Revisiting noninteracting string partition functions in Rindler space

TL;DR

This paper analyzes non-interacting string partition functions in Rindler space, clarifying their structure, edge states, and divergences, with implications for black hole physics and string theory consistency.

Contribution

It provides a detailed examination of string partition functions in Rindler space, highlighting the role of edge states, surface terms, and the impact of excluding these terms on modular invariance and divergences.

Findings

Partition functions split into surface and non-surface parts.

Excluding surface terms yields modular invariant partition functions.

IR divergences suggest a maximal acceleration near black hole horizons.

Abstract

We revisit non-interacting string partition functions in Rindler space by summing over fields in the spectrum. In field theory, the total partition function splits in a natural way in a piece that does not contain surface terms and a piece consisting of solely the so-called edge states. For open strings, we illustrate that surface contributions to the higher spin fields correspond to open strings piercing the Rindler origin, unifying the higher spin surface contributions in string language. For closed strings, we demonstrate that the string partition function is not quite the same as the sum over the partition functions of the fields in the spectrum: an infinite overcounting is present for the latter. Next we study the partition functions obtained by excluding the surface terms. Using recent results of JHEP 1505 (2015) 106, this construction, first done by Emparan, can be put on much firmer ground. We generalize to type II and heterotic superstrings and demonstrate modular invariance. All of these exhibit an IR divergence that can be interpreted as a maximal acceleration close to the black hole horizon. Ultimately, since these partition functions are only part of the full story, divergences here should not be viewed as a failure of string theory: maximal acceleration is a feature of a faulty treatment of the higher spin fields in the string spectrum. We comment on the relevance of this to Solodukhin's recent proposal. A possible link with the firewall paradox is apparent.