Abstract
By incorporating higher-form symmetries, we propose a refined definition of the theories obtained by compactification of the 6d (2, 0) theory on a three-manifold M3. This generalization is applicable to both the 3d mathcal{N} = 2 and mathcal{N} = 1 supersymmetric reductions. An observable that is sensitive to the higher-form symmetries is the Witten index, which can be computed by counting solutions to a set of Bethe equations that are determined by M3. This is carried out in detail for M3 a Seifert manifold, where we compute a refined version of the Witten index. In the context of the 3d-3d correspondence, we complement this analysis in the dual topological theory, and determine the refined counting of flat connections on M3, which matches the Witten index computation that takes the higher-form symmetries into account.
Highlights
The compactification of higher dimensional Quantum Field Theories has led to a deeper understanding of the physical properties of the lower dimensional theories, especially their dualities and symmetries
By incorporating higher-form symmetries, we propose a refined definition of the theories obtained by compactification of the 6d (2, 0) theory on a three-manifold M3
These theories are often referred to as T [M3]; more explicitly, we can write TN =1,2[M3, g], when we need to specify the additional data. These theories depend on the topological manifold M3, and many of their detailed properties can be understood in terms of the topology of M3
Summary
The compactification of higher dimensional Quantum Field Theories has led to a deeper understanding of the physical properties of the lower dimensional theories, especially their dualities and symmetries. We find that the refined Witten index of T [M3, g] in a sector of fixed 1-form symmetry charges maps to the number of flat GC connections on M3 with prescribed values for their second Stiefel-Whitney class and behavior under large gauge transformations This can be understood by reversing the order of compactification and studying the 4d N = 4 Super Yang-Mills theory with gauge group G. We begin our study of the theories T [M3] obtained by compactification of the 6d (2, 0) theory on a Seifert, or more generally, a graph manifold, M3.5 We introduce some of the basic concepts, such as the topological twist, the quiver description of the resulting 3d theories, and dualities. We illustrate these in the case of the simplest non-abelian groups, U(2) and SU(2)
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