Representations of Motion Groups of Links via Dimension Reduction of TQFTs

Motion groups of links in the three sphere S-3 are generalizations of the braid groups, which are motion groups of points in the disk D-2. Representations of motion groups can be used to model statistics of extended objects such as closed strings in physics. Each 1-extended (3+ 1)-topological quantum field theory (TQFT) will provide representations of motion groups, but it is difficult to compute such representations explicitly in general. In this paper, we compute representations of the motion groups of links in S3 with generalized axes from Dijkgraaf-Witten (DW) TQFTs inspired by dimension reduction. A succinct way to state our result is as a step toward a twisted generalization (Conjecture 1) of a conjecture for DW theories of dimension reduction from (3+1) to (2+1): DWG(3)(+1) congruent to circle plus[ g]epsilon[G]DWC(g)2+1, where the sum runs over all conjugacy classes [g] epsilon [G] of G and C(g) the centralizer of any element g epsilon [g]. We prove a version of Conjecture 1 for the mapping class groups of closed manifolds and the case of torus links labeled by pure fluxes.

Automated summary

In this paper, representations of motion groups of links in S3 with generalized axes are computed using Dijkgraaf-Witten TQFTs inspired by dimension reduction. The results represent a step towards a twisted generalization of DW theories, as stated in Conjecture 1.