Color characters for white hot string bits

摘要

The state space of a generic string bit model is spanned by N × 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 species of fermionic bits. The string, emerging in the N → ∞ limit, identifies P~+ = mM2~(1/2) where M is the bit number operator and P~- = H2~(1/2) where H is 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 Tre~(-βmM) can be identified, putting x = e~(-βm), with a generating function x_0~(bf)(x), for which the coefficient of xn 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 = ∞ 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 = ∞ 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 → 1 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.