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The Ryu-Takayanagi conjecture predicts a holographic dual of the entanglemententropy of a CFT. It proposes that the entanglement entropy is given by thearea of the minimal surface in the dual spacetime. In the semi-classical limit,this conjecture is supported by the saddle point approximation. If there aremultiple classical solutions, it is assumed that only the minimal actioncontributes to the entanglement entropy. However, we will point out that thesesaddles equally contribute to the entanglement entropy in some cases.Therefore, the derivation of the conjecture is incomplete if there are multipleextremal surfaces that extend from a sub-system on the AdS boundary. We willconsider two disjoint intervals in CFT$_{1+1}$ as the simplest but non-trivialexample, and propose another candidate for a holographic dual of theentanglement entropy of this system, which is the sum of all the signed areasof extremal surfaces in the dual spacetime. After that, we will derive it fromthe CFT calculations and propose the corresponding gravity side action.

Holographic entanglement entropy of two disjoint intervals in AdS$_3$/CFT$_2$