Yang-Mills Flows on Nearly Kähler Manifolds and G 2-Instantons

We consider Lie(G)-valued G-invariant connections on bundles over spaces ${G/H,\, \mathbb{R}\times G/H\, {\rm and}\, \mathbb{R}^2\times G/H}We give a geometric construction of the ${\mathcal{W}_{1+\infty}}We consider Lie(G)-valued G-invariant connections on bundles over spaces G/H, \mathbbR×G/H and \mathbbR²×G/H{G/H,\, \mathbb{R}\times G/H\, {\rm and}\, \mathbb{R}^2\times G/H}, where G/H is a compact nearly K?hler six-dimensional homogeneous space, and the manifolds \mathbbR×G/H{\mathbb{R}\times G/H} and \mathbbR²×G/H{\mathbb{R}^2\times G/H} carry G ₂- and Spin(7)-structures, respectively. By making a G-invariant ansatz, Yang-Mills theory with torsion on \mathbbR×G/H{\mathbb{R}\times G/H} is reduced to Newtonian mechanics of a particle moving in a plane with a quartic potential. For particular values of the torsion, we find explicit particle trajectories, which obey first-order gradient or hamiltonian flow equations. In two cases, these solutions correspond to anti-self-dual instantons associated with one of two G ₂-structures on \mathbbR×G/H{\mathbb{R}\times G/H}. It is shown that both G ₂-instanton equations can be obtained from a single Spin(7)-instanton equation on \mathbbR²×G/H{\mathbb{R}^2\times G/H}.     

Diffusion Limited Aggregation on a Cylinder

We consider the DLA process on a cylinder . It is shown that this process “grows arms”, provided that the base graph G has small enough mixing time. Specifically, if the mixing time of G is at most , the time it takes the cluster to reach the m ^th layer of the cylinder is at most of order . In particular we get examples of infinite Cayley graphs of degree 5, for which the DLA cluster on these graphs has arbitrarily small density. In addition, we provide an upper bound on the rate at which the “arms” grow. This bound is valid for a large class of base graphs G, including discrete tori of dimension at least 3. It is also shown that for any base graph G, the density of the DLA process on a G-cylinder is related to the rate at which the arms of the cluster grow. This implies that for any vertex transitive G, the density of DLA on a G-cylinder is bounded by 2/3.     

RenormalizedG-convolution ofN-point functions in quantum field theory: Convergence in the Euclidean case II

The notion of Feynman amplitude associated with a graphG in perturbative quantum field theory admits a generalized version in which each vertexv ofG is associated with ageneral (non-perturbative)n _v -point functionH ⁿ _v , n_vdenoting the number of lines which are incident tov inG. In the case where no ultraviolet divergence occurs, this has been performed directly in complex momentum space through Bros-Lassalle'sG-convolution procedure.     

Existence and invariance of twisted products on a symplectic manifold

It is now well-known [1] that the twisted product on the functions defined on a symplectic manifold, play a fundamental role in an invariant approach of quantum mechanics. We prove here a general existence theorem of such twisted products. If a Lie group G acts by symplectomorphisms on a symplectic manifold and if there is a G-invariant symplectic connection, the manifold admits G-invariant Vey twisted products. In particular, if a homogeneous space G/H admits an invariant linear connection, T ^*(G/H) admits a G-invariant Vey twisted product. For the connected Lie group G, the group T ^*G admits a symplectic structure, a symplectic connection and a Vey twisted product which are bi-invariant under G.     

Instantons and Yang–Mills Flows on Coset Spaces

We consider the Yang–Mills flow equations on a reductive coset space G/H and the Yang–Mills equations on the manifold \mathbbR×G/H{\mathbb{R}\times G/H}. On non-symmetric coset spaces G/H one can introduce geometric fluxes identified with the torsion of the spin connection. The condition of G-equivariance imposed on the gauge fields reduces the Yang–Mills equations to f⁴{\phi^4}-kink equations on \mathbbR{\mathbb{R}}. Depending on the boundary conditions and torsion, we obtain solutions to the Yang–Mills equations describing instantons, chains of instanton–anti-instanton pairs or modifications of gauge bundles. For Lorentzian signature on \mathbbR×G/H{\mathbb{R}\times G/H}, dyon-type configurations are constructed as well. We also present explicit solutions to the Yang–Mills flow equations and compare them with the Yang–Mills solutions on \mathbbR×G/H{\mathbb{R}\times G/H}.     

Symmetry of the physical probability function implies modularity of the lattice of decision effects

This paper states two equivalent conditions from which modularity of the latticeG of decision effectsE can be derived. It may be seen as a supplement to Ludwig's approach [5] to an axiomatic foundation of physical theories. As a consequence of these conditions every filterT _E is a self-adjoint projector on the Hilber spaceB generated by the decision effects.This paper extends a final report presented to and supported by the Deutsche Forschungsgemeinschaft.     

Photoconductivity derived from an exact modelling of the uncompensated one-donor model. II—computer simulation of the steady-state exact model

Abstract

Stationary photoconductivity is treated with the model of Part I, free from any of the usual simplifying approximations of the related equations. This simulation leads to set forth a new concept of characteristic temperature T ₀, at which the donor population is independent of the illumination intensity G. T ₀ separates two intervals of temperature over which G either partially empties (T< T ₀) or fills (T > T ₀) the level. Also T ₀ has, in particular, an influence on the ratio n/p of free electrons and holes. The effective isothermal behaviour of n(G) shows that n(G) ∞ G ^1/2 on the lower side of the G range, and n(G) ∞ G at higher G. Variations of n(T) at constant G also display original, T ₀ dependent, characteristics. Finally, a qualitative comparison is made of the 1D model with the two 1D_ai approximate models described in I, in order to distinguish their most prominent behaviour differences.     

Symmetry-adapted linear combinations for the eigenvalues and eigenvectors of reciprocal graphs

ABSTRACT

Three classes of reciprocal graphs, viz. monocycle (G^C_n), linear chain (G^L_n) and star (G^K_n) with reciprocal pairs of eigenvalues (λ, 1/λ), are well known. Reciprocal graphs of monocycle (G^C_n) and linear chain (G^L_n) are obtained by putting a pendant vertex to each vertex of simple monocycle (C_n) and simple linear chain (L_n), respectively. A star graph of such kind is obtained by attaching a pendant vertex to the central vertex and to each of the (n ? 1) peripheral vertices of the star graph (K_{1, (n?1)}). An n-fold rotational axis of symmetry for G^C_n and (n ? 1)-fold rotational axis of symmetry for G^K_n have been exploited for obtaining their respective condensed graphs. The condensed graph for G^L_n has been generated from that of G^C_n incorporating proper boundary conditions. Condensed graphs are lower dimensional graphs and are capable of keeping all eigeninformation in condensed form. Thus the eigensolutions (i.e. the eigenvalues and the eigenvectors) in analytical forms for such graphs are obtained by solving 2 × 2 or 4 × 4 determinants that in turn result in the charge densities and bond orders of the corresponding molecules in analytical forms. Some mathematical properties of the eigenvalues of such graphs have also been explored.     

The Monodromy Rings of the Necklace Graphs

A necklace graph is a Feynman graph obtained from a single loop graph by replacing each internal line by a multiplet (i.e. a set of one or more internal lines joining the same two vertices). In this paper the monodromy rings of the necklace graphs are determined.     

Coincidence theorem for the direct correlation function of hard-particle fluids

The Mayerf-function for purely hard particles of arbitrary shape satisfiesf ²(1, 2)=–f(1, 2). This relation can be introduced into the graphical expansion of the direct correlation functionc(1, 2) to obtain a graphical expression for the case of exact coincidence, in position and orientation, of two identical hard cores. The resulting expression forc(1, 1)+1 contains only graphsG fromc(1), the sum of irreducible graphs with one labeled point. Relative to its coefficient inc(1),G occurs inc(1, 1) with an additional factorR _c which is 1 for the leading graph in the expansion and of the form 2–2L(G) for all other graphs. HereL(G)=0, 1, 2,..., is a nonnegative integer. Topological analysis is used to derive an expression forL(G) in terms of the connectivity properties ofG.     

Families of Line-Graphs and Their Quantization

Any directed graph G with N vertices and J edges has an associated line-graph L(G) where the J edges form the vertices of L(G). We show that the non-zero eigenvalues of the adjacency matrices are the same for all graphs of such a family L ⁿ (G). We give necessary and sufficient conditions for a line-graph to be quantisable and demonstrate that the spectra of associated quantum propagators follow the predictions of random matrices under very general conditions. Line-graphs may therefore serve as models to study the semiclassical limit (of large matrix size) of a quantum dynamics on graphs with fixed classical behaviour.     

Minimal orthomodular lattices

The lattices calledminimal orthomodular (MOL) arise in a special exclusion problem concerning the class of all orthomodular lattices (OML) and the subclass of all modular orthocomplemented lattices. This problem was given in G. Kalmbach's book,Orthomodular Lattices. We prove that an exclusion system necessarily must contain an infinite lattice. We prove that, except one, all the finite, irreducible MOLs have only blocks with eight elements. We characterize finite MOLs by a covering property related to equational classes generated by the modular ortholattices MOn.     

The Massless Higher-Loop Two-Point Function

We introduce a new method for computing massless Feynman integrals analytically in parametric form. An analysis of the method yields a criterion for a primitive Feynman graph G to evaluate to multiple zeta values. The criterion depends only on the topology of G, and can be checked algorithmically. As a corollary, we reprove the result, due to Bierenbaum and Weinzierl, that the massless 2-loop 2-point function is expressible in terms of multiple zeta values, and generalize this to the 3, 4, and 5-loop cases. We find that the coefficients in the Taylor expansion of planar graphs in this range evaluate to multiple zeta values, but the non-planar graphs with crossing number 1 may evaluate to multiple sums with 6^th roots of unity. Our method fails for the five loop graphs with crossing number 2 obtained by breaking open the bipartite graph K _3,4 at one edge. CNRS.     

On the Becker/Döring Theory of Nucleation of Liquid Droplets in Solids

Nucleation of liquid precipitates in semi-insulating GaAs is accompanied by deviatoric stresses resulting from the liquid/solid misfit. A competition of surface tension and stress deviators at the interface determines the nucleation barrier.The evolution of liquid precipitates in semi-insulating GaAs is due to diffusional processes in the vicinity of the droplet. The diffusion flux results from a competition of chemical and mechanical driving forces.The size distribution of the precipitates is determined by a Becker--Dö-ring system. The study of its properties in the presence of deviatoric stresses is the subject of this study. The main tasks of this study are: (i) We propose a new Becker/Döring model that takes thermomechanical coupling into account. (ii) We compare the current model with already existing models from the literature. Irrespective of the incorporation of mechanical stresses, the various models differ due to different environments where the evolution of precipitates takes place. (iii) We determine the structure of equilibrium solutions according to the Becker/Döring model, and we compare these solutions with those that result from equilibrium thermodynamics.     

Boundary Scattering,Symmetric Spaces and the Principal Chiral Model on the Half-Line

We investigate integrable boundary conditions (BCs) for the principal chiral model on the half-line, and rational solutions of the boundary Yang-Baxter equation (BYBE). In each case we find a connection with (type I, Riemannian, globally) symmetric spaces G/H: there is a class of integrable BCs in which the boundary field is restricted to lie in a coset of H; these BCs are parametrized by G/H×G/H; there are rational solutions of the BYBE in the defining representations of all classical G parametrized by G/H; and using these we propose boundary S-matrices for the principal chiral model, parametrized by G/H×G/H, which correspond to our boundary conditions.An erratum to this article can be found at     

On interesting walks in a graph

Notions of interesting walks and of their equivalence are introduced. A general formula for the number _l, of equivalence classes of interesting walks of lengthl in a given graphG is derived and applied forl 5 so as to express _l in terms of the adjacency matrix ofG.     

Exact Potts Model Partition Functions for Strips of the Honeycomb Lattice

We present exact calculations of the Potts model partition function Z(G,q,v) for arbitrary q and temperature-like variable v on strip graphs G of the honeycomb lattice for a variety of transverse widths equal to L _y vertices and for arbitrarily great length, with free longitudinal boundary conditions and free and periodic transverse boundary conditions. These partition functions have the form , where m denotes the number of repeated subgraphs in the longitudinal direction. We give general formulas for N _Z,G,j for arbitrary L _y . We also present plots of zeros of the partition function in the q plane for various values of v and in the v plane for various values of q. Plots of specific heat for infinite-length strips are also presented, and, in particular, the behavior of the Potts antiferromagnet at is investigated.     

Some Graded Bialgebras Related to Borcherds Superalgebras

In present paper we define a new kind of weak quantized enveloping algebra of Borcherds superalgebras. We denote this algebra by wU_q^t(G)wU_{q}^{\tau}(\mathcal{G}). It is a noncommutative and noncocommutative weak graded Hopf algebra under some additional condition. It has a homomorphic image which is isomorphic to the usual quantum enveloping algebra U_q(G)U_{q}(\mathcal{G}) of G\mathcal{G}.     

Non-Abelian Vortices,Super Yang–Mills Theory and Spin(7)-Instantons

We consider a complex vector bundle E{\mathcal{E}} endowed with a connection A{\mathcal{A}} over the eight-dimensional manifold \mathbbR²×G/H{\mathbb{R}^2\times G/H}, where G/H = SU(3)/U(1) × U(1) is a homogeneous space provided with a never-integrable almost-complex structure and a family of SU(3)-structures. We establish an equivalence between G-invariant solutions A{\mathcal{A}} of the Spin(7)-instanton equations on \mathbbR²×G/H{\mathbb{R}^2\times G/H} and general solutions of non-Abelian coupled vortex equations on \mathbbR²{\mathbb{R}^2}. These vortices are BPS solitons in a d = 4 gauge theory obtained from N = 1{\mathcal{N} =1} supersymmetric Yang–Mills theory in ten dimensions compactified on the coset space G/H with an SU(3)-structure. The novelty of the obtained vortex equations lies in the fact that Higgs fields, defining morphisms of vector bundles over \mathbbR²{\mathbb{R}^2}, are not holomorphic in the generic case. Finally, we introduce BPS vortex equations in N = 4{\mathcal{N} =4} super Yang–Mills theory and show that they have the same feature.