Berry Curvature and Riemann Curvature in Kinematic Space with Spherical Entangling Surface

Abstract

We discover the connection between the Berry curvature and the Riemann curvature tensor in any kinematic space of minimal surfaces anchored on spherical entangling surfaces. This new holographic principle establishes the Riemann geometry in kinematic space of arbitrary dimensions from the holonomy of modular Hamiltonian, which in the higher dimensions is specified by a pair of time‐like separated points as in CFT1 and CFT2. The Berry curvature that we constructed also shares the same property of the Riemann curvature for all geometry: internal symmetry; skew symmetry; first Bianchi identity. We derive the algebra of the modular Hamiltonian and its deformation, the latter of which can provide the maximal modular chaos to the modular scrambling modes. The algebra also dictates the parallel transport, which leads to the Berry curvature exactly matching to the Riemann curvature tensor. Finally, we compare CFT1 to higher dimensional CFTs and show the difference from the OPE block.

Berry phases, wormholes and factorization in AdS/CFT

“…Then, we insert this solution into (4.9) to obtain the modular Berry curvature operator. Note that this commutator can be simplified using standard identities such that the Berry curvature is simply given by (see also [52] for a derivation)…”

Berry phases, wormholes and factorization in AdS/CFT

“…Then, we insert this solution into (4.9) to obtain the modular Berry curvature operator. Note that this commutator can be simplified using standard identities such that the Berry curvature is simply given by (see also [51] for a derivation)…”

Emergence of Kinematic Space from Quantum Modular Geometric Tensor

“…Recently, one provided a modular extension to a Berry curvature [12,13], similar to replacing a Hamiltonian by H mod . One could also show the equivalence between the modular Berry geometry and Riemann geometry on KS [14,15]. Therefore, we expect to obtain KS and modular Berry curvature (MBC) from Quantum Modular Geometric Tensor [11] g…”

“…where T 00 is the (00)-component of a stress tensor, and R is the radius of the (d − 1)dimensional sphere centered at x. When d = 1, we can the OPE block to define the modular Hamiltonian, but the integration variable becomes time [14,15]. When the radius R approaches zero, the modular Hamiltonian becomes:…”