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1+1 Dimensional Compactifications of String Theory

Naureen Goheer, Matthew Kleban, Leonard Susskind

TL;DR

The paper argues that stable, maximally symmetric string theory compactifications to 1+1 dimensions with macroscopic horizon area are incompatible with holography. By combining thermofield dynamics with the 1+1D Poincaré and AdS2 algebras, it shows that finite horizon entropy would require a discrete spectrum for the Rindler Hamiltonian, but the thermofield double construction yields a non-countable spectrum for the full thermofield Hamiltonian, leading to a contradiction. The authors also discuss IR instabilities from massless moduli in 1+1D, and they conclude that observer complementarity and finite-entropy holography generally demand infinite horizon area, with certain loopholes noted. Overall, the work narrows the allowed backgrounds for holographic theories in 1+1D and highlights the tension between horizon entropy and symmetry in low-dimensional compactifications.

Abstract

We argue that stable, maximally symmetric compactifications of string theory to 1+1 dimensions are in conflict with holography. In particular, the finite horizon entropies of the Rindler wedge in 1+1 dimensional Minkowski and anti de Sitter space, and of the de Sitter horizon in any dimension, are inconsistent with the symmetries of these spaces. The argument parallels one made recently by the same authors, in which we demonstrated the incompatibility of the finiteness of the entropy and the symmetries of de Sitter space in any dimension. If the horizon entropy is either infinite or zero the conflict is resolved.

1+1 Dimensional Compactifications of String Theory

TL;DR

The paper argues that stable, maximally symmetric string theory compactifications to 1+1 dimensions with macroscopic horizon area are incompatible with holography. By combining thermofield dynamics with the 1+1D Poincaré and AdS2 algebras, it shows that finite horizon entropy would require a discrete spectrum for the Rindler Hamiltonian, but the thermofield double construction yields a non-countable spectrum for the full thermofield Hamiltonian, leading to a contradiction. The authors also discuss IR instabilities from massless moduli in 1+1D, and they conclude that observer complementarity and finite-entropy holography generally demand infinite horizon area, with certain loopholes noted. Overall, the work narrows the allowed backgrounds for holographic theories in 1+1D and highlights the tension between horizon entropy and symmetry in low-dimensional compactifications.

Abstract

We argue that stable, maximally symmetric compactifications of string theory to 1+1 dimensions are in conflict with holography. In particular, the finite horizon entropies of the Rindler wedge in 1+1 dimensional Minkowski and anti de Sitter space, and of the de Sitter horizon in any dimension, are inconsistent with the symmetries of these spaces. The argument parallels one made recently by the same authors, in which we demonstrated the incompatibility of the finiteness of the entropy and the symmetries of de Sitter space in any dimension. If the horizon entropy is either infinite or zero the conflict is resolved.

Paper Structure

This paper contains 8 sections, 23 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Thermofield dynamics
  3. Rindler space
  4. Continuity of the spectrum
  5. Representations of the Poincaré algebra
  6. String theory arguments
  7. Conclusion
  8. Acknowledgements

Figures (1)

  • Figure 1: On the left, the conformal diagram for Minkowski space, showing the Rindler wedges I and III. In Rindler coordinates, continuing from region I to region III involves shifting time by $i \beta /2$, in accord with \ref{['timecont']}. On the right, the "Rindler" wedge of AdS$_2$.