Color Characters for White Hot String Bits

The state space of a generic string bit model is spanned by $N\times N$ matrix creation operators acting on a vacuum state. Such creation operators transform in the adjoint representation of the color group $U(N)$ (or $SU(N)$ if the matrices are traceless). We consider a system of $b$ species of bosonic bits and $f$ speciesof fermionic bits. The string, emerging in the $N\to\infty$ limit, identifies $P^+=mM\sqrt{2}$ with $M$ the bit number operator and $P^-=H\sqrt{2}$ with $H$ the system Hamiltonian. We study the thermal properties of this string bit system in the case $H=0$, which can be considered the tensionless string limit: the only dynamics is restricting physical states to color singlets. Then the thermal partition function ${\rm Tr} e^{-\beta mM}$ can be identified, putting $x=e^{-\beta m}$, with a generating function $\chi_0^{bf}(x)$, for which the coefficient of $x^n$ in its expansion about $x=0$ is the number of color singlets with bit number $M=n$. This function is a purely group theoretic object, which is well-studied in the literature. We show that at $N=\infty$ this system displays a Hagedorn divergence at $x=1/(b+f)$ with ultimate temperature $T_H=m/\ln(b+f)$. The corresponding function for finite $N$ is perfectly finite for $0<x<1$, so the $N=\infty$ system exhibits a phase transition at temperature $T_H$ which is absent for any finite $N$. We demonstrate that the low temperature phase is unstable above $T_H$. The lowest-order $1/N$ asymptotic correction, for $x\to1$ in the high temperature phase, is computed for large $N$. Remarkably, this is related to the number of labeled Eulerian digraphs with $N$ nodes. Systematic methods to extend our results to higher orders in $1/N$ are described.