A Note on the Stringy Embeddings of Certain N = 2 Dualities

Seiberg-Witten theory can be embedded in F-theory using D3 branes probing an orientifold geometry. The non-perturbative corrections in the orientifold picture map directly to the instanton corrections in the corresponding gauge theory that convert the classical moduli space to the quantum one. In this short review we argue that the recently proposed class of conformal Gaiotto models may also be embedded in F-theory. The F-theory constructions help us not only to understand the Gaiotto dualities but also to extend to the non-conformal cases with and without cascading behaviors. For the conformal cases, the near horizon geometries in F-theory capture both the UV and IR behaviors succinctly.Comment: 6 pages, LaTeX, Based on the talk given by K. D at the Theory Canada Conference June 2012; v2: Typos corrected and references adde

An Improved Algorithm for Incremental DFS Tree in Undirected Graphs

Depth first search (DFS) tree is one of the most well-known data structures for designing efficient graph algorithms. Given an undirected graph $G=(V,E)$ with $n$ vertices and $m$ edges, the textbook algorithm takes $O(n+m)$ time to construct a DFS tree. In this paper, we study the problem of maintaining a DFS tree when the graph is undergoing incremental updates. Formally, we show: Given an arbitrary online sequence of edge or vertex insertions, there is an algorithm that reports a DFS tree in $O(n)$ worst case time per operation, and requires $O\left(\min\{m \log n, n^2\}\right)$ preprocessing time. Our result improves the previous $O(n \log^3 n)$ worst case update time algorithm by Baswana et al. and the $O(n \log n)$ time by Nakamura and Sadakane, and matches the trivial $\Omega(n)$ lower bound when it is required to explicitly output a DFS tree. Our result builds on the framework introduced in the breakthrough work by Baswana et al., together with a novel use of a tree-partition lemma by Duan and Zhan, and the celebrated fractional cascading technique by Chazelle and Guibas

Metastable Vacua in Warped Throats at Non-Isolated Singularities

We study the existence of metastable vacua in cascades based on fractional brane configurations at non-isolated singularities preserving N = 1 supersymmetry. We find that in a large class of models the extra moduli typically generated along such cascades may be stabilized by utilizing special monopole points found recently. We illustrate this in detail for cascades based on the SPP singularity. The supergravity interpretation of these constructions in terms of warped throats with supersymmetry breaking localized near their tips as well as applications to string compactification is discussed. Our constructions are designed to realize a large class of warped throats with supersymmetry breaking localized inside of a highly curved tip region.Comment: 18 pages, added clarifications, published version; 1 figur

The pp-Center Problem in Tree Networks Revisited

We present two improved algorithms for weighted discrete $p$-center problem for tree networks with $n$ vertices. One of our proposed algorithms runs in $O(n \log n + p \log^2 n \log(n/p))$ time. For all values of $p$, our algorithm thus runs as fast as or faster than the most efficient $O(n\log^2 n)$ time algorithm obtained by applying Cole's speed-up technique [cole1987] to the algorithm due to Megiddo and Tamir [megiddo1983], which has remained unchallenged for nearly 30 years. Our other algorithm, which is more practical, runs in $O(n \log n + p^2 \log^2(n/p))$ time, and when $p=O(\sqrt{n})$ it is faster than Megiddo and Tamir's $O(n \log^2n \log\log n)$ time algorithm [megiddo1983]

IIA Perspective On Cascading Gauge Theory

We study the N=1 supersymmetric cascading gauge theory found in type IIB string theory on p regular and M fractional D3-branes at the tip of the conifold, using the T-dual type IIA description. We reproduce the supersymmetric vacuum structure of this theory, and show that the IIA analog of the non-supersymmetric state found by Kachru, Pearson and Verlinde in the IIB description is metastable in string theory, but the barrier for tunneling to the supersymmetric vacuum goes to infinity in the field theory limit. We also comment on the N=2 supersymmetric gauge theory corresponding to regular and fractional D3-branes on a near-singular K3, and clarify the origin of the cascade in this theory.Comment: 43 pages, 18 figures, harvma

Cascades with Adjoint Matter: Adjoint Transitions

A large class of duality cascades based on quivers arising from non-isolated singularities enjoy adjoint transitions - a phenomenon which occurs when the gauge coupling of a node possessing adjoint matter is driven to strong coupling in a manner resulting in a reduction of rank in the non-Abelian part of the gauge group and a subsequent flow to weaker coupling. We describe adjoint transitions in a simple family of cascades based on a Z2-orbifold of the conifold using field theory. We show that they are dual to Higgsing and produce varying numbers of U(1) factors, moduli, and monopoles in a manner which we calculate. This realizes a large family of cascades which proceed through Seiberg duality and Higgsing. We briefly describe the supergravity limit of our analysis, as well as a prescription for treating more general theories. A special role is played by N=2 SQCD. Our results suggest that additional light fields are typically generated when UV completing certain constructions of spontaneous supersymmetry breaking into cascades, potentially leading to instabilities.Comment: 29 pages, a few typos fixed, improved discussion, added figure; now there is 1 figur

The energy partitioning of non-thermal particles in a plasma: or the Coulomb logarithm revisited

The charged particle stopping power in a highly ionized and weakly to moderately coupled plasma has been calculated to leading and next-to-leading order by Brown, Preston, and Singleton (BPS). After reviewing the main ideas behind this calculation, we use a Fokker-Planck equation derived by BPS to compute the electron-ion energy partitioning of a charged particle traversing a plasma. The motivation for this application is ignition for inertial confinement fusion -- more energy delivered to the ions means a better chance of ignition, and conversely. It is therefore important to calculate the fractional energy loss to electrons and ions as accurately as possible, as this could have implications for the Laser Megajoule (LMJ) facility in France and the National Ignition Facility (NIF) in the United States. The traditional method by which one calculates the electron-ion energy splitting of a charged particle traversing a plasma involves integrating the stopping power dE/dx. However, as the charged particle slows down and becomes thermalized into the background plasma, this method of calculating the electron-ion energy splitting breaks down. As a result, the method suffers a systematic error of order T/E0, where T is the plasma temperature and E0 is the initial energy of the charged particle. In the case of DT fusion, for example, this can lead to uncertainties as high as 10% or so. The formalism presented here is designed to account for the thermalization process, and in contrast, it provides results that are near-exact.Comment: 10 pages, 3 figures, invited talk at the 35th European Physical Society meeting on plasma physic

Cascading Cosmology

We develop a fully covariant, well-posed 5D effective action for the 6D cascading gravity brane-world model, and use this to study cosmological solutions. We obtain this effective action through the 6D decoupling limit, in which an additional scalar degree mode, \pi, called the brane-bending mode, determines the bulk-brane gravitational interaction. The 5D action obtained this way inherits from the sixth dimension an extra \pi self-interaction kinetic term. We compute appropriate boundary terms, to supplement the 5D action, and hence derive fully covariant junction conditions and the 5D Einstein field equations. Using these, we derive the cosmological evolution induced on a 3-brane moving in a static bulk. We study the strong- and weak-coupling regimes analytically in this static ansatz, and perform a complete numerical analysis of our solution. Although the cascading model can generate an accelerating solution in which the \pi field comes to dominate at late times, the presence of a critical singularity prevents the \pi field from dominating entirely. Our results open up the interesting possibility that a more general treatment of degravitation in a time-dependent bulk, or taking into account finite brane-thickness effects, may lead to an accelerating universe without a cosmological constant.Comment: [v2] 27 pages, 2 figures, corrected typos, expanded discussion of late-time cosmological behavio