Abstract
We consider a nearly-AdS2 gravity theory on the two-sided wormhole geometry. We construct three gauge-invariant operators in NAdS2 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.
Highlights
Introduction and motivationAny black hole with finite temperature has a near horizon geometry that can be approximated by flat space
The boost symmetry of this flat space region corresponds to the full modular Hamiltonian of the outside region of the black hole, and it is an exact symmetry of the full wormhole geometry
AdS2 (NAdS2) gravity [1,2,3,4] captures the gravitational dynamics of near extremal black holes after a Kaluza-Klein reduction
Summary
Any black hole with finite temperature has a near horizon geometry that can be approximated by flat space. We relate the generators we defined to other operators which are well defined for finite N , but agree with the generators in the semiclassical limit This allows us to identify operators in both a gravity theory and the SYK model which approximately obey an SL(2) algebra and should be identified with the symmetries of AdS2. These approximate symmetries behave as SL(2)u transformations of the physical boundary time of a pair of NCFT1s, and we give an approximate state-operator map that organizes the NCFT1 Hilbert space into primaries and descendants, in analogy with higher dimensions.
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