Heat Flow for the Yang–Mills–Higgs Field and the Hermitian Yang–Mills–Higgs Metric

For a parameter > 0, we study a type of vortex equations, which generalize the well-known Hermitian–Einstein equation, for a connection A and a section of a holomorphic vector bundle E over a Kähler manifold X. We establish a global existence of smooth solutions to heat flow for a self-dual Yang–Mills–Higgs field on E. Assuming the -stability of (E, ), we prove the existence of the Hermitian Yang–Mills–Higgs metric on the holomorphic bundle E by studying the limiting behaviour of the gauge flow.     

Renormalizability and Unitarity of the Englert–Broute–Higgs–Kibble Model

We show that the Englert–Broute–Higgs–Kibble model is renormalizable and unitary.     

On the constraints problem for the Einstein–Yang–Mills–Higgs system

We provide proofs of some key propositions that were used in previous work by Dossa and Tadmon dealing with the characteristic initial value problem for the Einstein–Yang–Mills–Higgs (EYMH) system. The aforesaid proofs were missing, making the considered work difficult to understand. This work is presented with a view to have an almost self-contained paper. With this respect we completely recall the process of constructing initial data for the EYMH system on two intersecting smooth null hypersurfaces as done in the work of Dossa and Tadmon mentioned above. This is achieved by successfully adapting the hierarchical method set up by Rendall to solve the same problem for the Einstein equations in vacuum and with perfect fluid source. Many delicate calculations and expressions are given in details so as to address, in a forthcoming work, the issue of global resolution of the characteristic initial value problem for the EYMH system. The method obviously applies to the Einstein–Maxwell and the Einstein-scalar field models as well.     

Low-energy vortex dynamics in the self-dual Chern–Simons–Higgs model

The relativistic Chern–Simons–Higgs theory finds application in anyonic superconductivity and contains topological vortices whose dynamics are poorly understood. The gauge fields are defined by a set of nonlinear constraint equations that can be accurately solved with effective Green’s functions, spectral methods, and a discretization scheme using lattice gauge techniques. Simulations show that low-energy two-vortex interactions are elastic with final scattering angles sensitive to vortex velocity; furthermore, vortex pairs form rotating breather states for certain impact parameters. In this study, a function that reproduces scattering angles in the adiabatic limit for nontangential collisions is presented. Simulation results are discussed in the context of analytical methods that extract vortex dynamics from low-energy effective Lagrangians, and a numerical method to calculate the effective Lagrangian is suggested. The numerical techniques used can be applied to the study of other Chern–Simon theories.     

On the Chern–Simons–Higgs vortex equation without gauge fields

This paper is concerned with the nonself-dual Chern–Simons–Higgs model on R²${\mathbb{R}}^2$ with vanishing gauge fields. We prove the existence of radial solutions with the topological boundary condition, and the nonexistence of radial solutions with the nontopological boundary condition. We also establish the asymptotic properties of solutions and derive the quantization of the potential energy.     

Multiple Existence Results for the Self-Dual Chern–Simons–Higgs Vortex Equation

We study the asymptotic behavior for the condensate solutions of the self-dual Chern–Simons–Higgs equation as the Chern–Simons parameter tends to zero. By using these estimates, we establish existence results for solutions of non-topological type.     

Minimizers near the first critical field for the nonself-dual Chern–Simons–Higgs energy

The Chern–Simons–Higgs energy serves as a model for high temperature superconductivity. We show the existence of weak solutions to the CSH equations that are minimizers of the CSH energy. The solutions are vortexless for an applied magnetic field h _ex below the critical field strength, whereas vortices appear when h _ex exceeds the critical field strength. D. Spirn was supported in part by NSF grants DMS-0510121 and DMS-0707714. X. Yan was supported in part by NSF grants DMS-0700966 and DMS-0401048.     

Topological multivortex solutions for the Chern–Simons system with two Higgs particles

In this paper, we prove the uniqueness of topological multivortex solutions to the self-dual abelian Chern–Simons model if either the Chern–Simons coupling parameter is sufficiently small or sufficiently large. In addition, we also establish the sharp region of the flux for nontopological solutions with a single vortex point.     

Geometry of Higgs bundles over elliptic curves related to automorphisms of simple Lie algebras,Calogero–Moser systems,and KZB equations

We construct twisted Calogero–Moser systems with spins as Hitchin systems derived from the Higgs bundles over elliptic curves, where the transition operators are defined by arbitrary finite-order automorphisms of the underlying Lie algebras. We thus obtain a spin generalization of the twisted D’Hoker–Phong and Bordner–Corrigan–Sasaki–Takasaki systems. In addition, we construct the corresponding twisted classical dynamical r-matrices and the Knizhnik–Zamolodchikov–Bernard equations related to the automorphisms of Lie algebras.     

Approximate Hermitian–Yang–Mills structures and semistability for Higgs bundles. I: generalities and the one-dimensional case

We review the notions of (weak) Hermitian–Yang–Mills structure and approximate Hermitian–Yang–Mills structure for Higgs bundles. Then, we construct the Donaldson functional for Higgs bundles over compact K?hler manifolds and we present some basic properties of it. In particular, we show that its gradient flow can be written in terms of the mean curvature of the Hitchin–Simpson connection. We also study some properties of the solutions of the evolution equation associated with that functional. Next, we study the problem of the existence of approximate Hermitian–Yang–Mills structures and its relation with the algebro-geometric notion of semistability and we show that for a compact Riemann surface, the notion of approximate Hermitian–Yang–Mills structure is in fact the differential- geometric counterpart of the notion of semistability. Finally, we review the notion of admissible Hermitian structure on a torsion-free Higgs sheaf and define the Donaldson functional for such an object.     

Higgs标量场的引力坍缩

本文在数值相对论框架中,研究了Higgs标量场的引力坍缩问题,它是由暴涨宇宙模型(Inflation universe)的提出而引起的。一般认为在宇宙的过冷膨胀阶段Higgs场的足够大的起伏会导致引力坍缩。本文用一个模拟模型对热化阶段的黑洞生成给出了一些估计。对于具有宇宙视界尺度的起伏来说,产生黑洞的条件是:起伏处的标量场的演化大约比背景滞后一个膨胀时间尺度。由此产生的黑洞的质量数量级为10⁴g。     

Higgs场的虫洞解

已有不少人讨论过引力场与各种物质场耦合下的虫洞解,有鉴于Ｈｉｇｇｓ场在现代场论中的极端重要性,探求引力场与Ｈｉｇｇｓ场耦合下是否存在虫洞解就是一个有意义的问题,该文指出在一定条件下,虫洞解是存在的。     

Euler, Pisot, Prouhet–Thue–Morse, Wallis and the duplication of sines

We all know Euler’s product ${\prod(1+X^{2^n}) = (1-X)^{-1}}$ and its companion ${\prod(1-X^{2^n}) = \sum \pm X^j}$ , where the sequence of signs is the so-called Prouhet–Thue–Morse automatic sequence. Discussing generalizations of these two formulae, we are led respectively (1) to Wallis’ famous infinite product for π, (2) to a characterization of Pisot numbers, (3) to multigrade equalities and the Prouhet–Tarry–Escott problem, (4) to the product ${\prod_{0\leq j \leq n}{\rm sin}(2^j x)}$ and its sequence of signs as x runs through the intervals ${(j \pi/2^n, (j+1) \pi/2^n)}$ , ${j \in [0, 2^n-1]}$ , (5) and finally to the Gelfond and Newman-Slater product and its generalization ${\prod \sin r^j x}$ , which plays a rôle in several papers when r = 2.     

Ornstein–Uhlenbeck process,Cauchy process,and Ornstein–Uhlenbeck–Cauchy process on a circle

By a simple mathematical method, we obtain the transition probability density functions of the Ornstein–Uhlenbeck process, Cauchy process, and Ornstein–Uhlenbeck–Cauchy process on a circle.     

KZ equation,G-opers,quantum Drinfeld–Sokolov reduction,and quantum Cayley–Hamilton identity

The Lax operator of Gaudin-type models is a 1-form at the classical level. In virtue of the quantization scheme proposed by D. Talalaev, it is natural to treat the quantum Lax operator as a connection; this connection is a partcular case of the Knizhnik–Zamolodchikov connection. In this paper, we find a gauge trasformation that produces the “second normal form,” or the “Drinfeld–Sokolov” form. Moreover, the differential operator nurally corresponding to this form is given precisely by the quantum characteristic polynomial of the Lax operator (this operator is called the G-oper or Baxter operator). This observation allows us to relate solutions of the KZ and Baxter equations in an obvious way, and to prove that the immanent KZ equation has only meromorphic solutions. As a corollary, we obtain the quantum Cayley–Hamilton identity for Gaudin-type Lax operators (including the general case). The presented construction sheds a new light on the geometric Langlands correspondence. We also discuss the relation with the Harish-Chandra homomorphism. Bibliography: 19 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 360, 2008, pp. 246–259.     

Fractional Sobolev,Moser–Trudinger,Morrey–Sobolev inequalities under Lorentz norms

We consider the Sobolev type inequalities under Lorentz norms on bounded open domains for fractional derivatives (−∆) ^s/2 in the following three cases: n > ps, n = ps, and n < ps, whence establishing the weak type Sobolev inequalities, Moser–Trudinger and Morrey–Sobolev inequalities for fractional derivatives in Lorentz norms. Applying these inequalities, we obtain the trace forms of six related functional inequalities. Bibliography: 44 titles.