Non-unitary Wightman CFTs and non-unitary vertex algebras

• Sebastiano Carpi ^[2] ; Christopher Raymond ^[3] ; Yoh Tanimoto ^[2] ; James E. Tener ^[1]
  1. [1] Australian National University

     Australian National University

     Australia

     
  2. [2] Dipartimento di Matematica, Università di Roma Tor Vergata, Italy
  3. [3] Department of Mathematics, University of Hamburg,Germany

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• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 31, Nº. 4, 2025
• Idioma: inglés
• DOI: 10.1007/s00029-025-01063-4
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• Resumen

  • We give an equivalence of categories between: (i) Möbius vertex algebras which are equipped with a choice of generating family of quasiprimary vectors, and (ii) (not-necessarily-unitary) Möbius-covariant Wightman conformal field theories on the unit circle. We do not impose any technical restrictions on the theories considered (such as finite-dimensional conformal weight spaces or simplicity), yielding the most general equivalence between these two axiomatizations of two-dimensional chiral conformal field theory.This provides new opportunities to study non-unitary vertex algebras using the lens of algebraic conformal field theory and operator algebras, which we demonstrate by establishing a non-unitary version of the Reeh-Schlieder theorem.

• Referencias bibliográficas
  • Adamo, M.S., Giorgetti, L., Tanimoto, Y.: Wightman fields for two-dimensional conformal field theories with pointed representation category....
  • Adamo, M. S., Moriwaki, Y., Tanimoto, Y.: Osterwalder–Schrader axioms for unitary full vertex operator algebras. arXiv:2407.18222 [math-ph],...
  • Allen, R., Wood, S.: Bosonic ghostbusting: The bosonic ghost vertex algebra admits a logarithmic module category with rigid fusion. Commun....
  • Buhl, G., Karaali, G.: Spanning sets for Möbius vertex algebras satisfying arbitrary difference conditions. J. Algebra 320(8), 3345–3364 (2008)
  • Carpi, S., Gaudio, T., Hillier, R.: From vertex operator superalgebras to graded-local conformal nets and back. arXiv:2304.14263 [math.OA],...
  • Carpi, S., Kawahigashi, Y., Longo, R., Weiner, M.: From vertex operator algebras to conformal nets and back. Mem. Amer. Math. Soc., 254(1213),...
  • Carpi, S., Weiner, M., Xu, F.: From vertex operator algebra modules to representations of conformal nets. In preparation
  • Evans, D.E., Gannon, T.: Non-unitary fusion categories and their doubles via endomorphisms. Adv. Math. 310, 1–43 (2017)
  • Frenkel, I.B., Huang, Y.-Z., Lepowsky, J.: On axiomatic approaches to vertex operator algebras and modules. Mem. Amer. Math. Soc. 104(494),...
  • Gui, B.: Unbounded field operators in categorical extensions of conformal nets. arXiv:2001.03095 [math.QA], (2020)
  • Gui, B.: Categorical extensions of conformal nets. Comm. Math. Phys. 383(2), 763–839 (2021)
  • Gui, B.: Convergence of sewing conformal blocks. Commun. Contemp. Math. 26(3), 2350007 (2024)
  • Gui, B.: Sewing and propagation of conformal blocks. New York J. Math. 30, 187–230 (2024)
  • Gui, B., Zhang, H.: Analytic Conformal Blocks of C2-cofinite Vertex Operator Algebras I: Propagation and Dual Fusion Products. arXiv:2305.10180...
  • Huang, Y.-Z., Lepowsky, J., Zhang, L.: Logarithmic tensor category theory for generalized modules for a conformal vertex algebra, I: introduction...
  • Huang, Y.-Z.: A functional-analytic theory of vertex (operator) algebras. I. Comm. Math. Phys. 204(1), 61–84 (1999)
  • Huang, Y.-Z.: A functional-analytic theory of vertex (operator) algebras. II. Comm. Math. Phys. 242(3), 425–444 (2003)
  • Huang, Y.-Z.: Generators, spanning sets and existence of twisted modules for a grading-restricted vertex (super)algebra. Selecta Math. (N.S.)...
  • Kac, V.: Vertex algebras for beginners, volume 10 of University Lecture Series. American Mathematical Society, Providence, RI, second edition,...
  • Li, H.S.: Symmetric invariant bilinear forms on vertex operator algebras. J. Pure Appl. Algebra 96(3), 279–297 (1994)
  • Moriwaki, Y.: Two-dimensional conformal field theory, full vertex algebra and current-current deformation. Adv. Math. 427, 109125 (2023)
  • Narici, L., Beckenstein, E.: Topological vector spaces, volume 296 of Pure and Applied Mathematics (Boca Raton). CRC Press, Boca Raton, FL,...
  • Reeh, H., Schlieder, S.: Bemerkungen zur Unitäräquivalenz von Lorentzinvarianten Felden. Nuovo Cimento 10(22), 1051–1068 (1961)
  • Raymond, C., Tanimoto, Y., Tener, J.E.: Unitary vertex algebras and Wightman conformal field theories. Comm. Math. Phys. 395(1), 299–330 (2022)
  • Segal, G.: The definition of conformal field theory. In Topology, geometry and quantum field theory, volume 308 of London Math. Soc. Lecture...
  • Strocchi, F.: Selected topics on the general properties of quantum field theory. World Scientific Lecture Notes in Physics, vol. 51. World...
  • Streater, R.F., Wightman, A.S.: PCT, spin and statistics, and all that. W. A. Benjamin Inc, New York-Amsterdam (1964)
  • Tener, J.E.: Geometric realization of algebraic conformal field theories. Adv. Math. 349, 488–563 (2019)
  • Tener, J.E.: Representation theory in chiral conformal field theory: from fields to observables. Selecta Math. (N.S.) 25(5), 76 (2019)
  • Tener, J.E.: Fusion and positivity in chiral conformal field theory. Geom. Funct. Anal. 34(4), 1226–1296 (2024)
  • Trèves, F.: Topological vector spaces, distributions and kernels. Academic Press, New York-London (1967)