AI 中文总结
该研究在$A_\tau$代数框架下引入全内积与循环陈-西蒙斯形式等概念，将其应用于开格罗莫夫-威滕理论，得到规范不变的超势与$\tau$-模，为后代开格罗莫夫-威滕不变量的定义奠定基础。
AI 中文摘要
我们引入了$A_\tau$代数上全内积的概念，以及与拓扑幂零元相关的循环陈-西蒙斯形式。将全内积应用于循环陈-西蒙斯形式，会得到一个超势函数，该函数在被称为有界上闭链的Maurer-Cartan方程解上是规范不变的，我们计算了该超势的导数。全内积的定义将之前同伦内积概念中出现的严格对称性替换为由无限族协调同伦构成的上同调。我们展示了如何将之前的同伦内积概念作为全内积的特例来恢复，全内积旨在促进后代开格罗莫夫-威滕不变量的定义。全内积与循环陈-西蒙斯形式的定义使用了循环余微分形式的全复形，它为循环同调提供了一个链模型，我们给出了与其他已知模型同伦等价的显式公式。我们还讨论了源于Connes循环复形的$\tau$-迹概念，以及相关的$\tau$-模，它为有界上闭链提供了另一个规范不变函数。在开格罗莫夫-威滕理论中，$\tau$-模用于归一化应用超势的有界上闭链。我们针对一般弯曲巴拿赫$A_\tau$代数给出了定义，并给出了主要结果的含单位与不含单位两种版本。在含单位情形下，我们处理弱有界上闭链，即涉及单位的非齐次Maurer-Cartan方程的解，弱有界上闭链出现在具有非零Maslov类的拉格朗日子流形的开格罗莫夫-威滕理论中，弱有界上闭链处超势的导数与$\tau$-模的导数相关。
英文摘要
We introduce the notion of a total inner product on an $A_\infty$-algebra, and a cyclic Chern-Simons form associated to a topologically nilpotent element. The total inner product applied to the cyclic Chern-Simons form gives a superpotential function that is gauge invariant on solutions of the Maurer-Cartan equation known as bounding cochains. The derivative of the superpotential is computed. The definition of total inner product replaces strict symmetries that appear in previous notions of homotopy inner products with symmetries up to an infinite family of coherent homotopies. We show how to recover previous notions of homotopy inner products as special cases of total inner products. Total inner products are designed to facilitate the definition of descendent open Gromov-Witten invariants. The definitions of the total inner product and the cyclic Chern-Simons form use the total complex of cyclic codifferential forms, which gives a chain model for cyclic homology. We give explicit formulas for homotopy equivalences with other known models. We discuss also the notion of an $\infty$-trace, which arises from Connes' cyclic complex, and the associated $\infty$-modulus, which gives another gauge-invariant function on bounding cochains. In open Gromov-Witten theory, the $\infty$-modulus is used to normalize the bounding cochains to which the superpotential is applied. We develop our definitions for general curved Banach $A_\infty$-algebras, and give both unital and non-unital versions of the main results. In the unital setting, we work with weak bounding cochains, solutions of an inhomogeneous Maurer-Cartan equation involving the unit. Weak bounding cochains arise in the open Gromov-Witten theory of Lagrangian submanifolds with non-vanishing Maslov class. The derivative of the superpotential at a weak bounding cochain is related to the derivative of the $\infty$-modulus.
Comments320 pages, 85 figures
