L_infinity algebras, the Batalin-Vilkovisky quantisation, and gauge theory

该项目涉及规范理论的基本数学结构。特别地，人们已经认识到，任何拉格朗日场论都有一个基本的L_infinity代数来控制它的动力学。L_infinity代数，又称强同伦李代数，是李代数的高级（在高级范畴论意义下）推广。例如，对于L_infinity代数，Jacobi恒等式只允许在高阶相干同伦下成立。这些基本L_infinity代数的起源在Batalin-Vilkovisky量子化中最容易理解，而Batalin-Vilkovisky量子化又是Becchi-Rouet-Stora-Tyrinsky量子化的推广。在这类代数的背景下，我们可以看到，物理等价的理论是由它们的基础L_infinity代数之间的L_infinity拟同构。该项目涉及规范理论散射振幅，可积性和重正化在这个框架内的工作，在超对称杨米尔斯理论的背景下，更高的类别理论的研究。

The project deals with the underlying mathematical structures of gauge theory. In particular, it has been realised that any Lagrangian field theory has an underlying L_infinity algebra governing its dynamics. L_infinity algebras, also known as strong homotopy Lie algebras, are higher (in the sense of higher category theory) generalisations of Lie algebras. For instance, for L_infinity algebras the Jacobi identity is allowed to hold only up to higher coherent homotopy. The origin of these underlying L_infinity algebras is most easily understood in the Batalin-Vilkovisky quantisation which, in turn, is a generalisation of the Becchi-Rouet-Stora-Tyutin quantisation. Within the context of such algebras it is seen that physically equivalent theories are related by L_infinity quasi-isomorphisms between their underlying L_infinity algebras. The project deals with studies of gauge theory scattering amplitudes, integrability, and renormalisation within this frame work of higher category theory in the context of supersymmetric Yang-Mills theories.