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Quantum group origins of edge states in double-scaled SYK

Andreas Belaey, Thomas G. Mertens, Thomas Tappeiner

Abstract

Double-scaled SYK (DSSYK) is known to have an underlying quantum group theoretical description. We precisely pinpoint the quantum group structure, improving upon earlier work in the literature. This allows us to utilize this framework for bulk gravitational applications. We explain bulk discretization in DSSYK from the underlying irreducibility of the representations. We derive trumpet and brane amplitudes using character insertions of the quantum group, simplifying earlier calculations. Most importantly, we factorize the bulk Hilbert space dual to DSSYK in the quantum group description using a complete set of edge degrees of freedom living at a bulk entangling surface. An analogous treatment for $\mathcal{N}=1$ DSSYK is provided in the same quantum group theoretical framework.

Quantum group origins of edge states in double-scaled SYK

Abstract

Double-scaled SYK (DSSYK) is known to have an underlying quantum group theoretical description. We precisely pinpoint the quantum group structure, improving upon earlier work in the literature. This allows us to utilize this framework for bulk gravitational applications. We explain bulk discretization in DSSYK from the underlying irreducibility of the representations. We derive trumpet and brane amplitudes using character insertions of the quantum group, simplifying earlier calculations. Most importantly, we factorize the bulk Hilbert space dual to DSSYK in the quantum group description using a complete set of edge degrees of freedom living at a bulk entangling surface. An analogous treatment for DSSYK is provided in the same quantum group theoretical framework.

Paper Structure

This paper contains 30 sections, 221 equations.

Table of Contents

  1. Introduction
  2. Observation: $q$-Hermite integral identity
  3. Refined quantum group structure of double-scaled SYK
  4. The quantum deformation of $U(\mathfrak{sl}_2)$
  5. Restriction to smooth geometries - the positive semigroup
  6. Gravitational real form
  7. Hopf duality and the quantum group $\text{ SL}^+_{0<q<1}(2,\mathbb{R})$
  8. Aside: Principal series (co)-representations from quantum Möbius transformations
  9. Gravitational matrix element
  10. Whittaker vectors; or gravitational boundary eigenstates
  11. Whittaker function; or two-boundary gravitational wavefunction
  12. Summary
  13. Gravitational applications
  14. Length discreteness in $U_q(\mathfrak{sl}(2,\mathbb{R}))$
  15. Length positivity for boundary amplitudes
  16. ...and 15 more sections