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arXiv:1904.12820 · Journal of High Energy Physics

Symmetries near the horizon

Henry W. Lin, Juan Maldacena, Ying Zhao

Published Aug 1, 2019• 90 citations
arXivHTMLPDF

Abstract

A bstract We consider a nearly-AdS 2 gravity theory on the two-sided wormhole geometry. We construct three gauge-invariant operators in NAdS 2 which move bulk matter relative to the dynamical boundaries. In a two-sided system, these operators satisfy an SL(2) algebra (up to non perturbative corrections). In a semiclassical limit, these generators act like SL(2) transformations of the boundary time, or conformal symmetries of the two sided boundary theory. These can be used to define an operator-state mapping. A particular large N and low temperature limit of the SYK model has precisely the same structure, and this construction of the exact generators also applies. We also discuss approximate, but simpler, constructions of the generators in the SYK model. These are closely related to the “size” operator and are connected to the maximal chaos behavior captured by out of time order correlators.

DOI
10.1007/jhep08(2019)049

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A bstract We consider a nearly-AdS 2 gravity theory on the two-sided wormhole geometry. We construct three gauge-invariant operators in NAdS 2 which move bulk matter relative to the dynamical boundaries. In a two-sided system, these operators satisfy an SL(2) algebra (up to non perturbative corrections). In a semiclassical limit, these generators act like SL(2) transformations of the boundary time, or conformal symmetries of the two sided boundary theory. These can be used to define an operator-state mapping. A particular large N and low temperature limit of the SYK model has precisely the same structure, and this construction of the exact generators also applies. We also discuss approximate, but simpler, constructions of the generators in the SYK model. These are closely related to the “size” operator and are connected to the maximal chaos behavior captured by out of time order correlators.
We consider a nearly-AdS_2 gravity theory on the two-sided wormhole geometry. We construct three gauge-invariant operators in NAdS_2 which move bulk matter relative to the dynamical boundaries. In a two-sided system, these operators satisfy an SL(2) algebra (up to non perturbative corrections). In a semiclassical limit, these generators act like SL(2) transformations of the boundary time, or conformal symmetries of the two sided boundary theory. These can be used to define an operator-state mapping. A particular large N and low temperature limit of the SYK model has precisely the same structure, and this construction of the exact generators also applies. We also discuss approximate, but simpler, constructions of the generators in the SYK model. These are closely related to the “size” operator and are connected to the maximal chaos behavior captured by out of time order correlators.
The double-scaled limit of the Sachdev-Ye-Kitaev (SYK) model takes the number of fermions and their interaction number to infinity in a coordinated way. In this limit, two entangled copies of the SYK model have a bulk description of sorts known as the “chord Hilbert space”. We analyze a symmetry algebra acting on this Hilbert space, generated by the two Hamiltonians together with a two-sided operator known as the chord number. This algebra is a deformation of the JT gravitational algebra, and it contains a subalgebra that is a deformation of the \mathfrak{sl}_2 𝔰 𝔩 2 near-horizon symmetries. The subalgebra has finite-dimensional unitary representations corresponding to matter moving around in a discrete Einstein-Rosen bridge. In a semiclassical limit the discreteness disappears and the subalgebra simplifies to \mathfrak{sl}_2 𝔰 𝔩 2 , but with a non-standard action on the boundary time coordinate. One can make the action of \mathfrak{sl}_2 𝔰 𝔩 2 algebra more standard at the cost of extending the boundary circle to include some “fake” portions. Such fake portions also accommodate certain subtle states that survive the semi-classical limit, despite oscillating on the scale of discreteness. We discuss applications of this algebra, including sub-maximal chaos, the traversable wormhole protocol, and a two-sided OPE.
A bstract We study a single-sided black hole with an end-of-the-world (EoW) brane behind the horizon in the double-scaled SYK (DSSYK). The new Hamiltonian is a deformation of the original DSSYK Hamiltonian with an extra exponential wormhole length operator, which leads to a new chord diagram rule. The boundary algebra is defined as generated by the new Hamiltonian and boundary matter. There is an alternative but equivalent definition with a q -coherent state due to a nontrivial isomorphism of the vN algebra of DSSYK. This isomorphism induces a unitary equivalence, which yields a surprising result that the boundary algebra of a single-sided black hole in DSSYK has a non-trivial commutant and is a type II 1 vN factor. It follows that the full bulk reconstruction from the boundary is impossible, and there is a “no man’s island” behind the horizon in the semiclassical JT limit. Inspired by the EoW brane, we construct a family of matter-brane states with an arbitrary number of matter chords and behaving like an EoW brane. They exactly solve the full spectrum of DSSYK. We take different ways to understand the nontrivial commutant. We argue that the commutant is complex on chord number basis and thus non-geometric. In the semiclassical JT limit, the commutant becomes the canonical purification of the boundary algebra and claims the no man’s island. In the context of Hawking radiation after Page time, the unitary equivalence is interpreted as encoding the canonical purification into the old Hawking radiation, and the no man’s island has the same essence as the island. Including the exponential wormhole length operator independently, the boundary algebra is extended to all bounded operators and reconstructs the no man’s island. This can be regarded as a different choice for the definition of boundary algebra. This type I ∞ algebra is closely related to the EoW brane in Kourkoulou-Maldacena.
Four coupled SYK models and Nearly AdS2_22​ gravities: Phase Transitions in Traversable wormholes and in Bra-ket wormholes
We study four coupled SYK models and nearly AdS2_22​ gravities. In the SYK model side, we construct a model that couples two copies of two coupled SYK models. In nearly AdS2_22​ gravity side, we entangle matter fields in two copies of traversable wormholes. In both cases, the systems show first order phase transitions at zero temperature by changing couplings, which is understood as the exchange of traversable wormhole configurations. In nearly AdS2_22​ gravity cases, by exchanging the role of space and time the wormholes are interpreted as bra-ket wormholes. In Lorentzian signature, these bra-ket wormholes lead to two closed universes that are entangled with each other as well as matter fields in the flat space without dynamical gravity. We study the effect of projection or entangling operation for matters on flat spaces and they cause phase transitions in bra-ket wormholes, which leads to the pair annihilation of closed universes. Using these bra-ket wormholes, we discuss the way to embed states in 2d holographic CFTs into Hilbert space of many 2d free fields.
A bstract In holography, gauging symmetries of the boundary theory leads to important modifications in the bulk. In this work, we study constraints to gauge symmetry sectors in the chord Hilbert space of the double-scaled SYK (DSSYK) with matter, and we connect them to different proposals of its bulk dual. These sectors include chord parity symmetry, corresponding to End-Of-The-World (ETW) branes and Euclidean wormholes in sine dilaton gravity; and relative time-translations in a doubled DSSYK model (resulting from a single DSSYK with an infinitely heavy matter chord) used in de Sitter holography. We define and evaluate partition functions and thermal correlation functions of the ETW brane and Euclidean wormhole systems in the boundary theory. We deduce the holographic dictionary by matching geodesic lengths in the bulk with the spread complexity of the parity-gauged DSSYK. The Euclidean wormholes of fixed size are perturbatively stable, and their baby universe Hilbert space is non-trivial only when matter is added. We conclude studying the constraints in the path integral of the doubled DSSYK. We derive the gauge invariant operator algebra of one of the DSSYKs dressed to the other one and discuss its holographic interpretation.
Abstract We study four coupled Sachdev–Ye–Kitaev (SYK) models and nearly AdS2 gravities. In the SYK model side, we construct a model that couples two copies of two coupled SYK models. In nearly AdS2 gravity side, we entangle matter fields in two copies of traversable wormholes. In both cases, the systems show first order phase transitions at zero temperature by changing couplings, which is understood as the exchange of traversable wormhole configurations. In nearly AdS2 gravity cases, by exchanging the role of space and time the wormholes are interpreted as bra-ket wormholes. In Lorentzian signature, these bra-ket wormholes lead to two closed universes that are entangled with each other as well as matter fields in the flat space where we do not have dynamical gravity. We study the effect of projection or entangling operation for matters on flat spaces and they cause phase transitions in bra-ket wormholes, which leads to the pair annihilation of closed universes. Using these bra-ket wormholes, we discuss the way to embed states in 2D holographic CFTs into Hilbert space of many 2D free fields.