We prove$L$^{$q$}-inequalities for the gradient of the Green potential ($Gf$) in bounded, connected NTA-domains in$R$^{$n$},$n$≥2. These domains may have a highly non-rectifiable boundary and in the plane the set of all bounded simply connected NTA-domains coincides with the set of all quasidiscs. We get a restriction on the exponent$q$for which our inequalities are valid in terms of the validity of a reverse Hölder inequality for the Green function close to the boundary.

In these notes we collect some results about finite-dimensional representations of $U_{q}(\mathfrak {gl}(1\mid1))$ and related invariants of framed tangles, which are well-known to experts but difficult to find in the literature. In particular, we give an explicit description of the ribbon structure on the category of finite-dimensional $U_{q}(\mathfrak {gl}(1\mid1))$ -representations and we use it to construct the corresponding quantum invariant of framed tangles. We explain in detail why this invariant vanishes on closed links and how one can modify the construction to get a non-zero invariant of framed closed links. Finally we show how to obtain the Alexander polynomial by considering the vector representation of $U_{q}(\mathfrak {gl}(1\mid1))$ .

Bosonic topological insulators (BTI) in three dimensions are symmetry-protected topological phases (SPT) protected by time-reversal and boson number conservation symmetries. BTI in three dimensions were first proposed and classified by the group cohomology theory which suggests two distinct root states, each carrying a Z2 index. Soon after, surface anomalous topological orders were proposed to identify different root states of BTI, which even leads to a new BTI root state beyond the group cohomology classification. In this paper, we propose a universal physical mechanism via vortex-line condensation from a 3d superfluid to achieve all three root states. It naturally produces bulk topological quantum field theory (TQFT) description for each root state. Topologically ordered states on the surface are rigorously derived by placing TQFT on an open manifold, which allows us to explicitly demonstrate the bulk-boundary correspondence. Finally, we generalize the mechanism to ZN symmetries and discuss potential SPT phases beyond the group cohomology classification.

No abstract uploaded!

No abstract uploaded!