The bulk Hilbert space of double scaled SYK

Abstract

The emergence of the bulk Hilbert space is a mysterious concept in holography. In [1], 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.

Algebras and States in JT Gravity

“…This role of the high temperature limit has also been noted in the context of a double-scaled version of the SYK model[13]; in that context, the algebra is of Type II1. A description of double-scaled SYK in which the high temperature limit is conveniently accessible had been developed in[18].…”

“…A consequence is that states Ψ S for S of the highly restricted form S = e −βH/2 Φe −βH/2 actually suffice to generate H. Indeed, we can choose Φ to generate any desired state of the matter system, multiplied by a function of χ that depends on β. 13 Taking linear combinations of the states we get for different values of β, we can approximate any desired function of χ; consequently, states Ψ S for S of this restricted form suffice to generate H. All of the other strings that we could have used, with more than one CFT operator, are therefore redundant in the sense that they do not enable us to produce any new states in H. So the map A 0 → H has a very large space N of null vectors, as asserted earlier.…”

“…In the present article, we will study JT gravity from the point of view of understanding the algebra of observables accessible to a boundary observer living on one side of the system. It is believed that in JT gravity with or without additional matter fields, it is not possible to define a one-sided black hole Hilbert space, but it is certainly possible to define a two-sided Hilbert space H, as studied for example in [5,6,8,[11][12][13]. We will analyze the algebra A of operators acting on H that can be defined on, say, the left boundary of the system.…”

Krylov complexity in large-$q$ and double-scaled SYK model

“…In such a case, we believe that a more systematic understanding from the boundary side is required, particularly in terms of the chord diagrams [60], where the auxiliary Hilbert space can be treated as a Krylov-like subspace [90]. Especially, an interesting direction we hope to return to is the bulk computation for the same, where the bulk Hilbert space is formed by a Krylov-like construction [77]. Moreover, the 1/q 2 correction is important since it might shed light on the contribution of the disconnected geometries [26], which provide the subleading corrections of Lanczos coefficients and the complexity from the gravity side.…”

Quantization of Jackiw-Teitelboim gravity with a massless scalar

“…An explicit evaluation of this two-point function does not seem to be so straightforward. Neither is it clear to us how the above correlation functions are related to the conventional correlation functions in literatures [34,36,38,39,40,41,3]. Further studies are required in this direction.…”