On the Holographic Dual of a Symmetry Operator at Finite Temperature

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.