The bulk Hilbert space of double scaled SYK

TL;DR

The paper constructs a concrete bulk Hilbert space for double-scaled SYK by slicing open chord diagrams, revealing a dynamical lattice-like bulk with chord number as the fundamental length. It provides an explicit bulk-to-boundary map that relates bulk chord states to boundary states, showing that chord number corresponds to operator size and that the map preserves inner products in the double scaling limit. A Type II$_1$ algebra of bulk observables is identified, including the Hamiltonian and matter operators, along with a gravitational subalgebra that reproduces the JT/Schwarzian structure in the appropriate limit. The framework extends to wormholes with arbitrary numbers of matter particles, yielding a generalized bulk Hilbert space and a lattice-like bulk theory with a well-defined bulk-boundary dictionary, and it discusses potential deformations away from triple scaling and connections to bulk symmetries.

Abstract

The emergence of the bulk Hilbert space is a mysterious concept in holography. In arXiv:1811.02584, the SYK model was solved in the double scaling limit by summing chord diagrams. Here, we explicitly construct the bulk Hilbert space of double scaled SYK by slicing open these chord diagrams; this Hilbert space resembles that of a lattice field theory where the length of the lattice is dynamical and determined by the chord number. Under a calculable bulk-to-boundary map, states of fixed chord number map to particular entangled 2-sided states with a corresponding size. This bulk reconstruction is well-defined even when quantum gravity effects are important. Acting on the double scaled Hilbert space is a Type II$_1$ algebra of observables, which includes the Hamiltonian and matter operators. In the appropriate quantum Schwarzian limit, we also identify the JT gravitational algebra including the physical SL(2,R) symmetry generators, and obtain explicit representations of the algebra using chord diagram techniques.

Figures (12)

• Figure 1: (a) intersection of two $H$ chords. This gives a factor of $\mathfrak{q}$. (b) Intersection of $H$ (black) and $M$ (green) chords. (c) Intersection of two $M$ chords.
• Figure 2: (a) The Hilbert space of JT gravity with no matter. The dynamical degree of freedom is the length $\ell$ between the two sides (pink curves). The disk partition function is obtained by imposing $\ell=0$ boundary conditions in the Euclidean past and future. (b) The Hilbert space of double scaled SYK with no matter insertions. The dynamical variable is $n$, the number of chords. In the microscopic description $q n$ is just the size of the density matrix. We have drawn the chords so that for any red curve, the chords which intersect that curve cross do not cross in the Euclidean past.
• Figure 3: Two processes that can happen when we act with $H_\mathsf{R}$. In (a) an extra chord is added. In (b) a chord is removed. It crosses 3 chords giving a factor of $\mathfrak{q}^3$. We could have also consider the action of $H_\mathsf{L}$. The same processes happen but we would draw the insertion on the left.
• Figure 4: Interpretation of the chord diagram as an overlap of a bra $\bra{\text{top}}$ and ket $\ket{\text{bottom}}$ defined in the bulk Hilbert space. The middle region ( pink) defines an inner product between states. It is defined as a 1-way region: all chords entering through $\gamma_n$ must exit through $\gamma_m$.
• Figure 5: (a) The Hilbert space of JT gravity with a particle (green) in the wormhole. The matter particle divides a geodesic from the left to right boundary into two pieces, with lengths $\ell_\mathsf{L}$ and $\ell_\mathsf{R}$. The disk partition function with two operator insertions is obtained by imposing $\ell_\mathsf{L}= \ell_\mathsf{R} = 0$ boundary conditions in the Euclidean past and future. (b) The Hilbert space of double scaled SYK with an operator insertion $M_s$ represented by a green chord. The Hilbert space is spanned by states labeled by two integers $n_\mathsf{L}, n_\mathsf{R}$ that are the left and right chord numbers.