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Open Strings On The Rindler Horizon

Edward Witten

TL;DR

This work investigates the entanglement entropy of Rindler space in string theory by applying a Z_N orbifold replica construction to open strings on a Dp-brane crossing the Rindler horizon. By formulating the 1-loop open-string partition function Z_N(τ) via a meromorphic continuation in N and evaluating the relevant residues, the paper identifies how analytic continuation to Re ${ m Re} olinebreak { m { olinebreak N}}>1$ can remove the closed-string tachyonic divergence in the open-string channel, while indicating a potential nonunitary, logarithmic CFT structure upon continuation. The analysis details the K1 (RR-like) and K2 (NS-NS-like) contributions to ${ m Tr} ho^{ olinebreak { m { olinebreak N}}}$, obtaining finiteness constraints (notably $p olinebreak leq 2$ for 1-loop entropy) and uncovering double-pole phenomena intrinsic to the analytic continuation. The results bridge open-string annulus amplitudes, residue calculus on meromorphic elliptic functions, and possible nonunitary CFT behavior, highlighting both the promise and the subtle pitfalls of extending replica-based entropies into string theory.

Abstract

It has been proposed that a certain Z_N orbifold, analytically continued in N, can be used to describe the thermodynamics of Rindler space in string theory. In this paper, we attempt to implement this idea for the open-string sector. The most interesting result is that, although the orbifold is tachyonic for positive integer N, the tachyon seems to disappear after analytic continuation to the region that is appropriate for computing ${\mathrm {Tr}} ρ^{\mathcal N}$, where $ρ$ is the density matrix of Rindler space and Re $\mathcal N$>1. Analytic continuation of the full orbifold conformal field theory remains a challenge, but we find some evidence that if such analytic continuation is possible, the resulting theory is a logarithmic conformal field theory, necessarily nonunitary.

Open Strings On The Rindler Horizon

TL;DR

This work investigates the entanglement entropy of Rindler space in string theory by applying a Z_N orbifold replica construction to open strings on a Dp-brane crossing the Rindler horizon. By formulating the 1-loop open-string partition function Z_N(τ) via a meromorphic continuation in N and evaluating the relevant residues, the paper identifies how analytic continuation to Re can remove the closed-string tachyonic divergence in the open-string channel, while indicating a potential nonunitary, logarithmic CFT structure upon continuation. The analysis details the K1 (RR-like) and K2 (NS-NS-like) contributions to , obtaining finiteness constraints (notably for 1-loop entropy) and uncovering double-pole phenomena intrinsic to the analytic continuation. The results bridge open-string annulus amplitudes, residue calculus on meromorphic elliptic functions, and possible nonunitary CFT behavior, highlighting both the promise and the subtle pitfalls of extending replica-based entropies into string theory.

Abstract

It has been proposed that a certain Z_N orbifold, analytically continued in N, can be used to describe the thermodynamics of Rindler space in string theory. In this paper, we attempt to implement this idea for the open-string sector. The most interesting result is that, although the orbifold is tachyonic for positive integer N, the tachyon seems to disappear after analytic continuation to the region that is appropriate for computing , where is the density matrix of Rindler space and Re >1. Analytic continuation of the full orbifold conformal field theory remains a challenge, but we find some evidence that if such analytic continuation is possible, the resulting theory is a logarithmic conformal field theory, necessarily nonunitary.

Paper Structure

This paper contains 11 sections, 93 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Strategy For Analytic Continuation
  3. Dp-Brane Partition Function
  4. Computing the Partition Function
  5. Residues
  6. Closed-String Sector
  7. Computation of ${\mathrm{Tr}}\,\rho^{\mathcal{N}}$
  8. The $K_1$ Contribution
  9. The $K_2$ Contribution
  10. A Reformulation Of The Problem Of Analytic Continuation
  11. An Alternative Formula For $G(z,\tau)$

Figures (1)

  • Figure 1: An annulus in which $\sigma_1$ parametrizes an interval and $\sigma_2$ parametrizes a circle describes an open string propagating in the $\sigma_2$ direction, or a closed string propagating in the $\sigma_1$ direction, known as the crossed channel. The closed string states that contribute all have $L_0=\overline L_0$, since the annulus is invariant under rotation of the circle parametrized by $\sigma_2$.