On the Holographic Dual of a Symmetry Operator at Finite Temperature

TL;DR

The paper constructs and tests a holographic dictionary between boundary topological zero-form symmetry operators and bulk dynamical branes in AdS/CFT. By comparing Euclidean CFT thermodynamics with bulk gravitational saddles containing branes, it shows that brane backreaction shifts the on-shell action in a controlled way, yielding explicit expressions for boundary operator expectation values in various settings. In particular, it reproduces conical-deficit and horizon-wrapping effects in AdS$_3$ and AdS$_5$ scenarios and demonstrates exact consistency between boundary index calculations for BPS black holes and bulk brane actions. The results illuminate how symmetry operators probe bulk geometries, hint at links to HRRT-like minimal-volume cycles, and invite generalizations to broader topologies and subleading $1/N$ corrections.

Abstract

Topological symmetry operators of holographic large $N$ CFT$_D$'s are dual to dynamical branes in the gravity dual AdS$_{D+1}$. We use this correspondence to establish a dictionary between thermal expectation values of symmetry operators in the Euclidean CFT$_D$ and the evaluation of gravitational saddles in the presence of a dynamical brane. Expectation values of $0$-form symmetry operators in the CFT$_D$ are then related to branes wrapped on volume minimizing cycles in the bulk, i.e., the Euclidean continuation of a black hole horizon. We illustrate with some representative examples, including gravity in AdS$_3$, duality / triality defects in 4D $\mathcal{N} = 4$ Super Yang-Mills theory, and the dual of R-symmetry operators probing 5D BPS black holes.

Figures (4)

• Figure 1: Sketch of the topological transition associated with the Hawking-Page transition. We label the topological models associated with the two phases and their interface. Reading left to right: Transition from thermal AdS, through an interface geometry, to an AdS black hole background. The two central figures are distinctly presented, yet equivalent, and related by an interchange $S^1\leftrightarrow S^{D-1}$. The conformal boundary to all geometries is $M_D=S^1\times S^{D-1}$. Thermal AdS contains a minimal dimension-1 bulk cycle $S^1_{\text{min}}$, in contrast, the AdS black hole background contains a minimal codimension-2 bulk cycle $\gamma=S^{D-1}_{\text{min}}$.
• Figure 2: Sketch of the Schwarzschild AdS black hole with Euclidean brane $\widetilde{\mathcal{U}}$ inserted (blue). In (i) we show a profile view, (ii) gives the same geometry head on. In both cases we sketch the $S^{D-1}$ as a circle and do not display $S^1_\beta$, where the brane sits at a point. The brane has finite tension and contracts until it stabilizes, wrapping the black hole horizon $\gamma$ at AdS radius $r=r_+$.
• Figure 3: The metrics of thermal AdS$_3$ and the BTZ black hole parametrize solid tori. We sketch 2-tori with coordinates $x,\tau$, and radial coordinate $z$. For thermal AdS$_3$ and the BTZ black hole the circles parametrized by $x$ and $\tau$ collapse at $z=R$ and $z=T$, leaving a single circle parametrized by $\tau$ and $x$ respectively (dashed line / thick line).
• Figure 4: We sketch a BPS black hole in AdS$_5$ with global time coordinate $\tau$. The spatial slice at time $\tau_*$ contains a Euclidean brane $\widetilde{\mathcal{U}}_{\alpha}$ linking the world line of the black hole. In particular, the topological worldvolume terms of this brane measure the black hole charges.