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.
","abstract_html":"<p>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.</p>","abstract_has_math":false,"creators":["Lee, Byeongho"],"institution":null,"degree_name":"Doctor of Philosophy (PhD)","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Ralph M. Kaufmann","Peter Albers","James E. McClure","Sai-Kee Yeung"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2015,"date_issued":"2015-04-01T07:00:00Z","date_published":"2015-04-01T07:00:00Z","updated_at":"2026-07-24T03:53:28Z","subjects":["Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://docs.lib.purdue.edu/open_access_dissertations/498","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Ralph M. Kaufmann","Peter Albers","James E. McClure","Sai-Kee Yeung"]},{"key":"dc:creator","label":"Author","values":["Lee, Byeongho"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://docs.lib.purdue.edu/open_access_dissertations/498"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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.
"]},{"key":"dc:title","label":"Title","values":["G-Frobenius manifolds"]}]}],"canonical_facts":{"dc:contributor":["Ralph M. Kaufmann","Peter Albers","James E. McClure","Sai-Kee Yeung"],"dc:creator":["Lee, Byeongho"],"dc:description.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.
"],"dc:identifier":["https://docs.lib.purdue.edu/open_access_dissertations/498"],"dc:subject":["Mathematics"],"dc:title":["G-Frobenius manifolds"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-24T03:53:28Z"}