On the holographic dual of a symmetry operator at finite temperature

Topological symmetry operators of holographic large <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>N</a:mi> </a:math> <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mrow> <c:msub> <c:mrow> <c:mi>CFT</c:mi> </c:mrow> <c:mrow> <c:mi>D</c:mi> </c:mrow> </c:msub> </c:mrow> </c:math> ’s are dual to dynamical branes in the gravity dual <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mrow> <e:msub> <e:mrow> <e:mi>AdS</e:mi> </e:mrow> <e:mrow> <e:mi>D</e:mi> <e:mo>+</e:mo> <e:mn>1</e:mn> </e:mrow> </e:msub> </e:mrow> </e:math> . We use this correspondence to establish a dictionary between thermal expectation values of symmetry operators in the Euclidean <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mrow> <g:msub> <g:mrow> <g:mi>CFT</g:mi> </g:mrow> <g:mrow> <g:mi>D</g:mi> </g:mrow> </g:msub> </g:mrow> </g:math> and the evaluation of gravitational saddles in the presence of a dynamical brane. Expectation values of 0-form symmetry operators in the <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mrow> <i:msub> <i:mrow> <i:mi>CFT</i:mi> </i:mrow> <i:mrow> <i:mi>D</i:mi> </i:mrow> </i:msub> </i:mrow> </i:math> 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 <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mrow> <k:msub> <k:mrow> <k:mi>AdS</k:mi> </k:mrow> <k:mrow> <k:mn>3</k:mn> </k:mrow> </k:msub> </k:mrow> </k:math> , duality/triality defects in four-dimensional <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mi mathvariant="script">N</m:mi> <m:mo>=</m:mo> <m:mn>4</m:mn> </m:math> super Yang-Mills theory, and the dual of R-symmetry operators probing five-dimensional Bogomol’nyi–Prasad–Sommerfield black holes.