Abstract
We obtain entanglement entropy on the noncommutative (fuzzy) two-sphere. To define a subregion with a well defined boundary in this geometry, we use the symbol map between elements of the noncommutative algebra and functions on the sphere. We find that entanglement entropy is not proportional to the length of the region's boundary. Rather, in agreement with holographic predictions, it is extensive for regions whose area is a small (but fixed) fraction of the total area of the sphere. This is true even in the limit of small noncommutativity. We also find that entanglement entropy grows linearly with N, where N is the size of the irreducible representation of SU(2) used to define the fuzzy sphere.
Highlights
(volume-law) holographic entanglement entropy, instead of the more usual area-law behaviour
We find that entanglement entropy grows linearly with N, where N is the size of the irreducible representation of SU(2) used to define the fuzzy sphere
To interpret these findings within field theory, we must answer the following questions: can one divide the Hilbert space of a field theory on some noncommutative geometry into two components associated with the inside and the outside of some geometric region? If not, what precisely is the meaning of holographic entanglement entropy in field theory, and if yes, is the volume-law behaviour observed through a holographic description a property associated with strong coupling or would it be seen at weak coupling? This last question is further motivated by the fact that, for example, the enhancement in thermalization timescale mentioned above is not seen in perturbation theory [2]
Summary
We review the concepts of a symbol and a corresponding star product as a way to encode the noncommutative structure of geometry. We begin by reviewing the better-known example of a noncommutative plane, and show how the same tools can be applied to treat the noncommutative sphere. Our general approach is similar to that in [17], though the details are somewhat different.
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