Abstract
We survey a recent progress on algebraic quantum field theory in connection with subfactor theory. We mainly concentrate on one-dimensional conformal quantum field theory.
Similar content being viewed by others
References
Haag, R.: “Local Quantum Physics”, Springer-Verlag, Berlin-Heidelberg-New York, (1996)
Brunetti, R., Guido, D., Longo, R.: Modular structure and duality in conformal quantum field theory. Commun. Math. Phys., 156 201–219, (1993)
Buchholz, D., Schulz-Mirbach, H.: Haag duality in conformal quantum field theory. Rev. Math. Phys., 2, 105–125 (1990)
Fröhlich, J., Gabbiani, F.: Operator algebras and conformal field theory. Commun. Math. Phys., 155, 569–640 (1993)
Guido, D., Longo, R.: The conformal spin and statistics theorem. Commun. Math. Phys., 181, 11–35 (1996)
Hislop, P., Longo, R.: Modular structure of the local algebras associated with the free massless scalar field theory. Commun. Math. Phys., 84, 71–85 (1982)
Kawahigashi, Y., Longo, R., M¨uger, M.: Multi-interval subfactors and modularity of representations in conformal field theory. Commun. Math. Phys., 219, 631–669 (2001)
Fredenhagen, K., Rehren, K. H., Schroer, B.: Superselection sectors with braid group statistics and exchange algebras I. Commun. Math. Phys., 125, 201–226 (1989), II. Rev. Math. Phys., Special issue, 113–157 (1992)
Doplicher, S., Haag, R., Roberts, J. E.: Local observables and particle statistics I. Commun. Math. Phys., 23, 199–230 (1971); II. 35, 49–85 (1974)
Jones, V. F. R.: Index for subfactors. Invent. Math., 72, 1–25 (1983)
Kosaki, H.: Extension of Jones’ theory on index to arbitrary factors. J. Funct. Anal., 66, 123–140 (1986)
Longo, R.: Index of subfactors and statistics of quantum fields I. Commun. Math. Phys., 126, II. 217–247 130, 285–309 (1989), (1990)
Doplicher, S., Roberts, J. E.: A new duality theory for compact groups. Invent. Math., 98, 157–218 (1989)
Turaev, V. G.: “Quantum Invariants of Knots and 3-Manifolds”. Walter de Gruyter, Berlin–New York, (1994)
Rehren, K. H.: Braid group statistics and their superselection rules. in: “The Algebraic Theory of Superselection Sectors”, D. Kastler ed., World Scientific, Singapore, 333–355 (1990)
Bakalov, B., Kirillov, Jr. A.: “Lectures on tensor categories and modular functors”. University Lecture Series 21, Amer. Math. Soc., (2001)
Buchholz, D., D’Antoni, C., Longo, R.: Nuclear maps and modular structures II. Commun. Math. Phys., 129, 115–138 (1990)
D'Antoni, C., Longo, R. Radulescu, F.: Conformal nets, maximal temperature and models from free probability. J. Operator Theory, 45, 195–208 (2001)
Evans, D. E., Kawahigashi, Y.: “Quantum Symmetries on Operator Algebras”, Oxford University Press, Oxford, (1998)
Longo. R., Rehren, K. H.: Nets of subfactors. Rev. Math. Phys., 7, 567–597 (1995)
Izumi, M.: The structure of sectors associated with the Longo-Rehren inclusions. Commun. Math. Phys., 213, 127–179 (2000)
Masuda, T.: An analogue of Longo’s canonical endomorphism for bimodule theory and its application to asymptotic inclusions. Internat. J. Math., 8, 249–265 (1997)
Wassermann, A.: Operator algebras and conformal field theory III: Fusion of positive energy representations of SU(N) using bounded operators. Invent. Math., 133, 467–538 (1998)
Xu, F.: Jones-Wassermann subfactors for disconnected intervals. Commun. Contemp. Math., 2, 307–347 (2000)
Di Francesco, P., Mathieu, P., Sénéchal, D.: “Conformal Field Theory”. Springer-Verlag, Berlin–Heidelberg– New York, (1996)
Xu, F.: Algebraic coset conformal field theories I. Commun. Math. Phys., 211, 1–44 (2000)
Xu, F.: On a conjecture of Kac-Wakimoto. Publ. RIMS, Kyoto Univ., 37, 165–190 (2001)
Xu, F.: 3-manifold invariants from cosets. preprint 1999, math.GT/9907077
Xu, F.: Algebraic orbifold conformal field theories. Proc. Nat. Acad. Sci. U.S.A., 97, 14069–14073 (2000)
Longo, R.: Conformal subnets and intermediate subfactors. math.OA/0102196, to appear in Commun. Math. Phys.
Roberts, J. E.: Local cohomology and superselection structure. Commun. Math. Phys., 51, 107–119 (1976)
Xu, F.: New braided endomorphisms from conformal inclusions. Commun. Math. Phys., 192, 347–403 (1998)
Xu, F.: Applications of braided endomorphisms from conformal inclusions. Internat. Math. Res. Notices, 5–23 (1998)
Böckenhauer, J., Evans, D. E.: Modular invariants, graphs and α-induction for nets of subfactors I. Commun. Math. Phys., 197, 361–386 (1998), II, 200, 57–103 (1999), III 205, 183–228 (1999)
Böckenhauer, J., Evans, D. E.: Modular invariants from subfactors: Type I coupling matrices and intermediate subfactors. Commun. Math. Phys., 213, 267–289 (2000)
Böckenhauer, J., Evans, D. E., Kawahigashi, Y.: On α-induction, chiral projectors and modular invariants for subfactors. Commun. Math. Phys., 208, 429–487 (1999)
Böckenhauer, J., Evans, D. E., Kawahigashi, Y.: Chiral structure of modular invariants for subfactors. Commun. Math. Phys., 210, 733–784 (2000)
Böckenhauer, J., Evans, D. E., Kawahigashi, Y.: Longo-Rehren subfactors arising from α-induction. Publ. RIMS, Kyoto Univ., 37, 1–35 (2001)
Ocneanu, A.: Paths on Coxeter diagrams: from Platonic solids and singularities to minimal models and subfactors, (Notes recorded by S. Goto), in Lectures on operator theory, (ed. B. V. Rajarama Bhat et al.), The Fields Institute Monographs, AMS Publications, 243–323 (2000)
Goodman, F., de la Harpe, P., Jones, V. F. R.: “Coxeter Graphs and Towers of Algebras” MSRI Publications, Springer, Berlin–Heidelberg–New York, 14, (1989)
Kawahigashi, Y.: Classification of paragroup actions on subfactors. Publ. RIMS, Kyoto Univ., 31, 481–517 (1995)
Cappelli, A., Itzykson, C., Zuber, J. B.: The A-D-E classification of minimal and A (1) 1 conformal invariant theories. Commun. Math. Phys., 113, 1–26 (1987)
Kato, A.: Classification of modular invariant partition functions in two dimensions. Modern Phys. Lett., A2, 585–600 (1987)
Gannon, T.: Modular data: the algebraic combinatorics of conformal field theory. preprint 2001, math. QA/0103044
Friedan, D., Qiu, Z., Shenker, S.: Details of the non-unitarity proof for highest weight representations of the Virasoro algebra. Commun. Math. Phys., 107, 535–542 (1986)
Goddard, P., Kent, A., Olive, D.: Unitary representations of the Virasoro and super-Virasoro algebras. Commun. Math. Phys., 103, 105–119 (1986)
Kawahigashi, Y., Longo, R.: Classification of Local Conformal Nets. Case c < 1. Ann. Math., to appear, math-ph/0201015
Loke, T.: Operator algebras and conformal field theory of the discrete series representations of Diff(S 1), Thesis, University of Cambridge, (1994)
Izumi, M., Longo, R., Popa, S.: A Galois correspondence for compact groups of automorphisms of von Neumann algebras with a generalization to Kac algebras. J. Funct. Anal., 10, 25–63 (1998)
Longo, R.: A duality for Hopf algebras and for subfactors. Commun. Math. Phys., 159, 133–150 (1994)
Kirillov, Jr. A., Ostrik, V.: On q-analog of McKay correspondence and ADE classification of sl (2) conformal field theories. Adv. Math., 171, 183–227 (2002)
Kawahigashi, Y.: Generalized Longo-Rehren subfactors and α-induction. Commun. Math. Phys., 226, 269–287 (2002)
Rehren, K. H.: Canonical tensor product subfactors. Commun. Math. Phys., 211, 395–406 (2000)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported in parts by the Grants-in-Aid for Scientific Research, JSPS
Rights and permissions
About this article
Cite this article
Kawahigashi, Y. Conformal Quantum Field Theory and Subfactors. Acta Math Sinica 19, 557–566 (2003). https://doi.org/10.1007/s10114-003-0270-6
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s10114-003-0270-6