A symmetry algebra in double-scaled SYK

Lin, Henry W.; Stanford, Douglas

doi:10.21468/SciPostPhys.15.6.234

SciPost Physics

A symmetry algebra in double-scaled SYK

Henry W. Lin, Douglas Stanford

SciPost Phys. 15, 234 (2023) · published 8 December 2023

doi: 10.21468/SciPostPhys.15.6.234
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Abstract

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 $\mathfrak{sl}_2$ 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 $\mathfrak{sl}_2$, but with a non-standard action on the boundary time coordinate. One can make the action of $\mathfrak{sl}_2$ 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.

TY  - JOUR
PB  - SciPost Foundation
DO  - 10.21468/SciPostPhys.15.6.234
TI  - A symmetry algebra in double-scaled SYK
PY  - 2023/12/08
UR  - https://scipost.org/SciPostPhys.15.6.234
JF  - SciPost Physics
JA  - SciPost Phys.
VL  - 15
IS  - 6
SP  - 234
A1  - Lin, Henry W.
AU  - Stanford, Douglas
AB  - 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 $\mathfrak{sl}_2$ 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 $\mathfrak{sl}_2$, but with a non-standard action on the boundary time coordinate. One can make the action of $\mathfrak{sl}_2$ 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.
ER  -

@Article{10.21468/SciPostPhys.15.6.234,
	title={{A symmetry algebra in double-scaled SYK}},
	author={Henry W. Lin and Douglas Stanford},
	journal={SciPost Phys.},
	volume={15},
	pages={234},
	year={2023},
	publisher={SciPost},
	doi={10.21468/SciPostPhys.15.6.234},
	url={https://scipost.org/10.21468/SciPostPhys.15.6.234},
}

Cited by 7

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¹ Henry W. Lin,
¹ Douglas Stanford

¹ Stanford University [SU]

Funders for the research work leading to this publication