# AdS/CFT and finiteness of entanglement entropy in CFT

AdS/CFT duality maps string theory to conformal field theory. String theory confirms Bekenstein-Hawking entropy, and thus the dual CFT must confirm it. However, CFT is still a quantum field theory, with entropic calculations technically turning out to be infinite without cutoff and scaling with volume.

I tried looking at some introductions, but in the middle of discussing holographic entanglement entropy, they all seem to introduce some "cutoff" without justifications, and thus I had to stop reading.

Also, some magic seems to occur by doing "holographic lift" that solves volume-scaling problem. Is this scaling problem resolved to area-scaling because of special nature of conformal field theory relative to quantum field theory?

If cut-off is necessary, how justified is that? Does this mean that nature described in CFT-or-QFT does have fundamental cutoff?

I hope what I share would be relevant to your question. Introducing a cut-off in the entanglement entropy $($EE$)$ is a way to regulate the counting of degrees of freedom. First, recall that for any co-dimension 2 space-like surface $B$ in a bulk spacetime, there's a Bousso bound $($with quantum corrections $S_{ent}(B)$$) on the integral of the entropy flux "s" on the light-sheet L(B) as [1]:$$\int_{L(B)}s \le \frac{Area(B)}{4G_{N}} + S_{ent}(B) $$Second, working in the framework of AdS/CFT, we assume the role of (H)EE as a measure of the degrees of freedom by the use of holographic c-theorem [2]. Now in general, the #d.o.f of the boundary theory is infinite, so is the area of the asymptotically AdS space. Therefore one can speak of some regulations in order to compare these infinities, by introducing a UV cut-off in the field theory which is equivalent to replacing the continuum by the discrete lattice of cells of the cut-off size. As discussed in [3], one can further assume that each of these cells contain one d.o.f, and also each d.o.f is capable of storing one bit of information. Hence in the calculation of holographic EE, the regulation in Area(B) means introducing an IR cut-off (z=\epsilon \to 0 in Poincare or \rho=\rho_{0} \to \infty in global coordinates) in the area functional$$ Area=\int d^{d-1}\sigma \, \sqrt{det(g_{ij})}$$Long story short, the cut-off is equivalent to the one bit information$($d.o.f$)\$ per Planck area bound. I hope it helps you.