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

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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.