Abstract
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.
Degree Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
Department
Mathematics
Committee Chair
Ralph M. Kaufmann
Date of Award
Spring 2015
Recommended Citation
Lee, Byeongho, "G-Frobenius manifolds" (2015). Open Access Dissertations. 498.
https://docs.lib.purdue.edu/open_access_dissertations/498
First Advisor
Ralph M. Kaufmann
Committee Member 1
Peter Albers
Committee Member 2
James E. McClure
Committee Member 3
Sai-Kee Yeung