Single-Sided Black Holes in Double-Scaled SYK Model and No Man's Island

TL;DR

We study a single-sided black hole with an end-of-the-world brane behind the horizon in the double-scaled SYK model, introducing a deformed right boundary Hamiltonian that includes an exponential wormhole-length term and yields a boundary von Neumann algebra of type II$_1$. This leads to a nontrivial commutant and the no-man's island in the semiclassical JT limit, while two equivalent descriptions—the $q$-coherent EoW-brane state and a family of matter-brane states—allow exact diagonalization of the full DSSYK spectrum. The commutant interpretation is explored via Tomita–Takesaki theory, modular purification, and an isomorphism to a left algebra; a minimal extension with the operator $q^{N_0} r^{N_1}$ recovers the full operator algebra, highlighting an ambiguity in boundary algebra definitions. In the JT limit, the no-man's island corresponds to the canonical purification of the boundary algebra, and including the length operator can render the boundary algebra global, relating the setup to island phenomena and Kourkoulou–Maldacena EoW branes. Overall, the work provides a UV-complete, algebraic framework for studying islands and their commutants in DSSYK, with precise solvability of the full spectrum and clear bulk–boundary dualities.

Abstract

We study a single-sided black hole with an end-of-the-world (EoW) brane behind the horizon in the double-scaled SYK (DSSYK). The new Hamiltonian is a deformation of the original DSSYK Hamiltonian with an extra exponential wormhole length operator, which leads to a new chord diagram rule. The boundary algebra is defined as generated by the new Hamiltonian and boundary matter. There is an alternative but equivalent definition with a $q$-coherent state due to a nontrivial isomorphism of the vN algebra of DSSYK. This isomorphism induces a unitary equivalence, which yields a surprising result that the boundary algebra of a single-sided black hole in DSSYK has a non-trivial commutant and is a type II$_1$ vN factor. It follows that the full bulk reconstruction from the boundary is impossible, and there is a ``no man's island" behind the horizon in the semiclassical JT limit. Inspired by the EoW brane, we construct a family of matter-brane states with an arbitrary number of matter chords and behaving like an EoW brane. They exactly solve the full spectrum of DSSYK. We take different ways to understand the nontrivial commutant. We show that the commutant is complex on chord number basis and thus non-geometric. In the semiclassical JT limit, the commutant becomes the canonical purification of the boundary algebra and claims the no man's island. In the context of Hawking radiation after Page time, the unitary equivalence is interpreted as encoding the canonical purification into the old Hawking radiation, and the no man's island has the same essence as the island. Including the exponential wormhole length operator independently, the boundary algebra is extended to all bounded operators and reconstructs the no man's island. This can be regarded as a different choice for the definition of boundary algebra. This type I$_\infty$ algebra is closely related to the EoW brane in Kourkoulou-Maldacena.

Figures (17)

• Figure 1: A single-sided black hole with an EoW brane behind the horizon in JT gravity. The dark green curve is the trajectory of the EoW brane; the blue dot is the horizon; the shaded region behind the horizon is the "no man's island"; the entanglement wedge of the boundary is the same as the causal wedge and is the dotted region outside the horizon.
• Figure 2: Topologically inequivalent chord diagrams contributing to $\langle \mathrm{tr}(H^6) \rangle$, respectively taking values $q^3$, $q$, $1$, $q^2$ and $q$. Note that multiplicity of each diagram has to be accounted for when calculating the trace.
• Figure 3: Examples of chord diagrams contributing to $\left\langle \text{tr}(H^2MH^2MHMHM)\right\rangle$, respectively taking values $q^2r^3$ and $q^2r^3q_m$. One needs to sum over all possible chord diagrams. $H$ and $M$ chords are respectively denoted by black solid lines and blue lines.
• Figure 4: Chord diagrams can be equivalently drawn horizontally, here we show the horizontal version of the chord diagram Figure \ref{['fig:chord1']}. We can trace the diagram from the left to the right, each Hamiltonian insertion on the boundary then either opens a new chord or annihilates an existing one.
• Figure 5: Inner products between two chord states are calculated by summing over all possible ways to pair up chords of the same type and counting crossings, with each crossing contributing a factor. Here we include some examples, Figure \ref{['fig:4a']}, \ref{['fig:4b']}, \ref{['fig:4c']} respectively contribute to inner products $\braket{0000}$, $\braket{001010}$ and $\braket{100001}{001010}$, with values $q^2$, $q^3r^4$ and $q^2r^4s$.

Theorems & Definitions (25)

• Lemma 1
• proof
• Definition 1
• Lemma 2
• Lemma 3
• proof
• Lemma 4
• proof
• Remark
• Theorem 5