[Paper Review] A symmetry algebra in double-scaled SYK

Henry W. Lin, Douglas Stanford|arXiv (Cornell University)|Jul 30, 2023

Black Holes and Theoretical PhysicsPhysics and Astronomy50 references3 citations

TL;DR

This paper identifies a deformed $υ_{q}$-deformed $υυ_{2}$ algebra as a symmetry algebra in the double-scaled Sachdev-Ye-Kitaev (SYK) model, arising from two Hamiltonians and a chord number operator. The algebra contains finite-dimensional unitary representations describing matter in a discrete Einstein-Rosen bridge; in the semiclassical limit, it reduces to $υυ_{2}$ with a non-standard time action, recoverable via a 'fake' extension of the boundary circle, enabling analysis of sub-maximal chaos and traversable wormhole protocols.

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.

Motivation & Objective

• To identify and characterize a new symmetry algebra in the double-scaled limit of the SYK model, where $N$ and $p$ scale together.
• To understand the role of the chord number operator and its interplay with Hamiltonians in generating a deformed $υυ_{2}$ algebra.
• To analyze finite-dimensional unitary representations of the algebra, corresponding to matter in a discrete Einstein-Rosen bridge.
• To explore the semiclassical limit and the emergence of standard $υυ_{2}$ symmetry via a 'fake' boundary extension.
• To apply the algebra to quantum chaos, traversable wormhole protocols, and two-sided OPE structures.

Proposed method

• The chord Hilbert space is constructed from entangled SYK copies, with states labeled by left and right chord counts $n_L, n_R$ and matter insertions.
• The symmetry algebra is generated by two Hamiltonians and the chord number operator ${\bar{n}}$, forming a $q$-deformation of the JT gravitational algebra.
• A subalgebra is identified that commutes with ${\bar{n}}$, isomorphic to $\mathrm{U}_{\!\sqrt{q}}(\mathfrak{sl}_{2})$, with generators $B, E, P$.
• Finite-dimensional unitary representations are constructed for single- and two-particle states, with Casimir invariants and $q$-deformed integers.
• The semiclassical limit is taken as $\lambda \to 0$, where $q = e^{-\lambda}$, leading to a non-standard action of $\mathfrak{sl}_{2}$ on boundary time.
• A 'fake' boundary circle is introduced to restore standard $\mathfrak{sl}_{2}$ action, accommodating subtle states oscillating at the scale of discreteness.

Experimental results

Research questions

• RQ1 How does the symmetry algebra in the double-scaled SYK model relate to JT gravity and near-horizon $\mathfrak{sl}_{2}$ symmetries?
• RQ2 What is the structure of the finite-dimensional unitary representations of the chord algebra, and how do they describe matter in a discrete Einstein-Rosen bridge?
• RQ3 How does the $q$-deformation of $\mathfrak{sl}_{2}$ emerge in the chord algebra, and what is its behavior in the semiclassical limit?
• RQ4 What is the role of the 'fake' boundary region in recovering standard $\mathfrak{sl}_{2}$ symmetry and hosting non-perturbative states?
• RQ5 How do the algebraic structures relate to out-of-time-order correlators, the traversable wormhole protocol, and the two-sided OPE?

Key findings

• The chord algebra is a $q$-deformation of the JT gravitational algebra, with a subalgebra isomorphic to $\mathrm{U}_{\!\sqrt{q}}(\mathfrak{sl}_{2})$.
• Finite-dimensional unitary representations exist for single-particle states, corresponding to matter moving in a discrete Einstein-Rosen bridge.
• In the semiclassical limit ($\lambda \to 0$), the algebra simplifies to $\mathfrak{sl}_{2}$, but with a non-standard action on the boundary time coordinate.
• Extending the boundary circle to include 'fake' regions restores the standard $\mathfrak{sl}_{2}$ action and accommodates states that oscillate at the scale of discreteness.
• The Lyapunov exponent correction from 'fake' effects at finite $p$ is approximately 17.5% of the total correction at $p=4$, indicating dominant non-perturbative contributions.
• The two-sided OPE and Streicher formula are derived using chord blocks, linking algebraic structures to correlation functions in the double-scaled SYK model.