Witten-Reshetikhin-Turaev invariants for \(3\)-manifolds from Lagrangian intersections in configuration spaces (Q6150022)

Summary: In this paper, we construct a topological model for the Witten-Reshetikhin-Turaev invariants for \(3\)-manifolds coming from the quantum group \(U_q(\mathrm{sl}(2))\), as graded intersection pairings of homology classes in configuration spaces. More precisely, for a fixed level \(\mathcal{N} \in \mathbb{N}\), we show that the level \(\mathcal{N}\) WRT invariant for a \(3\)-manifold is a state sum of Lagrangian intersections in a covering of a \textit{fixed} configuration space in the punctured disc. This model brings a new perspective on the structure of the level \(\mathcal{N}\) Witten-Reshetikhin-Turaev invariant, showing that it is completely encoded by the intersection points between certain Lagrangian submanifolds in a fixed configuration space, with additional gradings which come from a particular choice of a local system. This formula provides a new framework for investigating the open question about categorifications of the WRT invariants.