BPS invariants for a Knot in Seifert manifolds

Abstract

We calculate homological blocks for a knot in Seifert manifolds when the gauge group is SU(N). We obtain the homological blocks with a given representation of the gauge group from the expectation value of the Wilson loop operator by analytically continuing the Chern-Simons level. We also obtain homological blocks with the analytically continued level and representation for a knot in the Seifert integer homology spheres.

Quantum Modular $\widehat Z{}^G$-Invariants

“…As mentioned in the introduction, Rademacher sum expressions are interesting for many purposes and are often available for holomorphic quantum modular forms of the kind we study here. It would be interesting to systematically develop the Rademacher sum (−2; 1/2, 1/2, 3/5) σ 40 = {1, −, 39} 4,5,7,11,13,14,16,20,22,23,25,29,31,32, 34} (−2; 1/2, 1/2, 4/5) techniques for general quantum modular forms. In terms of the physics on the field theory side, we wish to compare the S 2 × S 1 superconformal indices of the 3d theory T [M 3 ], conjectured to be related to Z by…”

“…Often, Z-invariants admit totally different expressions, arising from realizing M 3 not by plumbing but by surgery along knots [32,40,41,46,64], or from alternative ways of expressing characters of logarithmic vertex algebras [24], leading to interesting q-series identities. While so far the analysis of quantum modularity relies mostly on the connection to lattice theta functions, it will be very interesting if modular properties can also be analyzed directly through these other expressions as well, as they are connected to yet different areas of mathematics and will lead to different applications.…”

“…This novel type of invariants has since been the subject of various recent studies in the literature. See [23,24,25,29,30,31,32,41,42,44,48,59,61,64,65] for some examples. In short, Z b (M 3 ) is physically defined as the half-index (also called vortex partition function) of the threedimensional N = 2 supersymmetric quantum field theory T [M 3 ] obtained by compactifying a six-dimensional N = (2, 0) superconformal field theory on the closed three-manifold M 3 .…”