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

TL;DR

This paper introduces a topological model for Witten-Reshetikhin-Turaev invariants of 3-manifolds using Lagrangian intersections in configuration spaces, offering a new perspective and potential for categorification.

Contribution

It constructs a novel geometric framework representing WRT invariants as graded intersection pairings in a fixed configuration space, linking quantum invariants to Lagrangian intersections.

Findings

WRT invariants are expressed as intersection pairings in configuration spaces.

The model encodes invariants via intersection points of Lagrangian submanifolds.

Provides a new approach for exploring categorifications of quantum invariants.

Abstract

In this paper we construct a topological model for the Witten-Reshetikhin-Turaev invariants for $3$-manifolds coming from the quantum group $U_q(sl(2))$, as graded intersection pairings of homology classes in configuration spaces. More precisely, for a fixed level $\cN \in \N$ we show that the level $\cN$ WRT invariant for a $3-$manifold is a state sum of Lagrangian intersections in a covering of a {\bf fixed} configuration space in the punctured disk. This model brings a new perspective on the structure of the level $\cN$ 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.