Conformal symmetry and its breaking in two dimensional Nearly Anti-de-Sitter space (1606.01857v2)

Abstract: We study a two dimensional dilaton gravity system, recently examined by Almheiri and Polchinski, which describes near extremal black holes, or more generally, nearly $AdS_2$ spacetimes. The asymptotic symmetries of $AdS_2$ are all the time reparametrizations of the boundary. These symmetries are spontaneously broken by the $AdS_2$ geometry and they are explicitly broken by the small deformation away from $AdS_2$. This pattern of spontaneous plus explicit symmetry breaking governs the gravitational backreaction of the system. It determines several gravitational properties such as the linear in temperature dependence of the near extremal entropy as well as the gravitational corrections to correlation functions. These corrections include the ones determining the growth of out of time order correlators that is indicative of chaos. These gravitational aspects can be described in terms of a Schwarzian derivative effective action for a reparametrization.

• The paper demonstrates that the interplay between spontaneous and explicit conformal symmetry breaking in nearly AdS2 space leads to an effective Schwarzian action capturing essential gravitational dynamics.
• It employs the Jackiw-Teitelboim dilaton gravity framework to uncover infrared corrections, notably affecting near-extremal entropy and correlation functions.
• The study explores implications for holographic duality and quantum chaos while addressing potential issues with ghost-like modes in the Schwarzian framework.

Conformal Symmetry and Its Breaking in Two-Dimensional Nearly AdS$_2$ Space

The paper provides an intricate analysis of the dynamics inherent in systems described by two-dimensional nearly anti-de Sitter (AdS$_2$) geometries, with a specific focus on the interplay between spontaneous and explicit breaking of conformal symmetry. By examining these processes, the paper sheds light on the gravitational backreaction occurring within such settings, with applications ranging from theoretical explorations to practical implications in examining nearly extremal black holes.

Central to the discussion is the concept of reparametrization symmetry in AdS$_2$ spacetimes, which is inherently tied to the boundary conditions of the system. In pure AdS$_2$ contexts, the reparametrization symmetry is spontaneously broken to $SL(2,R)$, the isometry group of the space, revealing an associated zero mode. When this symmetry is explicitly broken, an effective action emerges, famously identified as the Schwarzian derivative. The authors argue that this action captures crucial gravitational properties of the system, such as corrections to the entropy near extremality and higher-order correlation functions, indicative of chaotic behavior.

An innovative aspect of this research is the integration of the dilaton gravity approach, leveraging the Jackiw-Teitelboim framework in two-dimensional scenarios. One principal finding is that gravitational corrections, although formally suppressed by the Newton constant, become remarkably relevant in the infrared (IR) regime due to the effective action. This insight is pivotal in understanding phenomena like the linear temperature dependence of near-extremal entropy and alterations to correlation functions, including the examination of out-of-time-ordered correlators, which act as markers for chaos.

The theoretical implications of this work extend beyond the scope of near-extremal black holes. The paper explores prospective applications to anti-de Sitter/conformal field theory (AdS/CFT) correspondence, indicating that reparametrization dynamics in two dimensions could serve as an analogous model to more complex gravitational interactions in higher-dimensional scenarios. Detecting analogous symmetry-breaking patterns may enrich our understanding of holographic dualities and refine our approaches to quantum gravity theories.

However, certain concerns remain unaddressed, notably the potential presence of ghost-like modes within the Schwarzian framework. Given the higher derivative nature of the Schwarzian action, this research circumvents these concerns by treating $SL(2,R)$ transformations as gauge symmetries. The work further explores how this perspective relates to charges appearing in dual boundary theories, akin to edge modes identified in electromagnetic discussions, harboring implications for gauge theory formulations in holography.

In conclusion, the paper elucidates a nuanced understanding of conformal symmetry breaking in nearly AdS$_2$ scenarios, offering robust theoretical tools to probe gravitational anomalies and chaotic dynamics in quantum gravity paradigms. The presented framework suggests future explorations in extending these insights to broader holographic setups, thus articulating the ongoing interplay between gravity, symmetry, and chaos in theoretical physics. This approach not only paves the way for future investigations into quantum gravity models but also enriches the complex tapestry of interactions within multidimensional spaces, ultimately fortifying our grasp on the foundational tenets of modern physics.