Towards the QCD String: 2+1 dimensional Yang-Mills theory in the planar limit

We study the large $N$ (planar) limit of pure SU(N) 2+1 dimensional Yang-Mills theory ($YM_{2+1}$) using a gauge-invariant matrix parameterization introduced by Karabali and Nair. This formulation crucially relies on the properties of local holomorphic gauge invariant collective fields in the Hamiltonian formulation of $YM_{2+1}$. We show that the spectrum in the planar limit of this theory can be explicitly determined in the $N=\infty$, low momentum (large 't Hooft coupling) limit, using the technology of the Eguchi-Kawai reduction and the existing knowledge concerning the one-matrix model. The dispersion relation describing the planar $YM_{2+1}$ spectrum reads as $\omega(\vec{k}) = \sqrt{{\vec{k}}^2 + m_n^2}$, where $n=1,2,...$ and $m_n = n m_r$, where $m_r$ denotes the renormalized mass, the bare mass m being determined by the planar 't Hooft coupling $g_{YM}^2 N$ via $m= \frac{g_{YM}^2 N}{2 \pi}$. The planar, low momentum limit, also captures the expected short and long distance physics of $YM_{2+1}$ and gives an interesting new picture of confinement. The computation of the spectrum is possible due to a reduction of the $YM_{2+1}$ Hamiltonian for the large 't Hooft coupling to the {\it singlet} sector of an effective one matrix model. The crucial observation is that the correct vacuum (the large N master field), consistent with the area law and the existence of a mass gap, is described by an effective quadratic matrix model, in the large N, large 't Hooft coupling limit