Guo Han-ying

On the gauge invariance and anomaly-free condition of the Wess-Zumino-Witten effective action

A global anomaly-free condition and a non-abelian gauge invariant Wess-Zumino-Witten effective action with less terms have been found by a systematical method rather than by trial and error. The condition requires the difference between the left- and right-handed Chern-Simon five-forms wrt the gauge group must vanish and it turns out to be the usual condition in the local sense.

Difference Discrete Variational Principle, Euler–Lagrange Cohomology and Symplectic, Multisymplectic Structures II: Euler–Lagrange Cohomology*

In this second paper of a series of papers, we explore the difference discrete versions for the Euler-Lagrange cohomology and apply them to the symplectic or multisymplectic geometry and their preserving properties in both the Lagrangian and Hamiltonian formalisms for discrete mechanics and field theory in the framework of multi-parameter differential approach. In terms of the difference discrete Euler-Lagrange cohomological concepts, we show that the symplectic or multisymplectic geometry and their difference discrete structure-preserving properties can always be established not only in the solution spaces of the discrete Euler-Lagrange or canonical equations derived by the difference discrete variational principle but also in the function space in each case if and only if the relevant closed Euler-Lagrange cohomological conditions are satisfied.

Noncommutative Differential Calculus on Discrete Abelian Groups and Its Applications

A new noncommutative differential calculus on function space of discrete Abelian groups is proposed. The derivatives are introduced with respect to the generators of the groups only. It is applied to discrete symplectic geometry and Hamiltonian systems with as well as the lattice gauge theory on regular lattice.

Difference Discrete Variational Principles, Euler–Lagrange Cohomology and Symplectic, Multisymplectic Structures I: Difference Discrete Variational Principle*

In this first paper of a series, we study the difference discrete variational principle in the framework of multi-parameter differential approach by regarding the forward difference as an entire geometric object in view of noncommutative differential geometry. Regarding the difference as an entire geometric object, the difference discrete version of Legendre transformation can be introduced. By virtue of this variational principle, we can discretely deal with the variation problems in both the Lagrangian and Hamiltonian formalisms to get difference discrete Euler-Lagrange equations and canonical ones for the difference discrete versions of the classical mechanics and classical field theory.

Difference Discrete Variational Principles, Euler?Lagrange Cohomology and Symplectic, Multisymplectic Structures III: Application to Symplectic and Multisymplectic Algorithms

In the previous papers I and II, we have studied the difference discrete variational principle and the Euler–Lagrange cohomology in the framework of multi-parameter differential approach. We have gotten the difference discrete Euler–Lagrange equations and canonical ones for the difference discrete versions of classical mechanics and field theory as well as the difference discrete versions for the Euler–Lagrange cohomology and applied them to get the necessary and sufficient condition for the symplectic or multisymplectic geometry preserving properties in both the Lagrangian and Hamiltonian formalisms. In this paper, we apply the difference discrete variational principle and Euler–Lagrange cohomological approach directly to the symplectic and multisymplectic algorithms. We will show that either Hamiltonian schemes or Lagrangian ones in both the symplectic and multisymplectic algorithms are variational integrators and their difference discrete symplectic structure-preserving properties can always be established not only in the solution space but also in the function space if and only if the related closed Euler–Lagrange cohomological conditions are satisfied.

A Note on Symplectic Algorithm

We present the symplectic algorithm in the Lagrangian formalism for the Hamiltonian systems by virtue of the noncommutative differential calculus with respect to the discrete time and the Euler-Lagrange cohomological concepts. We also show that the trapezoidal integrator is symplectic in certain sense.

On Torsion and Nieh–Yan Form*

Using the well-known Chern–Weil formula and its generalization, we systematically construct the Chern–Simons forms and their generalization induced by torsion as well as the Nieh–Yan (N–Y) forms. We also give an argument on the vanishing of integration of N–Y form on any compact manifold without boundary. A systematic construction of N–Y forms in dimensions is also given.Using the well-known Chern-Weil formula and its generalization, we systematically construct the Chern-Simons forms and their generalization induced by torsion as well as the Nieh-Yan (N-Y) forms. We also give an argument on the vanishing of integration of N-Y form on any compact manifold without boundary. A systematic construction of N-Y forms in D=4n dimension is also given.

Connection Theory of Fibre Bundle and Prolongation Structures of Nonlinear Evolution Equations

It is shown that the proper geometrical framework for the nonlinear evolution equations (NEEs) and the soliton equations should be the fibre bundle theory, the principal bundle and its associated bundle and their connection theory. Based upon the requirement of covariance of the geometrical quantities, a covariant generic geometry theory for the prolongation structures of the NEEs is proposed and the fundamental equations for the prolongation structures are presented. From the fundamental equations it immediately follows that the connections corresponding to these NEEs always flat but with torsion and the covariant formulae satisfied by the conservation quantities associated with these NEEs are obtained. The prolongation structure of the MKdV equation, as an example, is concretely worked out by means of the covariant theory of the prolongation structure presented in this paper.

A Formulation of Nonlinear Gauge Theory and its Applications

In the lecture notes[1] one of us (LU) has studied the relations between the theory of connections and Gauge theory. But the formulation of the connection theory or Gauge theory in the lecture notes is linear in nature, i.e., we mainly studied the covariant derivative of a vector, tensor or spinor fields. More precisely, given a differentiable fibre bundle with base space , fibre , structure group G and transition functions , where and belong to an open covering we say a connection defined on is linear if is a linear space and acts linearly on otherwise we say is non-linear.

CHERN-SIMONS TYPE CHARACTERISTIC CLASSES

It is shown that one can generalize the concept of well-known Chern-Simons secondary characteristic classes to introducing a sequence of new characteristic classes named the Chern-Simons type characteristic classes. One of the important properties is given by a theorem which can be interpreted as that the exterior differential of a th Chern-Simons type characteristic class is exactly equal to the coboundary of the cochain of the -1st Chern-Simons type characteristic classes.