1,350 research outputs found
Structure Constants and Conformal Bootstrap in Liouville Field Theory
An analytic expression is proposed for the three-point function of the
exponential fields in the Liouville field theory on a sphere. In the classical
limit it coincides with what the classical Liouville theory predicts. Using
this function as the structure constant of the operator algebra we construct
the four-point function of the exponential fields and verify numerically that
it satisfies the conformal bootstrap equations, i.e., that the operator algebra
thus defined is associative. We consider also the Liouville reflection
amplitude which follows explicitly from the structure constants.Comment: 31 pages, 2 Postscript figures. Important note about existing (but
unfortunately previously unknown to us) paper which has significant overlap
with this work is adde
N=1 SUSY Conformal Block Recursive Relations
We present explicit recursive relations for the four-point superconformal
block functions that are essentially particular contributions of the given
conformal class to the four-point correlation function. The approach is based
on the analytic properties of the superconformal blocks as functions of the
conformal dimensions and the central charge of the superconformal algebra. The
results are compared with the explicit analytic expressions obtained for
special parameter values corresponding to the truncated operator product
expansion. These recursive relations are an efficient tool for numerically
studying the four-point correlation function in Super Conformal Field Theory in
the framework of the bootstrap approach, similar to that in the case of the
purely conformal symmetry.Comment: 12 pages, typos corrected, reference adde
On sigma model RG flow, "central charge" action and Perelman's entropy
Zamolodchikov's c-theorem type argument (and also string theory effective
action constructions) imply that the RG flow in 2d sigma model should be
gradient one to all loop orders. However, the monotonicity of the flow of the
target-space metric is not obvious since the metric on the space of
metric-dilaton couplings is indefinite. To leading (one-loop) order when the RG
flow is simply the Ricci flow the monotonicity was proved by Perelman
(math.dg/0211159) by constructing an ``entropy'' functional which is
essentially the metric-dilaton action extremised with respect to the dilaton
with a condition that the target-space volume is fixed. We discuss how to
generalize the Perelman's construction to all loop orders (i.e. all orders in
\alpha'). The resulting ``entropy'' is equal to minus the central charge at the
fixed points, in agreement with the general claim of the c-theorem.Comment: 14 pages. v2: minor comments added, misprints correcte
Conserved charges in the chiral 3-state Potts model
We consider the perturbations of the 3-state Potts conformal field theory
introduced by Cardy as a description of the chiral 3-state Potts model. By
generalising Zamolodchikov's counting argument and by explicit calculation we
find new inhomogeneous conserved currents for this theory. We conjecture the
existence of an infinite set of conserved currents of this form and discuss
their relevance to the description of the chiral Potts models
Superconducting Proximity Effect on the Edge of Fractional Topological Insulators
We study the superconducting proximity effect on the helical edge states of
time-reversal-symmetric fractional topological insulators(FTI). The Cooper
pairing of electrons results in many-particle condensation of the
fractionalized excitations on the edge. We find in the strong-coupling phase,
localized zero-energy modes emerge on interfaces between superconducting
regions and magnetically insulating regions, which are responsible for
topological degeneracy of the ground states. By mapping the low-energy
effective Hamiltonian to quantum Potts model, we determine the operator algebra
of the zero modes and show that they exhibit nontrivial braiding properties. We
then demonstrate that Josephson current in the junction between superconductors
mediated by the edge states of the FTI exhibit fractional Josephson effect with
period that is multiples of
Critical interfaces and duality in the Ashkin Teller model
We report on the numerical measures on different spin interfaces and FK
cluster boundaries in the Askhin-Teller (AT) model. For a general point on the
AT critical line, we find that the fractal dimension of a generic spin cluster
interface can take one of four different possible values. In particular we
found spin interfaces whose fractal dimension is d_f=3/2 all along the critical
line. Further, the fractal dimension of the boundaries of FK clusters were
found to satisfy all along the AT critical line a duality relation with the
fractal dimension of their outer boundaries. This result provides a clear
numerical evidence that such duality, which is well known in the case of the
O(n) model, exists in a extended CFT.Comment: 5 pages, 4 figure
Scaling Solutions in Continuous Dimension
We study scaling solutions of the RG flow equation for the Z_2-effective
potential in continuous dimension. As the dimension is lowered from d=4 we
first observe the appearance of the Ising scaling solution and successively the
apparence of multi-critical scaling solutions of arbitrary order. Approaching
d=2 these multi-critical scaling solutions converge to the unitary minimal
models found in CFT.Comment: 5 pages, 5 figures, published versio
The Renormalization Group Limit Cycle for the 1/r^2 Potential
Previous work has shown that if an attractive 1/r^2 potential is regularized
at short distances by a spherical square-well potential, renormalization allows
multiple solutions for the depth of the square well. The depth can be chosen to
be a continuous function of the short-distance cutoff R, but it can also be a
log-periodic function of R with finite discontinuities, corresponding to a
renormalization group (RG) limit cycle. We consider the regularization with a
delta-shell potential. In this case, the coupling constant is uniquely
determined to be a log-periodic function of R with infinite discontinuities,
and an RG limit cycle is unavoidable. In general, a regularization with an RG
limit cycle is selected as the correct renormalization of the 1/r^2 potential
by the conditions that the cutoff radius R can be made arbitrarily small and
that physical observables are reproduced accurately at all energies much less
than hbar^2/mR^2.Comment: 11 pages, 4 figure
Differential equation for four-point correlation function in Liouville field theory and elliptic four-point conformal blocks
Liouville field theory on a sphere is considered. We explicitly derive a
differential equation for four-point correlation functions with one degenerate
field . We introduce and study also a class of four-point
conformal blocks which can be calculated exactly and represented by finite
dimensional integrals of elliptic theta-functions for arbitrary intermediate
dimension. We study also the bootstrap equations for these conformal blocks and
derive integral representations for corresponding four-point correlation
functions. A relation between the one-point correlation function of a primary
field on a torus and a special four-point correlation function on a sphere is
proposed
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