Higher-form symmetries, Bethe vacua, and the 3d-3d correspondence

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 M 3 . This generalization is applicable to both the 3d N = 2 and 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 M 3 . This is carried out in detail for M 3 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 M 3 , which matches the Witten index computation that takes the higher-form symmetries into account.for which such 3d-3d duals have been discussed are the squashed three-sphere S 3 b [1, 2, 6], the superconformal index on S 2 × S 1 [3], and the twisted index on Σ g × S 1 [7]. The special case W 3 = T 3 was considered in [5,7] and computes the (regularized) Witten index [8] I = Tr(−1) F . 1 However, an important characteristic of the theories has so far been largely ignored: their higher-form symmetries [12]. Namely, the theory T [M 3 ] has a higher-form symmetry, which, as with other properties of T [M 3 ], is determined by the topology of M 3 . In fact, the theory T [M 3 ] is not fully specified by the manifold, M 3 , but requires additional topological data. This is related to the fact that the 6d N = (2, 0) theory itself is a relative QFT, i.e., it is only well-defined as the boundary of a 7d TQFT [13,14]. Equivalently, its observables depend on a choice of polarization, i.e., a choice of maximal isotropic subgroup of H 3 (M 3 , Z G ), where Z G is the center of the simply connected group, G, with Lie algebra g. Naturally, we expect that T [M 3 ] also depends on the polarization, as is the case for 4d theories [15,16]. In fact, we will see that this additional information translates into the residual 0-and 1-form symmetry of the 3d theory. We propose therefore a refined definition of the theories, which specifies this dataThis theory has a discrete 0-form (ordinary) symmetry group H. Its residual 1-form symmetry is given by the complementary subgroup, Υ H , 2 inside H 1 (M 3 , Z G ). We show that the choice of H can indeed be detected by the Witten index or, more generally, by the partition function on any W 3 with non-trivial homology. Thus, the different theories in (1.2) are indeed physically distinct.The main interest of this paper is to develop a sound definition of the theories T [M 3 , g, H]in (1.2) for M 3 a graph manifold [17], a class of three-manifolds we review in section 2.1. These manifolds, which also occur as the boundary of plumbed four-manifolds, are sometimes called plumbed three-manifolds, a special case of which are Seifert manifolds. Similar Lagrangians for the 3d theories associated to these manifolds were studied in [6,[18][19][20][21][22][23].In the following we point out new feat...

Duality and mock modularity

“…Note that when Λ is not self-dual, Γ is not self-dual even for a closed four-manifold. This implies that the effective 2d theory is not absolute but relative, meaning that instead of a single partition function it has a vector of partition functions labelled by the elements of the coset [42][43][44][45]…”

3d-3d correspondence for mapping tori

“…they preserve only 3d N = 2 supersymmetry. 33 It would be interesting to understand such N = 2 walls and, possibly, make contact with interesting recent work [107], where a particular choice was used. Arrangements of walls should obey equivalence relations (Kirby moves) represented by either known or new dualities of 3d N = 2 theories.…”

Star-shaped quiver theories with flux

“…Gauging SUð2Þ with level one Chern-Simons term then in the dual theory corresponds to gauging this symmetry for one of the T½SUð2Þ s. It is easier to perform this gauging in the mirror of T½SUð2Þ which has manifest SUð2Þ flavor symmetry. Using a known USpð2Þ duality [50,71,72], this can be shown to be a topological theory equivalent to a contact term for the other SUð2Þ with level one, tensored with a decoupled Uð1Þ 2 topological CS theory (see also [44]), as in Fig. 8.…”

“…Thus, if one believes the duality between the free trifundamental chiral field and the star-shaped quiver one derives the dual description with manifest symmetry. There is also a geometric way to understand this statement following the work of [21,74], and this can also be embedded in a more general set of dualities following [44]. The T½SUð2Þ theory is the three-dimensional model living on the duality wall of N ¼ 4 SUð2Þ gauge theory.…”