25 research outputs found
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
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 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