Generalised twisted partition functions

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

Conformal Boundary Conditions and what they teach us

The question of boundary conditions in conformal field theories is discussed, in the light of recent progress. Two kinds of boundary conditions are examined, along open boundaries of the system, or along closed curves or ``seams''. Solving consistency conditions known as Cardy equation is shown to amount to the algebraic problem of finding integer valued representations of (one or two copies of) the fusion algebra. Graphs encode these boundary conditions in a natural way, but are also relevant in several aspects of physics ``in the bulk''. Quantum algebras attached to these graphs contain information on structure constants of the operator algebra, on the Boltzmann weights of the corresponding integrable lattice models etc. Thus the study of boundary conditions in Conformal Field Theory offers a new perspective on several old physical problems and offers an explicit realisation of recent mathematical concepts.Comment: Expanded version of lectures given at the Summer School and Conference Nonperturbative Quantum Field Theoretic Methods and their Applications, August 2000, Budapest, Hungary. 35 page

Fusion Rings Related to Affine Weyl Groups

The construction of the fusion ring of a quasi-rational CFT based on $\hat{sl}(3)_k$ at generic level $k\not \in {\Bbb Q}$ is reviewed. It is a commutative ring generated by formal characters, elements in the group ring ${\Bbb Z}[\tilde{W}]$ of the extended affine Weyl group $\tilde{W}$ of $\hat{sl}(3)_k$. Some partial results towards the $\hat{sl}(4)_k$ generalisation of this character ring are presented.Comment: 13 pages; two figures. Talk at ``Lie Theory and Its Applications in Physics III'', Clausthal, 11-14 July, 1999, to appear in the Proceedings, eds. H.-D. Doebner et a

An Extension of the Character Ring of sl(3) and Its Quantisation

We construct a commutative ring with identity which extends the ring of characters of finite dimensional representations of sl(3). It is generated by characters with values in the group ring $Z[\tilde{W}]$ of the extended affine Weyl group of $\hat{sl}(3)_k$ at $k\not \in Q$. The `quantised' version at rational level $k+3=3/p$ realises the fusion rules of a WZW conformal field theory based on admissible representations of $\hat{sl}(3)_k$.Comment: contains two TeX files: main file using harvmac.tex, amssym.def, amssym.tex, 35p.; file with figures using XY-pic package, 4p; v2: minor corrections, Note adde