Abstract
String bit systems exhibit a Hagedorn transition in the $N\to\infty$ limit. However, there is no phase transition when $N$ is finite (but still large). We calculate two-loop, finite $N$ corrections to the partition function in the low temperature regime. The Haar measure in the singlet-restricted partition function contributes pieces to loop corrections that diverge as $\mathcal{O}(N)$ when summed over the mode numbers. We study how these divergent pieces cancel each other out when combined. The properly normalized two loop corrections vanish as $\mathcal{O}(N^{-1})$ for all temperatures below the Hagedorn temperature. The coefficient of this $1/N$ dependence decreases with temperature and diverges at the Hagedorn pole.
Highlights
One can study a light-cone-quantized string as the continuum limit of a polymer of point masses called string bits [1,2]
The longitudinal coordinate is recovered in the large N [4] limit and the continuum limit of such a polymer
Such systems exhibit a Hagedorn transition from a low-temperature phase that consists of closed chains to a high-temperature phase consisting of liberated bits
Summary
One can study a light-cone-quantized string as the continuum limit of a polymer of point masses called string bits [1,2]. These bits move in transverse space, enjoy nearest-neighbor interactions and transform adjointly under a global UðNÞ symmetry. It is instructive to study the behavior of such a system at finite temperature [6,7,8] Such systems exhibit a Hagedorn transition from a low-temperature phase that consists of closed chains to a high-temperature phase consisting of liberated bits. In this paper we shall present finite N corrections to the following partition function in the low-temperature regime:. Lk1;...;kp e2πiðn1k1þÁÁÁþnpkpÞ=N ð7Þ are the coupling constants in “Fourier space.” L21⁄2θ0 turns out to be a circulant matrix in the “position indices,” i.e., Lm;n1⁄2θ0 1⁄4 Fðjm − njÞ, and can be naturally diagonalized via the Fourier transform [8]
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