The self-dual string soliton in AdS<Subscript>4</Subscript>×S<Superscript>7</Superscript> spacetime

We construct self-dual string soliton solutions in AdS₄×S⁷ spacetime, starting from the covariant equations of motion of the M5-brane. We study the properties of the solutions and find that their actions are linearized, indicating the BPS nature of the solutions, and we find that they have the same electric and magnetic charges. The straight string soliton solution represents the configuration of the membranes ending on a M5-brane with a straight string intersection, and it behaves like the spiky solution in flat spacetime. The spherical string soliton solution, which could be related to the straight one by a conformal transformation, represents the membranes ending on a M5-brane with a spherical intersection.     

A note on supergravity solutions for partially localized intersecting branes

Using the method developed by Cherkis and Hashimoto we construct partially localized D3⊥D5(2), D4⊥D4(2) and M5⊥M5(3) supergravity solutions where one of the harmonic functions is given in an integral form. This is a generalization of the already known near-horizon solutions. The method fails for certain intersections such as D1⊥D5(1) which is consistent with the previous no-go theorems. We point out some possible ways of bypassing these results.     

M‐theory solutions invariant under D(2,1; γ) ⊕ D(2,1;γ)

We simplify and extend the construction of half‐BPS solutions to 11‐dimensional supergravity, with isometry superalgebra D(2,1;γ) ⊕ D(2,1;γ). Their space‐time has the form AdS₃× S³× S³ warped over a Riemann surface Σ. It describes near‐horizon geometries of M2 branes ending on, or intersecting with, M5 branes along a common string. The general solution to the BPS equations is specified by a reduced set of data (γ, h, G), where γ is the real parameter of the isometry superalgebra, and h and G are functions on Σ whose differential equations and regularity conditions depend only on the sign of γ. The magnitude of γ enters only through the map of h,G onto the supergravity fields, thereby promoting all solutions into families parametrized by |γ|. By analyzing the regularity conditions for the supergravity fields, we prove two general theorems: (i) that the only solution with a 2‐dimensional CFT dual is AdS₃× S³× S³× ℝ², modulo discrete identifications of the flat ℝ², and (ii) that solutions with γ < 0 cannot have more than one asymptotic higher‐dimensional AdS region. We classify the allowed singularities of h and G near the boundary of Σ, and identify four local solutions: asymptotic AdS₄/Z₂ or AdS₇^′ regions; highly‐curved M5‐branes; and a coordinate singularity called the “cap”. By putting these “Lego” pieces together we recover all known global regular solutions with the above symmetry, including the self‐dual strings on M5 for γ <0, and the Janus solution for γ > 0, but now promoted to families parametrized by |γ|. We also construct exactly new regular solutions which are asymptotic to AdS₄/Z₂ for γ < 0, and conjecture that they are a different superconformal limit of the self‐dual string. Finally, we construct exactly γ > 0 solutions with highly curved M5‐brane regions, which are the formal continuation of the self‐dual string solutions across the decompactification point at γ = 0.     

D-brane solutions in a light-like linear dilaton background

The light-like linear dilaton background presents a simple time dependent solution of type II supergravity equations of motion that preserves 1/2 supersymmetry in ten dimensions. We construct supergravity D-brane solutions in a linear dilaton background starting from the known intersecting brane solutions in string theory. By applying a Penrose limit on the intersecting (NS1–NS5–NS5′)-brane solution, we find out a D5-brane in a linear dilaton background. We solve the Killing spinor equations for the brane solutions explicitly, and show that they preserve 1/4 supersymmetry. We also find a M5-brane solution in eleven-dimensional supergravity.     

Fluxes,bundle gerbes and 2-Hilbert spaces

We elaborate on the construction of a prequantum 2-Hilbert space from a bundle gerbe over a 2-plectic manifold, providing the first steps in a programme of higher geometric quantisation of closed strings in flux compactifications and of M5-branes in C-fields. We review in detail the construction of the 2-category of bundle gerbes and introduce the higher geometrical structures necessary to turn their categories of sections into 2-Hilbert spaces. We work out several explicit examples of 2-Hilbert spaces in the context of closed strings and M5-branes on flat space. We also work out the prequantum 2-Hilbert space associated with an M-theory lift of closed strings described by an asymmetric cyclic orbifold of the \(\mathsf {S}\mathsf {U}(2)\) WZW model, providing an example of sections of a torsion gerbe on a curved background. We describe the dimensional reduction of M-theory to string theory in these settings as a map from 2-isomorphism classes of sections of bundle gerbes to sections of corresponding line bundles, which is compatible with the respective monoidal structures and module actions.     

{\mathcal{N} = 2} Supersymmetric AdS4 Solutions of M-theory

We analyse the most general ${\mathcal{N} = 2}$ supersymmetric solutions of D = 11 supergravity consisting of a warped product of four-dimensional anti-de-Sitter space with a seven-dimensional Riemannian manifold Y ₇. We show that the necessary and sufficient conditions for supersymmetry can be phrased in terms of a local SU(2)-structure on Y ₇. Solutions with non-zero M2-brane charge also admit a canonical contact structure, in terms of which many physical quantities can be expressed, including the free energy and the scaling dimensions of operators dual to supersymmetric wrapped M5-branes. We show that a special class of solutions is singled out by imposing an additional symmetry, for which the problem reduces to solving a second order non-linear ODE. As well as recovering a known class of solutions, that includes the IR fixed point of a mass deformation of the ABJM theory, we also find new solutions which are dual to cubic deformations. In particular, we find a new supersymmetric warped AdS₄ × S ⁷ solution with non-trivial four-form flux.     

Homogeneous Statistical Solutions and Local Energy Inequality for 3D Navier-Stokes Equations

We are interested in space-time spatially homogeneous statistical solutions of Navier-Stokes equations in space dimension three. We first review the construction of such solutions, and introduce convenient tools to study the pressure gradient. Then we show that given a spatially homogeneous initial measure with finite energy density, one can construct a homogeneous statistical solution concentrated on weak solutions which satisfy the local energy inequality.     

Stilbazolium merocyanine dye determination in different solutions,concentrations and colloids using SERS

Surface‐enhanced Raman scattering (SERS) measurements were carried out on stilbazolium merocyanine dye in methanol and pyridine solvents. Both solutions were measured in a series of concentrations covering a range of 5 × 10⁻⁵ M to 5 × 10⁻⁸ M . In these measurements, Ag and Au colloids were used, and the results have shown that Ag colloids yield better enhancement in the Raman spectra of this dye. Moreover, the effect of adding NaCl solution to the SERS samples was also studied. All measurements were carried out using the state‐of‐the‐art ChiralRaman instrument, which utilizes a 532 nm laser source. We report here on the success of using SERS to obtain Raman spectra of merocyanine dye at very low concentrations in an attempt to find a new approach that can be used for further investigations of the dye. The SERS spectra are reported here, and the results from different solutions, colloids, concentrations and pH values are compared. Copyright © 2006 John Wiley & Sons, Ltd.     

Anomaly free effective action for the elementary M5-brane

We construct an effective action describing an elementary M5-brane interacting with dynamical eleven-dimensional supergravity, which is free from gravitational anomalies. The current associated to the elementary brane is taken as a distribution valued δ-function on the support of the 5-brane itself. Crucial ingredients of the construction are the consistent inclusion of the dynamics of the chiral two-form on the 5-brane, and the use of an invariant Chern-kernel allowing to introduce a D=11 three-form potential which is well-defined on the worldvolume of the 5-brane.     

Hierarchical reconstruction for discontinuous Galerkin methods on unstructured grids with a WENO-type linear reconstruction and partial neighboring cells

The hierarchical reconstruction (HR) [Y.-J. Liu, C.-W. Shu, E. Tadmor, M.-P. Zhang, Central discontinuous Galerkin methods on overlapping cells with a non-oscillatory hierarchical reconstruction, SIAM J. Numer. Anal. 45 (2007) 2442-2467] is applied to the piecewise quadratic discontinuous Galerkin method on two-dimensional unstructured triangular grids. A variety of limiter functions have been explored in the construction of piecewise linear polynomials in every hierarchical reconstruction stage. We show that on triangular grids, the use of center biased limiter functions is essential in order to recover the desired order of accuracy. Several new techniques have been developed in the paper: (a) we develop a WENO-type linear reconstruction in each hierarchical level, which solves the accuracy degeneracy problem of previous limiter functions and is essentially independent of the local mesh structure; (b) we find that HR using partial neighboring cells significantly reduces over/under-shoots, and further improves the resolution of the numerical solutions. The method is compact and therefore easy to implement. Numerical computations for scalar and systems of nonlinear hyperbolic equations are performed. We demonstrate that the procedure can generate essentially non-oscillatory solutions while keeping the resolution and desired order of accuracy for smooth solutions.     

New compactifications of Chiral N = 2, d = 10 supergravity

We present a number of new compactifying solutions of chiral N = 2 ten-dimensional supergravity to five dimensions. Several are of the standard Freund-Rubin type; we give a complete classification of such compactifications for which the internal space M⁵ is a coset manifold. In another type of solution M⁵ is a non-Einstein U(1) bundle over a four-dimensional Kähler space, and the complex three-index field strength is nonvanishing in the internal directions. The latter construction gives a solution with SU(3) symmetry when M⁵ is taken to be a stretched five-sphere.     

Holographic Uniformization

We derive and study supergravity BPS flow equations for M5 or D3 branes wrapping a Riemann surface. They take the form of novel geometric flows intrinsically defined on the surface. Their dual field-theoretic interpretation suggests the existence of solutions interpolating between an arbitrary metric in the ultraviolet and the constant-curvature metric in the infrared. We confirm this conjecture with a rigorous global existence proof.     

On compatibility of tensor products on solutions to commutativity and WDVV equations

We show that topological conformal theory contains solutions for WDVV equations for genus zero amplitudes and solutions to Commutativity equations from its quantum mechanics on the annulus. We explain behavior of these solutions under the tensor product of the theories. We make a conjecture that it is compatible with the construction of solutions to WDVV equations from a solution to the Commutativity equation and explicitly check it in the first nontrivial case.     

Integrability analysis of a conformal equation in relativity

In 1987 C. C. Dyer, G. C. McVittie, and L. M. Oattes derived the (two) field equations for shear-free, spherically symmetric perfect fluid spacetimes which admit a conformai symmetry. We use the techniques of the Lie and Painlevé analyses of differential equations to find solutions of these equations. The concept of a pseudo-partial Painlevé property is introduced for the first time which could assist in finding solutions to equations that do not possess the Painlevé property. The pseudo-partial Painlevé property throws light on the distinction between the classes of solutions found independently by P. Havas and M. Wyman. We find a solution for all values of a particular parameter for the first field equation and link it to the solution of the second equation. We indicate why we believe that the first field equation cannot be solved in general. Both techniques produce similar results and demonstrate the close relationship between the Lie and Painlevé analyses. We also show that both of the field equations of Dyeret al. may be reduced to the same Emden-Fowler equation of index two.     

Sequential Bethe vectors and the quantum Ernst system

We give a brief review on the use of Bethe ansatz techniques to construct solutions of recursive functional equations which emerged in a bootstrap approach to the quantum Ernst system. The construction involves two particular limits of a rational Bethe ansatz system with complex inhomogeneities. First, we pinch two insertions to the critical value. This links Bethe systems with different number of insertions and leads to the concept of sequential Bethe vectors. Second, we study the semiclassical limit of the system in which the scale parameter of the insertions tends to infinity. Presented at the 9th Colloquium “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000 UMR 8548: Unité Mixte de Recherche du CNRS et de l’ENS. The work of M. N. was supported by NSF grant 97-22097. The work of H.S. was supported by EU contract ERBFMRX-CT96-0012.     

DNA methylation analysis: speedup of bisulfite-mediated deamination of cytosine in the genomic sequencing procedure

Understanding the biological consequences of DNA methylation is a current focus of intensive studies. A standard method for analyzing the methylation at position 5 of cytosines in genomic DNA involves chemical modification of the DNA with bisulfite, followed by PCR amplification and sequencing. Bisulfite deaminates cytosine, but it deaminates 5-methylcytosine only very slowly, thereby allowing determination of the methylated sites. The deamination is usually performed using sodium bisulfite solutions of 3–5 M concentration with an incubation period of 12–16 hr at 50 °C. We demonstrate here that this deamination can be speeded up significantly. We prepared a solution of 10 M bisulfite concentration of pH 5.4 and used it to treat DNA at temperatures up to 90 °C. In an experiment, in which denatured DNA was treated with 9 M bisulfite for 10 min at 90 °C, deamination of cytosines occurred to an extent of 99.6%, while 5-methylcytosine residues in the DNA were deaminated at less than 10%. Using a plasmid DNA fragment, we observed that the DNA can serve as a template for PCR amplification after the bisulfite treatment. This new procedure is expected to offer a significantly improved genomic sequencing method, leading to the promotion of research on understanding the biological and medical significance of DNA methylation.     

Singularity formation for time-like extremal hypersurfaces

We derive self-similar string solutions in a graph representation, near the point of singularity formation, which can be shown to extend to point-like singularities on M(em)branes, as well as to the radially symmetric case.     

Localized solutions of (N+1)-dimensional evolution equations

We give a method of constructing nonlinear Schrödinger equations which are similar to the Davey-Stewartson equation, but in N+1 dimensions, and posses, by construction, solutions describing interacting soliton-like objects.     

Metastable kinks in the orbifold

We consider static configurations of bulk scalar fields in extra-dimensional models in which the fifth dimension is an S1/Z2 orbifold. There may exist a finite number of such configurations, with the total number depending on the size of the orbifold interval. We perform a detailed Sturm-Liouville stability analysis that demonstrates that all but the lowest-lying configurations--those with no nodes in the interval--are unstable. We also present a powerful general criterion with which to determine which of these nodeless solutions are stable. The detailed analysis underlying the results presented in this Letter, and applications to specific models, are presented in a comprehensive companion paper [M. Toharia and M. Trodden, arXiv:hep-ph/0708.4008].