Double-scaled SYK, Chords and de Sitter Gravity

Double-scaled SYK, Chords and de Sitter Gravity

1 Feb 2024 | Herman Verlinde
This paper explores the connection between 3D de Sitter gravity and the double scaled SYK model. The authors study the partition function of 3D de Sitter gravity, defined as the trace over the Hilbert space obtained by quantizing the phase space of non-rotating Schwarzschild-de Sitter spacetime. They identify the Hamiltonian with the gravitational Wilson-line that measures the conical deficit angle. The Hamiltonian is expressed in terms of canonical variables and is shown to lead to the same chord rules and energy spectrum as the double scaled SYK model. The authors use this match to compute the partition function and scalar two-point function in 3D de Sitter gravity. The paper begins with an introduction that highlights the correspondence between the double scaled SYK model and 3D de Sitter gravity. The SYK model is defined by a Hamiltonian for N Majorana fermions with a p-body interaction. The authors consider the model in the double scaling limit with fixed λ = 2p²/N. They show that the computation of moments of the Hamiltonian reduces to the counting problem of chord diagrams weighted by a factor q^#intersections with q = e^{-λ}. The authors then describe the quantization of 3D Schwarzschild-de Sitter spacetime, focusing on its symmetries, phase space, and quantization. They show that the deficit angle α becomes an operator and that the phase space associated with the spacetime is two-dimensional. The authors introduce the holonomy variables L_A and L_Z, which span the phase space of Schwarzschild-de Sitter spacetime. They compute the Poisson bracket between these variables and show that the two lines intersect, leading to a non-zero bracket. The authors then quantize the phase space variables L_A and L_Z, showing that they satisfy a q-deformed oscillator algebra. They identify the A-cycle holonomy with the Hamiltonian of 3D de Sitter gravity and show that the Hamiltonian acts on the eigenbasis of the holonomy as follows: H|n> = |n+1> + [n]_q |n-1>, where [n]_q ≡ (1 - q^n)/(1 - q). This equation takes the exact same form as the action of the Hamiltonian of double scaled SYK on the chord number eigenbasis. The authors then compute the energy spectrum and partition function of the de Sitter gravity model, showing that they match with the spectral density of DSSYK. They also compute the scalar two-point function in 3D de Sitter gravity, showing that it can be expressed in terms of the gravitational Wilson line operator. The authors conclude that there is a holographic correspondence between the double scaled SYK model and 3D de Sitter gravity.This paper explores the connection between 3D de Sitter gravity and the double scaled SYK model. The authors study the partition function of 3D de Sitter gravity, defined as the trace over the Hilbert space obtained by quantizing the phase space of non-rotating Schwarzschild-de Sitter spacetime. They identify the Hamiltonian with the gravitational Wilson-line that measures the conical deficit angle. The Hamiltonian is expressed in terms of canonical variables and is shown to lead to the same chord rules and energy spectrum as the double scaled SYK model. The authors use this match to compute the partition function and scalar two-point function in 3D de Sitter gravity. The paper begins with an introduction that highlights the correspondence between the double scaled SYK model and 3D de Sitter gravity. The SYK model is defined by a Hamiltonian for N Majorana fermions with a p-body interaction. The authors consider the model in the double scaling limit with fixed λ = 2p²/N. They show that the computation of moments of the Hamiltonian reduces to the counting problem of chord diagrams weighted by a factor q^#intersections with q = e^{-λ}. The authors then describe the quantization of 3D Schwarzschild-de Sitter spacetime, focusing on its symmetries, phase space, and quantization. They show that the deficit angle α becomes an operator and that the phase space associated with the spacetime is two-dimensional. The authors introduce the holonomy variables L_A and L_Z, which span the phase space of Schwarzschild-de Sitter spacetime. They compute the Poisson bracket between these variables and show that the two lines intersect, leading to a non-zero bracket. The authors then quantize the phase space variables L_A and L_Z, showing that they satisfy a q-deformed oscillator algebra. They identify the A-cycle holonomy with the Hamiltonian of 3D de Sitter gravity and show that the Hamiltonian acts on the eigenbasis of the holonomy as follows: H|n> = |n+1> + [n]_q |n-1>, where [n]_q ≡ (1 - q^n)/(1 - q). This equation takes the exact same form as the action of the Hamiltonian of double scaled SYK on the chord number eigenbasis. The authors then compute the energy spectrum and partition function of the de Sitter gravity model, showing that they match with the spectral density of DSSYK. They also compute the scalar two-point function in 3D de Sitter gravity, showing that it can be expressed in terms of the gravitational Wilson line operator. The authors conclude that there is a holographic correspondence between the double scaled SYK model and 3D de Sitter gravity.
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[slides and audio] Double-scaled SYK%2C chords and de Sitter gravity