12 research outputs found
A Constant-Factor Approximation for Multi-Covering with Disks
We consider variants of the following multi-covering problem with disks. We
are given two point sets (servers) and (clients) in the plane, a
coverage function , and a constant . Centered at each server is a single disk whose radius we are free to
set. The requirement is that each client be covered by at least
of the server disks. The objective function we wish to minimize is
the sum of the -th powers of the disk radii. We present a polynomial
time algorithm for this problem achieving an approximation
On Geometric Priority Set Cover Problems
We study the priority set cover problem for simple geometric set systems in the plane. For pseudo-halfspaces in the plane we obtain a PTAS via local search by showing that the corresponding set system admits a planar support. We show that the problem is APX-hard even for unit disks in the plane and argue that in this case the standard local search algorithm can output a solution that is arbitrarily bad compared to the optimal solution. We then present an LP-relative constant factor approximation algorithm (which also works in the weighted setting) for unit disks via quasi-uniform sampling. As a consequence we obtain a constant factor approximation for the capacitated set cover problem with unit disks. For arbitrary size disks, we show that the problem is at least as hard as the vertex cover problem in general graphs even when the disks have nearly equal sizes. We also present a few simple results for unit squares and orthants in the plane
The Refined Topological Vertex
We define a refined topological vertex which depends in addition on a
parameter, which physically corresponds to extending the self-dual graviphoton
field strength to a more general configuration. Using this refined topological
vertex we compute, using geometric engineering, a two-parameter (equivariant)
instanton expansion of gauge theories which reproduce the results of Nekrasov.
The refined vertex is also expected to be related to Khovanov knot invariants.Comment: 70 Pages, 23 Figure
Computing Brane and Flux Superpotentials in F-theory Compactifications
In four-dimensional F-theory compactifications with N=1 supersymmetry the
fields describing the dynamics of space-time filling 7-branes are part of the
complex structure moduli space of the internal Calabi-Yau fourfold. We
explicitly compute the flux superpotential in F-theory depending on all complex
structure moduli, including the 7-brane deformations and the field
corresponding to the dilaton-axion. Since fluxes on the 7-branes induce 5-brane
charge, a local limit allows to effectively match the F-theory results to a
D5-brane in a non-compact Calabi-Yau threefold with threeform fluxes. We
analyze the classical and instanton contributions to the F-theory
superpotential using mirror symmetry for Calabi-Yau fourfolds. The F-theory
compactifications under consideration also admit heterotic dual descriptions
and we discuss the identification of the moduli in this non-perturbative
duality.Comment: 75 pages, 1 figure; typos corrected, references adde
Half-Integer Winding Number Solutions to the Landau-Ginzburg-Higgs Equations and Instability of the Abrikosov-Nielsen-Olesen Vortex
New solutions to the abelian U(1) Higgs model, corresponding to vortices of
integer and half-integer winding number bound onto the edges of domain walls
and possibly surrounded by annular current flows, are described, based on a
fine-grained analysis of the topology of such configurations in spacetime. The
existence of these states, which saturate BPS bounds in specific limits and are
quite reminiscent of D-branes and membranes in general, could have interesting
and some important consequences in a wide range of physical contexts. For
instance, they raise the possibility that for some regimes of couplings the
usual vortex of unit winding number would split into two vortices each of
one-half winding number bound by a domain wall. A similar approach may also be
relevant to other known topological states of field theory.Comment: 52 pages (Latex) + 2 postscript figure
Braiding Knots with Topological Strings
For an arbitrary knot in a three-sphere, the Ooguri-Vafa conjecture associates to it a unique stack of branes in type A topological string on the resolved conifold, and relates the colored HOMFLY invariants of the knot to the free energies on the branes. For torus knots, we use a modified version of the topological recursion developed by Eynard and Orantin to compute the free energies on the branes from the Aganagic-Vafa spectral curves of the branes, and find they are consistent with the known colored HOMFLY knot invariants Ă la the Ooguri-Vafa conjecture. In addition our modified topological recursion can reproduce the correct closed string free energies, which encode the information of the background geometry. We conjecture the modified topological recursion is applicable for branes associated to hyperbolic knots as well, encouraged by the observation that the modified topological recursion yields the correct planar closed string free energy from the Aganagic-Vafa spectral curves of hyperbolic knots. This has implications for the knot theory concerning distinguishing mutant knots with colored HOMFLY invariants. Furthermore, for hyperbolic knots, we present methods to compute colored HOMFLY invariants in nonsymmetric representations of U(N). The key step in this computation is computing quantum 6j-symbols in the quantum group U_q(sl_N)
Landscaping with fluxes and the E8 Yukawa Point in F-theory
Integrality in the Hodge theory of Calabi-Yau fourfolds is essential to find
the vacuum structure and the anomaly cancellation mechanism of four dimensional
F-theory compactifications. We use the Griffiths-Frobenius geometry and
homological mirror symmetry to fix the integral monodromy basis in the
primitive horizontal subspace of Calabi-Yau fourfolds. The Gamma class and
supersymmetric localization calculations in the 2d gauged linear sigma model on
the hemisphere are used to check and extend this method. The result allows us
to study the superpotential and the Weil-Petersson metric and an associated tt*
structure over the full complex moduli space of compact fourfolds for the first
time. We show that integral fluxes can drive the theory to N=1 supersymmetric
vacua at orbifold points and argue that fluxes can be chosen that fix the
complex moduli of F-theory compactifications at gauge enhancements including
such with U(1) factors. Given the mechanism it is natural to start with the
most generic complex structure families of elliptic Calabi-Yau 4-fold
fibrations over a given base. We classify these families in toric ambient
spaces and among them the ones with heterotic duals. The method also applies to
the creating of matter and Yukawa structures in F-theory. We construct two
SU(5) models in F-theory with a Yukawa point that have a point on the base with
an -type singularity on the fiber and explore their embeddings in the
global models. The explicit resolution of the singularity introduce a higher
dimensional fiber and leads to novel features.Comment: 150 page
String dualities and superpotential
The main objective of this thesis is the computation of the superpotential induced by D5-branes in the type IIB string theory and by five-branes in the heterotic string theory. Both superpotentials have the same functional form which is the chain integral of the holomorphic three-form. Using relative (co)homology we can unify the flux and brane superpotential. The chain integral can be seen as an example of the Abel-Jacobi map. We discuss many structures such as mixed Hodge structure which allows for the computation of Picard-Fuchs differential equations crucial for explicit computations. We blow up the Calabi-Yau threefold along the submanifold wrapped by the brane to obtain geometrically more appropriate configuration. The resulting geometry is non-Calabi-Yau and we have a canonically given divisor. This blown-up geometry makes it possible to restrict our attention to complex structure deformations. However, the direct computation is yet very difficult, thus the main tool for computation will be the lift of the brane configuration to a F-theory compactification. In F-theory, since complex structure, brane and, if present, bundle moduli are all contained in the complex structure moduli space of the elliptic Calabi-Yau fourfold, the computation can be dramatically simplified. The heterotic/F-theory duality is extended to include the blow-up geometry and thereby used to give the blow-up geometry a more physical meaning