A symmetry algebra in double-scaled SYK
Henry W. Lin, Douglas Stanford
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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mrow> <mml:mi>𝔰</mml:mi> <mml:mi>𝔩</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mrow> <mml:mi>𝔰</mml:mi> <mml:mi>𝔩</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> </mml:math> , but with a non-standard action on the boundary time coordinate. One can make the action of \mathfrak{sl}_2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mrow> <mml:mi>𝔰</mml:mi> <mml:mi>𝔩</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> </mml:math> 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.