论文标题
正则分解几何形状和迈向浮动理论的第一步
Regularized polysymplectic geometry and first steps towards Floer theory for covariant field theories
论文作者
论文摘要
本文的目的是介绍定义哈密顿浮子理论的协变田理论的第一步，从而相对地对待时间和空间。尽管已经存在许多用于概括象征性几何形状的协变场理论的竞争几何框架，但由于相应的动作函数太变态了，因此没有一个很容易适合于诸如Hamiltonian浮动理论的变异技术。取而代之的是，我们展示了桥梁引入的正规化过程如何导致一个新的几何框架，我们可以证明，有限的能量$ l^2 $ - 级别的速率线（称为浮子曲线）渐近地收敛到时空周期性解决方案。作为一个具体的示例，我们证明了对于本新框架中定义的一类耦合的粒子场系统，我们证明了浮曲线的存在，也是时空周期性解决方案的存在。
It is the goal of this paper to present the first steps for defining the analogue of Hamiltonian Floer theory for covariant field theory, treating time and space relativistically. While there already exist a number of competing geometric frameworks for covariant field theory generalizing symplectic geometry, none of them are readily suitable for variational techniques such as Hamiltonian Floer theory, since the corresponding action functionals are too degenerate. Instead, we show how a regularization procedure introduced by Bridges leads to a new geometric framework for which we can show that the finite energy $L^2$-gradient lines of the corresponding action functional, called Floer curves, converge asymptotically to space-time periodic solutions. As a concrete example we prove the existence of Floer curves, and hence also of space-time periodic solutions, for a class of coupled particle-field systems defined in this new framework.
