3-manifolds and 3d indices
On 100 pages the space of 3-dimensional superconformal field theories (SCFT) is explored. The main concern is with the case of \(N=2\) SCFTs. This class of theories includes the effective theories of M5-branes wrapped on 3-dimensional manifolds, as discussed by the authors in their previous work. Most SCFTs can be defined as the IR fixed points of general supersymmetric gauge theories coupled to matter and more generally comprises theories admitting a UV description of abelian Chern-Simons theories with superpotential. One result of this paper is to establish an invariant of SCFTs, the refined index, an index which is defined by some deformation of the Lagrangian, allowing a supersymmetric compactification on the 2-sphere. The article has six parts. Section 1 provides us with a careful introduction. Section 2 describes the general form of the index for SCFTs with a \(U(1)^N\) global symmetry and introduces line operators that play some special role in this article. An action of the group \(\mathrm{Sp}(2N, \mathbb Z)\) on SCFTs ins introduced. Section 3 the authors specialize their analysis to the so-called tetrahedron index. In Section 4 they proceed to write down gluing rules explicitly. The index from four dimensions and line operators is the subject of Section 5. Here the group \(\mathrm{Sp}(2N, \mathbb Z)\) is identified as the group of electromagnetic dualities. There are various hints that the discussed indices may have some relation to complex Chern-Simons theory with complex gauge group \(\mathrm{SL}(2,\mathbb C)\) and so the goal of Section 6 is to make this relation as precise as possible. The entire article is well-structured and clearly written, addressing pertinent issues of many previous papers (78 of them are cited).
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