{
  "abstract": "We determine exactly the range of the modular Hamiltonian of an interval in a chiral conformal field theory over states of bounded energy. For an interval of length $R$ in a theory of central charge $c$, and states of mean energy at most $E$, the supremum of the modular energy is $M_{\\max} = \\frac{πR}{2}E + \\frac{c}{6} - \\frac{πc^{2}}{288\\,RE} + O((RE)^{-2})$, and neither constant can be improved. The infimum is logarithmic, $M_{\\min} \\simeq -\\frac{c}{12}[\\ln(24πRE/c)-1]$, so the range is linear above and logarithmic below, the two meeting as $RE \\to 0$; it is strictly wider above, by more than $\\frac{2}{5}πRE$ at every energy, which reduces after a Stieltjes substitution to the positivity of a variance. Both bounds are attained, the one-parameter family of weights proving them being the Legendre structure of the answer. The two ingredients separate cleanly. The coefficient $πR/2$ is Möbius representation theory alone: the complementary weight is a translate of the special conformal generator, hence positive, so for the full modular Hamiltonian the bound needs no additive constant. The constant $c/6$ is precisely the price of truncating that weight at the endpoints, supplied by the quantum energy inequality of Fewster and Hollands; the saturating state places its negative energy exactly there. The constant is accordingly additive over entangling points, being $nc/6$ for $n$ disjoint intervals of equal length. Composing with positivity of relative entropy gives a sharp Bekenstein bound, $ΔS_I \\leq \\frac{πR}{2}E + \\frac{c}{6}$, recovering the original form of that inequality from the modular form that by itself imposes no constraint.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27830",
  "author_total": 2,
  "authors": [
    {
      "name": "Samuel L. Braunstein"
    },
    {
      "name": "Zhi-Wei Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f883pwej4zhmq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27830",
  "fulltext_url": null,
  "id": "f883pwej4zhmq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27830",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "hep-th"
    ],
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "openalex",
      "3106|Black Holes and Theoretical Physics"
    ]
  ],
  "openalex": "W7214225989",
  "pdf_url": "https://arxiv.org/pdf/2609.27830",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f883pwej4zhmq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:56Z",
      "native_id": "2609.27830",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27830"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:hep-th",
      "subfield": "high-energy"
    }
  ],
  "title": "The modular energy range of an interval in chiral conformal field theory",
  "title_text": "The modular energy range of an interval in chiral conformal field theory",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/f883pwej4zhmq",
  "venue": "arXiv",
  "versions": []
}