A note on the Wodzicki residue

In this note we explain the relationship of the Wodzicki residue of (certain powers of) an elliptic differential operator P acting on sections of a complex vector bundle E over a closed compact manifold M and the asymptotic expansion of the trace of the corresponding heat operator e^−tP. In the special case of a generalized laplacian Δ and dim M > 2, we thereby obtain a simple proof of the fact already shown in [KW], that the Wodzicki residue res(Δ^−n/2+1) is the integral of the second coefficient of the heat kernel expansion of Δ up to a proportional factor.     

Gravity in non-commutative geometry

We study general relativity in the framework of non-commutative differential geometry. As a prerequisite we develop the basic notions of non-commutative Riemannian geometry, including analogues of Riemannian metric, curvature and scalar curvature. This enables us to introduce a generalized Einstein-Hilbert action for non-commutative Riemannian spaces. As an example we study a space-time which is the product of a four dimensional manifold by a two-point space, using the tools of non-commutative Riemannian geometry, and derive its generalized Einstein-Hilbert action. In the simplest situation, where the Riemannian metric is taken to be the same on the two copies of the manifold, one obtains a model of a scalar field coupled to Einstein gravity. This field is geometrically interpreted as describing the distance between the two points in the internal space.Dedicated to H. ArakiSupported in part by the Swiss National Foundation (SNF)     

The Cartan algebra of exterior differential forms as a supermanifold: morphisms and manifolds associated with them

A supermanifold, M^m/n, can be caracterired by its smooth superfunctions which constitute an algebra A (Leites, Kostant). We associate canonically a la Gelfand certain fibred manifolds on which the automorphisms (the Jordan automorphisms) of A act as diffeomorphisms. For example, the kernels of all homomorphisms from the algebra of superfunctions onto the Grassmann algebra of dimension n form naturally a manifold of dimension m2^n-1 if n is even. To be more specific we explain this and similar constructions in the case of the algebra of smooth exterior differential forms defined on a smooth manifold. This algebra defines a particular supermanifold M^m/m.     

Curvature dependent lower bounds for the first eigenvalue of the Dirac operator

Using Weitzenböck techniques on any compact Riemannian spin manifold we derive inequalities that involve a real parameter and join the eigenvalues of the Dirac operator with curvature terms. The discussion of these inequalities yields vanishing theorems for the kernel of the Dirac operator D and lower bounds for the spectrum of D² if the curvature satisfies certain conditions.     

晶体离子质量差异对激活离子声子参助能量传递几率的影响

在已有理论模型的基础上讨论了基质晶体离子质量差异对激活离子声子参助能量传递几率的影响。结果表明基质离子的质量差异除了导致声子频率的变化从而影响其声子参助能量传递几率,还直接改变其传递几率的显式。对单声子参助能量传递过程,其传递几率必须乘以质量差异因子D²,对双声子参助能量传递过程其传递几率则乘以因子D⁴。文中给出了两种不同质量离子组成的晶体的因子D的表示式和p种不同质量的离子组成的晶体的因子D的一般表达式。     

The algebraic Bethe ansatz for rational braid-monoid lattice models

In this paper we study isotropic integrable systems based on the braid-monoid algebra. These systems constitute a large family of rational multistate vertex models and are realized in terms of the B_n, C_nand D_n Lie algebra and by the superalgebra Osp(n||2m). We present a unified formulation of the quantum inverse scattering method for many of these lattice models. The appropriate fundamental commutation rules are found, allowing us to construct the eigenvectors and the eigenvaluesof the transfer matrix associated to the B_n, C_n, D_n, Osp(2nt-1||2), Osp(2||2nt-2), Osp(2nt-2||2) and Osp(1||2n) models. The corresponding Bethe ansatz equations can be formulated in terms of the root structure of the underlying algebra.     

Kähler-Dirac ghosts for self-dual fields

We present the generalization to spacetime dimension D=4n+2 of the Lorentz covariant quadratic lagrangian for pairs of (anti)self-dual fields previously obtained by the authors in D=2. In the process BRST quantizing this lagrangian a first-order quadratic lagrangian for ghost (anti)self-dual fields is found which, after gauge fixing, can be written in terms of bispinors and it turns out to be a Kähler-Dirac lagrangian. The coupling to gravity is straightforward and the gravitational anomaly due to (anti)self-dual fields is obtained directly from an action principle.     

Connes’ spectral triple and U(1) gauge theory on finite sets

We first apply Connes’ noncommutative geometry to a finite point set. The explicit form of the action functional of U(1) gauge field on this n-point set is obtained. We then construct the U(1) gauge theory on a disconnected manifold consisting of n copies of a given manifold. In this case, the explicit action functional of U(1) gauge field is also obtained.     

1-forms over the moduli space of irreducible connections defined by the spectrum of Dirac operators

Let (P) be the moduli space of irreducible connections of a G-principal bundle P over a closed Riemannian spin manifold M. Let D_A be the Dirac operator of M coupled to a connection A of P and f a smooth function on M. We consider a smooth variation A(u) of A with tangent vector ω and denote T_ω:= (D_A(u)−f) (u=0. The coefficients of the asymptotic expansion of trace (T_ω · e^-t(D_A−f)2) near t=0 define 1-forms a^(k)_f, K=0, 1, 2, … on (P). In this paper we calculate aa⁽⁰⁾_f, a⁽¹⁾_f, a⁽²⁾_f and study some of their properties. For instance using the 1-form a⁽²⁾_f for suitable functions f we obtain a foliation of codimension 5 of the space of G-instantons of S⁴.     

Generalized Einstein manifolds

A Finslerian manifold is called a generalized Einstein manifold (GEM) if the Ricci directional curvature R(u,u) is independent of the direction. Let F⁰(M, g_t) be a deformation of a compact n-dimensional Finslerian manifold preserving the volume of the unitary fibre bundle W(M). We prove that the critical points g₀ F⁰(g_t) of the integral I(g_t) on W(M) of the Finslerian scalar curvature (and certain functions of the scalar curvature) define a GEM. We give an estimate of the eigenvalues of Laplacian Δ defined on W(M) operating on the functions coming from the base when (M, g) is of minima fibration with a constant scalar curvature H admitting a conformal infinitesimal deformation (CID). We obtain λ ≥ H/(n − 1) (Δf = λf). If M is simply connected and λ = H/(n − 1), then (M, g) is Riemannian and is isometric to an n-sphere. We first calculate, in the general case, the formula of the second variationals of the integral I (g_t) for G = g₀, then for a CID we show that for certain Finslerian manifolds, I″(g₀) > 0. Applications to the gravitation and electromagnetism in general relativity are given. We prove that the spaces characterizing Einstein-Maxwell equations are GEMs.     

Graded non-commutative geometries

We present a short exposition of graded finite non-commutative geometries. The theory that serves as an example is based on the algebra of matrices M_n . This non-commutative algebra replaces the algebra of functions on a manifold. Consequently, vector fields (differentiations), forms and connections are constructed. The gauge theory can be introduced without the notion of internal manifold. We discuss some physical application, the similarities with the standard model, and the graded version of this geometry.     

A Construction of Critical GJMS Operators Using Wodzicki's Residue

For an even dimensional, compact, conformal manifold without boundary we construct a conformally invariant differential operator of order the dimension of the manifold. In the conformally flat case, this operator coincides with the critical GJMS operator of Graham-Jenne-Mason-Sparling. We use the Wodzicki residue of a pseudo-differential operator of order −2, originally defined by A. Connes, acting on middle dimension forms.     

Differential cohomology of C (a)

Let be a finite dimensional real Lie algebra and ^* its dual. ^* is a Poisson manifold. Thus the space C^∞( ^*) of C^∞ functions on ^* has an associative and a Lie algebra structure. The problem of formal deformations of such a structure needs the determination of some cohomology groups of C^∞( ^*), considered as a module on itself for left multiplication or adjoint representation. We determine here these groups. The result is very similar to the case of C^∞(W), where W is a symplectic manifold except for the Lie algebras h_r × ^m, direct products of Heisenberg and abelian Lie algebras.     

Gravity coupled with matter and the foundation of non-commutative geometry

We first exhibit in the commutative case the simple algebraic relations between the algebra of functions on a manifold and its infinitesimal length elementds. Its unitary representations correspond to Riemannian metrics and Spin structure whileds is the Dirac propagatords=x−x=D ⁻¹, whereD is the Dirac operator. We extend these simple relations to the non-commutative case using Tomita's involutionJ. We then write a spectral action, the trace of a function of the length element, which when applied to the non-commutative geometry of the Standard Model will be shown ([CC]) to give the SM Lagrangian coupled to gravity. The internal fluctuations of the non-commutative geometry are trivial in the commutative case but yield the full bosonic sector of SM with all correct quantum numbers in this slightly non-commutative case. The group of local gauge transformations appears spontaneously as a normal subgroup of the diffeomorphism group.     

Intersecting M-branes

We present the magnetic duals of Güven's electric-type solutions of D = 11 supergravity preserving 1/4 or 1/8 of the D = 11 supersymmetry. We interpret the electric solutions as n orthogonal intersecting membranes and the magnetic solutions as n orthogonal intersecting 5-branes, with n = 2, 3; these cases obey the general rule that p-branes can self-intersect on (p − 2)-branes. On reduction to D = 4 these solutions become electric or magnetic dilaton black holes with dilaton coupling constant a = 1 (for n = 2) or (for n = 3). We also discuss the reduction to D = 10.     

N=2 6-dimensional supersymmetric E6 breaking

We study the N=2 supersymmetric E₆ models on the 6-dimensional space–time where the supersymmetry and gauge symmetry can be broken by the discrete symmetry. On the space–time M⁴×S¹/(Z₂×Z₂′)×S¹/(Z₂×Z₂′), for the zero modes, we obtain the 4-dimensional N=1 supersymmetric models with gauge groups SU(3)×SU(2)×SU(2)×U(1)², SU(4)×SU(2)×SU(2)×U(1), and SU(3)×SU(2)×U(1)³ with one extra pair of Higgs doublets from the vector multiplet. In addition, considering that the extra space manifold is the annulus A² and disc D², we list all the constraints on constructing the 4-dimensional N=1 supersymmetric SU(3)×SU(2)×U(1)³ models for the zero modes, and give the simplest model with Z₉ symmetry. We also comment on the extra gauge symmetry breaking and its generalization.     

Statistical analysis of the Perseus-Pisces redshift survey: spatial and luminosity properties

We analyze the spatial and the luminosity properties of the Perseus-Pisces redshift survey. We find that the two point correlation function (CF) Γ(r) is a power law up to the sample effective depth ( 30 h⁻¹ Mpc), showing the fractal nature of the galaxy distribution in this catalog. The fractal dimension turns out to be D 2. We also consider the CF ξ(r) and in particular the behavior of the “correlation length” r₀ (ξ(r₀)1) as function of the sample size. In this respect we find, unambiguously, that the luminosity segregation effect is not supported by any experimental evidence. In addition we have studied the galaxian number-density (n(r)) and number-counts (N(m)) in the VL subsamples finding a good agreement with the properties of a fractal distribution. In particular our conclusion is that the n(r) relation permits to extend the analysis of the fractal nature up to a deeper depth than that reached by the CF analysis, and, we find evidence for fractal properties up to the limiting depth of 130 h⁻¹ Mpc. We clarify the role of the small-scale fluctuations in the determination of the galaxy counts. Even in this case the results are in agreement with the previous ones. Finally we have considered the correlations between galaxy positions and luminosities by means of the multifractal analysis. We find clear evidence for self-similar behavior of the whole luminosity-space distribution. These results confirm and extend those of Coleman and Pietronero (1992).     

Hamilton formalism in non-cummutative geometry

We study the Hamilton formalism for Connes-Lott models, i.e. for Yang-Mills theory in non-commutative geometry. The starting point is an associative *-algebra A which is of the form A = C (I, A_s), where A_s is itself an associative *-algebra. With appropriate choice of a K-cycle over A it is possible to identify the time-like part of the generalized differential algebra constructed out of A. We define the non-commutative analogue of integration on space-like surfaces via the Dixmier trace restricted to the representation of the space-like part A_s of the algebra. Due to this restriction it is possible to define the Lagrange function resp. Hamilton function also for Minkowskian space-time. We identify the phase-space and give a definition of the Poisson bracket for Yang-Mills theory in non-commutative geometry. This general formalism is applied to a model on a two-point space and to a model on Minkowski space-time x two-point space.     

Universal properties of self-avoiding walks from two-dimensional field theory

We use the recently conjectured exact S-matrix of the massive O(n) model to derive its form factors and ground state energy. This information is then used in the limit n → 0 to obtain quantitative results for various universal properties of self-avoiding chains and loops. In particular, we give the first theoretical prediction of the amplitude ratio C/D which relates the mean square end-to-end distance of chains to the mean square radius of gyration of closed loops. This agrees with the results from lattice enumeration studies to within their errors, and gives strong support for the various assumptions which enter into the field theoretic derivation. In addition, we obtain results for the scaling function of the structure factor of long loops, and for various amplitude ratios measuring the shape of self-avoiding chains. These quantities are all related to moments of correlation functions which are evaluated as a sum over m-particle intermediate states in the corresponding field theory. We show that in almost all cases, the restriction to m 2 gives results which are accurate to at least one part in 10³. This remarkable fact is traced to a softening of the m > 2 branch cuts relative to their behaviour based on phase space arguments alone, a result which follows from the threshold behaviour of the two-body S-matrix, S(O) = −1. Since this is a general property of interacting 2D field theories, it suggests that similar approximation may well hold for other models. However, we also study the moments of the area of self-avoiding loops, and show that, in this case, the two-particle approximation is not valid.     

Remarks on stochastic non-linear birth and death models

The first part of this paper is an attempt to formulate and motivate additional work on the important problem of obtaining global bounds applicable to the controlled truncation of the paper relates specifically to the linear birth, quadratic death model. Asymptotic results are given for the first finite difference ΔT_m where T_m is the exactly known mean time to extinction starting from state m (m= 0,1,…). These results are in terms of the environmental carrying capacity n^* taken to be large. For m near zero ΔT_me^n*/(n^*)²; whereas, for m near n^*ΔT_m ≈ (π/2)^1/2/(n^*)^3/2. This indicates the vastly different time scales in those two regions of state space - with considerably slower action near extinction than near n^*.