Abstract
We consider the q-nonabelianization map, which maps links L in a 3-manifold M to combinations of links tilde{L} in a branched N -fold cover tilde{M} . In quantum field theory terms, q-nonabelianization is the UV-IR map relating two different sorts of defect: in the UV we have the six-dimensional (2, 0) superconformal field theory of type mathfrak{gl} (N ) on M × ℝ2,1, and we consider surface defects placed on L × {x4 = x5 = 0}; in the IR we have the (2, 0) theory of type gl (1) on tilde{M} × ℝ2,1, and put the defects on tilde{L} × {x4 = x5 = 0}. In the case M = ℝ3, q-nonabelianization computes the Jones polynomial of a link, or its analogue associated to the group U(N ). In the case M = C × ℝ, when the projection of L to C is a simple non-contractible loop, q-nonabelianization computes the protected spin character for framed BPS states in 4d mathcal{N} = 2 theories of class S. In the case N = 2 and M = C × ℝ, we give a concrete construction of the q-nonabelianization map. The construction uses the data of the WKB foliations associated to a holomorphic covering tilde{C}to C .
Highlights
This paper concerns a geometric construction which we call q-nonabelianization
In quantum field theory terms, q-nonabelianization is the UV-IR map relating two different sorts of defect: in the UV we have the six-dimensional (2, 0) superconformal field theory of type gl(N ) on M × R2,1, and we consider surface defects placed on L × {x4 = x5 = 0}; in the IR we have the (2, 0) theory of type gl(1) on M × R2,1, and put the defects on L × {x4 = x5 = 0}
In the case N = 2 and M = C × R, we give a concrete construction of the q-nonabelianization map
Summary
This paper concerns a geometric construction which we call q-nonabelianization. In short, q-nonabelianization is an operation which maps links on a 3-manifold M to links on a branched N -fold cover M. A bit more precisely, q-nonabelianization is a map of linear combinations of links modulo certain skein relations, encoded in skein modules associated to M and M. In this introduction we formulate the notion of q-nonabelianization rather generally: in particular, we discuss arbitrary N , and a general 3-manifold M. The q-nonabelianization map we describe is related to many previous constructions in the literature, as we discuss in the rest of this introduction, and close to the works [1,2,3] which provided important inspirations for our approach; this paper was motivated by the problem of understanding their constructions in a more covariant and local way. Approach to the Jones polynomial described in [4], section 6.7 of that paper; from that point of view, what we are doing in this paper is explaining a way to replace R3 by C × R
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