Large-small dualities between periodic collapsing/expanding branes and brane funnels

We consider space and time dependent fuzzy spheres $S^{2p}$ arising in $D1-D(2p+1)$ intersections in IIB string theory and collapsing D(2p)-branes in IIA string theory. In the case of $S^2$, where the periodic space and time-dependent solutions can be described by Jacobi elliptic functions, there is a duality of the form $r$ to ${1 \over r}$ which relates the space and time dependent solutions. This duality is related to complex multiplication properties of the Jacobi elliptic functions. For $S^4$ funnels, the description of the periodic space and time dependent solutions involves the Jacobi Inversion problem on a hyper-elliptic Riemann surface of genus 3. Special symmetries of the Riemann surface allow the reduction of the problem to one involving a product of genus one surfaces. The symmetries also allow a generalisation of the $r$ to ${1 \over r}$ duality. Some of these considerations extend to the case of the fuzzy $S^6$.Comment: Latex, 50 pages, 2 figures ; v2 : a systematic typographical error corrected + minor change

Analysis of fuzzy clustering and a generic fuzzy rule-based image segmentation technique

Many fuzzy clustering based techniques when applied to image segmentation do not incorporate spatial relationships of the pixels, while fuzzy rule-based image segmentation techniques are generally application dependent. Also for most of these techniques, the structure of the membership functions is predefined and parameters have to either automatically or manually derived. This paper addresses some of these issues by introducing a new generic fuzzy rule based image segmentation (GFRIS) technique, which is both application independent and can incorporate the spatial relationships of the pixels as well. A qualitative comparison is presented between the segmentation results obtained using this method and the popular fuzzy c-means (FCM) and possibilistic c-means (PCM) algorithms using an empirical discrepancy method. The results demonstrate this approach exhibits significant improvements over these popular fuzzy clustering algorithms for a wide range of differing image types

Non-Abelian BIonic Brane Intersections

We study "fuzzy funnel" solutions to the non-Abelian equations of motion of the D-string. Our funnel describes n^6/360 coincident D-strings ending on n^3/6 D7-branes, in terms of a fuzzy six-sphere which expands along the string. We also provide a dual description of this configuration in terms of the world volume theory of the D7-branes. Our work makes use of an interesting non-linear higher dimensional generalization of the instanton equations.Comment: 17 pages uses harvmac; v2: small typos corrected, refs adde

String Webs from Field Theory

The spectrum of stable electrically and magnetically charged supersymmetric particles can change discontinuously as one changes the vacuum on the Coulomb branch of gauge theories with extended supersymmetry in four dimensions. We show that this decay process can be understood and is well described by semiclassical field configurations purely in terms of the low energy effective action on the Coulomb branch even when it occurs at strong coupling. The resulting picture of the stable supersymmetric spectrum is a generalization of the ``string web'' picture of these states found in string constructions for certain theories.Comment: 53 pages, 6 figures; more references adde

Gravitational non-commutativity and G\"odel-like spacetimes

We derive general conditions under which geodesics of stationary spacetimes resemble trajectories of charged particles in an electromagnetic field. For large curvatures (analogous to strong magnetic fields), the quantum mechanicical states of these particles are confined to gravitational analogs of {\it lowest Landau levels}. Furthermore, there is an effective non-commutativity between their spatial coordinates. We point out that the Som-Raychaudhuri and G\"odel spacetime and its generalisations are precisely of the above type and compute the effective non-commutativities that they induce. We show that the non-commutativity for G\"odel spacetime is identical to that on the fuzzy sphere. Finally, we show how the star product naturally emerges in Som-Raychaudhuri spacetimes.Comment: Two sections added (Relation to the fuzzy sphere, Emergence of the star product). 10 pages, Revtex. To appear in General Relativity and Gravitatio

Giant Gravitons in type IIA PP-wave Background

We examine giant gravitons with a worldvolume magnetic flux $q$ in type IIA pp-wave background and find that they can move away from the origin along $x^4$ direction in target space satisfying $Rx^4=-q$. This nontrivial relation can be regarded as a complementary relation of the giant graviton on IIA pp-wave and is shown to be connected to the spacetime uncertainty principle. The giant graviton is also investigated in a system of N D0-branes as a fuzzy sphere solution. It is observed that $x^4$ enters into the fuzzy algebra as a deformation parameter. Such a background dependent Myers effect guarantees that we again get the crucial relation of our giant graviton. In the paper, we also find a BIon configuration on the giant graviton in this background.Comment: 10 pages, no figure, content added, typo corrected, reference adde

Noncommutative geometry, Quantum effects and DBI-scaling in the collapse of D0-D2 bound states

We study fluctuations of time-dependent fuzzy two-sphere solutions of the non-abelian DBI action of D0-branes, describing a bound state of a spherical D2-brane with N D0-branes. The quadratic action for small fluctuations is shown to be identical to that obtained from the dual abelian D2-brane DBI action, using the non-commutative geometry of the fuzzy two-sphere. For some of the fields, the linearized equations take the form of solvable Lam\'e equations. We define a large-N DBI-scaling limit, with vanishing string coupling and string length, and where the gauge theory coupling remains finite. In this limit, the non-linearities of the DBI action survive in both the classical and the quantum context, while massive open string modes and closed strings decouple. We describe a critical radius where strong gauge coupling effects become important. The size of the bound quantum ground state of multiple D0-branes makes an intriguing appearance as the radius of the fuzzy sphere, where the maximal angular momentum quanta become strongly coupled.Comment: 34 pages, Latex; v2: Minor correction in conformal transformation of couplings, references adde

Electrified Fuzzy Spheres and Funnels in Curved Backgrounds

We use the non-Abelian DBI action to study the dynamics of $N$ coincident $Dp$-branes in an arbitrary curved background, with the presence of a homogenous world-volume electric field. The solutions are natural extensions of those without electric fields, and imply that the spheres will collapse toward zero size. We then go on to consider the $D1-D3$ intersection in a curved background and find various dualities and automorphisms of the general equations of motion. It is possible to map the dynamical equation of motion to the static one via Wick rotation, however the additional spatial dependence of the metric prevents this mapping from being invertible. Instead we find that a double Wick rotation leaves the static equation invariant. This is very different from the behaviour in Minkowski space. We go on to construct the most general static fuzzy funnel solutions for an arbitrary metric either by solving the static equations of motion, or by finding configurations which minimise the energy. As a consistency check we construct the Abelian $D3$-brane world-volume theory in the same generic background and find solutions consistent with energy minimisation. In the $NS$5-brane background we find time dependent solutions to the equations of motion, representing a time dependent fuzzy funnel. These solutions match those obtained from the $D$-string picture to leading order suggesting that the action in the large $N$ limit does not need corrections. We conclude by generalising our solutions to higher dimensional fuzzy funnels.Comment: 38 pages, Latex; references adde