Bubbling wormholes and matrix models

Imagine the universe as a giant, complex video game. In this game, there are two main ways to describe reality: one is the "code" (a mathematical theory called a Quantum Field Theory), and the other is the "graphics" (a theory of gravity and space-time called General Relativity). Usually, these two look completely different, but a famous idea called Holography says they are actually two sides of the same coin.

This paper explores a new, strange way to connect these two sides using a concept called "entanglement," but with a twist.

The Setup: Two Separate Worlds

Imagine you have two identical, separate rooms (let's call them Room A and Room B). Inside each room, there is a complex machine (a Matrix Model) that generates the physics of that room. Normally, these rooms don't talk to each other.

In standard physics, if you want to connect these rooms, you usually link them by matching their energy levels. This creates a "wormhole"—a tunnel connecting the two rooms.

The New Idea: Linking by "Identity"

The authors of this paper asked: What if we don't link them by energy, but by their "identity"?

Think of the machines in the rooms as having a set of dials (eigenvalues). Usually, the dials in Room A are set randomly, and the dials in Room B are set randomly. The authors propose a special "glue" (a mathematical operator they call a delta operator) that forces the dials in Room A to match the dials in Room B perfectly.

It's like taking two separate orchestras and forcing every violinist in Orchestra A to play the exact same note at the exact same time as a specific violinist in Orchestra B. This isn't just a gentle handshake; it's a rigid, mathematical lock.

The Result: The "Bubbling Wormhole"

When they apply this "glue," something weird happens to the space-time graphics.

Instead of two separate rooms connected by a simple tunnel, the universe transforms into a "Bubbling Wormhole."

The Shape: Imagine two soap bubbles that have merged. They share a common rim (a circle). The paper suggests that the "boundary" of our universe (where the rules are written) is no longer two separate spheres, but two spheres that touch and share a single circle.
The "Bubble" Effect: The space inside isn't just a smooth tunnel. It's a "multi-cover." Imagine taking a map of the world and folding it over itself twice (for a "two-cover") or four times (for a "four-cover"). The geometry looks locally like normal space, but globally, it's a complex, folded structure.
The "Bubbling": In the middle of this folded space, there are "bubbles" or singularities. Think of these as pinched points where the fabric of space is stretched tight. To make the math work, these points need a source of "negative energy" (like a cosmic tension that pulls inward rather than pushing out).

The "Glue" Explained Simply

The "glue" they use is a sum over all possible "patterns" (representations) the machines can make.

Analogy: Imagine you have two decks of cards. You shuffle them separately. Then, you take a magical rule that says: "For every card you pull from Deck A, you must pull the exact matching card from Deck B."
The Effect: This rule acts like a Delta Function. In math, a delta function is like a spike that says "Only these specific values are allowed; everything else is zero." This forces the two separate systems to behave as one single, entangled entity.

What They Found

The Geometry: They calculated exactly what this folded, multi-layered space looks like. It has specific "conical singularities" (sharp points) where the geometry is slightly distorted, requiring that negative energy source to hold it together.
The Cost: They calculated the "energy cost" (free energy) of creating this connection. It turns out to be a negative number, which makes sense because the "glue" is attractive—it pulls the two worlds together.
The Probe: They tested this new universe by sending a "probe" (a tiny string) through it. They found that the string behaves in a way that perfectly matches the predictions of the "glued" card decks (the matrix models). This confirms that their mathematical glue successfully creates the physical wormhole.

Summary

The paper proposes a new way to build a wormhole. Instead of connecting two universes by their energy, they connect them by forcing their internal "dials" to match perfectly. This creates a strange, folded universe where two boundaries share a common edge, held together by a mysterious negative energy source. It's a mathematical demonstration that if you entangle two quantum systems in a very specific, rigid way, the holographic result is a "bubbling" wormhole geometry.

Technical Summary: Bubbling Wormholes and Matrix Models

Problem Statement
The paper addresses the holographic realization of entanglement between multiple copies of a Conformal Field Theory (CFT). While the thermofield double state entangles two CFT copies via a sum over energy eigenstates, dual to a two-sided eternal black hole, this work explores an analogous construction where entanglement occurs over gauge group representations rather than energy eigenstates. Specifically, the authors investigate the holographic duals of "entangled" sums of half-BPS Wilson loops in multiple copies of $U(N)$ $\mathcal{N}=4$ super Yang-Mills (SYM) theory. The central question is whether such sums, which act as delta-function-like operators correlating the eigenvalues of matrix models, correspond to connected bulk geometries distinct from standard Euclidean wormholes.

Methodology
The authors employ a combination of supersymmetric localization, matrix model analysis, and type IIB supergravity solutions:

Matrix Model Construction:
- Starting with two (or four) decoupled $U(N)$ $\mathcal{N}=4$ SYM theories, the authors insert a "supersymmetric delta operator" $\hat{\delta}_{12}$ (Eq. 1.6). This operator is constructed as a ratio of determinants involving traces over irreducible representations and their transposes, effectively acting as a delta function on the group manifold that correlates the eigenvalues of the Hermitian matrices $M_1$ and $M_2$ (or $M_1, \dots, M_4$ ).
- They analyze the saddle point equations of the resulting multi-matrix models. By assuming a bound-state configuration where eigenvalues from different matrices form pairs (or quadruplets) with parametrically small separations ( $\sim 1/N$ ) compared to the spectral width ( $\sim \sqrt{\lambda}/N$ ), they derive the spectral curves of these coupled systems.
- The authors identify that the spectral curves of these coupled models match those of specific Gaussian-Penner matrix models (Appendix A), which serve as effective descriptions of the bulk geometries.
Supergravity Analysis:
- The authors construct "bubbling wormhole" (BW) geometries using the half-BPS ansatz of Lin, Lunin, and Maldacena. These solutions are determined by harmonic functions $h_1$ and $h_2$ on a Riemann surface $\Sigma$ .
- They explicitly construct two-cover (BW2) and four-cover (BW4) solutions. These geometries are multi-covers of $AdS_5 \times S^5$ that differ globally but are locally identical.
- The conformal boundaries of these geometries consist of multiple four-spheres ( $S^4$ ) that intersect on a common circle ( $S^1$ ), rather than being completely disconnected.
- The solutions feature codimension-2 conical singularities in the interior of $\Sigma$ (at fixed points of the covering map) with a conical excess of $2\pi$ (total angle $4\pi$ for the two-cover).
Free Energy and Probe Calculations:
- The free energy of the delta operator is computed in the matrix model, yielding a leading order behavior of $F_\delta \sim -N^2/\sqrt{\lambda}$ .
- On the bulk side, the authors calculate the on-shell action of the conical singularity and the Einstein-Hilbert action. They find that the leading $O(N^2)$ contributions from the negative-tension cosmic brane source and the conical excess cancel exactly.
- To match the subleading $O(N^2/\sqrt{\lambda})$ scaling of the matrix model free energy, they propose a Dvali-Gabadadze-Porrati (DGP) term (induced gravity) on the worldvolume of the bulk source.
- Probe fundamental strings are analyzed in these backgrounds to compute Wilson loop expectation values, comparing bulk minimal surface areas with matrix model results.

Key Contributions and Results

Bubbling Wormhole Geometries: The paper proposes a new class of holographic geometries called "bubbling wormholes." These are multi-covers of $AdS_5 \times S^5$ where the conformal boundary is a union of multiple $S^4$ s intersecting along a common $S^1$ . Unlike standard Euclidean wormholes with disconnected boundaries, these geometries possess a single connected conformal boundary.
Delta Operator as a Gluing Mechanism: The authors demonstrate that the insertion of a supersymmetric delta operator (sum over representations) into the path integral of decoupled matrix models effectively "glues" the eigenvalue distributions of the separate theories. This results in a single spectral curve that corresponds to the multi-cover geometry.
Spectral Curve Matching: A precise mapping is established between the resolvents of the coupled multi-matrix models (and their effective Gaussian-Penner counterparts) and the harmonic functions defining the bubbling wormhole supergravity solutions.
- For the two-cover case, the map is a two-to-one transformation $w(z) = z - e_1e_2/z$ , mapping two cuts on the $z$ -plane to a single cut on the $w$ -plane.
- For the four-cover case, a four-to-one transformation maps four cuts to a single cut.
Negative Energy Sources: The geometries require codimension-2 sources of negative tension to support the conical excess. The authors model this as a cosmic brane. The cancellation of the leading $O(N^2)$ terms in the on-shell action suggests that the physics of the delta operator is captured by dynamics intrinsic to the source (induced gravity) rather than its tension alone.
Probe Loops: The expectation values of probe Wilson loops in the bubbling wormhole backgrounds are calculated. The results show that probe strings localized at the conical singularities dominate over those on the branch cuts, and their actions match specific "diagonal" combinations of Wilson loops in the dual matrix models.

Significance and Claims
The paper claims to provide a concrete holographic dictionary for entanglement over gauge representations. It suggests that:

Entanglement over representations generates connected bulk geometries (bubbling wormholes) that are multi-covers of the standard vacuum, distinct from the two-sided black hole dual to energy entanglement.
Matrix models with delta operators offer a microscopic description of these geometries, where the "gluing" of eigenvalues corresponds to the global identification of the bulk spacetime.
Conical excesses in half-BPS solutions can be consistently realized with negative-tension sources, potentially realized via smeared orientifolds or effective induced gravity terms (DGP), providing a mechanism for the subleading free energy scaling.

The authors maintain a modest tone regarding the microscopic origin of the negative tension source, noting that while smeared orientifolds are a candidate, a fully localized string-theoretic completion compatible with charge quantization remains an open question. Similarly, the identification of the "diagonal" Wilson loops dual to probe strings at conical singularities is presented as a speculation reflecting an ambiguity in defining directions on the covering geometry.