$G$-Frobenius Manifolds

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 $G = \mathbb{Z}/2\mathbb{Z}$ and prove the structure theorem for (pre-)$\mathbb{Z}/2\mathbb{Z}$-Frobenius manifolds. We also construct an example of a $\mathbb{Z}/2\mathbb{Z}$-Frobenius manifold using this theorem, that arises in singularity theory in the hypothetical context of orbifolding.