Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Reparametrization modes, shadow operators, and quantum chaos in higher-dimensional CFTs

Felix M. Haehl, Wyatt Reeves, Moshe Rozali

TL;DR

This work presents two complementary frameworks to encode universal conformal constraints in higher-dimensional CFTs and connect them to chaotic dynamics: a reformulated shadow operator formalism featuring a negative-dimension soft mode ${\cal E}$ and an effective theory of reparametrization modes derived from the conformal anomaly. These approaches unify conformal blocks, shadow blocks, kinematic-space OPE blocks, and monodromy projections, enabling compact calculations of global stress-tensor exchanges via bilinear couplings ${\cal B}_{\cal E,V}^{(1)}$ and their propagators. By mapping stress-tensor exchanges to reparametrization-mode exchanges, the paper provides a transparent route to compute conformal partial waves in arbitrary even dimensions and to isolate physical blocks from shadow contributions. In thermal states realized on hyperbolic space, the reparametrization-mode dynamics predict maximal chaos with Lyapunov exponent $\lambda_L = \frac{2\pi}{\beta}$ and butterfly velocity $v_B = \frac{1}{d-1}$, and pole-skipping phenomena in the energy-energy correlator corroborate this chaotic behavior, linking conformal representation theory to quantum chaos in $d>2$.

Abstract

We study two novel approaches to efficiently encoding universal constraints imposed by conformal symmetry, and describe applications to quantum chaos in higher dimensional CFTs. The first approach consists of a reformulation of the shadow operator formalism and kinematic space techniques. We observe that the shadow operator associated with the stress tensor (or other conserved currents) can be written as the descendant of a field ${\cal E}$ with negative dimension. Computations of stress tensor contributions to conformal blocks can be systematically organized in terms of the "soft mode" ${\cal E}$, turning them into a simple diagrammatic perturbation theory at large central charge. Our second (equivalent) approach concerns a theory of reparametrization modes, generalizing previous studies in the context of the Schwarzian theory and two-dimensional CFTs. Due to the conformal anomaly in even dimensions, gauge modes of the conformal group acquire an action and are shown to exhibit the same dynamics as the soft mode ${\cal E}$ that encodes the physics of the stress tensor shadow. We discuss the calculation of the conformal partial waves or the conformal blocks using our effective field theory. The separation of conformal blocks from shadow blocks is related to gauging of certain symmetries in our effective field theory of the soft mode. These connections explain and generalize various relations between conformal blocks, shadow operators, kinematic space, and reparametrization modes. As an application we study thermal physics in higher dimensions and argue that the theory of reparametrization modes captures the physics of quantum chaos in Rindler space. This is also supported by the observation of the pole skipping phenomenon in the conformal energy-energy two-point function on Rindler space.

Reparametrization modes, shadow operators, and quantum chaos in higher-dimensional CFTs

TL;DR

This work presents two complementary frameworks to encode universal conformal constraints in higher-dimensional CFTs and connect them to chaotic dynamics: a reformulated shadow operator formalism featuring a negative-dimension soft mode and an effective theory of reparametrization modes derived from the conformal anomaly. These approaches unify conformal blocks, shadow blocks, kinematic-space OPE blocks, and monodromy projections, enabling compact calculations of global stress-tensor exchanges via bilinear couplings and their propagators. By mapping stress-tensor exchanges to reparametrization-mode exchanges, the paper provides a transparent route to compute conformal partial waves in arbitrary even dimensions and to isolate physical blocks from shadow contributions. In thermal states realized on hyperbolic space, the reparametrization-mode dynamics predict maximal chaos with Lyapunov exponent and butterfly velocity , and pole-skipping phenomena in the energy-energy correlator corroborate this chaotic behavior, linking conformal representation theory to quantum chaos in .

Abstract

We study two novel approaches to efficiently encoding universal constraints imposed by conformal symmetry, and describe applications to quantum chaos in higher dimensional CFTs. The first approach consists of a reformulation of the shadow operator formalism and kinematic space techniques. We observe that the shadow operator associated with the stress tensor (or other conserved currents) can be written as the descendant of a field with negative dimension. Computations of stress tensor contributions to conformal blocks can be systematically organized in terms of the "soft mode" , turning them into a simple diagrammatic perturbation theory at large central charge. Our second (equivalent) approach concerns a theory of reparametrization modes, generalizing previous studies in the context of the Schwarzian theory and two-dimensional CFTs. Due to the conformal anomaly in even dimensions, gauge modes of the conformal group acquire an action and are shown to exhibit the same dynamics as the soft mode that encodes the physics of the stress tensor shadow. We discuss the calculation of the conformal partial waves or the conformal blocks using our effective field theory. The separation of conformal blocks from shadow blocks is related to gauging of certain symmetries in our effective field theory of the soft mode. These connections explain and generalize various relations between conformal blocks, shadow operators, kinematic space, and reparametrization modes. As an application we study thermal physics in higher dimensions and argue that the theory of reparametrization modes captures the physics of quantum chaos in Rindler space. This is also supported by the observation of the pole skipping phenomenon in the conformal energy-energy two-point function on Rindler space.

Paper Structure

This paper contains 55 sections, 112 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Summary.
  3. Outline.
  4. Global conformal blocks and the shadow of the stress tensor
  5. Review of kinematic space and OPE blocks
  6. Kinematic space.
  7. OPE blocks as kinematic space operators.
  8. Conformal blocks, shadow blocks, and partial waves
  9. Conformal partial waves.
  10. The stress tensor block.
  11. A reformulation of the shadow operator formalism
  12. The stress tensor shadow as a total derivative.
  13. Coupling to external operators.
  14. Application to four-point functions.
  15. Conformal partial waves in arbitrary even dimensions
  16. ...and 40 more sections

Figures (2)

  • Figure 1: Out-of-time-order configuration in Rindler space. Inside the red shaded Rindler wedge we show two Rindler time trajectories in blue, which are at constant geodesic separation. The four operators are inserted at the green dots. We show the out-of-time-ordered configuration by including a small amount of imaginary time.
  • Figure 2: Plots of $\log | {\cal G}_{00,00}(\omega_E,k) |$ in $d=5$ (left) and $d=4$ (right). The "warm" lines (orange/red) correspond to poles. The "cold" lines (blue) correspond to zeros. The four vertical zero lines are due to the polynomial prefactors in \ref{['eq:AnyDimRes']} and \ref{['eq:G0000even']}. Pole skipping occurs when zero lines and pole lines intersect. In order to make a connection with exponentially growing modes in quantum chaos, one should focus on the upper half plane, where pole skipping is observed at precisely two locations (black circles): $(\omega_E , k)_\text{skip} = (1, \pm i \frac{d}{2})$. Note that in the even dimensional case (right figure) the precise shape of the zero lines (but not the pole skipping locations) depend on ambiguous contact terms.