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Splitting and gluing in sine-dilaton gravity: matter correlators and the wormhole Hilbert space

Chuanxin Cui, Moshe Rozali

Abstract

Sine-dilaton gravity has been proposed as the holographic dual of the double scaled SYK (DSSYK) model. In this work, we examine this duality by deriving general matter correlation functions directly from the bulk perspective. A novel technique we develop is to treat matter lines in the bulk as end-of-the-world (EOW) branes, allowing us to implement a splitting and gluing procedure on the bulk spacetime. This geometric procedure gives rise to the structure of two-sided multi-particle wormhole Hilbert space. In contrast to previous Hilbert space constructions in DSSYK, the length basis in our case factorizes across the subregions, whereas the energy basis acquires a non-local, state-dependent structure determined by the EOW brane quantization in each subregion. Different choices of splitting correspond to distinct representations of the same Hilbert space, all of which are equivalent for physical observables. Based on this framework, we compute general correlation functions, including the OTOC, and show that they exactly reproduce the DSSYK results obtained from chord diagrams. By calculating the OTOC in different representations, we uncover a new identity for the 6j-symbol of the quantum group $\mathcal{U}_q(su(1,1))$. Finally, the resulting wormhole Hilbert space enables us to go beyond the disk to compute matter correlation functions on the double trumpet and to include bulk matter loops.

Splitting and gluing in sine-dilaton gravity: matter correlators and the wormhole Hilbert space

Abstract

Sine-dilaton gravity has been proposed as the holographic dual of the double scaled SYK (DSSYK) model. In this work, we examine this duality by deriving general matter correlation functions directly from the bulk perspective. A novel technique we develop is to treat matter lines in the bulk as end-of-the-world (EOW) branes, allowing us to implement a splitting and gluing procedure on the bulk spacetime. This geometric procedure gives rise to the structure of two-sided multi-particle wormhole Hilbert space. In contrast to previous Hilbert space constructions in DSSYK, the length basis in our case factorizes across the subregions, whereas the energy basis acquires a non-local, state-dependent structure determined by the EOW brane quantization in each subregion. Different choices of splitting correspond to distinct representations of the same Hilbert space, all of which are equivalent for physical observables. Based on this framework, we compute general correlation functions, including the OTOC, and show that they exactly reproduce the DSSYK results obtained from chord diagrams. By calculating the OTOC in different representations, we uncover a new identity for the 6j-symbol of the quantum group . Finally, the resulting wormhole Hilbert space enables us to go beyond the disk to compute matter correlation functions on the double trumpet and to include bulk matter loops.

Paper Structure

This paper contains 34 sections, 122 equations, 16 figures, 1 table.

Table of Contents

  1. Introduction
  2. Summary
  3. Splitting and gluing procedure
  4. One-particle Hilbert space
  5. Multi-particle Hilbert space
  6. General correlation functions
  7. Sine-dilaton gravity setup
  8. Matter fields in sine-dilaton gravity
  9. Splitting a particle worldline in $\mathrm{AdS}_2$
  10. EOW brane quantization
  11. Fully-open channel
  12. Semi-open channel
  13. Partition functions of the left and right regions
  14. Gluing condition and the two-point function
  15. Wormhole Hilbert space with matter
  16. ...and 19 more sections

Figures (16)

  • Figure 1: (a) Matter geodesic as a thin shell in the bulk. (b) Splitting the shell into two parts.
  • Figure 2: Fully-open channel
  • Figure 3: Semi-open channel
  • Figure 4: Bulk pictures in the fully-open channel: (a) Identity operator as a density matrix; (b) Gluing of two regions with each having an EOW brane boundary.
  • Figure 5: Bulk pictures in the semi-open channel: (a) $|n_M=0\rangle$ state; (b) State on the wormhole slice.
  • ...and 11 more figures