Fuzzy Objects and their applications to gravity

I have been thinking of fuzzy objects in noncommutative geometry. The trigger was the offer to give a seminar at Nagoya university, where I was questioned about our recent progress in the study of noncommutative geometry, in particular its application to gravity.
To tell the truth, we are studying the algebraic structure of some fuzzy objects as quantum group and have been away from their applications.
However, the question gave me a good opportunity to consider such a kind of thing, and actually my collaborator and I started to investigate that. At this moment, we are interested in finding or giving something like identification with a black hole counterpart in noncommutative geometry.
To this end, we are now treating a (2+1)-dimensional system. In our previous paper, we mentioned the usefulness of the fuzzy disc, which is a finite, disc-shaped region in the Moyal plane (two-dimensional noncommutative space). We showed that it can be cut radially, which corresponds to switching to the angle-base picture instead of the number-base picture. This way of looking is directly related to the bulk-boundary correspondence that we do not have to say how it is interesting.
We want to obtain a black hole or a black-hole-like object in noncommutative geometry, and have already grasped the clue for that. I hope I will be able to talk about that in the seminar tomorrow...

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