2,453 research outputs found
Quivers, Tilings, Branes and Rhombi
We describe a simple algorithm that computes the recently discovered brane
tilings for a given generic toric singular Calabi-Yau threefold. This therefore
gives AdS/CFT dual quiver gauge theories for D3-branes probing the given
non-compact manifold. The algorithm solves a longstanding problem by computing
superpotentials for these theories directly from the toric diagram of the
singularity. We study the parameter space of a-maximization; this study is made
possible by identifying the R-charges of bifundamental fields as angles in the
brane tiling. We also study Seiberg duality from a new perspective.Comment: 36 pages, 40 figures, JHEP
An Algorithmic Metatheorem for Directed Treewidth
The notion of directed treewidth was introduced by Johnson, Robertson,
Seymour and Thomas [Journal of Combinatorial Theory, Series B, Vol 82, 2001] as
a first step towards an algorithmic metatheory for digraphs. They showed that
some NP-complete properties such as Hamiltonicity can be decided in polynomial
time on digraphs of constant directed treewidth. Nevertheless, despite more
than one decade of intensive research, the list of hard combinatorial problems
that are known to be solvable in polynomial time when restricted to digraphs of
constant directed treewidth has remained scarce. In this work we enrich this
list by providing for the first time an algorithmic metatheorem connecting the
monadic second order logic of graphs to directed treewidth. We show that most
of the known positive algorithmic results for digraphs of constant directed
treewidth can be reformulated in terms of our metatheorem. Additionally, we
show how to use our metatheorem to provide polynomial time algorithms for two
classes of combinatorial problems that have not yet been studied in the context
of directed width measures. More precisely, for each fixed , we show how to count in polynomial time on digraphs of directed
treewidth , the number of minimum spanning strong subgraphs that are the
union of directed paths, and the number of maximal subgraphs that are the
union of directed paths and satisfy a given minor closed property. To prove
our metatheorem we devise two technical tools which we believe to be of
independent interest. First, we introduce the notion of tree-zig-zag number of
a digraph, a new directed width measure that is at most a constant times
directed treewidth. Second, we introduce the notion of -saturated tree slice
language, a new formalism for the specification and manipulation of infinite
sets of digraphs.Comment: 41 pages, 6 figures, Accepted to Discrete Applied Mathematic
Dimer Models from Mirror Symmetry and Quivering Amoebae
Dimer models are 2-dimensional combinatorial systems that have been shown to
encode the gauge groups, matter content and tree-level superpotential of the
world-volume quiver gauge theories obtained by placing D3-branes at the tip of
a singular toric Calabi-Yau cone. In particular the dimer graph is dual to the
quiver graph. However, the string theoretic explanation of this was unclear. In
this paper we use mirror symmetry to shed light on this: the dimer models live
on a T^2 subspace of the T^3 fiber that is involved in mirror symmetry and is
wrapped by D6-branes. These D6-branes are mirror to the D3-branes at the
singular point, and geometrically encode the same quiver theory on their
world-volume.Comment: 55 pages, 27 figures, LaTeX2
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Dimer models from mirror symmetry and quivering amoebae
Dimer models are 2-dimensional combinatorial systems that have been shown to encode the gauge groups, matter content and tree-level superpotential of the world-volume quiver gauge theories obtained by placing D3-branes at the tip of a singular toric Calabi-Yau cone. In particular the dimer graph is dual to the quiver graph. However, the string theoretic explanation of this was unclear. In this paper we use mirror symmetry to shed light on this: the dimer models live on a T^2 subspace of the T^3 fiber that is involved in mirror symmetry and is wrapped by D6-branes. These D6-branes are mirror to the D3-branes at the singular point, and geometrically encode the same quiver theory on their world-volume
Charting Class Territory
We extend the investigation of the recently introduced class of
4d SCFTs, by considering a large family of quiver gauge
theories within it, which we denote . These theories admit a
realization in terms of orbifolds of Type IIA configurations of
D4-branes stretched among relatively rotated sets of NS-branes. This fact
permits a systematic investigation of the full family, which exhibits new
features such as non-trivial anomalous dimensions differing from free field
values and novel ways of gluing theories. We relate these ingredients to
properties of compactification of the 6d (1,0) superconformal
theories on spheres with different kinds of punctures. We describe the
structure of dualities in this class of theories upon exchange of punctures,
including transformations that correspond to Seiberg dualities, and exploit the
computation of the superconformal index to check the invariance of the theories
under them.Comment: 44 pages, 24 figure
Network error correction with unequal link capacities
This paper studies the capacity of single-source single-sink noiseless
networks under adversarial or arbitrary errors on no more than z edges. Unlike
prior papers, which assume equal capacities on all links, arbitrary link
capacities are considered. Results include new upper bounds, network error
correction coding strategies, and examples of network families where our bounds
are tight. An example is provided of a network where the capacity is 50%
greater than the best rate that can be achieved with linear coding. While
coding at the source and sink suffices in networks with equal link capacities,
in networks with unequal link capacities, it is shown that intermediate nodes
may have to do coding, nonlinear error detection, or error correction in order
to achieve the network error correction capacity
Ideals generated by the inner 2-minors of collections of cells
In 2012 Ayesha Asloob Qureshi connected collections of cells to Commutative Algebra assigning to every collection of cells the ideal of inner 2-minors, denoted by , in the polynomial ring . Investigating the main algebraic properties of depending on the shape of is the purpose of this research. Many problems are still open and they seem to be fascinating and exciting challenges.\\
In this thesis we prove several results about the primality of and the algebraic properties of like Cohen-Macaulyness, normality and Gorensteiness, for some classes of non-simple polyominoes. The study of the Hilbert-Poincar\'e series and the related invariants as Krull dimension and Castelnuovo-Mumford regularity are given. Finally we provide the code of the package \texttt{PolyominoIdeals} developed for \texttt{Macaulay2}
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