D3-branes in NS5-brane backgrounds

We study D3-branes in an NS5-branes background defined by an arbitrary 4d harmonic function. Using a gauge-invariant formulation of Born-Infeld dynamics as well as the supersymmetry condition, we find the general solution for the $\omega$-field. We propose an interpretation in terms of the Myers effect.Comment: 7 pages, 1 figure, version published in JHE

Lax matrix solution of c=1 Conformal Field Theory

To a correlation function in a two-dimensional conformal field theory with the central charge $c=1$, we associate a matrix differential equation $\Psi' = L \Psi$, where the Lax matrix $L$ is a matrix square root of the energy-momentum tensor. Then local conformal symmetry implies that the differential equation is isomonodromic. This provides a justification for the recently observed relation between four-point conformal blocks and solutions of the Painlev\'e VI equation. This also provides a direct way to compute the three-point function of Runkel-Watts theory -- the common $c\rightarrow 1$ limit of Minimal Models and Liouville theory.Comment: 20 pages, v3: Corrected sign mistakes in eqs. (4.35), (4.37), (4.42), (4.45) and (4.52). Conclusions unchange

AdS2 D-branes in AdS3 spacetime

I review some recent progress in understanding the properties of AdS2 branes in AdS3. Different methods - classical string motion, Born-Infeld dynamics, boundary states - are evocated and compared.Comment: 7 pages, to appear in the proceedings of the workshop "Across the present energy frontier : probing the origin of mass", Corfu, Greece, Sept. 200

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

Conformal field theory on the plane

We review conformal field theory on the plane in the conformal bootstrap approach. We introduce the main ideas of the bootstrap approach to quantum field theory, and how they apply to two-dimensional theories with local conformal symmetry. We describe the mathematical structures that appear in such theories, from the Virasoro algebra and its representations, to BPZ equations and conformal blocks. Examples include Liouville theory, (generalized) minimal models, free bosonic theories, the $H_3^+$ model, and the $SU_2$ and $\widetilde{SL}_2(\mathbb{R})$ WZW models. We also discuss relations between some of these models, and limits of these models when the central charge and/or conformal dimensions tend to particular values.Comment: 142 pages. Public domain. Suggest modifications at https://github.com/ribault/CFT-Review . Version 5: More on the fusing matrix, limits of Liouville theory and minimal models, Runkel--Watts-type CFTs. Improved notations and normalization

Diagonal fields in critical loop models

In critical loop models, there exist diagonal fields with arbitrary conformal dimensions, whose $3$-point functions coincide with those of Liouville theory at $c\leq 1$. We study their $N$-point functions, which depend on the $2^{N-1}$ weights of topologically inequivalent loops on a sphere with $N$ punctures. Using a numerical conformal bootstrap approach, we find that $4$-point functions decompose into infinite but discrete linear combinations of conformal blocks. We conclude that diagonal fields belong to an extension of the $O(n)$ model.Comment: 7 page