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

A bstract 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 $$ {\mathcal{U}}_q\left( su\left(1,1\right)\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>q</mml:mi> </mml:msub> <mml:mfenced> <mml:mrow> <mml:mi>su</mml:mi> <mml:mfenced> <mml:mn>1</mml:mn> <mml:mn>1</mml:mn> </mml:mfenced> </mml:mrow> </mml:mfenced> </mml:math> . 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.