73 research outputs found
Local logarithmic correlators as limits of Coulomb gas integrals
We will describe how logarithmic singularities arise as limits of Coulomb Gas
integrals. Our approach will combine analytic properties of the time-like
Liouville structure constants, together with the recursive formula of the
Virasoro conformal blocks. Although the Coulomb Gas formalism forces a diagonal
coupling between the chiral and antichiral sectors of the Conformal Field
Theory (CFT), we present new results for the multi-screening integrals which
are potentially interesting for applications to critical statistical systems
described by Logarithmic CFTs. In particular our findings extend and complement
previous results, derived with Coulomb Gas methods, at and .Comment: 38 pages, 12 figure
Moore-Read Fractional Quantum Hall wavefunctions and SU(2) quiver gauge theories
We identify Moore-Read wavefunctions, describing non-abelian statistics in
fractional quantum Hall systems, with the instanton partition of N=2
superconformal quiver gauge theories at suitable values of masses and
\Omega-background parameters. This is obtained by extending to rational
conformal field theories the SU(2) gauge quiver/Liouville field theory duality
recently found by Alday-Gaiotto-Tachikawa. A direct link between the Moore-Read
Hall -body wavefunctions and Z_n-equivariant Donaldson polynomials is
pointed out.Comment: 5 pages, 4 figure
Universal width distributions in non-Markovian Gaussian processes
We study the influence of boundary conditions on self-affine random functions
u(t) in the interval t/L \in [0,1], with independent Gaussian Fourier modes of
variance ~ 1/q^{\alpha}. We consider the probability distribution of the mean
square width of u(t) taken over the whole interval or in a window t/L \in [x,
x+\delta]. Its characteristic function can be expressed in terms of the
spectrum of an infinite matrix. This distribution strongly depends on the
boundary conditions of u(t) for finite \delta, but we show that it is universal
(independent of boundary conditions) in the small-window limit. We compute it
directly for all values of \alpha, using, for \alpha<3, an asymptotic expansion
formula that we derive. For \alpha > 3, the limiting width distribution is
independent of \alpha. It corresponds to an infinite matrix with a single
non-zero eigenvalue. We give the exact expression for the width distribution in
this case. Our analysis facilitates the estimation of the roughness exponent
from experimental data, in cases where the standard extrapolation method cannot
be usedComment: 15 page
AGT, N-Burge partitions and W_N minimal models
Let be a
conformal block, with consecutive channels \chi_{\i}, \i = 1, \cdots,
n, in the conformal field theory , where is a minimal model, generated by chiral fields
of spin , and labeled by two co-prime integers and
, , while is a free
boson conformal field theory. is the expectation value of vertex operators between an
initial and a final state. Each vertex operator is labelled by a charge vector
that lives in the weight lattice of the Lie algebra , spanned by
weight vectors . We restrict our attention to
conformal blocks with vertex operators whose charge vectors point along
. The charge vectors that label the initial and final states can
point in any direction.
Following the AGT correspondence, and using Nekrasov's
instanton partition functions without modification, to compute , leads to ill-defined expressions. We
show that restricting the states that flow in the channels \chi_{\i}, \i =
1, \cdots, n, to states labeled by partitions that satisfy conditions that
we call -Burge partitions, leads to well-defined expressions that we
identify with . We
check our identification by showing that a specific non-trivial conformal block
that we compute, using the -Burge conditions satisfies the expected
differential equation.Comment: 34 pages. More references, same conten
Notes on the solutions of Zamolodchikov-type recursion relations in Virasoro minimal models
We study Virasoro minimal-model 4-point conformal blocks on the sphere and
0-point conformal blocks on the torus (the Virasoro characters), as solutions
of Zamolodchikov-type recursion relations. In particular, we study the
singularities due to resonances of the dimensions of conformal fields in
minimal-model representations, that appear in the intermediate steps of solving
the recursion relations, but cancel in the final results.Comment: 26 pages, 1 figure minor modification
Critical Casimir Force between Inhomogeneous Boundaries
To study the critical Casimir force between chemically structured boundaries
immersed in a binary mixture at its demixing transition, we consider a strip of
Ising spins subject to alternating fixed spin boundary conditions. The system
exhibits a boundary induced phase transition as function of the relative amount
of up and down boundary spins. This transition is associated with a sign change
of the asymptotic force and a diverging correlation length that sets the scale
for the crossover between different universal force amplitudes. Using conformal
field theory and a mapping to Majorana fermions, we obtain the universal
scaling function of this crossover, and the force at short distances.Comment: 5 pages, 3 figure
A conformal bootstrap approach to critical percolation in two dimensions
We study four-point functions of critical percolation in two dimensions, and
more generally of the Potts model. We propose an exact ansatz for the spectrum:
an infinite, discrete and non-diagonal combination of representations of the
Virasoro algebra. Based on this ansatz, we compute four-point functions using a
numerical conformal bootstrap approach. The results agree with Monte-Carlo
computations of connectivities of random clusters.Comment: 16 pages, Python code available at
https://github.com/ribault/bootstrap-2d-Python, v2: some clarifications and
minor improvement
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