Regge trajectories from the adjoint sector of Matrix Quantum Mechanics

Abstract

We reexamine the large $N$ limit of SU$(N)$ symmetric quantum mechanics of a Hermitian matrix whose singlet sector is well known to be exactly solvable via free fermions. When the Fermi level approaches a maximum of the potential, there is critical behavior corresponding to string theory in two dimensions. We uncover new phenomena in the adjoint sector by solving the Marchesini-Onofri equation both numerically and analytically using semiclassical approximations: at criticality, the spectrum is governed by Regge trajectories with energy eigenvalues growing according to $Δ^2 \sim n/ α'$. In the dual 2D string theory, we interpret these states as oscillatory excitations of a ``short'' folded open string. Up to sub-leading corrections, this Regge behavior is essentially universal and is insensitive to the particular potential we choose to approach criticality. Slightly away from criticality, the highly excited states transition into ``long strings'' that extend far into the Liouville direction.

Figures (4)

• Figure 1: Left: A cartoon of the oscillatory folded string motion. The string extends from the "UV wall" at $\phi=0$ to some $\phi_\text{tip}$ (the dotted blue line). As long as the tip reaches sufficiently far, the dynamics of the string is universal. For most of the trajectory the tip of the string travels nearly at the speed of light, but near the turning point it has nonzero acceleration for a few string lengths. Right: Two Liouville regions attached at $\phi=0$, and the $\mathbb{Z}_2$ symmetry acts as $\phi\rightarrow -\phi$. The string endpoints are fixed at $\phi=0$, while the fold oscillates between the two regions.
• Figure 2: Left: the MO spectrum for the quartic theory near criticality $g\to g_c$. The $\mathbb{Z}_2$ even (odd) states are shown with blue (orange) dots. Here $n_{\rm max}$ is given in \ref{['eq:nmax-phi4']}, the approximate value where the red curve (high energy WKB) touches the green curve (the Regge trajectory). In the inset panels, we show the first few MO eigenvalues, which closely follow the Regge formula \ref{['eq:regge-phi4']}. The deviation at small $n$, which is nearly imperceptible, is presumably due to corrections to the Regge formula and non-universal features of the "UV wall." Right: spectrum for the cubic theory near criticality. It is qualitatively similar but there is no $\mathbb{Z}_2$ symmetry. The transition value $n_\text{max}$ between the Regge behavior and the high-energy WKB is given by \ref{['eq:nmax-phi3']}.
• Figure 3: Eigenvalues from diagonalizing the discretized MO equation for quartic theory with $M=36000$. On the right panel, we see that $\left(\Delta+\tfrac{2 \sqrt{2}}{\pi}\right)^2$ becomes evenly spaced for small $\mu$, which is the Regge behavior $\Delta^2 \sim n$. (We have included the constant offset in \ref{['eq:regge-phi4']} so that the spacings look more even for small $n$). On the left panel, we see that at large $\mu$, $\Delta$ becomes evenly spaced as expected from a weakly coupled theory ($g\to 0$). At energies much bigger than the crossover energy $\Delta \gg \Delta_\text{max}$, we expect the $\Delta$'s to be evenly spaced but with a much finer spacing \ref{['eq:WKB']} than the spacings in the weak coupling theory. Note that $\mu$ needs to be quite small $\mu \lesssim 10^{-4}$ in order to resolve several levels of the Regge trajectory. The transition between long and short strings takes place approximately at $\Delta_\text{max}$ which is given in \ref{['eq:nmax-phi4']}.
• Figure 4: Left: the effective potential $\eta$ in the MO equation \ref{['MO-1']} for the quartic potential \ref{['eq:quartpot']}, using the time of flight variable $\tau$. We use $\mu=10^{-14}$, which shows a good separation between the UV region near $\tau=0$ and the two "Liouville walls" at $\tau=\pm \tau_{\rm max}$. Right: the MO potential for the cubic using the time-of-flight variable $\tau$. At $\tau = 0$ there is the UV wall; at $\tau = \tau_{\rm max}$ there is the Liouville wall.