedinger eqn (presumably b/c they want to restrict themselves to dealing with simple local ‘probing’ operators like tr(dbar j dbar j).

They feel this is morally equivalent to some sort of expansion alla string theory, where higher order terms end up probing spatial extent. But, I don’t quite see why that has to be the case, and you might worry that theres a lot of physics in there thats getting chopped off.

Still the numbers are eerily close to lattice results