Communications in Mathematical Physics

, Volume 134, Issue 1, pp 139–196 | Cite as

Conformal field algebras with quantum symmetry from the theory of superselection sectors

  • Gerhard Mack
  • Volker Schomerus


According to the theory of superselection sectors of Doplicher, Haag, and Roberts, field operators which make transitions between different superselection sectors—i.e. different irreducible representations of the observable algebra—are to be constructed by adjoining localized endomorphisms to the algebra of local observables. We find the relevant endomorphisms of the chiral algebra of observables in the minimal conformal model with central chargec=1/2 (Ising model). We show by explicit and elementary construction how they determine a representation of the braid groupB which is associated with a Temperley-Lieb-Jones algebra. We recover fusion rules, and compute the quantum dimensions of the superselection sectors. We exhibit a field algebra which is quantum group covariant and acts in the Hilbert space of physical states. It obeys local braid relations in an appropriate weak sense.


Hilbert Space Field Operator Irreducible Representation Ising Model Quantum Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Wick, G.G., Wigner, E.P., Wightman, A.S.: Intrinsic parity of elementary particles, Phys. Rev.88, 101 (1952)Google Scholar
  2. 2.
    Haag, R., Kastler, D.: An algebraic approach to field theory. M. Math. Phys.5, 848 (1964)Google Scholar
  3. 3.
    Doplicher, S., Haag, R., Roberts, J.E.: Local observables and particle statistics I, II. Commun. Math. Phys.23, 199 (1971) and35, 49 (1974)Google Scholar
  4. 4.
    Takesaki, H., Winnik, W.: Local normality in quantum statistical mechanics. Commun. Math. Phys.30, 129 (1973)Google Scholar
  5. 5.
    Buchholz, D., Mack, G., Todorov, I.T.: The current algebra on the circle as a germ of local field theories. Nucl. Phys. B (Proc. Suppl.)5B, 20 (1988)Google Scholar
  6. 6.
    Pressley, A., Segal, G.: Loop groups. Oxford: Oxford Science Publications 1986Google Scholar
  7. 7.
    Rehren, K.H., Schroer, B.: Einstein causality and artin braids. Nucl. Phys.B 312, 715 (1989) Rehren, K.H.: Locality of conformal fields in two dimensions. Exchange algebras on the light cone. Commun. Math. Phys.116, 675 (1988)Google Scholar
  8. 8.
    Fredenhagen, K., Rehren, K.H., Schroer, B.: Superselection sectors with braid group statistics and exchange algebras, I: General theory. Commun. Math. Phys.125, 201–226 (1989)Google Scholar
  9. 9.
    Fredenhagen, K., Rehren, K.H., Schroer, B.: As reported in B. Schroer: New kinematics (statistics and symmetry) in low-dimensionalQFT with applications to conformalQFT 2, to be published in Proc. VXIIth Int. Conf. on Differential Geometric Methods in Theoretical Physics, 1989Google Scholar
  10. 10.
    Frenkel, I., Lepowsky, J., Meurman, A.: Vertex operators and the monster. New York: Academic Press 1988Google Scholar
  11. 11.
    Fröhlich, J., Gabbiani, F., Marchetti, P.A.: Superselection structure and statistics in three-dimensional local quantum theory, preprint ETH-TH/89-22Google Scholar
  12. 11a.
    Fröhlich, J., Gabbiani, F., Marchetti, P.A.: Braid statistics in three-dimensional local quantum theory, ETH-TH/89-36, to be publ. in Proceedings “Physics, Geometry, and Topology,” Banff 1989Google Scholar
  13. 12.
    Fredenhagen, K.: Structure of superselection sectors in low-dimensional quantum field theory, to be publ. in: XVII International conference on Differential Geometric Methods in Theoretical Physics: Physics and Geometry, Davis 1989Google Scholar
  14. 13.
    Doplicher, S., Roberts, J.E.: Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics, June 1989Google Scholar
  15. 13a.
    Doplicher, S., Roberts, J.E.:C *-algebras and duality for compact groups: Why there is a compact group of internal gauge symmetries in particle physics. In: VIIIth International Congress on Mathematical Physics, Marseille 1986, Mebkhout, M., Sénéor, R. (eds.)Google Scholar
  16. 13b.
    Doplicher, S., Roberts, J.E.: Endomorphisms ofC *-algebras, cross products and duality for compact groups. Ann. Math. (to appear)Google Scholar
  17. 14.
    Drinfel'd, V.G.: Quantum groups. Proc. ICM 798 (1987)Google Scholar
  18. 15.
    Fröhlich, J.: Statistics of fields, the Yang-Baxter equation and the theory of knots and links. In: Nonperturbative quantum field theory. t'Hooft, G. et al. (eds.). New York: Plenum Press 1988Google Scholar
  19. 16.
    Moore, G., Reshetikhin, N.: A comment on quantum symmetry in conformal field theory. Nucl. Phys. B328, 557 (1989)Google Scholar
  20. 17.
    Buchholz, D., Mack, G., Todorov, I.T.: As reported in I. Todorov. In: Proceedings of Conf. on Quantum Groups, Clausthal Zellerfeld, July 1989Google Scholar
  21. 18.
    Alvarez-Gaumé, L., Gomez, C., Sierra, G.: Hidden quantum symmetry in rational conformal field theories. Nucl. Phys. B310 (1989)Google Scholar
  22. 18a.
    Alvarez-Gaumé, L., Gomez, C., Sierra, G.: Quantum group interpretation of some conformal field theories. Phys. Lett.220B, 142 (1989)Google Scholar
  23. 18b.
    Alvarez-Gaumé, L., Gomez, C., Sierra, G.: Duality and quantum groups, CERN-TH-5369/89Google Scholar
  24. 19.
    Felder, G., Fröhlich, J., Keller, G.: On the structure of unitary conformal field theory II: Representation theoretic approach ETH-TH/89-12Google Scholar
  25. 20.
    Tsuchiya, A., Kanie, Y.: Vertex operators in the conformal field theory ofP 1 and monodromy representations of the braid group. Lett. Math. Phys.13, 303 (1987)Google Scholar
  26. 21.
    Temperley, H., Lieb, E.: Relation between the percolation and the colouring problem. Proc. Roy. Soc. (London) 251 (1971)Google Scholar
  27. 22.
    Jones, V.: Index for subfactors. Invent. Math.72, 1 (1982) and in: Braid group, knot theory and statistical mechanics. Yang, C.N., Ge, M.L. (eds.). Singapore: World Scientific 1989Google Scholar
  28. 23.
    Hamermesh, N.: Group theory, Reading, MA: Addison Wesley 1962, Chap. 3–17Google Scholar
  29. 24.
    Lüscher, M., Mack, G.: Global conformal invariance in quantum field theory. Commun. Math. Phys.41, 203 (1975)Google Scholar
  30. 25.
    Mack, G.: Introduction to conformal invariant quantum field theories in two dimensions. In: Nonperturbative quantum field theory. t'Hooft, G. et al. (eds.). New York: Plenum Press 1988Google Scholar
  31. 26.
    Schomerus, V.: Diplomarbeit, Hamburg, 1989Google Scholar
  32. 27.
    Buchholz, D., Schulz-Mirbach, H.: Work in preparationGoogle Scholar
  33. 28.
    Gelfand, I.M., Shilov, G.E.: Generalized functions vol. I. New York: Academic Press 1960Google Scholar
  34. 29.
    Connes, A., Evans, D.E.: Embedding ofU(1)-current algebras of classical statistical mechanics. Commun. Math. Phys.121, 507 (1988)Google Scholar
  35. 30.
    Longo, R.: Index of subfactors and statistics of quantum fields, I, II. Commun. Math. Phys.126, 217 (1989) and130, 285–309 (1990)Google Scholar
  36. 31.
    Kirillov, A.N., Reshetikhin, N.: Representations of the algebraU q (sl(2)),q-orthogonal polynomials and invariants of links, preprint LOMIE-9-88, Leningrad 1988Google Scholar
  37. 32.
    Segal, I.: Causally ordered manifolds and groups. Bull. Am. Math. Soc.77, 958 (1971)Google Scholar
  38. 33.
    Lusztig, G.: Modular representations of quantum groups. Contemp. Math.82, 59 (1989)Google Scholar
  39. 34.
    Ganchev, A., Petkova, V.:U q (sl(2)) invariant operators and minimal theories fusion matrices. Trieste preprint, IC/89/158 (June 89)Google Scholar
  40. 35.
    Pasquier, V.: Etiology of IRF models, Commun. Math. Phys.118, 365 (1988)Google Scholar
  41. 36.
    Moore, G., Seiberg, N.: Polynomial equations for rational conformal field theories. Phys. Lett.212B, 451 (1988)Google Scholar
  42. 37.
    Dotsenko, V.S., Fateev, V.A.: Four point correlation functions and the operator algebra in the 2-dimensional conformal quantum field theories with the central chargec<1. Nucl. Phys.B251 [FS 13], 691 (1985)Google Scholar
  43. 37a.
    Dotsenko, V.S., Fateev, V.A.: Conformal algebra and multipoint correlation functions in 2-dimensional statistical models. Nucl. Phys.B240 [FS 12], 312 (1984)Google Scholar
  44. 38.
    Pasquier, V., Saleur, H.: Common structures between finite systems and conformal field theories through quantum groups, submitted to Nucl. Phys. B [FS]Google Scholar
  45. 39.
    Wenzl, H.: Hecke algebras of typeA n and subfactors. Inv. Math.92, 349 (1988)Google Scholar
  46. 39a.
    Wenzl, H.: Quantum groups and subfactors of type B, C, E. UC San Diego preprint, July 1989Google Scholar
  47. 40.
    Bouwknegt, P., McCarthy, J., Pilch, K.: Free field realization of WZNW models, BRST complex and its quantum group structure, submitted to Phys. Lett. B (1989)Google Scholar
  48. 41.
    Fröhlich, J., King, C.: Two-dimensional conformal field theory and 3-dimensional topology, Commun. Math. Phys. (to appear)Google Scholar
  49. 41a.
    Felder, J., Fröhlich, J., Keller, G.: Braid matrices and structure constants for minimal conformal models. Commun. Math. Phys. (to appear)Google Scholar
  50. 42.
    Mack, G.: Duality in quantum field theory. Nucl. Phys.B118, 445 (1977), esp. p. 456Google Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • Gerhard Mack
    • 1
  • Volker Schomerus
    • 2
  1. 1.Mathematics DepartmentHarvard UniversityCambridgeUSA
  2. 2.II. Institut für Theoretische PhysikUniversität HamburgHamburg 50Germany

Personalised recommendations