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

The boundary states and correlation functions of the tricritical Ising model from the Coulomb-gas formalism

We consider the minimal conformal model describing the tricritical Ising model on the disk and on the upper half plane. Using the coulomb-gas formalism we determine its consistents boundary states as well as its 1-point and 2-point correlation functions.Comment: 20 pages, no figure. Version 2:A paragraph for the calculation of the 2-point correlators was added. Some typos and garammatical errors were corrected.Version 3: Equations 24 are modified. Version 4 : new introduction and minor correction

From modular invariants to graphs: the modular splitting method

We start with a given modular invariant M of a two dimensional su(n)_k conformal field theory (CFT) and present a general method for solving the Ocneanu modular splitting equation and then determine, in a step-by-step explicit construction, 1) the generalized partition functions corresponding to the introduction of boundary conditions and defect lines; 2) the quantum symmetries of the higher ADE graph G associated to the initial modular invariant M. Notice that one does not suppose here that the graph G is already known, since it appears as a by-product of the calculations. We analyze several su(3)_k exceptional cases at levels 5 and 9.Comment: 28 pages, 7 figures. Version 2: updated references. Typos corrected. su(2) example has been removed to shorten the paper. Dual annular matrices for the rejected exceptional su(3) diagram are determine

The Boundary Conformal Field Theories of the 2D Ising critical points

We present a new method to identify the Boundary Conformal Field Theories (BCFTs) describing the critical points of the Ising model on the strip. It consists in measuring the low-lying excitation energies spectra of its quantum spin chain for different boundary conditions and then to compare them with those of the different boundary conformal field theories of the $(A_2,A_3)$ minimal model.Comment: 7 pages, no figures. Talk given at the XXth International Conference on Integrable Systems and Quantum Symmetries (ISQS-20). Prague, June 201

Graphs and Reflection Groups

It is shown that graphs that generalize the ADE Dynkin diagrams and have appeared in various contexts of two-dimensional field theory may be regarded in a natural way as encoding the geometry of a root system. After recalling what are the conditions satisfied by these graphs, we define a bilinear form on a root system in terms of the adjacency matrices of these graphs and undertake the study of the group generated by the reflections in the hyperplanes orthogonal to these roots. Some ``non integrally laced " graphs are shown to be associated with subgroups of these reflection groups. The empirical relevance of these graphs in the classification of conformal field theories or in the construction of integrable lattice models is recalled, and the connections with recent developments in the context of ${\cal N}=2$ supersymmetric theories and topological field theories are discussed.Comment: 42 pages TEX file, harvmac and epsf macros, AMS fonts optional, uuencoded, 8 figures include

Bosonization and Scale Invariance on Quantum Wires

We develop a systematic approach to bosonization and vertex algebras on quantum wires of the form of star graphs. The related bosonic fields propagate freely in the bulk of the graph, but interact at its vertex. Our framework covers all possible interactions preserving unitarity. Special attention is devoted to the scale invariant interactions, which determine the critical properties of the system. Using the associated scattering matrices, we give a complete classification of the critical points on a star graph with any number of edges. Critical points where the system is not invariant under wire permutations are discovered. By means of an appropriate vertex algebra we perform the bosonization of fermions and solve the massless Thirring model. In this context we derive an explicit expression for the conductance and investigate its behavior at the critical points. A simple relation between the conductance and the Casimir energy density is pointed out.Comment: LaTex 31+1 pages, 2 figures. Section 3.6 and two references added. To appear in J. Phys. A: Mathematical and Theoretica

Vacuum Energy and Renormalization on the Edge

The vacuum dependence on boundary conditions in quantum field theories is analysed from a very general viewpoint. From this perspective the renormalization prescriptions not only imply the renormalization of the couplings of the theory in the bulk but also the appearance of a flow in the space of boundary conditions. For regular boundaries this flow has a large variety of fixed points and no cyclic orbit. The family of fixed points includes Neumann and Dirichlet boundary conditions. In one-dimensional field theories pseudoperiodic and quasiperiodic boundary conditions are also RG fixed points. Under these conditions massless bosonic free field theories are conformally invariant. Among all fixed points only Neumann boundary conditions are infrared stable fixed points. All other conformal invariant boundary conditions become unstable under some relevant perturbations. In finite volumes we analyse the dependence of the vacuum energy along the trajectories of the renormalization group flow providing an interesting framework for dark energy evolution. On the contrary, the renormalization group flow on the boundary does not affect the leading behaviour of the entanglement entropy of the vacuum in one-dimensional conformally invariant bosonic theories.Comment: 10 pages, 1 eps figur