Berry phases, wormholes and factorization in AdS/CFT

TL;DR

Addressing the factorization puzzle in AdS/CFT, the paper shows that wormholes are encoded as Berry phases in 2D holographic CFTs. It develops a tripartite classification of Berry phases—Virasoro, gauge, and modular—each linked to a different bulk diffeomorphism and wormhole geometry. The Virasoro sector is realized as holonomies of coadjoint orbits; gauge Berry phases arise from asymptotic symmetries with a boundary time shift; modular Berry phases for subregions connect to the Crofton form on kinematic space and exhibit entanglement-entropy transitions in thermal states. Together these results illuminate how bulk non-factorization manifests as boundary geometric phases and suggest connections to subregion complexity.

Abstract

For two-dimensional holographic CFTs, we demonstrate the role of Berry phases for relating the non-factorization of the Hilbert space to the presence of wormholes. The wormholes are characterized by a non-exact symplectic form that gives rise to the Berry phase. For wormholes connecting two spacelike regions in gravitational spacetimes, we find that the non-exactness is linked to a variable appearing in the phase space of the boundary CFT. This variable corresponds to a loop integral in the bulk. Through this loop integral, non-factorization becomes apparent in the dual entangled CFTs. Furthermore, we classify Berry phases in holographic CFTs based on the type of dual bulk diffeomorphism involved. We distinguish between Virasoro, gauge and modular Berry phases, each corresponding to a spacetime wormhole geometry in the bulk. Using kinematic space, we extend a relation between the modular Hamiltonian and the Berry curvature to the finite temperature case. We find that the Berry curvature, given by the Crofton form, characterizes the topological transition of the entanglement entropy in presence of a black hole.

Figures (7)

• Figure 1: L.h.s.: fixed time slice of the annulus geometry with the inner and outer boundaries $\Sigma_i$ and $\Sigma_o$, respectively. Along the dashed line, exemplary for a non-contractable circle in $\varphi$ direction, the holonomy $k_0$ is accumulated. The chiral bosons $\Phi$ and $\Psi$ are defined on the boundaries with action given in \ref{['eq:action_split']}, or after employing \ref{['eq:map_coadjoint_orbit']}, the geometric action in \ref{['eq:geometric_action']}. R.h.s.: to interpret the annulus setup in terms of the factorization map, the inner and outer boundaries represent the left and right boundaries. The holonomy, represented again by the red dashed line, is understood as inserting a defect operator which defines the factorization map.
• Figure 2: $H_L - H_R$ is the symmetry of the TFD state.
• Figure 3: The figure in the left panel shows the holographic dual to the TFD state time-evolved by $H_L + H_R$ while the figure in the right panel shows the time-evolution by $H_L - H_R$. The figure in the middle shows the holographic dual of the original TFD state.
• Figure 4: Kruskal diagram of an eternal AdS black hole, jagged lines representing the singularities and $H$ indicating the horizon. The blue and green lines describe the same point in phase space. These two are related by a trivial bulk diffeomorphism which does not induce a time shift at the boundary. The red line is a different point in phase space, corresponding to a different equivalence class of bulk diffeomorphisms.
• Figure 5: Illustration of modular transport. The base space consists of the modular Hamiltonians satisfying \ref{['eq:ModularHamiltonian']}; the fibers are modular zero-mode frames. Due to the zero-mode ambiguity of $\partial_\lambda U^\dagger U$ in \ref{['eq:deformation_H']}, parallel transport around a closed loop leads to a holonomy, known as the modular Berry phase.