Single-Sided DSSYK Black Hole

• The paper introduces a novel Hamiltonian deformation in DSSYK that integrates an end-of-the-world brane to model a single-sided black hole.
• It employs modified chord rules and boundary algebra analysis to reveal a nontrivial commutant encoding the 'no man’s island' behind the horizon.
• The study draws parallels to island prescriptions in evaporating black holes, offering insights into operator reconstruction and holographic duality.

A single-sided black hole in the double-scaled Sachdev-Ye-Kitaev (DSSYK) model is realized by deforming the original DSSYK Hamiltonian to incorporate an end-of-the-world (EoW) brane behind the horizon. This construction modifies the standard structure of boundary operator algebras and introduces a nontrivial commutant, leading to fundamental implications for bulk reconstruction and the entanglement structure of the system. In the appropriate semiclassical limit, the region behind the black hole horizon, bounded by the EoW brane—termed “no man’s island”—is unambiguously encoded within the commutant of the boundary algebra, and exhibits a direct correspondence with island regions diagnosed in black hole evaporation scenarios.

1. Hamiltonian Deformation and Modified Chord Rules

The foundational object in gravity-only (zero-matter) DSSYK is the $q$-deformed harmonic oscillator,

$$H_{bulk} = a + a^\dagger, \qquad [a, a^\dagger]_q = 1, \qquad a|0\rangle = 0.$$

To implement an EoW brane (a defect representing a geometrical boundary inside the bulk) behind the black hole horizon, the Hamiltonian is deformed by adding an "exponential wormhole-length" operator, resulting in

$$\tilde{H}_{R,0} = H_{R,0} + \mu\, q^{N_0}\, r^{N_1}, \quad \tilde{H}_{R,1} = H_{R,1},$$

where $H_{R,0} = a_{R,0} + a_{R,0}^\dagger$, $H_{R,1} = a_{R,1} + a_{R,1}^\dagger$, $q = e^{-\lambda_{HH}}$, $r = e^{-\lambda_{HM}}$, and $N_0, N_1$ are gravity and matter chord number operators, respectively. Here, $\mu$ encodes the coupling strength to the EoW brane.

Under this deformation, the chord diagram rules acquire a new feature: whenever a gravity chord is closed on the EoW brane, the configuration is weighted by a factor $\mu q^{N_0} r^{N_1}$. The remaining chord rules (including crossing weights and matter insertions) are unchanged, but a new chord type associated with the EoW defect is present.

2. Structure and Properties of the Boundary Algebra

The right boundary von Neumann algebra for the single-sided system is generated by the deformed operators: $$\tilde{A}_R = \{ \tilde{H}_{R,0}, \tilde{H}_{R,1} \}''.$$ The vacuum $|0\rangle$ is both cyclic and separating for $\tilde{A}_R$ on the total Hilbert space $\mathcal{H}$. The algebra is closed (in the weak operator topology) under polynomials in the $\tilde{H}_{R,i}$.

Unlike the original two-sided DSSYK, $\tilde{A}_R$ has a nontrivial commutant, denoted $\tilde{A}_L = \tilde{A}_R'$ (its set of bounded operators commuting with all elements of $\tilde{A}_R$). This fundamental algebraic structure precludes complete bulk reconstruction from the right boundary alone, as discussed further below.

In the matter-free sector, $\tilde{A}_R$ is a type $\mathrm{II}_1$ von Neumann factor, inheriting this property by unitary equivalence from the $q$-Gaussian algebra of standard DSSYK.

3. $q$-Coherent State Realization and Algebra Isomorphism

An alternative approach defines the single-sided EoW state as a right $q$-coherent state,

$$|B_\mu\rangle = \sum_{n\ge 0} \frac{(1-q)^{n/2}\, \mu^n}{[n]_q!}\;|n\rangle, \qquad [n]_q! = (1-q)\cdots(1-q^n)/(1-q)^n.$$

The right annihilation operators act as $$a_{R,0}|B_\mu\rangle = \mu|B_\mu\rangle,\, a_{R,1}|B_\mu\rangle = 0$$. One introduces “tilde” annihilators,

$$\tilde{a}_{R,0} = a_{R,0} + \mu q^{N_0} r^{N_1}, \quad \tilde{a}_{R,1} = a_{R,1},$$

which obey the same $q$-oscillator algebra as their untilde counterparts and annihilate the vacuum: $\tilde{a}_{R,i}|0\rangle = 0$.

A key identity relates vacua and $q$-coherent states: $$\langle 0|\tilde{O}_1\cdots\tilde{O}_k|0\rangle = \langle B_\mu| O_1\cdots O_k |B_\mu\rangle,$$ for any string of tilde/untilde operators. This yields a (weakly continuous) algebra isomorphism

$$\mathrm{Td}^{-1}: \tilde{A}_R \to A_R \equiv \{ H_{R,0}, H_{R,1} \}'',$$

and a unitary operator $U$ with $U|0\rangle = |B_\mu\rangle$ such that $\tilde{A}_R = U A_R U^\dagger$.

4. Commutant, Factoriality, and No Man’s Island

$\tilde{A}_R$ is type $\mathrm{II}_1$, and has a nontrivial commutant $\tilde{A}_L = \tilde{A}_R'$, as implied by the tracial state and established $q$-Gaussian operator algebra structure. Using Tomita-Takesaki theory, with cyclic/separating vector $|\psi\rangle = U|0\rangle = B_\mu^{-1/2}|0\rangle$, the modular conjugation $J_\psi$ satisfies $\tilde{A}_L = J_\psi \tilde{A}_R J_\psi$.

Explicit generators for $\tilde{A}_L$ commute with all of $\tilde{A}_R$. For example, in the pure-gravity subsector,

$$\tilde{H}_{L,0} = a_{L,0} + a_{L,0}^\dagger - \mu(1-q) a_{L,0}^\dagger a_{L,0} + \mu I,$$

while $\tilde{H}_{L,1}$ is a more involved operator. Notably, the commutant is non-geometric in the chord basis; its action cannot be interpreted simply in terms of geometric bulk operators.

In the semiclassical Jackiw-Teitelboim (JT) scaling limit ($q=e^{-\lambda}\to 1$, $\lambda\to 0$, $\mu\sim e^{-\mu_r/\lambda}$), the algebra $\tilde{A}_R$ descends to the GNS algebra of single-trace matter operators. The algebra $K_R$ becomes type $\mathrm{III}_1$, and only encodes the causal wedge exterior to the EoW brane. The region between the horizon and the brane—the “no man’s island”—is only captured by the commutant $K_L$, which serves as the canonical purification of $K_R$.

Concretely, in the (normalized) TFD state at inverse temperature $\beta_{BH}$, the two-point function is: $$\langle M_L(t_1) M_R(t_2) \rangle_{\mathrm{TFD}} = \left( \frac{\pi/\beta_{BH}}{\cosh[\pi(t_1 + t_2)/\beta_{BH}]} \right)^2,$$ with $M_R(t) = e^{i h_R t} M_R e^{-i h_R t}$, $h_R = H_R/\sqrt{1-q}$. The "no man’s island" (the region behind the horizon and before the EoW brane) is encoded in the mirror commutant $K_L$.

5. Relation to Islands in Black Hole Evaporation and Encoding Map

There is a precise analogy to the standard island prescription in semiclassical gravity. In the traditional Page-time scenario describing black hole evaporation, the radiation algebra eventually develops an island (an entanglement wedge containing a region behind the horizon). Any operator associated to this island can be represented on the radiation Hilbert space by a complex, state-dependent mapping; equivalently, by a unitary mapping to a second, fictitious black hole in a two-sided TFD setup, in which the island operators become simple mirror operators.

In the single-sided DSSYK construction, the commutant $K_L$ (initially non-geometric in the chord basis) becomes the simple "left" algebra under the unitary isomorphism $U: \tilde{A}_R \to A_R$. Thus, $U$ acts as an encoding map, translating complicated radiation-side operators into simple mirror operators, and realizing the “no man’s island” as the analog of the standard gravitational island.

6. Full Operator Algebra and Traversable Wormholes

To generate the full bounded operator algebra $B(\mathcal{H})$, it suffices to add the exponential wormhole-length operator,

$e^{-\bar{b}} \equiv q^{N_0} r^{N_1},$

to the generating set $\{ \tilde{H}_{R,0}, \tilde{H}_{R,1} \}$. Thus,

$$\{ \tilde{H}_{R,0}, \tilde{H}_{R,1}, e^{-\bar{b}} \}'' = B(\mathcal{H}).$$

All creation and annihilation operators $a_{R,i}^\dagger$, $a_{R,i}$ can be recovered by projecting in the joint spectrum of $H$ and $e^{-\bar{b}}$.

In the semiclassical JT limit, this additional operator implements a traversable-wormhole coupling or an averaged measurement protocol, permitting reconstruction of the “no man’s island” region from the single boundary system, completing the algebraic construction.

7. Connections to Symmetry Sectors and Relational Holography

Work on symmetry sectors in chord space provides complementary perspectives on the realization of single-sided EoW brane physics in DSSYK, as detailed in (Aguilar-Gutierrez, 26 Jun 2025). By imposing chord-parity constraints in the chord Hilbert space, one constructs reduced systems that correspond to AdS$_2$ with an EoW brane. The parity-gauged Hamiltonians take an Al-Salam–Chihara (ASC) form, and the boundary-to-bulk dictionary is established via the spread-complexity to wormhole length map, in which

$$C(t) = \frac{1}{\lambda} \log e^{-\ell(t)} \quad \Leftrightarrow \quad \lambda C(t) = L_{ETW}(u).$$

The presence of the EoW brane is encoded in the operator content and is reflected in the underlying chord sector constraints. The spectrum, stability, and correlation functions are computable in closed form, with the interpretation that matter chords correspond to ETW-brane insertions in the bulk and their backreaction on the geometry.

A plausible implication is that these algebraic and geometrical constructions are mutually compatible and illuminate the universality of "island" phenomena in single-sided holographic systems.

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In summary, the single-sided black hole in DSSYK, realized via a Hamiltonian deformation introducing an EoW brane, yields a boundary algebra with a nontrivial commutant. In the semiclassical limit, this structure gives rise to a region behind the horizon (no man's island) that is not reconstructable from the single boundary algebra but is encoded in the commutant, paralleling the island prescription in evaporating black holes. The algebraic framework unifies the description of single-sided bulk geometries, their associated operator algebras, and their boundary-to-bulk holographic dictionaries.