Consider a smooth projective curve C (or a Riemann surface) of genus at least two, which supports an automorphism of odd prime order. We define the equivariant Hitchin index to be the index of the induced action of this automorphism on the cohomology groups of the determinant bundle on the moduli space of stable rank two Higgs bundles on the curve C (with fixed determinant of odd degree). Motivated by TQFT axioms we prove that this index is an invariant of the 3-manifold obtained as the mapping torus of the automorphism, thus establishing a new link between algebraic geometry and quantum topology. This is joint work with Tamas Hausel and Jørgen E. Andersen.