Quasiseparable Hessenberg reduction of real diagonal plus low rank matrices and applications

We present a novel algorithm to perform the Hessenberg reduction of an $n\times n$ matrix $A$ of the form $A = D + UV^*$ where $D$ is diagonal with real entries and $U$ and $V$ are $n\times k$ matrices with $k\le n$. The algorithm has a cost of $O(n^2k)$ arithmetic operations and is based on the quasiseparable matrix technology. Applications are shown to solving polynomial eigenvalue problems and some numerical experiments are reported in order to analyze the stability of the approac

Double quiver gauge theory and nearly Kahler flux compactifications

We consider G-equivariant dimensional reduction of Yang-Mills theory with torsion on manifolds of the form MxG/H where M is a smooth manifold, and G/H is a compact six-dimensional homogeneous space provided with a never integrable almost complex structure and a family of SU(3)-structures which includes a nearly Kahler structure. We establish an equivalence between G-equivariant pseudo-holomorphic vector bundles on MxG/H and new quiver bundles on M associated to the double of a quiver Q, determined by the SU(3)-structure, with relations ensuring the absence of oriented cycles in Q. When M=R^2, we describe an equivalence between G-invariant solutions of Spin(7)-instanton equations on MxG/H and solutions of new quiver vortex equations on M. It is shown that generic invariant Spin(7)-instanton configurations correspond to quivers Q that contain non-trivial oriented cycles.Comment: 42 pages; v2: minor corrections; Final version to be published in JHE

Group theoretic, Lie algebraic and Jordan algebraic formulations of the SIC existence problem

Although symmetric informationally complete positive operator valued measures (SIC POVMs, or SICs for short) have been constructed in every dimension up to 67, a general existence proof remains elusive. The purpose of this paper is to show that the SIC existence problem is equivalent to three other, on the face of it quite different problems. Although it is still not clear whether these reformulations of the problem will make it more tractable, we believe that the fact that SICs have these connections to other areas of mathematics is of some intrinsic interest. Specifically, we reformulate the SIC problem in terms of (1) Lie groups, (2) Lie algebras and (3) Jordan algebras (the second result being a greatly strengthened version of one previously obtained by Appleby, Flammia and Fuchs). The connection between these three reformulations is non-trivial: It is not easy to demonstrate their equivalence directly, without appealing to their common equivalence to SIC existence. In the course of our analysis we obtain a number of other results which may be of some independent interest.Comment: 36 pages, to appear in Quantum Inf. Compu

On The Structure Of The Chan-Paton Factors For D-Branes In Type II Orientifolds

We determine the structure of the Chan-Paton factors of the open strings ending on space filling D-branes in Type II orientifolds. Through the analysis, we obtain a rule concerning possible distribution of O-plane types. The result is applied to classify the topology of D-branes in terms of Fredholm operators and K-theory, deriving a proposal made earlier and extending it to more general cases. It is also applied to compactifications with N=1 supersymmetry in four-dimensions. We adapt and develop the language of category in this context, and use it to describe some decay channels.Comment: 137 page

Kirchhoff's Rule for Quantum Wires

In this article we formulate and discuss one particle quantum scattering theory on an arbitrary finite graph with $n$ open ends and where we define the Hamiltonian to be (minus) the Laplace operator with general boundary conditions at the vertices. This results in a scattering theory with $n$ channels. The corresponding on-shell S-matrix formed by the reflection and transmission amplitudes for incoming plane waves of energy $E>0$ is explicitly given in terms of the boundary conditions and the lengths of the internal lines. It is shown to be unitary, which may be viewed as the quantum version of Kirchhoff's law. We exhibit covariance and symmetry properties. It is symmetric if the boundary conditions are real. Also there is a duality transformation on the set of boundary conditions and the lengths of the internal lines such that the low energy behaviour of one theory gives the high energy behaviour of the transformed theory. Finally we provide a composition rule by which the on-shell S-matrix of a graph is factorizable in terms of the S-matrices of its subgraphs. All proofs only use known facts from the theory of self-adjoint extensions, standard linear algebra, complex function theory and elementary arguments from the theory of Hermitean symplectic forms.Comment: 40 page