Abstract
This article reviews some recent progress in our understanding of the structure of rational conformal field theories, based on ideas that originate for a large part in the work of A. Ocneanu. The consistency conditions that generalize modular invariance for a given RCFT in the presence of various types of boundary conditions—open, twisted—are encoded in a system of integer multiplicities that form matrix representations of fusion-like algebras. These multiplicities are also the combinatorial data that enable one to construct an abstract “quantum” algebra, whose 6j-and 3j-symbols contain essential information on the operator product algebra of the RCFT and are part of a cell system, subject to pentagonal identities. It looks quite plausible that the classification of a wide class of RCFT amounts to a classification of “Weak C*- Hopf algebras”.
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References
R.E. Behrend, P.A. Pearce, V.B. Petkova and J.-B. Zuber, On the classification of bulk and boundary conformal field theories, Phys. Lett. B 444: 163–166, 1998, hep-th/9809097.
R.E. Behrend, P.A. Pearce, V.B. Petkova and J.-B. Zuber, Boundary conditions in rational conformal theories, Nucl. Phys., B 579: 707–773, 2000, hep-th/9908036.
J. Böckenhauer and D.E. Evans,Modular invariants, graphs and oz-induction for nets of subfactors, II: Comm. Math. Phys. 200: 57–103, 1999, hep-th/ 98 0502 3; III: Comm. Math. Phys. 205: 183–228, 1999, hep-th/9812110; Modular invariants from subfactors: Type I coupling matrices and intermediate subfactors, math. OA/9911239.
J. Böckenhauer, D. E. Evans and Y. Kawahigashi, On a-induction, chiral generators and modular invariants for subfactors Comm. Math. Phys. 208: 429–487, 1999, math.OA/9904109; Chiral structure of modular invariants for subfactors, Comm. Math. Phys. 210: 733–784, 2000, math. OA/ 9 9 0 714 9.
G. Böhm and K. Szlachányi, A coassociative C*-quantum group with non-integral dimensions, Lett. Math. Phys. 200: 437–456 1996, q-a l g/ 9 5 0 9 0 0 8; Weak C*-Hopf algebras: The coassociative symmetry of non-integral dimensions, in: Quantum Groups and Quantum Spaces, Banach Center Publ. v. 40, (1997) 9–19; G. Böhm, Weak C* -Hopf Algebras and their application to spin models, Ph.D. Thesis, Budapest 1997.
J.L. Cardy, Operator content of two-dimensional conformally invariant theories, Nucl. Phys. B 270: 186–204, 1986.
J.L. Cardy, Fusion rules and the Verlinde Formula, Nucl. Phys. B 324: 581–596, 1989.
J.L. Cardy and D.C. Lewellen, Bulk and boundary operators in conformal field theory, Phys. Lett. B 259: 274–278, 1991; D.C. Lewellen, Sewing constraints for conformal field theories on surfaces with boundaries, Nucl. Phys. B 372: 654–682, 1992.
C.H.O. Chui, Ch. Mercat, W. Orrick and P.A. Pearce, Integrable lattice realizations of conformal twisted boundary conditions, hep-th/0106182.
C.H.O. Chui, Ch. Mercat, W. Orrick and P.A. Pearce, private communication and to appear.
R. Coquereaux, Notes on the quantum tetrahedron, math-ph/0011006; R. Coquereaux and G. Schieber, Twisted partition functions for ADE boundary conformal field theories and Ocneanu algebras of quantum symmetries, hep-th/0107001.
P. Di Francesco and J.-B. Zuber, SU (N) lattice integrable models associated with graphs, Nucl. Phys. B 338: 602–646, 1990.
P. Di Francesco and J.-B. Zuber, SU(N) lattice integrable models and modular invariance, in Recent Developments in Conformal Field Theories, Trieste Conference, (1989), S. Randjbar-Daemi, E. Sezgin and J.-B. Zuber eds., World Scientific (1990); P. Di Francesco, Integrable lattice models, graphs and modular invariant conformal field theories, Int. J. Math. Phys. A 7: 407–500, 1992.
R. Dijkgraaf, C. Vafa, E. Verlinde and H. Verlinde, The Operator algebra of orbifold models, Comm. Math. Phys. 123: 485–526, 1989.
G. Felder, J. Fröhlich, J. Fuchs and C. Schweigert, Conformal boundary conditions and three-dimensional topological field theory, Phys.Rev.Lett. 84: 1659–1662, 2000, hep-th/9909140.
J. Fuchs, L.R. Huiszoon, A.N. Schellekens, C. Schweigert and J. Walcher, Boundaries, crosscaps and simple currents, Phys. Lett. B 495: 427–434, 2000, hep-th/0007174.
J. Fuchs and C. Schweigert, A classifying algebra for boundary conditions, Phys. Lett. B 414: 251–259, 1997, hep-th/9708141.
J. Fuchs and C. Schweigert, Category theory for conformal boundary conditions, math.CT/0106050.
T. Gannon, The Classification of affine SU(3) modular Invariants, Comm. Math. Phys. 161: 233–263, 1994; The Classification of SU(3) modular invariants revisited, hep-th/ 9404185; The level two and three modular invariants of SU(n), Lett. Math. Phys. 39: 289–298, 1997.
T. Gannon, Boundary conformal field theory and fusion ring representations, hep-th/0106105.
V.G. Kac and D.H. Peterson, Infinite-Dimensional Lie Algebras, Theta Functions and Modular Forms, Adv. Math. 53: 125–264, 1984.
A. Kirillov, Jr and V. Ostrik, On q-analog of McKay correspondence and ADE classification of:s72 conformal field theories, math. QA/ 0101219.
I. Kostov, Free field presentation of the A,z coset models on the torus, Nucl. Phys. B 300: 559–587, 1988.
A. Ocneanu, Quantum cohomology, quantum groupoids and subfactors, unpublished lectures given at the First Carribean School of Mathematics and Theoretical Physics, Saint François, Guadeloupe 1993.
A. Ocneanu, Paths on Coxeter diagrams: From Platonic solids and singularities to minimal models and subfactors, Lectures on Operator Theory, Fields Institute, Waterloo, Ontario, April 26–30, 1995, (Notes taken by S. Goto), Fields Institute Monographs, AMS 1999, Rajarama Bhat et al, eds.
A. Ocneanu, Quantum symmetries for SU(3) CFT Models, Lectures at Bariloche Summer School, Argentina, Jan 2000, to appear in AMS Contemporary Mathematics„ R. Coquereaux, A. Garcia and R. Trinchero, eds.
V. Pasquier, Two-dimensional critical systems labelled by Dynkin diagramsNucl. Phys. B 285: 162–172, 1987.
V. Pasquier, Operator content of the ADE lattice modelsJ. Phys. A 20: 57075717, 1987.
V.B. Petkova and J.-B Zuber, On structure constants of sl(2) theories, Nucl. Phys. B 438: 347–372, 1995, hep-th/9410209.
V.B. Petkova and J.-B Zuber, From CFT to graphs, Nucl. Phys. B 463: 161–193, 1996, hep-th/9510175; Conformal field theory and graphs, GROUP21 Physical Applications and Mathematical Aspects of Geometry, Groups, and Algebras v. 2, (1997), p. 627, Goslar Conference (1996), eds. H.-D. Doebner et al, World Scientific, Singapore, hep-th/ 9701103.
V.B. Petkova and J.-B Zuber, Generalized twisted partition functions, Phys. Lett. B 504: 157–164, 2000, hep-th/0011021.
V.B. Petkova and J.-B Zuber, The many faces of Ocneanu Cells, Nucl. Phys. B 603: 449–496, 2001, hep-th/ 0101151.
G. Pradisi, A. Sagnotti and Ya.S. Stanev, Completeness conditions for boundary operators in 2D conformal field theory, Nucl. Phys. B 381: 97–104, 1996, hep-th/9603097.
A. Recknagel, V. Schomerus, Boundary Deformation Theory and Moduli Spaces of D-Branes, Nucl.Phys. B 545: 233–282, 1999, hep-th/9811237.
I. Runkel, Boundary structure constants for the A-series Virasoro minimal models, Nucl. Phys. B 549: 563–578, 1999, hep-th/9811178;
I. Runkel, Structure constants for the D-series Virasoro minimal models, Nucl. Phys. B 579: 561–589, 2000, hep-th/9908046.
R. Talbot, Pasquier models at c = 1: Cylinder partition functions and the role of the affine Coxeter element, J. Phys. A 33: 9101–9118, 2000, hep-th/9911058.
E. Verlinde, Fusion rules and modular transformations in 2-D conformal field theory, Nucl. Phys. B 300: 360–376, 1988.
F. Xu, New braided endomorphisme from conformal inclusions. Comm. Math. Phys. 192: 349–403, 1998.
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Petkova, V., Zuber, JB. (2002). Conformal Field Theories, Graphs and Quantum Algebras. In: Kashiwara, M., Miwa, T. (eds) MathPhys Odyssey 2001. Progress in Mathematical Physics, vol 23. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0087-1_15
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