We compute the
PSL(2,C) Chern-Simons partition function of a closed 3-manifold obtained from Dehn fillings of the link complement
S3\L, where $\mathcal{L}=\mathcal{K}# H$ is the connected sum of the knot
K with the Hopf link
H. Motivated by our earlier work on topological entanglement and the reduced density matrix
σ for such link complements, we wanted to determine a choice of Dehn filling so that the trace of the matrix
σ becomes equal to the
PSL(2,C) partition function of the closed 3-manifold. We use the SnapPy program and numerical techniques to show this equivalence up to the leading order. We have given explicit results for all hyperbolic knots
K up to six crossings.
Quantum disordering a discrete-symmetry-breaking state by condensing domain walls can lead to a trivial symmetric insulator state. In this work, we show that if we bind a one-dimensional representation of the symmetry (such as a charge) to the intersection point of several domain walls, condensing such modified domain walls can lead to a nontrivial symmetry-protected topological (SPT) state. This result is obtained by showing that the modified domain-wall condensed state has a nontrivial SPT invariant, the symmetry-twist-dependent partition function. We propose two different kinds of field theories that can describe the above-mentioned SPT states. The first one is a Ginzburg-Landau--type nonlinear sigma model theory, but with an additional multikink domain-wall topological term. Such theory has an anomalous
Uk(1) symmetry but an anomaly-free
ZNk symmetry. The second one is a gauge theory, which is beyond Abelian Chern-Simons/BF gauge theories. We argue that the two field theories are equivalent at low energies. After coupling to the symmetry twists, both theories produce the desired SPT invariant.