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Holography on the Quantum Disk

Ahmed Almheiri, Fedor K. Popov

TL;DR

This work presents a concrete framework for holography on a noncommutative hyperbolic disk, the quantum disk, whose isometries are governed by the quantum group $SU_q(1,1)$. It develops the algebraic construction, coordinates, and a $q$-calculus to formulate wave equations, propagators, and boundary correlators, and then analyzes a putative boundary theory with $SU_q(1,1)$ symmetry, including two-point and OPE structures. The results illuminate how noncommutativity and a nontrivial coproduct imprint boundary dynamics and offer a controlled toy model that connects to the DSSYK program and its bulk dual. The study advances understanding of $q$-deformed holography, UV/IR aspects, and possible $q$-Schwarzian/CFT structures relevant for quantum gravity in noncommutative settings.

Abstract

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 $U_q(\mathfrak{su}_{1,1})$. 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.

Holography on the Quantum Disk

TL;DR

This work presents a concrete framework for holography on a noncommutative hyperbolic disk, the quantum disk, whose isometries are governed by the quantum group . It develops the algebraic construction, coordinates, and a -calculus to formulate wave equations, propagators, and boundary correlators, and then analyzes a putative boundary theory with symmetry, including two-point and OPE structures. The results illuminate how noncommutativity and a nontrivial coproduct imprint boundary dynamics and offer a controlled toy model that connects to the DSSYK program and its bulk dual. The study advances understanding of -deformed holography, UV/IR aspects, and possible -Schwarzian/CFT structures relevant for quantum gravity in noncommutative settings.

Abstract

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 . We review how this algebra is defined and its associated group 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.
Paper Structure (30 sections, 1 theorem, 148 equations)

This paper contains 30 sections, 1 theorem, 148 equations.

Table of Contents

  1. Introduction
  2. A review of the quantum disk
  3. $SU_q(1,1)$: the symmetry group of the quantum disk
  4. $U_q(\mathfrak{su}_{1,1})$: generators of $SU_q(1,1)$
  5. Coordinates on the quantum disk and their algebra
  6. Disk coordinates
  7. Differential $q$-calculus
  8. Holography
  9. Asymptotic analysis
  10. Propagators
  11. Boundary two point function
  12. $q$-Conformal Quantum Mechanics
  13. Correlation functions
  14. Two point function
  15. Operator Product Expansion
  16. ...and 15 more sections

Key Result

Theorem 1

If $I_1 = \sum_i a_i \otimes b_i, \ I_2 = \sum_j c_j \otimes d_j$ are $U_q(\mathfrak{su}_{1,1})$ invariant, then $I_{12} \equiv I_1 \times_{\mathrm{op}}I_2 \equiv \sum_{i,j} c_j a_i \otimes b_i d_j$ is also $U_q(\mathfrak{su}_{1,1})$ invariant.

Theorems & Definitions (1)

  • Theorem 1