Holography on the quantum disk
Ahmed Almheiri, Fedor K. Popov
Abstract
A bstract Motivated by recent study of DSSYK and the non-commutative nature of its bulk dual, we review and analyze an example of a non-commutative spacetime known as the quantum disk proposed by L. Vaksman. The quantum disk is defined as the space whose isometries are generated by the quantum algebra $$ {U}_q\left(\mathfrak{s}{\mathfrak{u}}_{1,1}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>q</mml:mi> </mml:msub> <mml:mfenced> <mml:mrow> <mml:mi>s</mml:mi> <mml:msub> <mml:mi>u</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> </mml:mfenced> </mml:math> . We review how this algebra is defined and its associated group SU q (1, 1) that it generates, highlighting its non-trivial coproduct that sources bulk non-commutativity. We analyze the structure of holography on the quantum disk and study the imprint of non-commutativity on the putative boundary dual.