The geometric Langlands twist in five and six dimensions

TL;DR

This paper explores a geometric twist of 6d (2,0) theory and its reduction to 5d, revealing new supersymmetric structures and a vanishing theorem for BPS contact instantons, extending the geometric Langlands program.

Contribution

It introduces a novel twist of 6d (2,0) theory compatible with five-manifolds and generalizes the geometric Langlands twist to five dimensions, including non-Abelian gauge groups.

Findings

Constructed a twisted 6d (2,0) theory on five-manifolds.

Derived a vanishing theorem for BPS contact instantons.

Extended the geometric Langlands twist to 5d SYM theories.

Abstract

Abelian 6d (2,0) theory has SO(5) R symmetry. We twist this theory by identifying the R symmetry group with the SO(5) subgroup of the SO(1,5) Lorentz group. This twisted theory can be put on any five-manifold M, times R, while preserving one scalar supercharge. We subsequently assume the existence of one unit normalized Killing vector field on M, and we find a corresponding SO(4) twist that preserves two supercharges and is a generalization of the geometric Langlands twist of 4d SYM. We generalize the story to non-Abelian gauge group for the corresponding 5d SYM theories on M. We derive a vanishing theorem for BPS contact instantons by identifying the 6d potential energy and its BPS bound, in the 5d theory. To this end we need to perform a Wick rotation that complexifies the gauge field.