We construct new topological theories related to sigma models whose
target space is a seven-dimensional manifold of G2 holonomy. We define
a new type of topological twist and identify the BRST operator and the
physical states. Unlike the more familiar six-dimensional case, our topological
model is defined in terms of conformal blocks and not in terms of
local operators of the original theory. We also present evidence that one
can extend this definition to all genera and construct a seven-dimensional
topological string theory. We compute genus zero correlation functions
and relate these to Hitchin’s functional for three-forms in seven dimensions.
Along the way we develop the analogue of special geometry for
G2 manifolds. When the seven-dimensional topological twist is applied
to the product of a Calabi–Yau manifold and a circle, the result is an
interesting combination of the six-dimensional A and B models.