Homological algebra of Feynman graphs

费曼图的同调代数

• 批准号: EP/J008451/1
• 负责人: Andrey Lazarev
• 金额: $ 26.7万
• 依托单位: Lancaster University
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2012
• 资助国家: 英国
• 起止时间: 2012 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FJ008451%2F1
• 关键词: Homological; algebra; Feynman; graphs

The area of the proposed research is at the junction of several branches of pure mathematics and mathematical physics. It follows the pattern of applying the physical intuition and ideas to solving mathematical problems which has been a characteristic feature of many groundbreaking developments in algebra in geometry in the last two decades.The project has two closely related themes. One purports to link two deep algebraic constructions which have been extensively studied in their own right. The first is the calculation of the Chevalley-Eilenberg cohomology of infinite matrices with values in an algebra and the second is the calculation of the Chevalley-Eilenberg cohomology of the infinite-dimensional algebra of noncommutative hamiltonians. One conjectural application is the construction of an algebraic version of chain level Gromov-Witten invariants.The second theme derives its motivation from the general problem of quantizing field theories defined classically, i.e. in terms of an action functional. More precisely, a classical field theory is modelled as a certain algebraic structure, called L-infinity algebra which is determined by a certain polynomial or power series function, and one studies other algebraic structures derived by integrating this function. The application will include topological theories of Chern-Simons type and Poisson sigma-models.

建议的研究领域是在纯数学和数学物理的几个分支的交界处。它遵循应用物理直觉和思想来解决数学问题的模式，这是过去二十年来代数几何中许多突破性发展的特征。一个旨在连接两个深代数结构已被广泛研究自己的权利。第一个是计算的Chevalley-Eilenberg上同调的无限矩阵的值在一个代数和第二个是计算的Chevalley-Eilenberg上同调的无限维代数的非交换汉密尔顿。第二个主题的动机来自于经典定义的量子化场论的一般问题，即用作用泛函来定义场论。更确切地说，经典场论被建模为某种代数结构，称为L-无穷代数，由某种多项式或幂级数函数确定，并研究通过积分此函数导出的其他代数结构。应用将包括陈-西蒙斯型和泊松σ模型的拓扑理论。

项目成果

Unimodular homotopy algebras and Chern-Simons theory

幺模同伦代数和 Chern-Simons 理论

• DOI: 10.1016/j.jpaa.2015.05.017
• 发表时间: 2015
• 期刊: Journal of Pure and Applied Algebra
• 影响因子: 0.8
• 作者: Braun C

Maurer-Cartan moduli and models for function spaces

功能空间的 Maurer-Cartan 模数和模型

• DOI: 10.1016/j.aim.2012.11.009
• 发表时间: 2013
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: Lazarev A

Cocommutative coalgebras: homotopy theory and Koszul duality

共交换余代数：同伦理论和 Koszul 对偶性

• DOI: 10.4310/hha.2016.v18.n2.a17
• 发表时间: 2016
• 期刊: Homology, Homotopy and Applications
• 作者: Chuang J

Maurer-Cartan moduli and theorems of Riemann-Hilbert type

Maurer-Cartan 模量和 Riemann-Hilbert 型定理

• DOI: 10.48550/arxiv.1802.02549
• 发表时间: 2018
• 期刊: arXiv e-prints
• 作者: Chuang Joseph

Homotopy BV algebras in Poisson geometry

泊松几何中的同伦 BV 代数

• DOI: 10.1090/s0077-1554-2014-00216-8
• 发表时间: 2014
• 期刊: Transactions of the Moscow Mathematical Society
• 作者: Braun C