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<p>We compute the modular flow and conjugation of time interval algebras of conformal Generalized Free Fields (GFF) in 0 + 1-dimensions in the 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<sub>2</sub> 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 <i>Twisted Modular Inclusion</i> and a <i>Twisted Modular Intersection</i> 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<i>,</i> ℝ), respectively. We discuss the implications of our result for the emergence of AdS<sub>2</sub> geometries in large <i>N</i> theories without a large gap. Our result applied to higher dimensional eternal AdS black holes explains the emergence of two copies of PSL(2<i>,</i> ℝ) on future and past Killing horizons.</p>

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Modular intersections, time interval algebras and emergent AdS2

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年10月16日

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  • Nima Lashkari,
  • Kwing Lam Leung,
  • Mudassir Moosa,
  • Shoy Ouseph

摘要

We compute the modular flow and conjugation of time interval algebras of conformal Generalized Free Fields (GFF) in 0 + 1-dimensions in the 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 AdS2 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 Twisted Modular Inclusion and a 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, ℝ), respectively. We discuss the implications of our result for the emergence of AdS2 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, ℝ) on future and past Killing horizons.