Abstract
One of the main challenges in 3d-3d correspondence is that no existent approach offers a complete description of 3d mathcal{N} = 2 SCFT T [M3] — or, rather, a “collection of SCFTs” as we refer to it in the paper — for all types of 3-manifolds that include, for example, a 3-torus, Brieskorn spheres, and hyperbolic surgeries on knots. The goal of this paper is to overcome this challenge by a more systematic study of 3d-3d correspondence that, first of all, does not rely heavily on any geometric structure on M3 and, secondly, is not limited to a particular supersymmetric partition function of T [M3]. In particular, we propose to describe such “collection of SCFTs” in terms of 3d mathcal{N} = 2 gauge theories with “non-linear matter” fields valued in complex group manifolds. As a result, we are able to recover familiar 3-manifold invariants, such as Turaev torsion and WRT invariants, from twisted indices and half-indices of T [M3], and propose new tools to compute more recent q-series invariants hat{Z} (M3) in the case of manifolds with b1> 0. Although we use genus-1 mapping tori as our “case study,” many results and techniques readily apply to more general 3-manifolds, as we illustrate throughout the paper.
Highlights
We propose to describe such “collection of SCFTs” in terms of 3d N = 2 gauge theories with “non-linear matter” fields valued in complex group manifolds
In the special case of 3-manifolds with non-empty toral boundaries, the construction [2] offers the best candidate for the Higgs branch of the 3d N = 2 theory T [M3]. (See [7,8,9,10,11,12] for a discussion in the class of manifolds, mostly with boundary, that will be close to our prime examples here.) This construction does not include abelian flat connections, which are crucial for computing WRT invariants and Floer homology of M3 from T [M3]
For G = U (1) and in many cases for G = SU (2), we find that Mvacua has the interpretation directly in three dimensions, as a target space of 3d N = 2 theory T [M3, G] itself, with non-trivial interacting SCFTs residing at singularities of Mvacua
Summary
By a theorem of Lickorish and Wallace [70, 71], any closed oriented 3-manifold can be obtained by performing an integral Dehn surgery on a link in S3. While the general formula (3.19) was obtained (on the 3-manifold side of 3d-3d correspondence) by extending the derivation in [54] to plumbings with loops, it has a clear interpretation as a half-index (3.1) of the theory T [M3]. It is easy to check that S2 = (ST ) is equivalent to the plumbing graph of I but with a half-twist (meaning that it is −I) Using this dictionary, it is easy to see that the element ST a1 · · · ST an is equivalent to the following tangle: ST a1 ST a2 ...ST an =. If one needs a surgery presentation inside S3, it can be obtained by including an additional 0-surgery as shown on figure 8 This simple dictionary between torus bundles and plumbing graphs was described in (2.11).. From this plumbing graph description of MU , it is straightforward to derive (3.10) and other formulas for genus 1 mapping tori given in subsection 3.1 from (3.19)
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