Role of the removed vertex in capturing reducible flat connections

Determine the role of the removed vertex in the Dimofte–Gaiotto–Gukov construction of punctured lens spaces, specifically how its removal enables the associated three-dimensional gauge theory to capture reducible $SL(2,\mathbb{C})$ flat connections.

Background

The Dimofte–Gaiotto–Gukov construction generally detects irreducible SL(2,C)SL(2,\mathbb{C}) flat connections while losing reducible ones. Although an unpunctured lens space has only reducible flat connections and the corresponding DGG partition function vanishes, the gauge theories constructed from punctured lens spaces have numbers of supersymmetric vacua matching the nontrivial reducible flat connections. The paper identifies the removed vertex as the relevant topological difference but does not explain its mechanism.

References

Before we begin, we remark on an aspect of the construction that remains unclear. The DGG construction only sees irreducible $SL(2,\mathbb{C})$ flat connections on the complex Chern-Simons theory side, loosing the reducible ones as pointed out in. Since the lens space has reducible flat connections only, the partition functions of the corresponding DGG theory vanish due to spontaneously broken supersymmetry. However, for the {\it punctured} lens space $L_{n,k}$, we find $=2$ gauge theories with the number of supersymmetric vacua exactly matches the number of non-trivial {\it reducible} flat connections. The only difference between the two three-manifolds is the removal of a vertex $v$, which is a universal feature in the context of layered triangulations. Thus, it would be interesting to study the role of this vertex in the DGG construction for capturing the reducible flat connections in the spirit of.

— 3d $\mathcal{N}=$ 4 rank-0 SCFT from punctured lens space  (2608.13300 - Kim, 13 Aug 2026) in Introduction