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The stated finite valueis the Casimir energy of a two-dimensional conformal field theory on the unit spatial circle. The plane-to-cylinder conformal map shifts the Hamiltonian by $-c/12$, so the global ${\mathrm{AdS}}_3$ vacuum is dual to the CFT cylinder vacuum rather than to a state of zero cylinder energy. The sign assigned directly to the Lorentzian on-shell action depends on whether it is identified with the action density or with the vacuum-energy generating functional; the universal boundary datum is the vacuum energy $E_0=-c/12$ supplied in the question.