987 research outputs found
On the unimodality of independence polynomials of some graphs
In this paper we study unimodality problems for the independence polynomial
of a graph, including unimodality, log-concavity and reality of zeros. We
establish recurrence relations and give factorizations of independence
polynomials for certain classes of graphs. As applications we settle some
unimodality conjectures and problems.Comment: 17 pages, to appear in European Journal of Combinatoric
Phases and geometry of the N=1 A_2 quiver gauge theory and matrix models
We study the phases and geometry of the N=1 A_2 quiver gauge theory using
matrix models and a generalized Konishi anomaly. We consider the theory both in
the Coulomb and Higgs phases. Solving the anomaly equations, we find that a
meromorphic one-form sigma(z)dz is naturally defined on the curve Sigma
associated to the theory. Using the Dijkgraaf-Vafa conjecture, we evaluate the
effective low-energy superpotential and demonstrate that its equations of
motion can be translated into a geometric property of Sigma: sigma(z)dz has
integer periods around all compact cycles. This ensures that there exists on
Sigma a meromorphic function whose logarithm sigma(z)dz is the differential. We
argue that the surface determined by this function is the N=2 Seiberg-Witten
curve of the theory.Comment: 41 pages, 2 figures, JHEP style. v2: references adde
Conifold Transitions in M-theory on Calabi-Yau Fourfolds with Background Fluxes
We consider topology changing transitions for M-theory compactifications on
Calabi-Yau fourfolds with background G-flux. The local geometry of the
transition is generically a genus g curve of conifold singularities, which
engineers a 3d gauge theory with four supercharges, near the intersection of
Coulomb and Higgs branches. We identify a set of canonical, minimal flux quanta
which solve the local quantization condition on G for a given geometry,
including new solutions in which the flux is neither of horizontal nor vertical
type. A local analysis of the flux superpotential shows that the potential has
flat directions for a subset of these fluxes and the topologically different
phases can be dynamically connected. For special geometries and background
configurations, the local transitions extend to extremal transitions between
global fourfold compactifications with flux. By a circle decompactification the
M-theory analysis identifies consistent flux configurations in four-dimensional
F-theory compactifications and flat directions in the deformation space of
branes with bundles.Comment: 93 pages; v2: minor changes and references adde
Holomorphic matrix integrals
We study a class of holomorphic matrix models. The integrals are taken over
middle dimensional cycles in the space of complex square matrices. As the size
of the matrices tends to infinity, the distribution of eigenvalues is given by
a measure with support on a collection of arcs in the complex planes. We show
that the arcs are level sets of the imaginary part of a hyperelliptic integral
connecting branch points.Comment: 9 pages, 1 figure, reference adde
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