q-nonabelianization for line defects
Andrew Neitzke, Fei Yan
Abstract
A bstract We consider the q - nonabelianization map, which maps links L in a 3-manifold M to combinations of links $$ \tilde{L} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>L</mml:mi> <mml:mo>˜</mml:mo> </mml:mover> </mml:math> in a branched N -fold cover $$ \tilde{M} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>M</mml:mi> <mml:mo>˜</mml:mo> </mml:mover> </mml:math> . 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 $$ \mathfrak{gl} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>gl</mml:mi> </mml:math> ( N ) on M × ℝ 2 , 1 , and we consider surface defects placed on L × { x 4 = x 5 = 0}; in the IR we have the (2 , 0) theory of type gl (1) on $$ \tilde{M} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>M</mml:mi> <mml:mo>˜</mml:mo> </mml:mover> </mml:math> × ℝ 2 , 1 , and put the defects on $$ \tilde{L} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>L</mml:mi> <mml:mo>˜</mml:mo> </mml:mover> </mml:math> × { x 4 = x 5 = 0}. In the case M = ℝ 3 , 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 $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 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 $$ \tilde{C}\to C $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>C</mml:mi> <mml:mo>˜</mml:mo> </mml:mover> <mml:mo>→</mml:mo> <mml:mi>C</mml:mi> </mml:math> .