Dynamical actions and q-representation theory for double-scaled SYK

Abstract

We show that DSSYK amplitudes are reproduced by considering the quantum mechanics of a constrained particle on the quantum group SU${}_q(1,1)$. We construct its left-and right-regular representations, and show that the representation matrices reproduce two-sided wavefunctions and correlation functions of DSSYK. We then construct a dynamical action and path integral for a particle on SU${}_q(1,1)$, whose quantization reproduces the aforementioned representation theory. By imposing boundary conditions or constraining the system we find the $q$-analog of the Schwarzian and Liouville boundary path integral descriptions. This lays the technical groundwork for identifying the gravitational bulk description of DSSYK. We find evidence the theory in question is a sine dilaton gravity, which interestingly is capable of describing both AdS and dS quantum gravity.

51 pages, v2: matches published version

References in corpus (57)

• Edge states and topological invariants of non-Hermitian systems
• Gapless Spin-Fluid Ground State in a Random Quantum Heisenberg Magnet
• Comments on the Sachdev-Ye-Kitaev model
• Pseudo-Hermiticity versus PT Symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian
• Chaos in AdS${}_2$ holography
• Two Dimensional Gauge Theories Revisited
• An Investigation of AdS${}_2$ Backreaction and Holography
• The asymptotic dynamics of three-dimensional Einstein gravity with a negative cosmological constant
• Liouville theory revisited
• Two-Dimensional Gravity and Nonlinear Gauge Theory
• Sachdev-Ye-Kitaev Model as Liouville Quantum Mechanics
• Solving the Schwarzian via the Conformal Bootstrap
• On the Liouville three-point function
• Clebsch-Gordan and Racah-Wigner coefficients for a continuous series of representations of U_q(sl(2,R))
• Lectures on 2D Yang-Mills Theory, Equivariant Cohomology and Topological Field Theories
• Solvable Models of Quantum Black Holes: A Review on Jackiw-Teitelboim Gravity
• The Quantum Gravity Dynamics of Near Extremal Black Holes
• The Bulk Dual of SYK: Cubic Couplings
• AdS${}_3$ gravity and random CFT
• Towards a full solution of the large N double-scaled SYK model
• The Schwarzian Theory - Origins
• The bulk Hilbert space of double scaled SYK
• Statistical mechanics of a two-dimensional black hole
• Unitary representations of $U_{q}(\mathfrak{sl}(2,\RR))$, the modular double, and the multiparticle q-deformed Toda chains
• The Schwarzian Theory - A Wilson Line Perspective
• Chord diagrams, exact correlators in spin glasses and black hole bulk reconstruction
• Conformal Bootstrap, Universality and Gravitational Scattering
• An exact quantization of Jackiw-Teitelboim gravity
• Three Dimensional View of the SYK/AdS Duality
• Liouville quantum gravity -- holography, JT and matrices
• Quantum Mechanics on the Noncommutative Torus
• Infrared Realization of dS${}_2$ in AdS${}_2$
• Menagerie of AdS${}_2$ boundary conditions
• Fine Structure of Jackiw-Teitelboim Quantum Gravity
• A symmetry algebra in double-scaled SYK
• Poisson sigma models and deformation quantization
• Three Dimensional View of Arbitrary $q$ SYK models
• General Form of Dilaton Gravity and Nonlinear Gauge Theory
• Krylov complexity in large-$q$ and double-scaled SYK model
• Edge Dynamics from the Path Integral: Maxwell and Yang-Mills
• Towards a bulk description of higher spin SYK
• A proposal for 3d quantum gravity and its bulk factorization
• The double scaled limit of Super--Symmetric SYK models
• From Quantum Groups to Liouville and Dilaton Quantum Gravity
• Gauge Theories for Gravity on a Line
• Edge State Quantization: Vector Fields in Rindler
• Constructing AdS_2 flow geometries
• Free-Field Representation of Group Element for Simple Quantum Group
• Supergroup Structure of Jackiw-Teitelboim Supergravity
• Correlators of double scaled SYK at one-loop
• Finite dimensional representations of the quantum group $G{L}_{p,q}(2)$ using the exponential map from $U_{p,q}(gl(2))$
• Exponential mapping for non semisimple quantum groups
• Hartle-Hawking wavefunction in double scaled SYK
• Duality in Quantum Liouville Theory
• Quantum exponentials for the modular double and applications in gravity models
• The Two-exponential Liouville Theory and the Uniqueness of the Three-point Function
• The exponential map for representations of $U_{p,q}(gl(2))$

Cited by in corpus (18)

• Krylov spread complexity as holographic complexity beyond JT gravity
• The dilaton gravity hologram of double-scaled SYK
• Tridiagonal Hamiltonians modeling the density of states of the Double-Scaled SYK model
• Wormholes, branes and finite matrices in sine dilaton gravity
• Quantum gravity of the Heisenberg algebra
• Complexity = Anything Can Grow Forever in de Sitter
• Branes in JT (super)gravity from group theory
• The complex Liouville string: the gravitational path integral
• Quantum group origins of edge states in double-scaled SYK
• Horizon causality from holographic scattering in asymptotically dS${}_3$
• Gravitational wavefunctions in JT supergravity
• Holographic central charge for sine dilaton gravity
• Holographic tensor network for double-scaled SYK
• Probing sine dilaton gravity with flow central charge
• Unveiling Novel Insights in Quantum Dynamics through Extended Sachdev-Ye-Kitaev Model
• Minimal Factorization of Chern-Simons Theory -- Gravitational Anyonic Edge Modes
• Fiducial observers and the thermal atmosphere in the black hole quantum throat
• Asymptotics of Weil-Petersson volumes and two-dimensional quantum gravities