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