A vanishing theorem for T-branes

Abstract We consider regular polystable Higgs pairs (E, ϕ) on compact complex manifolds. We show that a non-trivial Higgs field ϕ ∈ H 0(End(E) ⊗ K S ) restricts the Ricci curvature of the manifold, generalising previous results in the literature. In particular ϕ must vanish for positive Ricci curvature, while for trivial canonical bundle it must be proportional to the identity. For Kähler surfaces, our results provide a new vanishing theorem for solutions to the Vafa-Witten equations. Moreover they constrain supersymmetric 7-brane configurations in F-theory, giving obstructions to the existence of T-branes, i.e. solutions with [ϕ, ϕ †] ≠ 0. When non-trivial Higgs fields are allowed, we give a general characterisation of their structure in terms of vector bundle data, which we then illustrate in explicit examples.