Maximally heavy dynamics in the causal diamond

Abstract

Correlation functions of CFT operators with infinite scaling dimension are rich, multifaceted objects that describe physics ranging across classical holography, black hole dynamics, and flat-space scattering amplitudes. In this work, we provide a rigorous framework for characterizing the space of four-point functions of identical operators with infinite dimension in terms of well-defined ``maximally heavy observables,'' which are akin to intrinsic quantities describing statistical systems in the thermodynamic limit. These observables are highly constrained by crossing symmetry and unitarity, and give novel insights into the locality of bulk states through the emergence of dynamical phase transitions. In certain cases, these results connect directly to the more familiar picture of torus partition functions at large central charge. We apply our framework to a number of illustrative examples including generalized free fields, chiral product correlators, and maximal giant gravitons in planar $\mathcal{N}=4$ SYM.

Figures (9)

• Figure 1: An illustration of the extended causal diamond $\widehat{\mathscr{C}} \cong T^2 \times\mathscr{C}$.
• Figure 2: Different regularization schemes access different heavy observables (accumulation points). While the set of classical measures $\{\nu\}$ are in bijection with their characteristic functions $\{\phi_\nu\}$ via Fourier transform $\mathcal{F}$, rate functions $\{\lambda\}$ and classical measures are generally not. That is, a heavying correlator sequence may have multiple rate functions, but only one classical measure, and vice-versa.
• Figure 3: Bounds on rate functions along the diagonal limit (left) and self-dual line (right), as given in proposition \ref{['universalratefunctionbound']}. The black line denotes the upper bound $\lambda_+$ and the colored lines represent the individual curves whose maximum over $\boldsymbol{x} \in\mathbb{R}^2$ gives the total lower bound $\lambda_-$: $(I) = -\Sigma$, $(II) = f(x) + f(\bar{x})$, $(III) = (x+\bar{x}) + f(-x)+f(-\bar{x})$, and $(IV) = (x+\bar{x}) - \Sigma$. The gray area between these curves is the allowed region for a rate function $\lambda$ with self-dual free energy density $\Sigma = 1$.
• Figure 4: The holographic interpretation of dynamical phase transitions in the Poincaré patch (restricted to the diagonal limit of the correlator). Left: for $z<1/2$, geodesic transport between heavy operators in the same causal diamond dominates. Center: for $z = 1/2$, both s and t-channel pairs of geodesics contribute equally, along with other non-universal contributions (not pictured). Right: for $z >1/2$, geodesic transport between pairs of heavy operators across the nested causal diamonds dominates.
• Figure 5: Top row: the light gray region is the convex hull of $\mathrm{supp}(\nu)$ and the dark gray region is the allowed region for non-identity contributions with classical twist gaps of $\eta_0 = \{0,\sqrt{2}/4,\sqrt{2}/2\}$. Bottom row: the coherent state rate functions computed from $\nu$ plotted over $\boldsymbol{x} \in [-1,1]\times[-1,1]$. The red region is the high-temperature t-channel phase, the blue region is the low-temperature s-channel phase, and the tan region is a non-universal chiral phase induced by the presence of off-diagonal modes in $\nu$.

Theorems & Definitions (74)

• Definition 3.1: Heavying sequence
• Definition 3.2: Regularization
• Definition 3.3: Good regularizations and maximally heavy observables
• Remark
• Remark
• Definition 3.4: Dynamical free energy density
• Definition 3.5: Rescaled OPE measure and characteristic function
• Definition 3.6: $\alpha$-local dynamical free energy density
• Remark
• Proposition 3.1: Goodness of the three regularization schemes