Modular Intersections, Time Interval Algebras and Emergent AdS$_2$
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866908598188113920 |
|---|---|
| author | Lashkari, Nima Leung, Kwing Lam Moosa, Mudassir Ouseph, Shoy |
| author_facet | Lashkari, Nima Leung, Kwing Lam Moosa, Mudassir Ouseph, Shoy |
| contents | We compute the modular flow and conjugation of time interval algebras of conformal Generalized Free Fields (GFF) in $(0+1)$-dimensions in vacuum. For non-integer scaling dimensions, for general time-intervals, the modular conjugation and the modular flow of operators outside the algebra are non-geometric. This is because they involve a Generalized Hilbert Transform (GHT) that treats positive and negative frequency modes differently. However, the modular conjugation and flows viewed in the dual bulk AdS$_2$ are local, because the GHT geometrizes as the local antipodal symmetry transformation that pushes operators behind the Poincaré horizon. These algebras of conformal GFF satisfy a $\textit{Twisted Modular Inclusion}$ and a $\textit{Twisted Modular Intersection}$ property. We prove the converse statement that the existence of a (twisted) modular inclusion/intersection in any quantum system implies a representation of the (universal cover of) conformal group $PSL(2,\mathbb{R})$, respectively. We discuss the implications of our result for the emergence of Stringy AdS$_2$ geometries in large $N$ theories without a large gap. Our result applied to higher dimensional eternal AdS black holes explains the emergence of two copies of $PSL(2,\mathbb{R})$ on future and past Killing horizons. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_19882 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Modular Intersections, Time Interval Algebras and Emergent AdS$_2$ Lashkari, Nima Leung, Kwing Lam Moosa, Mudassir Ouseph, Shoy High Energy Physics - Theory General Relativity and Quantum Cosmology Mathematical Physics We compute the modular flow and conjugation of time interval algebras of conformal Generalized Free Fields (GFF) in $(0+1)$-dimensions in vacuum. For non-integer scaling dimensions, for general time-intervals, the modular conjugation and the modular flow of operators outside the algebra are non-geometric. This is because they involve a Generalized Hilbert Transform (GHT) that treats positive and negative frequency modes differently. However, the modular conjugation and flows viewed in the dual bulk AdS$_2$ are local, because the GHT geometrizes as the local antipodal symmetry transformation that pushes operators behind the Poincaré horizon. These algebras of conformal GFF satisfy a $\textit{Twisted Modular Inclusion}$ and a $\textit{Twisted Modular Intersection}$ property. We prove the converse statement that the existence of a (twisted) modular inclusion/intersection in any quantum system implies a representation of the (universal cover of) conformal group $PSL(2,\mathbb{R})$, respectively. We discuss the implications of our result for the emergence of Stringy AdS$_2$ geometries in large $N$ theories without a large gap. Our result applied to higher dimensional eternal AdS black holes explains the emergence of two copies of $PSL(2,\mathbb{R})$ on future and past Killing horizons. |
| title | Modular Intersections, Time Interval Algebras and Emergent AdS$_2$ |
| topic | High Energy Physics - Theory General Relativity and Quantum Cosmology Mathematical Physics |
| url | https://arxiv.org/abs/2412.19882 |