Quantum groups, non-commutative AdS 2, and chords in the double-scaled SYK model
Abstract We study the double-scaling limit of SYK (DS-SYK) model and elucidate the underlying quantum group symmetry. The DS-SYK model is characterized by a parameter q, and in the q → 1 and low-energy limit it goes over to the familiar Schwarzian theory. We relate the chord and transfer-matrix pict...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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SpringerOpen
2023-08-01
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| Series: | Journal of High Energy Physics |
| Subjects: | |
| Online Access: | https://doi.org/10.1007/JHEP08(2023)076 |
| _version_ | 1827781319823720448 |
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| author | Micha Berkooz Misha Isachenkov Prithvi Narayan Vladimir Narovlansky |
| author_facet | Micha Berkooz Misha Isachenkov Prithvi Narayan Vladimir Narovlansky |
| author_sort | Micha Berkooz |
| collection | DOAJ |
| description | Abstract We study the double-scaling limit of SYK (DS-SYK) model and elucidate the underlying quantum group symmetry. The DS-SYK model is characterized by a parameter q, and in the q → 1 and low-energy limit it goes over to the familiar Schwarzian theory. We relate the chord and transfer-matrix picture to the motion of a “boundary particle” on the Euclidean Poincaré disk, which underlies the single-sided Schwarzian model. AdS 2 carries an action of sl $$ \mathfrak{sl} $$ (2, ℝ) ≃ su $$ \mathfrak{su} $$ (1, 1), and we argue that the symmetry of the full DS-SYK model is a certain q-deformation of the latter, namely U q $$ {\mathcal{U}}_{\sqrt{q}} $$ ( su $$ \mathfrak{su} $$ (1, 1)). We do this by obtaining the effective Hamiltonian of the DS-SYK as a (reduction of) particle moving on a lattice deformation of AdS 2, which has this U q $$ {\mathcal{U}}_{\sqrt{q}} $$ ( su $$ \mathfrak{su} $$ (1, 1)) algebra as its symmetry. We also exhibit the connection to non-commutative geometry of q-homogeneous spaces, by obtaining the effective Hamiltonian of the DS-SYK as a (reduction of) particle moving on a non-commutative deformation of AdS 3. There are families of possibly distinct q-deformed AdS 2 spaces, and we point out which are relevant for the DS-SYK model. |
| first_indexed | 2024-03-11T15:17:12Z |
| format | Article |
| id | doaj.art-cec8e4d0fa864e388991db38188e375a |
| institution | Directory Open Access Journal |
| issn | 1029-8479 |
| language | English |
| last_indexed | 2024-03-11T15:17:12Z |
| publishDate | 2023-08-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of High Energy Physics |
| spelling | doaj.art-cec8e4d0fa864e388991db38188e375a2023-10-29T12:11:43ZengSpringerOpenJournal of High Energy Physics1029-84792023-08-012023816210.1007/JHEP08(2023)076Quantum groups, non-commutative AdS 2, and chords in the double-scaled SYK modelMicha Berkooz0Misha Isachenkov1Prithvi Narayan2Vladimir Narovlansky3Department of Particle Physics and Astrophysics, Weizmann Institute of ScienceInstitute of Physics, University of AmsterdamDepartment of Physics, Indian Institute of TechnologyDepartment of Physics, Princeton UniversityAbstract We study the double-scaling limit of SYK (DS-SYK) model and elucidate the underlying quantum group symmetry. The DS-SYK model is characterized by a parameter q, and in the q → 1 and low-energy limit it goes over to the familiar Schwarzian theory. We relate the chord and transfer-matrix picture to the motion of a “boundary particle” on the Euclidean Poincaré disk, which underlies the single-sided Schwarzian model. AdS 2 carries an action of sl $$ \mathfrak{sl} $$ (2, ℝ) ≃ su $$ \mathfrak{su} $$ (1, 1), and we argue that the symmetry of the full DS-SYK model is a certain q-deformation of the latter, namely U q $$ {\mathcal{U}}_{\sqrt{q}} $$ ( su $$ \mathfrak{su} $$ (1, 1)). We do this by obtaining the effective Hamiltonian of the DS-SYK as a (reduction of) particle moving on a lattice deformation of AdS 2, which has this U q $$ {\mathcal{U}}_{\sqrt{q}} $$ ( su $$ \mathfrak{su} $$ (1, 1)) algebra as its symmetry. We also exhibit the connection to non-commutative geometry of q-homogeneous spaces, by obtaining the effective Hamiltonian of the DS-SYK as a (reduction of) particle moving on a non-commutative deformation of AdS 3. There are families of possibly distinct q-deformed AdS 2 spaces, and we point out which are relevant for the DS-SYK model.https://doi.org/10.1007/JHEP08(2023)0761/N ExpansionAdS-CFT CorrespondenceField Theories in Lower DimensionsNon-Commutative Geometry |
| spellingShingle | Micha Berkooz Misha Isachenkov Prithvi Narayan Vladimir Narovlansky Quantum groups, non-commutative AdS 2, and chords in the double-scaled SYK model Journal of High Energy Physics 1/N Expansion AdS-CFT Correspondence Field Theories in Lower Dimensions Non-Commutative Geometry |
| title | Quantum groups, non-commutative AdS 2, and chords in the double-scaled SYK model |
| title_full | Quantum groups, non-commutative AdS 2, and chords in the double-scaled SYK model |
| title_fullStr | Quantum groups, non-commutative AdS 2, and chords in the double-scaled SYK model |
| title_full_unstemmed | Quantum groups, non-commutative AdS 2, and chords in the double-scaled SYK model |
| title_short | Quantum groups, non-commutative AdS 2, and chords in the double-scaled SYK model |
| title_sort | quantum groups non commutative ads 2 and chords in the double scaled syk model |
| topic | 1/N Expansion AdS-CFT Correspondence Field Theories in Lower Dimensions Non-Commutative Geometry |
| url | https://doi.org/10.1007/JHEP08(2023)076 |
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