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A symmetry algebra in double-scaled SYK

Henry W. Lin, Douglas Stanford

TL;DR

This work constructs a symmetry algebra acting on the chord Hilbert space of double-scaled SYK, revealing a deformation of the JT gravity algebra with a finite-dimensional, unitary subalgebra that commutes with the chord number. The authors develop the chord algebra, its coproduct, and Casimir, and classify its irreps, including 0-, 1-, and 2-particle sectors, with a detailed decomposition into near-horizon sl2-like structures. In the λ→0 semiclassical limit, the subalgebra contracts to a finite-temperature sl2 realized on a fake disk, enabling concrete calculations of conformal blocks, OPEs, Lyapunov growth via the scramblon P−, and a traversable wormhole protocol. The paper further discusses chord-block decomposition, the relation between chords and bulk geometry, the nature of the fake region, and open questions about the algebra’s full representation and gauge structure. Overall, the work provides a robust algebraic framework for sub-maximal chaos, two-sided OPEs, and wormhole phenomena in double-scaled SYK with potential links to q-deformed SL2 holography and Liouville-type theories.

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.

A symmetry algebra in double-scaled SYK

TL;DR

This work constructs a symmetry algebra acting on the chord Hilbert space of double-scaled SYK, revealing a deformation of the JT gravity algebra with a finite-dimensional, unitary subalgebra that commutes with the chord number. The authors develop the chord algebra, its coproduct, and Casimir, and classify its irreps, including 0-, 1-, and 2-particle sectors, with a detailed decomposition into near-horizon sl2-like structures. In the λ→0 semiclassical limit, the subalgebra contracts to a finite-temperature sl2 realized on a fake disk, enabling concrete calculations of conformal blocks, OPEs, Lyapunov growth via the scramblon P−, and a traversable wormhole protocol. The paper further discusses chord-block decomposition, the relation between chords and bulk geometry, the nature of the fake region, and open questions about the algebra’s full representation and gauge structure. Overall, the work provides a robust algebraic framework for sub-maximal chaos, two-sided OPEs, and wormhole phenomena in double-scaled SYK with potential links to q-deformed SL2 holography and Liouville-type theories.

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 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 , but with a non-standard action on the boundary time coordinate. One can make the action of 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.
Paper Structure (57 sections, 324 equations, 1 figure, 1 table)

This paper contains 57 sections, 324 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Different regimes of SYK
  3. Review of the JT gravitational algebra
  4. Summary
  5. The chord algebra
  6. Review of the chord Hilbert space
  7. The chord algebra
  8. The chord coproduct
  9. A subalgebra that commutes with $\bar{n}$
  10. Casimir
  11. Representations of the chord algebra
  12. 0-particle irrep
  13. 1-particle irreps
  14. 2-particle representations
  15. Decomposition into irreps
  16. ...and 42 more sections

Figures (1)

  • Figure 1: We plot $|x|_\infty$ for different values of $q$ with $q^{n/2} = 1/2$ and $\Delta = 1$. The maximum eigenvalue of $x$ is constant and equal to $\log 2 \approx 0.693$. As $q\to 1$, we expect $|x|_\infty \ge \text{arccosh}(2) \approx 1.317$.