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We introduce supergroup analogues of 3-manifold invariants $hat{Z}$, also known as homological blocks, which were previously considered for ordinary compact semisimple Lie groups. We focus on superunitary groups, and work out the case of SU(2|1) in details. Physically these invariants are realized as the index of BPS states of a system of intersecting fivebranes wrapping a 3-manifold in M-theory. As in the original case, the homological blocks are q-series with integer coefficients. We provide an explicit algorithm to calculate these q-series for a class of plumbed 3-manifolds and study quantum modularity and resurgence properties for some particular 3-manifolds. Finally, we conjecture a formula relating the $hat{Z}$ invariants and the quantum invariants constructed from a non-semisimple category of representation of the unrolled version of a quantum supergroup.
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We study Chern-Simons theory on 3-manifolds M that are circle-bundles over 2-dimensional orbifolds S by the method of Abelianisation. This method, which completely sidesteps the issue of having to integrate over the moduli space of non-Abelian flat connections, reduces the complete partition function of the non-Abelian theory on M to a 2-dimensional Abelian theory on the orbifold S which is easily evaluated.
Motivated by physical constructions of homological knot invariants, we study their analogs for closed 3-manifolds. We show that fivebrane compactifications provide a universal description of various old and new homological invariants of 3-manifolds. In terms of 3d/3d correspondence, such invariants are given by the Q-cohomology of the Hilbert space of partially topologically twisted 3d N=2 theory T[M_3] on a Riemann surface with defects. We demonstrate this by concrete and explicit calculations in the case of monopole/Heegaard Floer homology and a 3-manifold analog of Khovanov-Rozansky link homology. The latter gives a categorification of Chern-Simons partition function. Some of the new key elements include the explicit form of the S-transform and a novel connection between categorification and a previously mysterious role of Eichler integrals in Chern-Simons theory.
We provide a physical definition of new homological invariants $mathcal{H}_a (M_3)$ of 3-manifolds (possibly, with knots) labeled by abelian flat connections. The physical system in question involves a 6d fivebrane theory on $M_3$ times a 2-disk, $D^2$, whose Hilbert space of BPS states plays the role of a basic building block in categorification of various partition functions of 3d $mathcal{N}=2$ theory $T[M_3]$: $D^2times S^1$ half-index, $S^2times S^1$ superconformal index, and $S^2times S^1$ topologically twisted index. The first partition function is labeled by a choice of boundary condition and provides a refinement of Chern-Simons (WRT) invariant. A linear combination of them in the unrefined limit gives the analytically continued WRT invariant of $M_3$. The last two can be factorized into the product of half-indices. We show how this works explicitly for many examples, including Lens spaces, circle fibrations over Riemann surfaces, and plumbed 3-manifolds.
We study 7D maximally supersymmetric Yang-Mills theory on 3-Sasakian manifolds. For manifolds whose hyper-Kahler cones are hypertoric we derive the perturbative part of the partition function. The answer involves a special function that counts integer lattice points in a rational convex polyhedral cone determined by hypertoric data. This also gives a more geometric structure to previous enumeration results of holomorphic functions in the literature. Based on physics intuition, we provide a factorisation result for such functions. The full proof of this factorisation using index calculations will be detailed in a forthcoming paper.
Using the ideas from the BPS/CFT correspondence, we give an explicit recursive formula for computing supersymmetric Wilson loop averages in 3d $mathcal{N}=2$ Yang-Mills-Chern-Simons $U(N)$ theory on the squashed sphere $S^3_b$ with one adjoint chiral and two antichiral fundamental multiplets, for specific values of Chern-Simons level $kappa_2$ and Fayet-Illiopoulos parameter $kappa_1$. For these values of $kappa_1$ and $kappa_2$ the north and south pole turn out to be completely independent, and therefore Wilson loop averages factorize into answers for the two constituent $D^2 times S^1$ theories. In particular, our formula provides results for the theory on the round sphere when the squashing is removed.
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