Asymptotic Heat Kernel Expansion in the Semi-Classical Limit

Let H_(^h/2p) = (^h/_2p)²L +V{H_\hbar = \hbar^{2}L +V}, where L is a self-adjoint Laplace type operator acting on sections of a vector bundle over a compact Riemannian manifold and V is a symmetric endomorphism field. We derive an asymptotic expansion for the heat kernel of H_(^h/2p){H_\hbar} as (^h/_2p) \searrow 0{\hbar \searrow 0}. As a consequence we get an asymptotic expansion for the quantum partition function and we see that it is asymptotic to the classical partition function. Moreover, we show how to bound the quantum partition function for positive (^h/_2p){\hbar} by the classical partition function.     

Octonionic Yang–Mills instanton on quaternionic line bundle of Spin(7) holonomy

The total space of the spinor bundle on the four-dimensional sphere S⁴ is a quaternionic line bundle that admits a metric of Spin(7) holonomy. We consider octonionic Yang–Mills instanton on this eight-dimensional gravitational instanton. This is a higher dimensional generalization of (anti-) self-dual instanton on the Eguchi-Hanson space. We propose an ansatz for Spin(7) Yang–Mills field and derive a system of non-linear ordinary differential equations. The solutions are classified according to the asymptotic behavior at infinity. We give a complete solution when the gauge group is reduced to a product of SU(2) subalgebras in Spin(7). The existence of more general Spin(7) valued solutions can be seen by making an asymptotic expansion.     

Non-Perturbative Heat Kernel Asymptotics on Homogeneous Abelian Bundles

We study the heat kernel for a Laplace type partial differential operator acting on smooth sections of a complex vector bundle with the structure group G × U(1) over a Riemannian manifold M without boundary. The total connection on the vector bundle naturally splits into a G-connection and a U(1)-connection, which is assumed to have a parallel curvature F. We find a new local short time asymptotic expansion of the off-diagonal heat kernel U(t|x, x′) close to the diagonal of M × M assuming the curvature F to be of order t ⁻¹. The coefficients of this expansion are polynomial functions in the Riemann curvature tensor (and the curvature of the G-connection) and its derivatives with universal coefficients depending in a non-polynomial but analytic way on the curvature F, more precisely, on tF. These functions generate all terms quadratic and linear in the Riemann curvature and of arbitrary order in F in the usual heat kernel coefficients. In that sense, we effectively sum up the usual short time heat kernel asymptotic expansion to all orders of the curvature F. We compute the first three coefficients (both diagonal and off-diagonal) of this new asymptotic expansion.     

Spontaneous Symmetry Breaking in Presence of Electric and Magnetic Charges

Starting with the definition of quaternion gauge theory, we have undertaken the study of SU(2) _e ×SU(2) _m ×U(1) _e ×U(1) _m in terms of the simultaneous existence of electric and magnetic charges along with their Yang-Mills counterparts. As such, we have developed the gauge theory in terms of four coupling constants associated with four-gauge symmetry SU(2) _e ×SU(2) _m ×U(1) _e ×U(1) _m . Accordingly, we have made an attempt to obtain the abelian and non-Abelian gauge structures for the particles carrying simultaneously the electric and magnetic charges (namely dyons). Starting from the Lagrangian density of two SU(2)×U(1) gauge theories responsible for the existence of electric and magnetic charges, we have discussed the consistent theory of spontaneous symmetry breaking and Higgs mechanism in order to generate the masses. From the symmetry breaking, we have generated the two electromagnetic fields, the two massive vector W ^± and Z ⁰ bosons fields and the Higgs scalar fields.     

The geometry of a covariant expansion in chiral theories with nucleons

We deal with the geometrical foundations of a covariant scheme, which is developed to give a co-ordinate independent perturbation expansion ofSU(2)×SU(2) chiral theories with pions and nucleons. The pion fields play the role of local co-ordinates on a 3-D manifold with constant curvature in isospace. The presence of the nucleon isospinor field forces us to deal with the problem of endowing the manifold with a spin structure. In this way the nucleon isospinor is accommodated in the fiber space of the principal fiber bundle of the tangent bundle of the manifold.     

The affine geometry of the Lanczos H-tensor formalism

We identify the fiber-bundle-with-connection structure that underlies the Lanczos H-tensor formulation of Riemannian geometrical structure. We consider linear connections to be type (1,2) affine tensor fields, and we sketch the structure of the appropriate fiber bundle that is needed to describe the differential geometry of such affine tensors, namely the affine frame bundleA ₁ ² M with structure groupA ₁ ² (4) =GL(4) T ₁ ^{2 4} over spacetimeM. Generalized affine connections on this bundle are in 1-1 correspondence with pairs(, K) onM, where thegl(4)-component denotes a linear connection and the T ₁ ^{2 4}-componentK is a type (1,3) tensor field onM. We show that the Lanczos H-tensor arises from a gauge fixing condition on this geometrical structure. The resulting translation gauge, theLanczos gauge, is invariant under the transformations found earlier by Lanczos. The other Lanczos variablesQ _mandq are constructed in terms of the translational component of the generalized affine connection in the Lanczos gauge. To complete the geometric reformulation we reconstruct the Lanczos Lagrangian completely in terms of affine invariant quantities. The essential field equations derived from ourA ₁ ² (4)-invariant Lagrangian are the Bianchi and Bach-Lanczos identities for four-dimensional Riemannian geometry.     

异质富勒烯C19M(M=Cr,Mo,W)的结构和电子性质的计算研究

采用密度泛函理论(DFT)中的B3LYP方法, 在LANL2DZ基组水平上, 优化了C₁₉M(M=Cr,Mo,W)团簇的几何结构, 得出了它们的基态构型, 并研究了基态结构的物理化学性能.结果表明: 不同的M原子替换后, C₁₉M的动力学稳定性相差不大, 而热力学稳定性随着M原子序数的增加而逐渐升高; 通过对C₁₉M的前线轨道分析发现, M原子对各个轨道均有一定的贡献, M原子对各个轨道的贡献大致随着M原子序数的增加而增加, C₁₉M中金属原子M(M=Cr,Mo,W)上集中了大量的正电荷; C₁₉M的芳香性随着M原子序数的增加而减弱.     

Asymptotics for the Covariance of the Airy<Subscript>2</Subscript> Process

In this paper we compute some of the higher order terms in the asymptotic behavior of the two point function \mathbbP(A₂(0) ￡ s₁,A₂(t) ￡ s₂)\mathbb{P}(\mathcal {A}_{2}(0)\leq s_{1},\mathcal{A}_{2}(t)\leq s_{2}), extending the previous work of Adler and van Moerbeke (; Ann. Probab. 33, 1326–1361, 2005) and Widom (J. Stat. Phys. 115, 1129–1134, 2004). We prove that it is possible to represent any order asymptotic approximation as a polynomial and integrals of the Painlevé II function q and its derivative q′. Further, for up to tenth order we give this asymptotic approximation as a linear combination of the Tracy-Widom GUE density function f ₂ and its derivatives. As a corollary to this, the asymptotic covariance is expressed up to tenth order in terms of the moments of the Tracy-Widom GUE distribution.     

(Ca3N2)n(n=1—4)团簇结构与性质的密度泛函理论研究

用密度泛函理论(DFT)的杂化密度泛函B3LYP方法在6-31G^*基组水平上对(Ca₃N₂)_n(n=1—4)团簇各种可能的构型进行几何结构优化,预测了各团簇的最稳定结构.并对最稳定结构的振动特性、成键特性、电荷特性和稳定性等进行了理论分析.结果表明,(Ca₃N₂)_n(n=1—4)团簇最稳定构型中N原子为3—5配位,Ca—N键长为0.231—0.251nm,Ca—Ca键长为0.295—0.358nm;N原子的自然电荷在-1.553e—-2.241e之间,Ca原子的自然电荷在1.035e—1.445e之间,Ca和N原子间相互作用呈现较强的离子性,Ca₃N₂和(Ca₃N₂)₃团簇有相对较高的动力学稳定性. 关键词： 3N₂)_n(n=1—4)团簇')" href="#">(Ca₃N₂)_n(n=1—4)团簇 密度泛函理论 结构与性质     

Quantum Group as Semi-infinite Cohomology

We obtain the quantum group SL _q (2) as semi-infinite cohomology of the Virasoro algebra with values in a tensor product of two braided vertex operator algebras with complementary central charges c+[`(c)]=26{c+\bar{c}=26}. Each braided VOA is constructed from the free Fock space realization of the Virasoro algebra with an additional q-deformed harmonic oscillator degree of freedom. The braided VOA structure arises from the theory of local systems over configuration spaces and it yields an associative algebra structure on the cohomology. We explicitly provide the four cohomology classes that serve as the generators of SL _q (2) and verify their relations. We also discuss the possible extensions of our construction and its connection to the Liouville model and minimal string theory.     

Spectral and Scattering Theory¶of Spatially Cut-Off P(ϕ)2 Hamiltonians

We study spatially cut-off P(ϕ)₂ Hamiltonians. We show the local finiteness of the pure point spectrum outside of thresholds, the limiting absorption principle and asymptotic completeness of scattering for such Hamiltonians. Our results imply the absence of singular continuous spectrum. Received: 25 June 1999 / Accepted: 9 February 2000     

A spin-statistics theorem for composites containing both electric and magnetic charges

The present paper states and proves an asymptotic spin-statistics theorem for composites consisting of electrically and magnetically charged particles. We work in the framework of a nonrelativistic theory, taking as the classical configuration space aU(1) bundle over the space of physical configurations, and as the quantum hilbert space the homogeneous square integrable functions on that bundle. The theorems are proved using a formalism we develop here for treating gauge spaces —U(1) bundles with connections; in particular, two products related to tensor products of vector bundles prove to be extremely useful in displaying the structure of the gauge spaces that naturally arise in this theory.Supported in part by the National Science Foundation under grant number PHY 77-07111Supported in part by the National Science Foundation under grant number PHY 78-24275     

Noncommutative Instantons on the 4-Sphere¶from Quantum Groups

We describe an approach to the noncommutative instantons on the 4-sphere based on quantum group theory. We quantize the Hopf bundle ?⁷→?⁴ making use of the concept of quantum coisotropic subgroups. The analysis of the semiclassical Poisson–Lie structure of U(4) shows that the diagonal SU(2) must be conjugated to be properly quantized. The quantum coisotropic subgroup we obtain is the standard SU _q (2); it determines a new deformation of the 4-sphere ∑⁴ _q as the algebra of coinvariants in ? _q ⁷. We show that the quantum vector bundle associated to the fundamental corepresentation of SU _q (2) is finitely generated and projective and we compute the explicit projector. We give the unitary representations of ∑⁴ _q , we define two 0-summable Fredholm modules and we compute the Chern–Connes pairing between the projector and their characters. It comes out that even the zero class in cyclic homology is non-trivial. Received: 3 January 2001 / Accepted: 14 November 2001     

Projective and volume-preserving bundle structures involved in the formulation ofA(4) gauge theories

The bundle structures required by volume-preserving and related projective properties are developed and discussed in the context ofA(4) gauge theories which may be taken as the proper framework for Poincaré gauge theories. The results of this paper include methods for extending both tensors and connections to a principal fiber bundle havingG1(4,R)xG1(4,R) as its structure group. This bundle structure is shown to be a natural arena for the generalized (±) covariant differentiation utilized by Einstein for his extended gravitational theories involving nonsymmetric connections. In particular, it is shown that this generalized (±) covariant differentiation is actually a special case of ordinary covariant differentiation with respect to a connection on theG1(4,R) xG1(4,R) bundle. These results are discussed in relation to certain properties of generalized gravitational theories based on a nonsymmetric connection which include the metric affine theories of Hehl et al. and the general requirement that it should be possible to formulate well-defined local conservation laws. In terms of the extended bundle structure considered in this paper, it is found that physically distinct particle number type conservation expressions could exist for certain given types of matter currents.     

Nuclear mean fields through self-consistent semiclassical calculations

Semiclassical expansions derived in the framework of the Extended Thomas-Fermi approach for the kinetic energy density τ( r) and the spin-orbit density J( r) as functions of the local density ρ( r) are used to determine the central nuclear potentials V _n( r) and V _p( r) of the neutron and proton distribution for effective interactions of the Skyrme type. We demonstrate that the convergence of the resulting semiclassical expansions for these potentials is fast and that they reproduce quite accurately the corresponding Hartree-Fock average fields. Received: 12 February 2000 / Accepted: 14 March 2002     

The electrostatics of Einstein’s unified field theory

When sources are added at their right-hand sides, and g_(ik) is a priori assumed to be the metric, the equations of Einstein’s Hermitian theory of relativity were shown to allow for an exact solution that describes the general electrostatic field of n point charges. Moreover, the injunction of spherical symmetry of g_(ik) in the infinitesimal neighbourhood of each of the charges was proved to yield the equilibrium conditions of the n charges in keeping with ordinary electrostatics. The tensor g_(ik), however, cannot be the metric of the theory, since it enters neither the eikonal equation nor the equation of motion of uncharged test particles. A physically correct metric that rules both the behaviour of wave fronts and of uncharged matter is the one indicated by Hély.In the present paper it is shown how the electrostatic solution predicts the structure of the n charged particles and their mutual positions of electrostatic equilibrium when Hély’s physically correct metric is adopted.     

An extension of the Borel-Weil construction to the quantum groupU q (n)

The Borel-Weil (BW) construction for unitary irreps of a compact Lie group is extended to a construction of all unitary irreps of the quantum groupU _q(n). Thisq-BW construction uses a recursion procedure forU _q(n) in which the fiber of the bundle carries an irrep ofU _q(n–1)×U _q(1) with sections that are holomorphic functions in the homogeneous spaceU _q(n)/U _q(n–1)×U _q(1). Explicit results are obtained for theU _q(n) irreps and for the related isomorphism of quantum group algebras.Supported in part by the National Science Foundation, No. PHY-9008007     

Operator coproduct-realization of quantum group transformations in two-dimensional gravity I

A simple connection between the universalR matrix ofU _q(sl(2)) (for spins 1/2 andJ) and the required form of the coproduct action of the Hilbert space generators of the quantum group symmetry is put forward. This leads us to an explicit operator realization of the coproduct action on the covariant operators. It allows us to derive the expected quantum group covariance of the fusion and braiding matrices, although it is of a new type: the generators depend upon worldsheet variables, and obey a new central extension of theU _q(sl(2)) algebra realized by (what we call) fixed point commutation relations. This is explained by showing on a general ground that the link between the algebra of field transformations and that of the coproduct generators is much weaker than previously thought. The central charges of our extendedU _q(sl(2)) algebra, which includes the Liouville zero-mode momentum in a non-trivial way, are related to Virasoro-descendants of unity. We also show how our approach can be used to derive the Hopf algebra structure of the extended quantum-group symmetry related to the presence of both of the screening charges of 2D gravity.Partially supported by the EC contracts CHRXCT920069 and CHRXCT920035.Unité Propre du Centre National de la Recherche Scientifique, associée à l'École Normale Supérieure et à l'Université de Paris-Sud.     

Quantum Group Symmetry in sine-Gordon and Affine Toda Field Theories on the Half-Line

 We consider the sine-Gordon and affine Toda field theories on the half-line with classically integrable boundary conditions, and show that in the quantum theory a remnant survives of the bulk quantized affine algebra symmetry generated by non-local charges. The paper also develops a general framework for obtaining solutions of the reflection equation by solving an intertwining property for representations of certain coideal subalgebras of U _q (ĝ). Received: 10 December 2001 / Accepted: 7 October 2002 Published online: 19 December 2002