非对易torus上的新孤子解

利用在非对易可积torus(环)上的算子都有约化矩阵这一特点, 孤子解的求解问题可以化为求满足代数方程Q(M)=0的有限维矩阵解问题. 本文研究了当矩阵M不可对角化时的情形, 分析这种情形,得到当势函数V(phi)具有三阶以上的极值点时, 有限维矩阵方程V′(M)=0存在不可对角化的矩阵解. 研究了这种解的一般形式, 并通过kq表象, 构造了非对易整环上以上述矩阵解为约化矩阵的新孤子解. 根据这种构造方法, 可以得到非对易orbifold上的新孤子解.     

Time Flow in a Noncommutative Regime

We develop an approach to dynamical and probabilistic properties of the model unifying general relativity and quantum mechanics, initiated in the paper (Heller et al. (2005) International Journal Theoretical Physics 44, 671). We construct the von Neumann algebra of random operators on a groupoid, which now is not related to a finite group G, but is the pair groupoid of the Lorentzian frame bundle E over spacetime M. We consider the time flow on depending on the state . The state defining the noncommutative dynamics is assumed to be normal and faithful. Then the pair is a noncommutative probabilistic space and can also be interpreted as an equilibrium thermal state, satisfying the Kubo-Martin-Schwinger condition. We argue that both the “time flow” and thermodynamics have their common roots in the noncommutative unification of dynamics and probability.     

c=3, N=2 超共形场论的Z2 Orbifold-Prime模型

讨论了二维环面上中心荷c=3, N=2 的超共形场论. 特别给出该理论的配分函数. 进一步,为了产生新的模型,回顾了一般的orbifold方法. 然后构造了模不变的Z2 Orbifold-Prime模型.     

Theta Functions on Noncommutative Tori

Ordinary theta functions can be considered as holomorphic sections of line bundles over tori. We show that one can define generalized theta functions as holomorphic elements of projective modules over noncommutative tori (theta vectors). The theory of these new objects is not only more general, but also much simpler than the theory of ordinary theta-functions. It seems that the theory of theta vectors should be closely related to Manin's theory of quantized theta functions, but we don't analyze this relation.     

Instantons on General Noncommutative R4

We study the U(1) and U(2) instanton solutions of gauge theory on general noncommutative R4. In allcases considered we obtain explicit results for the projection operators. In some cases we computed numerically theinstanton charge and found that it is an integer independent of the noncommutative parameters θ1,2.     

Gauge Theory on a Discrete Noncommutative Space

In this paper, we apply Connes' noncommutative geometry and the Seiberg—Witten map to a discrete noncommutative space consisting of n copies of a given noncommutative space R ^m . The explicit action functional of gauge fields on this discrete noncommutative space is obtained.     

Spinor Representation of Electromagnetic Fields on Noncommutative Spaces

We apply Campolattaro's spinor representation of the electromagnetic field to noncommutative spaces. The spinor representation of the (self-dual) electromagnetic field on noncommutative spaces is obtained.     

On Spin Structures and Dirac Operators on the Noncommutative Torus

We find and classify possible equivariant spin structures with Dirac operators on the noncommutative torus, proving that, similarly as in the classical case, the spectrum of the Dirac operator depends on the spin structure.     

Functional Integrals and Quantum Fluctuations on Two-Dimensional Noncommutative Space-Time

The generalized Thirring model with impurity coupling is defined on two-dimensional noncommutative space-time, a modified propagator and free energy are derived by means of functional integrals method. Moreover, quantum fluctuations and excitation energies are calculated on two-dimensional black hole and soliton background.     

Integrable Noncommutative Equations on Quad-graphs. The Consistency Approach

We extend integrable systems on quad-graphs, such as the Hirota equation and the cross-ratio equation, to the noncommutative context, when the fields take values in an arbitrary associative algebra. We demonstrate that the three-dimensional consistency property remains valid in this case. We derive the noncommutative zero curvature representations for these systems, based on the latter property. Quantum systems with their quantum zero curvature representations are particular cases of the general noncommutative ones.     

Vacuum Solutions of Classical Gravity on Cyclic Groups from Noncommutative Geometry

Based on the observation that the moduli of a link variable on a cyclic group modify Connes‘ distance on this group,we construct several action functionals for this link variable within the framework of noncommutative geometry.After solving the equations of motion,we find that one type of action gives nontrivial vacuum solution for gravity on this cyclic group in a broad range of coupling constants and that such a solution can be expressed with Chebyshev‘s polynomials.     

Noncommutative Ricci Curvature and Dirac Operator on C q [SL2] at Roots of Unity

We find a unique torsion free Riemannian spin connection for the natural Killing metric on the quantum group C _q [ SL₂], using a recent frame bundle formulation. We find that its covariant Ricci curvature is essentially proportional to the metric (i.e. an Einstein space). We compute the Dirac operator and find for q an odd rth root of unity that its eigenvalues are given by q-integers [m] _q for m=0,1...,r–1 offset by the constant background curvature. We fully solve the Dirac equation for r=3.     

双芯耦合光纤中高阶色散对光孤子相互作用的影响

利用变分原理对耦合纤芯中传输光孤子之间的相互作用和形成束缚孤子态的条件进行了研究，发现高阶色散减弱甚至可以消除光孤子之间的相互作用。这一结果为耦合光纤器件的设计提供了一种实用消除光孤子之间相互作用的方法。     

Noncommutative Cohomology and Electromagnetism on $$\mathbb{C}$$ q [SL2] at Roots of Unity

We compute the noncommutative de Rham cohomology for the finite dimensional q-deformed coordinate ring _q [SL₂] at odd roots of unity and with its standard four-dimensional differential structure. We find that H ⁱ is highly nontrivial compared to case of generic q and has as many modes 1 : 4 : 6 : 4 : 1 as the dimension of the exterior algebra. We solve the spin-0 and Maxwell theory on _q [SL₂], including a complete picture of the self-dual and anti-self dual solutions and of Lorentz and temporal gauge fixing. The system behaves in fact like a noncompact space with self-propagating modes (i.e., in the absence of sources). We also solve with examples of electric and magnetic sources including the biinvariant element H ¹ which we find can be viewed as a source in the local (Minkowski) time-direction (i.e. a uniform electric charge density).     

Z2-Graded Cocycles in Higher Dimensions

Current superalgebras and corresponding Schwinger terms in 1 and 3 space dimensions are studied. This is done by generalizing the quantization of chiral fermions in an external Yang–Mills potential to the case of a Z₂-graded potential coupled to bosons and fermions.     

Solitons and their resonances on two-dimensional superfluid films

The dynamics of saturated two-dimensional superfluid⁴He films is shown to be governed by the Kadomtsev-Petviashvili equation with negative dispersion. It is established that the phenomena of soliton resonance could be observed in such films. Under the lowest order nonlinearity, such resonance would happen only if two dimensional effects are taken into account. The amplitude and velocity of the resonant soliton are obtained.     

二维三角光学格子中的PT光孤子研究

研究了基于自散焦克尔非线性的PT对称三角格子中双峰光孤子的存在性及稳定性。采用改进的平方算子法(MSOM)迭代计算出孤子的数值解,发现双峰之间具有相同相位的双峰光孤子存在于第一带隙,并且可以在某个范围内稳定传输。由傅里叶配点法得到的线性稳定性与非线性模拟传输的结果是一致的。此外,这种PT对称的双峰光孤子对入射角度非常敏感,从不同角度入射的光孤子具有不同的传输特性。