# Langlands duality for skein modules of 3-manifolds

## TL;DR

This paper proposes new Langlands duality conjectures for skein modules of 3-manifolds, supported by special case proofs using double affine Hecke algebra representation theory and a novel 1-form symmetry structure.

## Contribution

It introduces the first conjectural framework linking Langlands duality with skein modules, combining geometric, algebraic, and physical insights.

## Key findings

- Special cases confirm the conjecture for certain 3-manifolds.
- Representation theory of double affine Hecke algebras is used in proofs.
- A new 1-form symmetry structure on skein modules is proposed.

## Abstract

I introduce new Langlands duality conjectures concerning skein modules of 3-manifolds, which we have made recently with David Ben-Zvi, Sam Gunningham, and Pavel Safronov. I recount some historical motivation and some recent special cases where the conjecture is confirmed. The proofs in these cases combine the representation theory of double affine Hecke algebras and a new 1-form symmetry structure on skein modules related to electric-magnetic duality. This note is an expansion of my talk given at String Math 2022 in Warsaw, and is submitted to the String Math 2022 Proceedings publication.