Partition Functions for the Rigid String and Membrane at Any Temperature

TL;DR

This paper derives exact partition functions for the rigid string and membrane at any temperature using hypergeometric functions and zeta regularization, enabling analysis of temperature effects and Hagedorn temperature determination.

Contribution

It provides new exact expressions for the partition functions of rigid strings and membranes at all temperatures, extending previous results and enabling systematic temperature analysis.

Findings

Exact partition functions expressed in hypergeometric functions.

Analytic continuation via zeta regularization for asymptotic sums.

Method to determine the Hagedorn temperature for membranes.

Abstract

Exact expressions for the partition functions of the rigid string and membrane at any temperature are obtained in terms of hypergeometric functions. By using zeta function regularization methods, the results are analytically continued and written as asymptotic sums of Riemann-Hurwitz zeta functions, which provide very good numerical approximations with just a few first terms. This allows to obtain systematic corrections to the results of Polchinski et al., corresponding to the limits $T\rightarrow 0$ and $T\rightarrow \infty$ of the rigid string, and to analyze the intermediate range of temperatures. In particular, a way to obtain the Hagedorn temperature for the rigid membrane is thus found.