Branes and supergroups
This bulky article evolved out of previous work by the second author [Quantum Topol. 3, No. 1, 1--137 (2012; Zbl 1241.57041)]. Adopting an elegant geometric approach, this advanced text describes methods for understanding the three-dimensional supergroup Chern-Simon theory and how dualities act on line and surface operators. Many mathematical details are supplied which make the paper reasonably self-contained. The authors consider what they call \textit{a two-sided problem}: a 5-brane of NS type with \(m\) D3-branes ending on it from the left and \(n\) D3-branes from the right. And so the paper is devoted primarily to the study of D3-branes interacting on both sides of a single fivebrane. Contrary to the \textit{one-sided} case, a supergroup \(\mathrm{U}(m|n)\) is associated to the Chern-Simons theory. A dual magnetic description is obtained by \(S\)-duality. The paper is organized as follows. Section 1 provides an introduction, Section 2 describes in detail the relation of the two-sided system to supergroups. In Section 3, line and surface operators are analyzed. Section 4 describes the \(S\)-duality. In Section 5, an O3 plane is incorporated and orthosymplectic groups and the novel duality are discussed. A symmetry-breaking process is analyzed in Section 6, while in Section 7 the magnetic description is lifted to a D4-D6 system and an analogue of Khovanov homology for supergroups is described. Many details are provided in five appendices. This paper is an outstanding resource for graduate students and researchers in mathematical physics.
scientific article