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We explore aspects of the correspondence between Seifert 3-manifolds and 3d $mathcal{N}=2$ supersymmetric theories with a distinguished abelian flavour symmetry. We give a prescription for computing the squashed three-sphere partition functions of such 3d $mathcal{N}=2$ theories constructed from boundary conditions and interfaces in a 4d $mathcal{N}=2^*$ theory, mirroring the construction of Seifert manifold invariants via Dehn surgery. This is extended to include links in the Seifert manifold by the insertion of supersymmetric Wilson-t Hooft loops in the 4d $mathcal{N}=2^*$ theory. In the presence of a mass parameter for the distinguished flavour symmetry, we recover aspects of refined Chern-Simons theory with complex gauge group, and in particular construct an analytic continuation of the $S$-matrix of refined Chern-Simons theory.
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One of the main challenges in 3d-3d correspondence is that no existent approach offers a complete description of 3d $N=2$ SCFT $T[M_3]$ --- 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 $M_3$ and, secondly, is not limited to a particular supersymmetric partition function of $T[M_3]$. In particular, 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. 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 $T[M_3]$, and propose new tools to compute more recent $q$-series invariants $hat Z (M_3)$ in the case of manifolds with $b_1 > 0$. 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.
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 $mathcal{N}=2$ 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 $M_3$. 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.
The 2d gauged linear sigma model (GLSM) gives a UV model for quantum cohomology on a Kahler manifold X, which is reproduced in the IR limit. We propose and explore a 3d lift of this correspondence, where the UV model is the N=2 supersymmetric 3d gauge theory and the IR limit is given by Giventals permutation equivariant quantum K-theory on X. This gives a one-parameter deformation of the 2d GLSM/quantum cohomology correspondence and recovers it in a small radius limit. We study some novelties of the 3d case regarding integral BPS invariants, chiral rings, deformation spaces and mirror symmetry.
We show that the four-dimensional Chern-Simons theory studied by Costello, Witten and Yamazaki, is, with Nahm pole-type boundary conditions, dual to a boundary theory that is a three-dimensional analogue of Toda theory with a novel 3d W-algebra symmetry. By embedding four-dimensional Chern-Simons theory in a partial twist of the five-dimensional maximally supersymmetric Yang-Mills theory on a manifold with corners, we argue that this three-dimensional Toda theory is dual to a two-dimensional topological sigma model with A-branes on the moduli space of solutions to the Bogomolny equations. This furnishes a novel 3d-2d correspondence, which, among other mathematical implications, also reveals that modules of the 3d W-algebra are modules for the quantized algebra of certain holomorphic functions on the Bogomolny moduli space.
Point signature, a representation describing the structural neighborhood of a point in 3D shapes, can be applied to establish correspondences between points in 3D shapes. Conventional methods apply a weight-sharing network, e.g., any kind of graph neural networks, across all neighborhoods to directly generate point signatures and gain the generalization ability by extensive training over a large amount of training samples from scratch. However, these methods lack the flexibility in rapidly adapting to unseen neighborhood structures and thus generalizes poorly on new point sets. In this paper, we propose a novel meta-learning based 3D point signature model, named 3Dmetapointsignature (MEPS) network, that is capable of learning robust point signatures in 3D shapes. By regarding each point signature learning process as a task, our method obtains an optimized model over the best performance on the distribution of all tasks, generating reliable signatures for new tasks, i.e., signatures of unseen point neighborhoods. Specifically, the MEPS consists of two modules: a base signature learner and a meta signature learner. During training, the base-learner is trained to perform specific signature learning tasks. In the meantime, the meta-learner is trained to update the base-learner with optimal parameters. During testing, the meta-learner that is learned with the distribution of all tasks can adaptively change parameters of the base-learner, accommodating to unseen local neighborhoods. We evaluate the MEPS model on two datasets, e.g., FAUST and TOSCA, for dense 3Dshape correspondence. Experimental results demonstrate that our method not only gains significant improvements over the baseline model and achieves state-of-the-art results, but also is capable of handling unseen 3D shapes.
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