Bubbling wormholes and matrix models

TL;DR

This work proposes a holographic framework in which entanglement across gauge representations in two copies of N=4 SYM is implemented by a delta operator built from sums over irreps, yielding bubbling wormhole geometries that connect multiple AdS5×S5 asymptotic regions along a common S1 boundary. The authors develop corresponding matrix models, including a two-matrix (and a four-matrix) construction using supersymmetric delta operators that glue eigenvalue densities to reproduce the BW2 and BW4 spectral curves, and they connect these curves to the harmonic data of the dual geometries. They compute the delta operator free energy, finding a negative leading N^2/√λ scaling consistent with an attractive bulk source whose leading contributions cancel against conical singularities, leaving subleading terms potentially described by induced gravity on the worldvolume. Probe computations with fundamental strings in BW2 and BW4 show saddles and actions that match diagonal Wilson loop operators on the field theory side, providing nontrivial checks of the proposed duality. Overall, the paper advances a concrete multi-boundary holographic program linking representation-theoretic entanglement, delta operators in Gaussian-like matrix models, and bubbling wormhole geometries with explicit spectral curves and probe diagnostics.

Abstract

The thermofield double state entangles two copies of a CFT via a sum over energy eigenstates and is dual to the two-sided eternal black hole. We explore an analogous construction using sums over gauge group representations of half-BPS Wilson loops in multiple copies of $U(N)$ $\mathcal{N}=4$ super Yang-Mills. These sums act as delta function-like operators that correlate the eigenvalues of the corresponding half-BPS matrix models. We suggest that the holographic duals are ''bubbling wormhole'' geometries: multi-covers of AdS$_5$ $\times S^5$ whose conformal boundary consists of multiple four-spheres intersecting on a common circle. We analyze the matrix model free energy, discuss its bulk interpretation, and study probe loops in these backgrounds.

Figures (8)

• Figure 1: The conformal boundary of the bubbling wormhole is given by the zero-length $\ell \to 0$ limit of the above "plumbed" geometry. The result is two four-spheres identified along a common $S^1$ on which the delta operator is placed.
• Figure 2: The bulk geometry is dictated by the Riemann surface $\Sigma$ and the number of cuts (blue) and poles (star) the functions $h_1$ and $h_2$ has on $\partial\Sigma$. Here we depict the usual case of a single pole (i.e. one asymptotically ${\rm AdS}_5 \times S^5$ region) and multiple cuts, which is a geometry dual to a single backreacted Wilson loop in a large representation $R$ with order $N^2$ number of boxes. $\mathcal{C}_D$ denote nontrivial cycles in the geometry, where $D$ is the dimension of the cycle.
• Figure 3: The Riemann surface $\Sigma$ describing ${\rm AdS}_5 \times S^5$ can be viewed as the lower half-plane with a single branch cut and a simple pole at infinity, which can be mapped to a disk containing one cut and one simple pole.
• Figure 4: The Riemann surface $\Sigma$ describing BW$_2$ can be viewed as the lower half-plane with two branch cuts and two simple poles, which can be mapped to a disk containing two cuts and two simple poles. The symbol $\otimes$ denotes the conical singularity (excess) on $\Sigma$.
• Figure 5: The lower half-plane with two cuts and two singularities (marked points) is equivalent to the disk with two cuts and two marked points or to a rectangle with two marked points under the map $z = \mathop{\mathrm{sn}}\nolimits(u,k)$. The harmonic function $h_2(u)$ vanishes at the edges of the rectangle that define the boundary of $\Sigma$. The blue cuts are the regions in which the $S^2$ vanishes and in the complementary boundary regions the $S^4$ vanishes.