A symmetry algebra in double-scaled SYK

A symmetry algebra in double-scaled SYK

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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 sophie allport bee curtains 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 tennessee titans dog bandana 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.