EAPSI: Exploring the relation between four-dimensional and two-dimensional quantum field theories

基本粒子的相互作用是用量子场论的框架来描述的。当粒子相互作用弱时，可以进行与实验结果一致的精确计算。然而，对于强相互作用粒子的动力学——比如夸克结合形成质子——仍然缺乏令人满意的理论理解。这个问题在具有超对称性的量子场论中变得更加容易处理，在超对称性中可以进行许多精确的计算。本研究将集中于最近发现的四维超对称量子场论与一类二维量子场论之间的关系。这种关系是非常有用的，因为在四维理论中非常困难的计算在二维理论中往往变得简单和算法化。本研究将对已知的4d/2d关系进行推广，从而产生新的4d/2d对。了解更多的例子——以及所涉及的数学结构——应该有助于构建4d/2d对的算法。这项研究将与日本东京大学该领域的主要专家立川裕二教授合作进行。具体来说，本课题将研究具有规范群SU(N)的4d N=2超对称规范理论中，在表面算子存在下，以N的配分标记的瞬时配分函数。我们推测，在纯规范理论中，瞬时配分函数等于w代数的Verma模中某相干态的范数，该模是由对应于N的同一配分的量子Drinfeld-Sokolov约简得到的。仿射sl(N)李代数。对于尚未研究的例子，这个猜想将通过显式计算加以验证。这个项目也将扩展对应到颤振规范理论。NSF EAPSI奖是与日本科学促进会合作资助的。

The interactions of elementary particles are described using a framework called quantum field theory. When particles interact weakly, precise calculations can be made which agree with experimental results. However, a satisfactory theoretical understanding of the dynamics of strongly-interacting particles - such as the binding of quarks to form a proton - is still lacking. This problem becomes much more tractable in quantum field theories with supersymmetry, in which many exact computations can be made. This research will focus on recently-discovered relations between supersymmetric quantum field theories in four dimensions and a class of quantum field theories in two dimensions. Such relations are very useful, as computations which are very difficult in the four-dimensional theory often become simple and algorithmic in the two-dimensional theory. This research will study a generalization of a well-known 4d/2d relation, which will lead to new 4d/2d pairs. Understanding more examples - and the mathematical structures involved - should help lead to an algorithm to construct 4d/2d pairs. This research will be done in collaboration with Professor Yuji Tachikawa, a leading expert in the field, at the University of Tokyo, Japan.Specifically, this project will study the instanton partition function of 4d N=2 supersymmetric gauge theory, with gauge group SU(N), in the presence of a surface operator, labeled by a partition of N. It has been conjectured that, in the pure gauge theory, the instanton partition function is equal to the norm of a certain coherent state in the Verma module of a W-algebra, obtained by the quantum Drinfeld-Sokolov reduction corresponding to the same partition of N, of the affine sl(N) Lie algebra. This conjecture will be checked, by explicit computation, for examples which have not yet been studied. This project will also extend the correspondence to quiver gauge theories. This NSF EAPSI award is funded in collaboration with the Japan Society for the Promotion of Science.