On the Holographic Dual of a Symmetry Operator at Finite Temperature

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.

On the Holographic Dual of a Symmetry Operator at Finite Temperature

The paper "On the Holographic Dual of a Symmetry Operator at Finite Temperature" investigates the holographic dual description of symmetry operators in large $N$ Conformal Field Theories (CFTs) and their relationship with dynamical branes in the bulk Anti-de Sitter (AdS) space. This study provides a framework for understanding how these topological symmetry operators, central to many quantum field systems, find their counterparts in the gravitational dual. The exploration is anchored in the AdS/CFT correspondence, wherein global symmetries in the boundary CFT correspond to gauge symmetries in the AdS bulk.

Key Contributions

The primary goal of this paper is to establish a dictionary for translating thermal expectation values of symmetry operators in a $D$-dimensional CFT to gravitational saddle evaluations in the AdS$_{D+1}$ space with a presence of dynamical branes. The research delineates how expectation values of zero-form symmetry operators in the boundary CFT relate to branes wrapped on volume-minimizing cycles—specifically, the Euclidean continuation of black hole horizons—in the bulk space.

Key aspects of this study include:

• Symmetry Operator and Brane Correspondence: The paper demonstrates that the topological symmetry operators in the boundary theory correspond directly to the dynamical branes in the bulk AdS. This bulk-brane perspective extends our understanding of the fusion rules and symmetry operations in CFTs and their gravitational duals.
• Thermal Expectation Values: The comprehensive analysis presented relates symmetry operator expectation values to gravitational path integrals in the presence of these branes, thus providing a robust method to compute such values by leveraging the AdS/CFT framework.
• Free Energy Evaluation: The study outlines how the addition of these dynamical branes deforms the free energy in the boundary CFT, interpreted via a gravitational lens through prescribed boundary conditions that manifest in on-shell action calculations.

Illustrative Examples and Results

The paper elucidates this framework with various illustrative examples:

• AdS3/CFT2 Setup: The authors explore the specific simplifications available in an AdS3/CFT2 setup, using the BTZ black hole as a backdrop to compute varied configurations and elucidate the correspondence principles between boundary operators and bulk geometries.
• Duality/Triality Defects: Utilizing the symmetries in 4D $\mathcal{N}=4$ Super Yang-Mills theory, the authors hitch on the symmetries associated with dynamical branes, reinforcing the correspondence by aligning duality defects in the boundary CFT with branes wrapped on bulk cycles.
• BPS Black Holes in AdS5: The exposition culminates with an analysis of BPS black holes within the context of $\mathcal{N}=8$ gauged supergravity in AdS5. The expectation values of R-charge topological symmetry operators offer evidence supporting the proposed bulk-boundary correspondence and highlight the intricate entwinement between boundary conditions and bulk geometry.

Implications and Speculation

Practically, this work translates quantum field theoretical challenges into gravitational computations, offering a new array of analytical tools applicable across high-energy physics, particularly in scenarios involving holographic duals and thermal field dynamics.

Looking forward, the implications of this research potentially stretch into broader classes of symmetries, nontopological configurations, and nonperturbative scenarios in holography. The framework set forth paves the way for further exploration beyond leading saddle approximations, extensions to more complex manifold configurations, and possibly even elucidating Lorentzian signatures of similar theoretical constructs.

The rigorous mathematical treatment and conceptual clarity of this paper significantly advance our understanding of symmetry in higher dimensions and furnish a powerful link between holographic theory and thermal fluctuations in quantum field theories. As holography continues to evolve, works of this nature will be foundational in bridging theoretical gaps and augmenting our insights into the fabric of spacetime and quantum symmetries.