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 $c=0$ and $c=-2$.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 $n$-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 ${\mathcal B}^{\, p, \, p^{\prime}, \, {\mathcal H}}_{N, n}$ be a conformal block, with $n$ consecutive channels \chi_{\i}, \i = 1, \cdots, n, in the conformal field theory $\mathcal{M}^{\, p, \, p^{\prime}}_N \! \times \! \mathcal{M}^{\mathcal{H}}$, where $\mathcal{M}^{\, p, \, p^{\prime}}_N$ is a $\mathcal{W}_N$ minimal model, generated by chiral fields of spin $1, \cdots, N$, and labeled by two co-prime integers $p$ and $p^{\prime}$, $1 < p < p^{\prime}$, while $\mathcal{M}^{\mathcal{H}}$ is a free boson conformal field theory. $\mathcal{B}^{\, p, \, p^{\prime}, \mathcal{H}}_{N, n}$ 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 $A_{N-1}$, spanned by weight vectors $\omega_1, \cdots, \omega_{N-1}$. We restrict our attention to conformal blocks with vertex operators whose charge vectors point along $\omega_1$. The charge vectors that label the initial and final states can point in any direction. Following the $\mathcal{W}_N$ AGT correspondence, and using Nekrasov's instanton partition functions without modification, to compute $\mathcal{B}^{\, p, \, p^{\prime}, \mathcal{H}}_{N, n}$, 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 $N$ partitions that satisfy conditions that we call $N$-Burge partitions, leads to well-defined expressions that we identify with $\mathcal{B}^{\, p, \, p^{\prime}, \, \mathcal{H}}_{N, n}$. We check our identification by showing that a specific non-trivial conformal block that we compute, using the $N$-Burge conditions satisfies the expected differential equation.Comment: 34 pages. More references, same conten

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

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