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We consider the set of partition functions that result from the insertion of twist operators compatible with conformal invariance in a given 2D Conformal Field Theory (CFT). A consistency equation, which gives a classification of twists, is written and solved in particular cases. This generalises old results on twisted torus boundary conditions, gives a physical interpretation of Ocneanu's algebraic construction, and might offer a new route to the study of properties of CFT.Comment: 12 pages, harvmac, 1 Table, 1 Figure . Minor typos corrected, the figure which had vanished reappears

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We define generalised chiral vertex operators covariant under the Ocneanu ``double triangle algebra'' {\cal A}, a novel quantum symmetry intrinsic to a given rational 2-d conformal field theory. This provides a chiral approach, which, unlike the conventional one, makes explicit various algebraic structures encountered previously in the study of these theories and of the associated critical lattice models, and thus allows their unified treatment. The triangular Ocneanu cells, the 3j-symbols of the weak Hopf algebra {\cal A}, reappear in several guises. With {\cal A} and its dual algebra {hat A} one associates a pair of graphs, G and {\tilde G}. While G are known to encode complete sets of conformal boundary states, the Ocneanu graphs {\tilde G} classify twisted torus partition functions. The fusion algebra of the twist operators provides the data determining {\hat A}. The study of bulk field correlators in the presence of twists reveals that the Ocneanu graph quantum symmetry gives also an information on the field operator algebra.Comment: 57 pages, 10 figures. Several misprints and the derivation of eq. (7.38) correcte

An expansion of the type (@(&ij &(&n~&() = (fr(' (& pl&(&2l&0(@(&p~' @(&n~&p (x, ) Xi is derived, where yi -fl, ci] are labels for infinite-dimensional symmetric tensor representations of the Euclidean conformal group 0 (2@+1, 1), X, =[ l, , -ci], the constants C (y i) are real, and Q)( and u)x have the properties of vacuum expectation values of field products. The starting point is an infinite set of coupled nonlinear integral equations for Euclidean Green's functions in 2h space-time dimensions of the type written some 15 years ago by Fradkin and Symanzik. The Green's functions of the corresponding Gell-Mann-Low limit theory are expanded in conformal partial waves. The dynamical equations imply the existence of poles and factorization of residues in the partial waves as functions of the representation parameters. In proving the validity of the expansion we use some differential relations between partially equivalent exceptional representations of 0& (2 h + 1, 1), established in an earlier paper. This work completes the group-theoretical derivation of the vacuum operator-product expansion undertaken by Mack in 1973.

We develop further the theory of Rational Conformal Field Theories (RCFTs) on a cylinder with specified boundary conditions emphasizing the role of a triplet of algebras: the Verlinde, graph fusion and Pasquier algebras. We show that solving Cardy's equation, expressing consistency of a RCFT on a cylinder, is equivalent to finding integer valued matrix representations of the Verlinde algebra. These matrices allow us to naturally associate a graph G to each RCFT such that the conformal boundary conditions are labelled by the nodes of G. This approach is carried to completion for s (2) theories leading to complete sets of conformal boundary conditions, their associated cylinder partition functions and the A-D-E classification. We also review the current status for WZW s (3) theories. Finally, a systematic generalization of the formalism of Cardy-Lewellen is developed to allow for multiplicities arising from more general representations of the Verlinde algebra. We obtain information on the bulk-boundary coefficients and reproduce the relevant algebraic structures from the sewing constraints.

We give function space realizations of all representations of the conformal superalgebra su(2,2/N) and of the supergroup s U ( 2 , 2 / N ) induced from irreducible finite-dimensional Lorentz and X U ( N ) representations realized without spin and isospin indices. We use the lowest weight module structure of our su(2,2/N) representations to present a general procedure (adapted from the semisimple Lie algebra case) for the canonical construction of invariant differential operators closely related to the reducible (indecomposable) representations. All conformal supercovariant derivatives are obtained in this way. Exa.mples of higher order invariant differential operators are given. l ) A. v. Humboldt Foundation Pellow. 2) On leave of absence from Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Sofia, Bulgaria. 538 V. K. DOBREV, V. B. PETKOVA, Group Theoretical Approach We construct and study all representations of su(2,2/N) and of the supergroup XU(2,Si N ) induced from irreducible finite-dimensional Lorentz and XU( N/O) representations.These are called (as in the semisimple case) elementary representations (ER). We give a more detailed derivation of the reducibility conditions of the E R for arbitrary N announced in [a] (for N = 1 see [9, 101). As for SU (2,2) [7] these are obtained purely algebraically since the E R contain the structure of lowest weight modules (LWM) over the complexification gc = s1(4/N; C) of g. Whenever such a reducibility condition for the LWM or E R x is satisfied there arises an invariant map from the representation space C, to the space C,) where x' is obtained by the action of a definite Weyl reflection on the lowest weight corresponding to x. The existence of such a map is equivalent to the partial equivalence of the LWM (or ER) x and 2'. These results were used in the purely algebraic setting in [4] for the classification of the physically important multiplets3) of LWM.The algebraic approach exploited in [a] is more economical and rather powerful (see . K. DOBREV and V. B. PETKOVA, in preparation. 1974, in Russian).

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