Witten–Reshetikhin–Turaev invariants for 3-manifolds from Lagrangian intersections in configuration spaces

Abstract

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 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.