University of Illinois – Urbana-Champaign
A construction of topological field theories.
Degree: PhD, Mathematics, 2016, University of Illinois – Urbana-Champaign
We give an explicit construction of extended topological field theories over a manifold taking values in the deloopings of U(1) from the data of differential forms on the manifold. More specifically, for a manifold M, using a version of the Dold-Kan construction, we create from the Deligne complex a Kan complex, |D_n|^+(M), and show that there is a natural map into (Fun)^⊗(Bord_n^or (M),B^nU(1)). The main theorem asserts that thismap becomes a weak equivalence after restricting to the framed bordisms. The construction is on the point- set level, so immediately can be refined to the setting where the Deligne complex is considered discretely, which is known as a model for the differential cohomology.
As a part of the proof we show that the geometric realization of the bordism (∞,n)-category is the n-fold delooping of the infinite loopspace of the Madsen-Weiss spectrum MTO(n). The direct attempts to generalize the Galatius-Madsen-Tillmann-Weiss proof of n = 1 case fail due the globularity restriction on the bordism (∞,n)-category. To overcome this, we use the abstract transversality argument of Bökstedt and Madsen, Gromov's theory of microflexible sheaves and Rezk's argument on realization fibrations.
Advisors/Committee Members: Ando, Matthew (advisor), Rezk, Charles (Committee Chair), McCarthy, Randy (committee member), Schenck, Henry (committee member).
Subjects/Keywords: topological field theories; cobordism hypothesis; Deligne complex; Dold-Kan construction
to Zotero / EndNote / Reference
APA (6th Edition):
Aramyan, N. (2016). A construction of topological field theories. (Doctoral Dissertation). University of Illinois – Urbana-Champaign. Retrieved from http://hdl.handle.net/2142/92901
Chicago Manual of Style (16th Edition):
Aramyan, Nerses. “A construction of topological field theories.” 2016. Doctoral Dissertation, University of Illinois – Urbana-Champaign. Accessed August 09, 2020.
MLA Handbook (7th Edition):
Aramyan, Nerses. “A construction of topological field theories.” 2016. Web. 09 Aug 2020.
Aramyan N. A construction of topological field theories. [Internet] [Doctoral dissertation]. University of Illinois – Urbana-Champaign; 2016. [cited 2020 Aug 09].
Available from: http://hdl.handle.net/2142/92901.
Council of Science Editors:
Aramyan N. A construction of topological field theories. [Doctoral Dissertation]. University of Illinois – Urbana-Champaign; 2016. Available from: http://hdl.handle.net/2142/92901