Entanglement is a physical phenomenon that each state cannot be described individually. Entanglement entropy gives quantitative understanding to the entanglement. We use decomposition of the Hilbert space to discuss properties of the entanglement. Therefore, partial trace operator becomes important to define the reduced density matrix from different centers, which commutes with all elements in the Hilbert space, corresponding to different entanglement choices or different observations on entangling surface. Entanglement entropy is expected to satisfy the strong subadditivity. We discuss decomposition of the Hilbert space for the strong subadditivity and other related inequalities. The entanglement entropy with centers can be computed from the Hamitonian formulations systematically, provided that we know wavefunctional. In the Hamitonian formulation, it is easier to obtain symmetry structure. We consider massless p-form theory as an example. The massless p-form theory in (2p + 2)-dimensions has global symmetry, similar to the electricmagnetic duality, connecting centers in ground state. This defines a duality structure in centers. Because it is hard to exactly compute the entanglement entropy from partial trace operator, we propose the Lagrangian formulation from the Hamitonian formulation to compute the entanglement entropy with centers. From the Lagrangian method and saddle point approximation, the codimension two surface term (leading order) in the Einstein gravity theory or holographic entanglement entropy should correspond to non-tensor product decomposition (center is not identity). Finally, we compute the entanglement entropy of the SU(N ) Yang-Mills lattice gauge theory in the fundamental representation using the strong coupling expansion in the extended lattice model to obtain spatial area term in total dimensions larger than two for N > 1.

We construct the low energy effective action for the bosonic sector on a Dp-brane in large constant RR (p − 1)-form field background. The action is invariant under both U (1) gauge symmetry and the volume-preserving diffeomorphism characterizing the RR-field background. Scalar fields representing transverse coordinates of the Dp-brane are included. It also respects T-duality and is consistent with the action for M5-brane in C-field background.

We establish the Lorentzian AdS 2 /CFT 1 correspondence from a reconstruction of all bulk points through the kinematic-space approach. The OPE block is exactly a bulk local operator. We formulate the correspondence between the bulk propagator in the non-interacting scalar field theory and the conformal block in CFT 1 . When we consider the stress tensor, the variation probes the variation of AdS 2 metric. The reparameterization provides the asymptotic boundary of the bulk spacetime as in the derivation of the Schwarzian theory from two-dimensional dilaton gravity theory. Finally, we find the AdS 2 Riemann curvature tensor based on the above consistent check.

We prove that in bosonic quantum mechanics the two-point spectral form factor can be obtained as an average of the two-point out-of-time ordered correlation function, with the average taken over the Heisenberg group. In quantum field theory, there is an analogous result with the average taken over the tensor product of many copies of the Heisenberg group, one copy for each field mode. The resulting formula is expressed as a path integral over two fields, providing a promising approach to the computation of the spectral form factor. We develop the formula that we have obtained using a coherent state description from the JC model and also in the context of the large-N limit of CFT and Yang-Mills theory from the large-N matrix quantum mechanics.