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.     

Connections on vector bundles over super Riemann surfaces

We show that the globally inequivalent off-shell N=1 super Yang-Mills theories in two dimensions classify the superholomorphic structures on vector bundles over super Riemann surfaces. More precisely, there is a one-to-one correspondence between superholomorphic structures on vector bundles over super Riemann surfaces and unitary connections satisfying certain curvature constraints. These curvature constraints are the canonical constraints used in superspace formulations of super Yang-Mills theories, but arise in our considerations as integrability requirements for the local existence of solutions to certain differential equations. Finally, we discuss the relationship of this work with some aspects of Witten's twistor-like transform.     

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.     

Mirror symmetry and generalized complex manifolds: Part II. Integrability and the transform for torus bundles

In this paper we continue the development of a relative version of T-duality in generalized complex geometry which we propose as a manifestation of mirror symmetry. We discuss the integrability of the transform from Part I in terms of data on the base manifold. We work with semi-flat generalized complex structures on real n-torus bundles with section over an n-dimensional base and use the transform on vector bundles developed in Part I of this paper to discuss the bijective correspondence between semi-flat generalized complex structures on pairs of dual torus bundles. We give interpretations of these results in terms of relationships between the cohomology of torus bundles and their duals. We comment on the ways in which our results generalize some well established aspects of mirror symmetry. Along the way, we give methods of constructing generalized complex structures on the total spaces of the bundles we consider.     

Measurements of fragment mass, charge and kinetic energy distributions in thermal neutron fission of U with a back-to-back ΔE-E detector system

A simultaneous measurement of mass (M), charge (Z) and kinetic energy (E_k) distributions of fragments in thermal-neutron fission of ²³⁵U has been carried out using a back-to-back ΔE-E detector system. A pair of gridded ionization chambers filled with P-5 gas measured the energy losses ΔE₁, ΔE₂ of the complementary fragments in the gas, and the residual fragment energies were measured with a pair of semiconductor detectors. The four-parameter data were analysed to obtain fragment Z-distributions using the mass-momentum conservation relations to obtain M and then using the dependence of ΔE on E/M and Z. The charge resolution for the light-fragment group was determined at the gas pressures of 44, 150, 270 Torr and the best Z-resolution was obtained at 270 Torr. The data at this latter gas pressure were analysed to obtain the variances σ²_z of the fragment charge distributions for different E_k windows and this was compared with the measured variances σ²_A of the fragment mass distribution as a function of E_k as obtained by twoparameter measurements of the correlated fragment energies. The dependence of the measured variances σ²_A, σ²_Z on E_k is considered with a view to learning about the degree of neutron-proton correlations in the nucleon exchanges taking place during the mass division in fission, in analogy with that used in the case of heavy-ion deep inelastic collisions (DIC). The variation of σ²_A/σ²_z with E_k observed in the present work, which is somewhat similar to that seen in heavy-ion DIC, implies that the fragment mass division in fission is governed by nucleon-exchange processes, where the degree of neutron-proton correlations is dependent on the total kinetic energy E_k. While for very large E_k the neutron-proton motion in the exchange processes is found to be uncorrelated, for lower E_k values the motion is seen to be correlated. Implications of these results regarding the applicability of fission models such as statistical model are pointed out.     

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.     

Shape coexistence in 76Se within the neutron–proton interacting boson model

We have investigated the low-lying energy spectrum and electromagnetic transition strengths in even–even ⁷⁶Se using the proton–neutron interacting boson model (IBM-2). The theoretical calculation for the energy levels and E2 and M1 transition strengths is in good agreement with the experimental data. Specifically, the excitation energy and E2 transition of ${0}_{2}^{+}$ state, which is intimately associated with shape coexistence, can be accurately reproduced. The analysis on low-lying states and the key structure indicators R₁, R₂, R₃ and R₄ and M1 transitions indicates that there is a coexistence between spherical shape and γ-soft shape in ⁷⁶Se.     

From Planck to GUT via dimensional transmutation

Consider a gauge singlet superfield S coupled to a pair of adjoint fields in a SUSY-GUT. If the tree-level vacuum is flat in S, the vev S which defines the GUT scale will be determined via dimensional transmutation at a scale M where the soft-breaking (mass)² vanishes as a result of running from M_P = (8πG_N)^−1/2. Because of the large number of adjoint fields N_A coupled to S, one finds that M can be generically close to M_GUT = 2 × 10¹⁶GeV:

, where λ is a Yukawa 0.7. This work examines the symmetries and dynamical constraints required in a SUSY-GUT in order that the desired flatness in S is achieved, and that this flatness may survive in a supergravity framework.     

Topology change and θ-vacua in 2D Yang-Mills theory

We discuss the existence of θ-vacua in pure Yang-Mills theory in two space-time dimensions. More precisely, a procedure is given which allows one to classify the distinct quantum theories possessing the same classical limit for an arbitrary connected gauge group G and compact space-time manifold M (possibly with boundary) possessing a special basepoint. For any such G and M it is shown that the above quantizations are in one-to-one correspondence with the irreducible unitary representations (IUR's) of π₁(G) if M is orientable, and with the IUR's of π₁(G)/2π₁(G) if M is non-orientable.     

Influence of alkali metal superoxides on structure,electronic,and optical properties of Be_(12)O_(12) nanocage:Density functional theory study

The effect of alkali metal superoxides M_3O(M = Li,Na,K) on the electronic and optical properties of a Be_(12)O_(12) nanocage was studied by density functional theory(DFT) and time-dependent density functional theory(TD-DFT).The energy gaps(Eg) of all configurations were calculated.Generally,the adsorption of alkali metal superoxides on the Be_(12)O_(12) nanocage causes a decrease of Eg.Electric dipole moment μ,polarizability α,and static first hyperpolarizability β were calculated and it was shown that the adsorption of alkali metal superoxides on Be_(12)O_(12) increases its polarizability.It was found that the absorption of M_3 O on Be_(12)O_(12) nanocluster improves its nonlinear optical properties.The highest first hyperpolarizability(β≈ 214000 a.u.) is obtained in the K_3O–Be_(12)O_(12)nanocluster.The TD-DFT calculations were performed to investigate the origin of the first hyperpolarizabilities and it was shown that a higher first hyperpolarizability belongs to the structure that has a lower transition energy.     

Flat connections and geometric quantization

Using the space of holomorphic symmetric tensors on the moduli space of stable bundles over a Riemann surface we construct a projectively flat connection on a vector bundle over Teichmüller space. The fibre of the vector bundle consists of the global sections of a power of the determinant bundle on the moduli space. Both Dolbeault and ech techniques are used.     

Tau functions and the twistor theory of integrable systems

We first define τ-functions as generalized cross-ratios of four points on a finite- or infinite-dimensional Grassmannian. We show how this definition can be used to construct a natural flat connection on a determinant line bundle associated with two equivariant holomorphic vector bundles over a twistor space, provided that the action of the symmetries on the bundles has the same normal form at the fixed points for the two bundles. The determinant line bundle has a natural meromorphic section of which the logarithmic covariant derivative is the logarithmic derivative of the τ-function. We establish a natural product formula for this τ-function; we show that it vanishes at the jumping lines of one bundle and has poles at the jumping lines of the other. We also show that this definition leads to standard expressions for the τ-functions of the KdV equation, the Ernst equation, and the isomonodromic deformation equations. We describe a new twistor treatment of the isomonodromic deformation equations.     

Flat Complex Vector Bundles,the Beltrami Differential and W-Algebras

Since the appearance of the paper by Bilal et al. in 1991, it has been widely assumed that W-algebras originating from the Hamiltonian reduction of an SL(n,C)-bundle over a Riemann surface give rise to a flat connection, in which the Beltrami differential may be identified. In this Letter, it is shown that the use of the Beltrami parametrization of complex structures on a compact Riemann surface over which flat complex vector bundles are considered, allows the construction of the above mentioned flat connection. It is stressed that the modulus of the Beltrami differential is of necessity less than one, and that solutions of the so-called Beltrami equation give rise to an orientation-preserving smooth change of local complex coordinates. In particular, the latter yields a smooth equivalence between flat complex vector bundles. The role of smooth diffeomorphisms which induce equivalent complex structures is specially emphasized. Furthermore, it is shown that, while the construction given here applies to the special case of the Virasoro algebra, the extension to flat complex vector bundles of arbitrary rank does not provide generalizations of the Beltrami differential usually considered as central objects for such non-linear symmetries.     

Hamiltonian circle actions on symplectic manifolds and the signature

Let M be a symplectic manifold with a Hamiltonian circle action with isolated fixed points. We prove that σ (M) = b₀(M) − b₂(M) + b₄(M) − b₆(M) + … where σ (M) is the signature of M and b_i(M) is the ith Betti number of M.     

基于咔唑-三嗪并吲哚的N-酰腙衍生物的NMR数据归属

N-酰腙类化合物在极性溶剂中存在着E/cis和E/trans两种异构体,本文利用元素分析、红外光谱、1D和2D核磁共振（NMR）波谱（包括¹H NMR、¹³C NMR、¹H-¹H COSY、¹H-¹³C HSQC、¹H-¹³C HMBC和NOESY）等技术,对基于咔唑-三嗪并吲哚的新型N-酰腙衍生物3,即2-（5H-[1,2,4]三嗪[5,6-b]吲哚-3-硫基）-N'-（9-乙基咔唑-3-亚甲基）乙酰肼的两种异构体（E/cis和E/trans）的¹H和¹³C NMR信号进行了全归属,测量了其偶合常数（J值）及两种异构体的含量,确定了其空间结构.     

MC与$lt;i$gt;M$lt;/i$gt;$lt;sub$gt;$lt;i$gt;n$lt;/i$gt;+1$lt;/sub$gt;$lt;i$gt;AC$lt;/i$gt;$lt;sub$gt;$lt;i$gt;n$lt;/i$gt;$lt;/sub$gt;稳定性与电子特征的第一性原理研究

研究MC与M_n+1AC_n(M=Sc, Ti, V, Cr, Mn; A=Al, Si, P, S; n=1, 2, 3)结构的稳定性与电子特征有利于探究三元层状结构M_n+1AC_n稳定性的内在原因和设计新型M_n+1AC_n结构. 第一性原理计算研究表明, M-3d与C-2p轨道间的电子转移对MC与M_n+1AC_n 的形成焓有较大影响. 供电子能力较强的前过渡金属可以形成稳定的MC结构. 计算结果显示, MC结构是缺电子体系, 其趋向于与具有一定供电子能力的MA结构结合形成M_n+1AC_n. 与M₂PC和M₂SC 相比, M₂AlC和M₂SiC可以更为容易地被分离成二维 M₂C结构. 关键词： MAX相结构 第一性原理 电子结构 过渡金属碳化物     

Décomposition du fibré des spineurs d''une variété spin Kähler-quanternionenienne sous l''action de la 4-forme fondamentale

On a spin quaternionic-Kähler manifold M^4m, we give the spectral decomposition of the spin bundle the action of the fundamental 4-form Ω. Moreover, we compute the eigenvalues of Ω which, in the compact case, play an essential role in the problem of estimating the eigenvalues of the Dirac operator. The proof is based on the decomposition of the spin representation into irreducible components under the action of the group Sp(1) × Sp(m). We show that this algebraic results induces a Whitney decomposition of the spin bundle which, when restricted to the fibres, is the spectral decomposition under the action of Ω.     

A Groupoid Approach to Noncommutative T-Duality

Topological T-duality is a transformation taking a gerbe on a principal torus bundle to a gerbe on a principal dual-torus bundle. We give a new geometric construction of T-dualization, which allows the duality to be extended in the following two directions. First, bundles of groups other than tori, even bundles of some nonabelian groups, can be dualized. Second, bundles whose duals are families of noncommutative groups (in the sense of noncommutative geometry) can be treated, though in this case the base space of the bundles is best viewed as a topological stack. Some methods developed for the construction may be of independent interest. These are a Pontryagin type duality that interchanges commutative principal bundles with gerbes, a nonabelian Takai type duality for groupoids, and the computation of certain equivariant Brauer groups. The research reported here was supported in part by National Science Foundation grants DMS-0703718 and DMS-0611653.