Revisiting noninteracting string partition functions in Rindler space

TL;DR

The paper investigates how to formulate and interpret one-loop string partition functions in Euclidean Rindler space by summing over field spectra and distinguishing edge (surface) terms from bulk contributions. It shows that open-string edge interactions reproduce higher-spin surface terms, while closed strings do not admit a simple sum-over-fields equivalence, necessitating a non-interacting construction that excludes surface terms and yields modular-invariant results for Type II and heterotic strings. The resulting non-interacting partition functions exhibit IR divergences associated with maximal acceleration near the horizon, prompting interpretations that these divergences are artifacts of treating higher-spin fields rather than fundamental inconsistencies of string theory, with implications for Solodukhin’s proposal and firewall discussions. Overall, the work strengthens the connection between string theory and field-theory spectra on conical spaces, clarifies the role of worldsheet duality, and highlights subtle aspects of horizon thermodynamics in string theory.

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.

Figures (12)

• Figure 1: Open and closed string diagrams at one loop in flat space (or its conical cousins) where the origin of the 2d plane is excised. Worldsheets are then automatically forced not to intersect the origin. Drawn here are the winding one graphs.
• Figure 3: String vacuum fluctuation that crosses the Rindler origin. This is seen by the Rindler observer as an open string with ending points fixed (and immobile) on the horizon.
• Figure 4: An open string piercing the Rindler origin, performing a loop around the thermal direction and forming a cylindrical worldsheet. Fixed timeslices are interpreted as an emission and reabsorption of an open string.
• Figure 5: An open string performing a loop around the thermal direction and forming a cylindrical worldsheet. Fixed timeslices are interpreted as a non-interacting open string.
• Figure 6: An open string worldsheet of wrapping number two can be equivalently seen as a twice wound closed string moving from the green to the red curve. All winding numbers of closed strings must be present simply because all open string wrapping numbers must be present in the non-interacting open string partition function in such a topologically non-trivial situation.