The goal of this dissertation 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 = Z/2Z and prove the structure theorem for (pre-)Z/2Z-Frobenius manifolds. We also construct an example of a Z/2Z-Frobenius manifold using this theorem, that arises in singularity theory in the hypothetical context of orbifolding.
Kaufmann, Purdue University.
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