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Multiway junction conditions: Jackiw-Teitelboim gravity

Jia-Yin Shen

Abstract

A booklet is a geometric structure formed by gluing multiple bulk spacetimes along a common interface and imposing gravitational consistency conditions at the junction. We have systematically investigated the properties of booklet structures, constructed the booklet geometry, and derived the multiway junction conditions applicable at the interface. In this work, we provide a complete solution to the multiway junction conditions for booklets composed of bulks governed by JT gravity. By constructing invariants of the dilaton solution space, we classify all dilaton configurations into three inequivalent types, each exhibiting attractive, repulsive, or neutral behavior. Through continuous isometric transformations, each type is fixed to a standard form characterized by a single physical parameter, effectively eliminating redundant degrees of freedom. This process selects a distinguished class of Poincaré coordinates for each type. Expanding the constraint equations order by order breaks coordinate invariance beyond the leading and subleading orders. By jointly solving the junction and continuity conditions up to subleading order, we find that junctions are only allowed when a sufficient number of bulks carry attractive dilatons, as captured quantitatively by a equilibrium condition. We further analyze all possible combinations of different dilaton types and determine the shape of the interface along with the explicit form of the dilaton defined on it.

Multiway junction conditions: Jackiw-Teitelboim gravity

Abstract

A booklet is a geometric structure formed by gluing multiple bulk spacetimes along a common interface and imposing gravitational consistency conditions at the junction. We have systematically investigated the properties of booklet structures, constructed the booklet geometry, and derived the multiway junction conditions applicable at the interface. In this work, we provide a complete solution to the multiway junction conditions for booklets composed of bulks governed by JT gravity. By constructing invariants of the dilaton solution space, we classify all dilaton configurations into three inequivalent types, each exhibiting attractive, repulsive, or neutral behavior. Through continuous isometric transformations, each type is fixed to a standard form characterized by a single physical parameter, effectively eliminating redundant degrees of freedom. This process selects a distinguished class of Poincaré coordinates for each type. Expanding the constraint equations order by order breaks coordinate invariance beyond the leading and subleading orders. By jointly solving the junction and continuity conditions up to subleading order, we find that junctions are only allowed when a sufficient number of bulks carry attractive dilatons, as captured quantitatively by a equilibrium condition. We further analyze all possible combinations of different dilaton types and determine the shape of the interface along with the explicit form of the dilaton defined on it.
Paper Structure (23 sections, 163 equations, 1 figure)

This paper contains 23 sections, 163 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Review of general results on multiway junction conditions
  3. Application: gluing multiple pages in JT gravity
  4. Junction condition for the extrinsic curvature
  5. Extrinsic curvature of the cutoff curve
  6. Near-boundary expansion of the junction condition for the curvature
  7. Junction condition for the dilaton
  8. Classification of the dilaton profiles
  9. Continuity conditions of the dilaton
  10. Near-boundary expansion of the dilaton's continuity conditions
  11. The junction condition of the dilaton
  12. Schwarzian derivative and $\mathfrak{sl}(2,\mathbb{R})$
  13. Solving the subleading-order conditions
  14. All pages taking Type $+$
  15. All pages taking Type $0$
  16. ...and 8 more sections

Figures (1)

  • Figure 1: Dilatons are classified based on the values of $A^2-4BC$, with Type $0$ and Type $-$ further divided into two disconnected parts. The topological disconnectedness of Type $0$ is due to the trivial solution $(0,0,0)$, which is inequivalent to all non-trivial dilatons