BPS D-branes from an Unstable D-brane in a Curved Background

We find exact tachyon kink solutions of DBI type effective action describing an unstable D5-brane with worldvolume gauge field turned on in a curved background. The background of interest is the ten-dimensional lift of the Salam-Sezgin vacuum and, in the asymptotic limit, it approaches ${\rm R}^{1,4}\times {\rm T}^2\times {\rm S}^3$. The solutions are identified as composites of lower-dimensional D-branes and fundamental strings, and, in the BPS limit, they become a D4D2F1 composite wrapped on ${\rm R}^{1,2}\times {\rm T}^2$ where ${\rm T}^2$ is inside ${\rm S}^3$. In one class of solutions we find an infinite degeneracy with respect to a constant magnetic field along the direction of NS-NS field on ${\rm S}^3$.Comment: 16 pages, 2 figures, a footnote added, typos corrected and a reference adde

Generalized Lee-Wick Formulation from Higher Derivative Field Theories

We study a higher derivative (HD) field theory with an arbitrary order of derivative for a real scalar field. The degree of freedom for the HD field can be converted to multiple fields with canonical kinetic terms up to the overall sign. The Lagrangian describing the dynamics of the multiple fields is known as the Lee-Wick (LW) form. The first step to obtain the LW form for a given HD Lagrangian is to find an auxiliary field (AF) Lagrangian which is equivalent to the original HD Lagrangian up to the quantum level. Till now, the AF Lagrangian has been studied only for N=2 and 3 cases, where $N$ is the number of poles of the two-point function of the HD scalar field. We construct the AF Lagrangian for arbitrary $N$. By the linear combinations of AF fields, we also obtain the corresponding LW form. We find the explicit mapping matrices among the HD fields, the AF fields, and the LW fields. As an exercise of our construction, we calculate the relations among parameters and mapping matrices for $N=2,3$, and 4 cases.Comment: 23 pages, version to appear in PRD, we improved the transformation from HD to LW in Subsection 3.1, added comments on gauge field related with AF Lagrangians in Conclusion, and added reference

Linear rank-width of distance-hereditary graphs II. Vertex-minor obstructions

In the companion paper [Linear rank-width of distance-hereditary graphs I. A polynomial-time algorithm, Algorithmica 78(1):342--377, 2017], we presented a characterization of the linear rank-width of distance-hereditary graphs, from which we derived an algorithm to compute it in polynomial time. In this paper, we investigate structural properties of distance-hereditary graphs based on this characterization. First, we prove that for a fixed tree $T$, every distance-hereditary graph of sufficiently large linear rank-width contains a vertex-minor isomorphic to $T$. We extend this property to bigger graph classes, namely, classes of graphs whose prime induced subgraphs have bounded linear rank-width. Here, prime graphs are graphs containing no splits. We conjecture that for every tree $T$, every graph of sufficiently large linear rank-width contains a vertex-minor isomorphic to $T$. Our result implies that it is sufficient to prove this conjecture for prime graphs. For a class $\Phi$ of graphs closed under taking vertex-minors, a graph $G$ is called a vertex-minor obstruction for $\Phi$ if $G\notin \Phi$ but all of its proper vertex-minors are contained in $\Phi$. Secondly, we provide, for each $k\ge 2$, a set of distance-hereditary graphs that contains all distance-hereditary vertex-minor obstructions for graphs of linear rank-width at most $k$. Also, we give a simpler way to obtain the known vertex-minor obstructions for graphs of linear rank-width at most $1$.Comment: 38 pages, 13 figures, 1 table, revised journal version. A preliminary version of Section 5 appeared in the proceedings of WG1

Late Time Behaviors of an Inhomogeneous Rolling Tachyon

We study an inhomogeneous decay of an unstable D-brane in the context of Dirac-Born-Infeld~(DBI)-type effective action. We consider tachyon and electromagnetic fields with dependence of time and one spatial coordinate, and an exact solution is found under an exponentially decreasing tachyon potential, $e^{-|T|/\sqrt{2}}$, which is valid for the description of the late time behavior of an unstable D-brane. Though the obtained solution contains both time and spatial dependence, the corresponding momentum density vanishes over the entire spacetime region. The solution is governed by two parameters. One adjusts the distribution of energy density in the inhomogeneous direction, and the other interpolates between the homogeneous rolling tachyon and static configuration. As time evolves, the energy of the unstable D-brane is converted into the electric flux and tachyon matter.Comment: 17 pages, 1 figure, version to appear in PR

An Upper Bound on the Size of Obstructions for Bounded Linear Rank-Width

We provide a doubly exponential upper bound in $p$ on the size of forbidden pivot-minors for symmetric or skew-symmetric matrices over a fixed finite field $\mathbb{F}$ of linear rank-width at most $p$. As a corollary, we obtain a doubly exponential upper bound in $p$ on the size of forbidden vertex-minors for graphs of linear rank-width at most $p$. This solves an open question raised by Jeong, Kwon, and Oum [Excluded vertex-minors for graphs of linear rank-width at most $k$. European J. Combin., 41:242--257, 2014]. We also give a doubly exponential upper bound in $p$ on the size of forbidden minors for matroids representable over a fixed finite field of path-width at most $p$. Our basic tool is the pseudo-minor order used by Lagergren [Upper Bounds on the Size of Obstructions and Interwines, Journal of Combinatorial Theory Series B, 73:7--40, 1998] to bound the size of forbidden graph minors for bounded path-width. To adapt this notion into linear rank-width, it is necessary to well define partial pieces of graphs and merging operations that fit to pivot-minors. Using the algebraic operations introduced by Courcelle and Kant\'e, and then extended to (skew-)symmetric matrices by Kant\'e and Rao, we define boundaried $s$-labelled graphs and prove similar structure theorems for pivot-minor and linear rank-width.Comment: 28 pages, 1 figur

Janus ABJM Models with Mass Deformation

We construct a large class of ${\cal N} = 3$ Janus ABJM models with mass deformation, where the mass depends on a spatial (or lightcone) coordinate. We also show that the resulting Janus model can be identified with an effective action of M2-branes in the presence of a background self-dual 4-form field strength varying along one spatial (or lightcone) coordinate.Comment: 17 pages, references added, published versio

A polynomial kernel for Block Graph Deletion

In the Block Graph Deletion problem, we are given a graph $G$ on $n$ vertices and a positive integer $k$, and the objective is to check whether it is possible to delete at most $k$ vertices from $G$ to make it a block graph, i.e., a graph in which each block is a clique. In this paper, we obtain a kernel with $\mathcal{O}(k^{6})$ vertices for the Block Graph Deletion problem. This is a first step to investigate polynomial kernels for deletion problems into non-trivial classes of graphs of bounded rank-width, but unbounded tree-width. Our result also implies that Chordal Vertex Deletion admits a polynomial-size kernel on diamond-free graphs. For the kernelization and its analysis, we introduce the notion of `complete degree' of a vertex. We believe that the underlying idea can be potentially applied to other problems. We also prove that the Block Graph Deletion problem can be solved in time $10^{k}\cdot n^{\mathcal{O}(1)}$.Comment: 22 pages, 2 figures, An extended abstract appeared in IPEC201