Taming supersymmetric defects in 3d–3d correspondence

Abstract

We study knots in 3d Chern-Simons theory with complex gauge group SL(N, C), in the context of its relation with 3d N = 2 theory (the so-called 3d-3d correspondence). The defect has either co-dimension 2 or co-dimension 4 inside the 6d (2, 0) theory, which is compactified on a 3-manifold M . We identify such defects in various corners of the 3d-3d correspondence, namely in 3d SL(N, C) Chern-Simons theory, in 3d N = 2 theory, in 5d N = 2 super Yang-Mills theory, and in the M-theory holographic dual. We can make quantitative checks of the 3d-3d correspondence by computing partition functions at each of these theories. This Letter is a companion to a longer paper [1], which contains more details and more results.Introduction.-One lesson from history is that physics and mathematics often develop hand in hand; the development on one side facilitates development in the other, creating a virtuous cycle of feedback. The recently-discovered 3d-3d correspondence [2-7] is a perfect example for this interplay. The correspondence states that there exists a surprising connection between 3d SL(N, C) Chern-Simons (CS) theory defined on a 3-manifold M on the one hand, and a 3d N = 2 supersymmetric gauge theory (which we call T N [M ]) on the other. Being a topological field theory, the SL(N, C) CS theory provides a functor equipped with a Hilbert space. For the N = 2 case, we have the SL(2, C) CS theory and the relevant Hilbert space corresponds to a quantization of the moduli space of SL(2, C)-flat connections on M , and it contains the space of hyperbolic structures on M when M is a hyperbolic manifold. The 3d-3d relation thus unifies and enriches the mathematics of both knot theory and 3d hyperbolic geometry.The physical origin of the 3d-3d correspondence is the compactification of the 6d (2, 0) theory on a closed 3-manifoldM , along which the theory is partially topologically twisted:

Aspects of defects in 3d-3d correspondence

“…We also highlight the non-Abelian description of the 3d N = 2 T N [M ] theory with defect included, as well as its Higgsing prescription and the resulting 'refinement' in complex CS theory. This paper is a companion to our shorter paper [1], which summarizes our main results. arXiv:1510.05011v1 [hep-th]…”

“…(4.22). These equations gives the gluing equations (3.39) for the ideal triangulation for the mapping torus with the identification We choose two central elements to be (1,2) . .…”

Refined 3d-3d correspondence

“…In terms of SL(N, C) Chern-Simons theory, ρ specifies a monodromy defect on C, while the weights λ,λ correspond to Wilson loops in irreducible representations of the subgroup of SL(N, C) left unbroken by the monodromy defect [48,49].…”

Cluster partition function and invariants of 3-manifolds

“…We would like to thank D. Gang, N. Kim and M. Yamazaki for collaboration in the works [3,53] on which this note is based. We also thank M. Gabella and S. Lee for useful discussions.…”