On the smoothness of multi-M2 brane horizons

We calculate the degree of horizon smoothness of multi- $M2$-brane solution with branes along a common axis. We find that the metric is generically only thrice continuously differentiable at any of the horizons. The four-form field strength is found to be only twice continuously differentiable. We work with Gaussian null-like co-ordinates which are obtained by solving geodesic equations for multi-$M2$ brane geometry. We also find different, exact co-ordinate transformations which take the metric from isotropic co-ordinates to co-ordinates in which metric is thrice differentiable at the horizon. Both methods give the same result that the multi-$M2$ brane metric is only thrice continuously differentiable at the horizon.Comment: 24 pages, reference added, modified equation for non-singularity of metri

Holographic Coulomb Branch Flows with N=1 Supersymmetry

We obtain a large, new class of N=1 supersymmetric holographic flow backgrounds with U(1)^3 symmetry. These solutions correspond to flows toward the Coulomb branch of the non-trivial N=1 supersymmetric fixed point. The massless (complex) chiral fields are allowed to develop vevs that are independent of their two phase angles, and this corresponds to allowing the brane to spread with arbitrary, U(1)^2 invariant, radial distributions in each of these directions. Our solutions are "almost Calabi-Yau:" The metric is hermitian with respect to an integrable complex structure, but is not Kahler. The "modulus squared" of the holomorphic (3,0)-form is the volume form, and the complete solution is characterized by a function that must satisfy a single partial differential equation that is closely related to the Calabi-Yau condition. The deformation from a standard Calabi-Yau background is driven by a non-trivial, non-normalizable 3-form flux dual to a fermion mass that reduces the supersymmetry to N=1. This flux also induces dielectric polarization of the D3-branes into D5-branes.Comment: 22 pages; harvmac. Typos corrected;small improvements in presentatio

Modular differential equations with movable poles and admissible RCFT characters

Studies of modular linear differential equations (MLDE) for the classification of rational CFT characters have been limited to the case where the coefficient functions (in monic form) have no poles, or poles at special points of moduli space. Here we initiate an exploration of the vast territory of MLDEs with two characters and any number of poles at arbitrary points of moduli space. We show how to parametrise the most general equation precisely and count its parameters. Eliminating logarithmic singularities at all the poles provides constraint equations for the accessory parameters. By taking suitable limits, we find recursion relations between solutions for different numbers of poles. The cases of one and two movable poles are examined in detail and compared with predictions based on quasi-characters to find complete agreement. We also comment on the limit of coincident poles. Finally we show that there exist genuine CFT corresponding to many of the newly-studied cases. We emphasise that the modular data is an output, rather than an input, of our approach

Classifying three-character RCFTs with Wronskian index equalling 3 or 4

In the Mathur-Mukhi-Sen (MMS) classification scheme for rational conformal field theories (RCFTs), a RCFT is identified by a pair of non-negative integers $\mathbf{[n, \ell]}$, with $\mathbf{n}$ being the number of characters and $\mathbf{\ell}$ the Wronskian index. The modular linear differential equation (MLDE) that the characters of a RCFT solve are labelled similarly. All RCFTs with a given $\mathbf{[n, \ell]}$ solve the modular linear differential equation (MLDE) labelled by the same $\mathbf{[n, \ell]}$. With the goal of classifying $\mathbf{[3,3]}$ and $\mathbf{[3,4]}$ CFTs, we set-up and solve those MLDEs, each of which is a three-parameter non-rigid MLDE, for character-like solutions. In the former case, we obtain four infinite families and a discrete set of $15$ solutions, all in the range $0 < c \leq 32$. Amongst these $\mathbf{[3,3]}$ character-like solutions, we find pairs of them that form coset-bilinear relations with meromorphic CFTs/characters of central charges $16, 24, 32, 40, 48, 56, 64$. There are six families of coset-bilinear relations where both the RCFTs of the pair are drawn from the infinite families of solutions. There are an additional $23$ coset-bilinear relations between character-like solutions of the discrete set. The coset-bilinear relations should help in identifying the $\mathbf{[3,3]}$ CFTs. In the $\mathbf{[3,4]}$ case, we obtain nine character-like solutions each of which is a $\mathbf{[2,2]}$ character-like solution adjoined with a constant character.Comment: 64 pages, 20 table