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

Non-critical string pentagon equations and their solutions

We derive pentagon type relations for the 3-point boundary tachyon correlation functions in the non-critical open string theory with generic c_{matter} < 1 and study their solutions in the case of FZZ branes. A new general formula for the Liouville 3-point factor is derived.Comment: 18 pages, harvmac; misprints corrected, section 3.2 extended, a new general formula for the Liouville 3-point factor adde