We study the dynamics of entanglement entropy for weakly excited states in conformal field theories by using the AdS/CFT. This is aimed at a first step to find a counterpart of Einstein equation in the CFT language. In particular, we point out that the entanglement entropy satisfies differential equations which directly correspond to the Einstein equation in several setups of AdS/CFT. We also define a quantity called entanglement density in higher dimensional field theories and study its dynamical property for weakly excited states in conformal field theories.

In the context of the surface-state correspondence we propose the geodesic curvature of a convex curve as a local measure of factorization of the dual Conformal Field Theory state. Its integral will be interpreted as computing the total bipartite entanglement among degrees of freedom with support on the chosen domain. We will derive results through application of the Gauss-Bonnet theorem and show quantitative agreement with computations using the Multiscale Entanglement Renormalization Ansatz tensor network and the formalism of entanglement density.

We compute an upper bound on the circuit complexity of quantum states in 3d Chern-Simons theory corresponding to certain classes of knots. Specifically, we deal with states in the torus Hilbert space of Chern-Simons that are the knot complements on the 3-sphere of arbitrary torus knots. These can be constructed from the unknot state by using the Hilbert space representation of the S and T modular transformations of the torus as fundamental gates. The upper bound is saturated in the semiclassical limit of Chern-Simons theory. The results are then generalized for a family of multi-component links that are obtained by "Hopf-linking" different torus knots. We also use the braid word presentation of knots to discuss states on the punctured sphere Hilbert space associated with 2-bridge knots and links. The calculations present interesting number theoretic features related with continued fraction representations of rational numbers. In particular, we show that the minimization procedure defining the complexity naturally leads to regular continued fractions, allowing a geometric interpretation of the results in the Farey tesselation of the upper-half plane. Finally, we relate our discussion to the framework of path integral optimization by generalizing the original argument to non-trivial topologies.

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