Antonini–Sasieta–Swingle–Rath Configuration

• The ASSR configuration is a holographic setup using a modified thermofield-double Euclidean path integral to create two asymptotically AdS regions and a causally disconnected closed universe.
• It presents competing bulk descriptions—one that includes a baby universe and one that projects it out—thereby raising fundamental questions about holographic encoding and bulk uniqueness.
• Swap tests and algebraic reformulations are employed to analyze entanglement, observer complementarity, and the duality puzzles inherent in the semiclassical ASSR geometry.

Searching arXiv for papers on the Antonini–Sasieta–Swingle–Rath configuration and related baby-universe discussions. The Antonini–Sasieta–Swingle–Rath (ASSR) configuration, also called the ASSR geometry, is a holographic setup in which a modified thermofield-double Euclidean path integral of two holographic CFTs, supplemented by a spherically symmetric heavy operator insertion on the Euclidean boundary, prepares a Lorentzian bulk spacetime with two approximately AdS universes and a large closed universe entangled with them. In the literature, it functions as a sharp laboratory for the Antonini–Rath puzzle: a single boundary CFT state appears to admit two semiclassical bulk descriptions, one with a baby universe and one without, thereby turning questions about bulk uniqueness, holographic encoding, and observer dependence into explicitly computable problems (Engelhardt et al., 8 Jul 2025, Higginbotham, 7 Jul 2025).

1. Geometric definition and kinematics

In the formulation used by Engelhardt, Gesteau, and Harlow, the configuration is introduced under the heading “Antonini-Sasieta-Swingle-Rath geometry.” The construction starts by modifying the thermofield double state of two holographic CFTs to include a spherically symmetric heavy operator $O$ midway along the Euclidean boundary. This produces a shell of dust in the dual geometry whose mass density is of order $1/G$. For sufficiently long Euclidean boundaries, the dominant Lorentzian continuation consists of two approximately AdS universes containing gases of matter particles that are jointly entangled with matter particles in a closed universe whose size in AdS units is large (Engelhardt et al., 8 Jul 2025).

The Lorentzian geometry has three components. First, there are two asymptotically AdS regions, each with its own boundary and its own gas of matter. Second, there is a closed universe, topologically compact and with no asymptotic region, which eventually undergoes a big crunch. Third, the relation between these components is set by the Euclidean preparation and by matter entanglement in the resulting state; the closed universe is causally disconnected from the AdS asymptotic regions, and the construction is not a traversable wormhole joining all three pieces (Engelhardt et al., 8 Jul 2025).

In the notation used for the ASSR geometry by Engelhardt, Gesteau, and Harlow, the effective bulk degrees of freedom in the AdS universes are denoted $a$, the effective bulk degrees of freedom in the closed universe are denoted $b$, and the boundary CFT degrees of freedom are denoted $A$. The same literature also uses a different notation, especially in discussions of the Antonini–Rath puzzle, where two disconnected AdS components are often called $a,b$ and the baby universe $i$. This suggests that the geometric content is more stable across the literature than the symbol choice.

Symbol	Meaning	Context
$H_a$	Effective Hilbert space of bulk degrees in the two AdS regions	ASSR notation
$H_b$	Effective Hilbert space of bulk degrees in the closed universe	ASSR notation
$H_A$	Fundamental Hilbert space of the two boundary CFTs	ASSR notation

A Penrose-like description of the geometry emphasizes a Euclidean preparation cap, two asymptotic AdS boundaries, a locus where the suppressed sphere shrinks to zero size, and a matter shell generated by $1/G$0 that collapses and sources the closed universe. In this sense, the ASSR configuration is not merely an entanglement pattern but a specific Euclidean-to-Lorentzian construction with definite bulk slices for the AdS sectors and for the closed FRW-like region (Engelhardt et al., 8 Jul 2025).

2. Holographic encoding and the two candidate bulk states

The bulk state prepared by the Euclidean path integral is written as

$1/G$1

with large but $1/G$2 matter entanglement between the AdS regions and the closed universe. The holographic encoding map is

$1/G$3

On the AdS degrees of freedom, the map uses the usual HKLL encoding isometry

$1/G$4

while the baby-universe sector is treated by a rank-one projection built from an orthogonal transformation $1/G$5 on $1/G$6, a fixed ancilla state $1/G$7, and projection onto a simple state $1/G$8. In schematic form, the baby-universe factor is acted on by

$1/G$9

The orthogonality of $a$0 is motivated by the gauging of CRT symmetry, which implies that the Hilbert space of a closed universe is real (Engelhardt et al., 8 Jul 2025).

A defining claim of the ASSR discussion is that there exists a pure AdS-only bulk state

$a$1

such that the boundary state is the same, up to normalization, as the full bulk state with the baby universe: $a$2 An explicit expression is

$a$3

Accordingly, $a$4 is the full ASSR bulk state on $a$5, while $a$6 is a bulk state only on the AdS regions, with the two AdS gases in a pure entangled state and no baby universe (Engelhardt et al., 8 Jul 2025).

This equivalence at the boundary is the core structural feature of the Antonini–Rath puzzle. One and the same boundary CFT state seems compatible with two distinct semiclassical narratives: a geometry containing a closed universe entangled with the AdS sectors, and a geometry in which the baby-universe degrees of freedom have been projected out and absorbed into a pure AdS state. A later formulation makes the same point through an explicitly non-isometric baby-universe map,

$a$7

so that the baby-universe degrees of freedom are post-selected rather than encoded as an independent boundary factor (Higginbotham, 7 Jul 2025).

3. Swap tests, purity, and the central dispute

The information-theoretic probe most closely associated with the ASSR configuration is the swap operator acting on two copies of the AdS gas degrees of freedom. In the effective bulk description, its expectation value satisfies

$a$8

If the AdS gases form a pure entangled state with no entanglement with the baby universe, then $a$9 and the expectation value is $b$0. If the AdS gases are mixed because they are entangled with the closed universe, then the expectation value is smaller than $b$1. In this way, the swap test becomes an operational criterion for deciding whether the effective bulk state should be regarded as $b$2 or $b$3 (Engelhardt et al., 8 Jul 2025).

One line of argument, developed by Engelhardt and Gesteau, constructs a low-energy and low-complexity boundary operator dual to a bulk swap of the asymptotic AdS sectors and concludes that the expectation values in the descriptions with and without the baby universe cannot match if the baby universe is semiclassical. In that analysis, the boundary state selects the spacetime without a semiclassical baby universe, and more generally forces any baby-universe Hilbert space to be effectively one-dimensional (Engelhardt et al., 20 Apr 2025).

A competing line of argument modifies the holographic map on the baby-universe sector by post-selection and then insists that the same boundary observable must be paired with the bulk operator induced by the appropriate, possibly non-isometric, map. In that formulation, the boundary operator $b$4 has dual $b$5 in the no-baby bulk and dual

$b$6

in the baby-universe bulk, leading to the relation

$b$7

On this view, the swap test cannot distinguish the two candidate bulk duals, because using the naive bulk swaperator $b$8 in the baby geometry amounts to evaluating the wrong bulk operator (Higginbotham, 7 Jul 2025).

The ASSR configuration is therefore the locus of a precise methodological disagreement. One side treats the asymptotic causal-wedge encoding and the extrapolate dictionary as sufficient to exclude a semiclassical baby universe; the other treats non-isometric post-selection on closed universes as consistent with the extrapolate dictionary and argues that boundary observables do not, by themselves, decide between the two bulk pictures (Engelhardt et al., 20 Apr 2025, Higginbotham, 7 Jul 2025).

4. Observer complementarity and observer-dependent codes

A further development places observers explicitly inside the holographic map. In the observer-rule framework adopted by Engelhardt, Gesteau, and Harlow, the effective Hilbert space is split into an observer sector and the rest, and the encoding need only preserve inner products after decohering the observer in a pointer basis: $b$9 with

$A$0

In the ASSR geometry, two observers are introduced: $A$1, located in one AdS region, and $A$2, located in the closed universe. The practical implementation clones the observer in the pointer basis into an external system $A$3 or $A$4, so that decoherence is realized by entanglement with an external record (Engelhardt et al., 8 Jul 2025).

With observer-dependent codes, Engelhardt, Gesteau, and Harlow compute the AdS swap expectation value separately for the two observers. For the AdS observer,

$A$5

matching the AdS-only state $A$6. For the closed-universe observer,

$A$7

so that $A$8 sees the AdS gases as mixed with the baby universe, up to errors exponentially small in the observer entropy. In their interpretation, this is a clean instance of observer complementarity: $A$9 concludes that the bulk state is $a,b$0, while $a,b$1 concludes that it is $a,b$2, and there is no operational contradiction because the two observers live in disconnected components of the universe (Engelhardt et al., 8 Jul 2025).

Later work revises this conclusion by redefining the observer swap operators through the full web of holographic maps in the Heisenberg and Schrödinger pictures. In that construction, the corrected baby-universe swap operator is

$a,b$3

and the observer operators are obtained by applying the induced maps $a,b$4 and, when available, $a,b$5. The resulting claim is that the AdS observer $a,b$6 cannot use the swap test to rule out the baby universe, while the closed-universe observer $a,b$7 can improve the accuracy of the description by choosing more appropriate reconstructed operators; in explicit toy models, the reconstruction can even be exact (Higginbotham, 19 Dec 2025).

The observer literature therefore preserves the ASSR configuration as a test case but changes what is taken to follow from it. In one reading, the closed universe is excised from the AdS observer’s semiclassical description; in another, the swap test is redefined so that no observer can exclude the baby universe on that basis alone.

5. Algebraic reformulation and the commutant picture

A distinct response to the Antonini–Rath puzzle abandons neither the baby universe nor the boundary description, but reframes the problem algebraically. In this approach, the relevant object is the pair $a,b$8, where $a,b$9 is an operator algebra and $i$0 a state on that algebra. For the AS$i$1 cosmology arising from partially entangled thermal states, the large-$i$2 GNS Hilbert space is identified with the full bulk Fock space,

$i$3

while the representation of the single-trace algebra is

$i$4

The baby universe is then identified with the commutant,

$i$5

rather than with an additional boundary Hilbert-space factor (Liu, 17 Sep 2025).

This algebraic description is paired with a modified large-$i$6 prescription. The proposal is that heavy-operator matrix elements in the low-energy sector take the oscillatory form

$i$7

so that pointwise large-$i$8 limits fail to exist, but an averaged large-$i$9 limit over $H_a$0 does exist. In that averaged limit, right and left two-point functions become thermal with inverse temperatures $H_a$1 and $H_a$2, cross-correlations vanish, and the second Rényi entropy reproduces the gravitational expression

$H_a$3

Within this framework, the Gesteau no-go theorem does not apply because its pointwise large-$H_a$4 assumption is violated (Liu, 17 Sep 2025).

The algebraic formulation also argues that there is no geometric entanglement wedge for the single right or left boundary at order $H_a$5. The ordinary quantum extremal surface logic would yield $H_a$6, while the exact entropy is instead encoded through

$H_a$7

That mismatch is used to motivate a description in which local baby-universe operators must be represented by operators involving both CFTs, rather than being attached to a single-boundary wedge (Liu, 17 Sep 2025).

6. Generalizations, saddle competition, and current status

The ASSR configuration belongs to a broader family of closed-universe constructions descending from the Antonini–Sasieta–Swingle setup. In three-dimensional gravity with heavy particles, a large “AS$H_a$8 menagerie” can be generated by starting with an arbitrary heavy-particle closed-universe cosmology and then gluing in an arbitrary number of AdS tubes between the past and future conformal boundaries of the associated Euclidean wormhole. This construction makes it straightforward to produce examples in which the cosmology is approximately homogeneous and isotropic, and it furnishes an infinite family of Euclidean manifolds whose Lorentzian continuations contain the same closed cosmology together with various asymptotic AdS regions (Raamsdonk et al., 15 Jan 2026).

At the same time, the menagerie analysis gives a necessary condition for a cosmological wormhole saddle to dominate the Euclidean path integral: in an appropriate vertical slicing, the corresponding two-CFT state must have left-right entanglement entropy of order $H_a$9. Applied to the original AS$H_b$0 setup, the conclusion is that the cosmological saddle is usually subdominant; local non-cosmological saddles often have smaller action, and the original AS$H_b$1 construction usually does not meet the stated dominance condition (Raamsdonk et al., 15 Jan 2026).

The present literature therefore contains several incompatible but sharply formulated positions on the ASSR configuration. One position treats the baby universe as effectively projected out or one-dimensional at the level of the fundamental holographic description (Engelhardt et al., 20 Apr 2025, Engelhardt et al., 8 Jul 2025). A second position treats non-isometric post-selection as the correct completion of the holographic map on closed universes and holds that the baby-universe geometry remains a valid semiclassical dual not excluded by swap tests (Higginbotham, 7 Jul 2025, Higginbotham, 19 Dec 2025). A third position reformulates the problem algebraically, with the closed universe encoded in the commutant of the single-trace algebra and with an averaged large-$H_b$2 limit built into the dictionary (Liu, 17 Sep 2025).

In that sense, the Antonini–Sasieta–Swingle–Rath configuration is best understood not as a single settled construction but as a controlled arena in which several competing conceptions of holography for closed universes can be compared. Across those conceptions, the stable core is the same: a Euclideanly prepared state of two holographic CFTs, a Lorentzian bulk containing two asymptotically AdS regions and a causally disconnected closed universe, and a boundary state whose interpretation tests how far bulk non-uniqueness, post-selection, algebraic duality, and observer dependence can be pushed within AdS/CFT (Engelhardt et al., 8 Jul 2025, Higginbotham, 7 Jul 2025, Liu, 17 Sep 2025).