The goal of this paper is to introduce the notion of G-Frobenius manifolds for any finite group G. This work is motivated by the fact that any G-Frobenius algebra yields an ordinary Frobenius algebra by taking its G-invariants. We generalize this on the level of Frobenius manifolds.

To define a G-Frobenius manifold as a braided-commutative generalization of the ordinary commutative Frobenius manifold, we develop the theory of G-braided spaces. These are defined as G-graded G-modules with certain braided-commutative “rings of functions”, generalizing the commutative rings of power series on ordinary vector spaces.

As the genus zero part of any ordinary cohomological field theory of Kontsevich–Manin contains a Frobenius manifold, we show that any G-cohomological field theory defined by Jarvis–Kaufmann–Kimura contains a G-Frobenius manifold up to a rescaling of its metric.

Finally, we specialize to the case of  and prove the structure theorem for (pre-)-Frobenius manifolds. We also construct an example of a -Frobenius manifold using this theorem, that arises in singularity theory in the hypothetical context of orbifolding.