Entanglement and geometry from subalgebras of the Virasoro algebra

TL;DR

We study generalized Virasoro coherent states generated from SL(2,$\mathbb{R}$) subalgebras of the Virasoro algebra in 2d CFTs. By constructing $|\Psi_k(\rho,\theta)\rangle$ and computing the stress-tensor expectation value, we express the result through a Schwarzian uniformization with a map $f_k(z)$ and parameter $\alpha_k$, connecting the CFT data to Banados bulk geometries. The holographic duals reproduce entanglement entropies via the Ryu-Takayanagi prescription, with explicit results for single-interval entanglement and its evolution for both $k=1$ and general $k\ge 2$, including HHLL block interpretations. The framework yields insights into operator growth (Krylov complexity) and inhomogeneous quenches, suggesting broad applicability to holographic bulk reconstruction and time-dependent CFT dynamics.

Abstract

In this work we study families of generalised coherent states constructed from SL(2,R) subalgebras of the Virasoro algebra in two-dimensional conformal field theories. We derive the energy density and entanglement entropy and discuss their equivalence with analogous quantities computed in locally excited states. Moreover, we analyze their dual, holographic geometries and reproduce entanglement entropies from the Ryu-Takayanagi prescription. Finally, we outline possible applications of this universal class of states to operator growth and inhomogeneous quenches.

Figures (8)

• Figure 1: The massive particle moves along the red dashed geodesic which connects the two operators at the asymptotic boundary. The blue region can be interpreted as Krylov complexity (see discussions in section \ref{['sec:SumDisc']}).
• Figure 2: The slice is a constant time slice $t=0$ in Poincar√© coordinates. $\mathcal{M}$ is the wedge being exercised, which is bounded by three surfaces, the cutoff surface $\Sigma$ and $\Sigma_{\pm}$.
• Figure 3: The projection of the excised wedge on the asymptotic boundary $z=0$ is the region within the red curve. The blue region is $\mathcal{M}_-$, which is evaluated by subtracting the green region from the corresponding portion of the sphere. Its mirroring region within the red curve is $\mathcal{M}_+$.
• Figure 4: The setup for the evolution of the entanglement entropy on the complex plane (left) and on the cylinder (right). On the complex plane, the two heavy operators are placed on the real axis, while the twist operators are placed on a circle centered around the origin, symmetrically along the real axis; on the cylinder, the two heavy operators are placed vertically, while the twist operators are placed on a time slice, here we illustrate on $\tau =0$ slice.
• Figure 5: Bulk interpretation in terms of geodesic crossing. The blue dashed geodesic connecting the two operators $\mathcal{O}$ is $\gamma_\mathcal{O}$ while the green one connecting the two twist operators is $\gamma_\sigma$. When the two operators $\mathcal{O}$ are moving towards each other, the blue geodesic shrinks, crossing the green one at the moment when the cross-ratio crosses the branch cut at $\eta=-1$.