Explicit Entropic Proofs of Irreversibility Theorems for Holographic RG Flows

We revisit the existence of monotonic quantities along renormalization group flows using only the Null Energy Condition and the Ryu-Takayanagi formula for the entanglement entropy of field theories with anti-de Sitter gravity duals. In particular, we consider flows within the same dimension and holographically reprove the $c$-, $F$-, and $a$-theorems in dimensions two, three, and four. We focus on the family of maximally spherical entangling surfaces, define a quasi-constant of motion corresponding to the breaking of conformal invariance, and use a properly defined distance between minimal surfaces to construct a holographic $c$-function that is monotonic along the flow. We then apply our method to the case of flows across dimensions: There, we reprove the monotonicity of flows from ${\mathrm{A}\mathrm{d}\mathrm{S}}_{D+1}$ to ${\mathrm{A}\mathrm{d}\mathrm{S}}_3$ and prove the novel case of flows from ${\mathrm{A}\mathrm{d}\mathrm{S}}_5$ to ${\mathrm{A}\mathrm{d}\mathrm{S}}_4$.