A Novel Representation for the Free Energy in String Theory at Non-Zero Temperature

TL;DR

This paper introduces a Laurent series representation for the free energy in string theory at non-zero temperature, linking the Hagedorn temperature to the series' convergence and exploring potential extensions beyond it.

Contribution

It presents a new Laurent series formalism for the free energy in string theory, explicitly deriving it for various string models and connecting the Hagedorn temperature to series convergence.

Findings

Laurent series representation for free energy obtained explicitly.

Hagedorn temperature linked to the convergence radius of the series.

Prospects for describing string theory above Hagedorn temperature via Borel continuation.

Abstract

A novel representation ---in terms of a Laurent series--- for the free energy of string theory at non-zero temperature is constructed. The examples of open bosonic, open supersymmetric and closed bosonic strings are studied in detail. In all these cases the Laurent series representation for the free energy is obtained explicitly. It is shown that the Hagedorn temperature arises in this formalism as the convergence condition (specifically, the radius of convergence) of the corresponding Laurent series. Some prospects for further applications are also discussed. In particular, an attempt to describe string theory above the Hagedorn temperature ---via Borel analytical continuation of the Laurent series representation--- is provided.