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Topological Field Theories with Defects

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2022, Doctor of Philosophy (PhD), Ohio University, Mathematics (Arts and Sciences).
Topological field theories are important due to their numerous applications in theoretical physics and modern mathematics. A useful way to study topological field theories is by using category theory. In this dissertation we considered different types of topological filed theories in dimension two. R.Dijkraaf found that the category $2\C ob_{c}$ of surfaces is free on a commutative Frobenius algebra. G.Segal and G.Moore showed that the category $2\C ob_{o\da c}$ of open surfaces is described by a Cardy pair. We extended these results to the following: the category $2\C ob_{2o}$ of two coloured ribbon graphs is free on a pair of Frobenius algebras and a Frobenius bimodule. In a separable case, a Frobenius bimodule and its dual form a Morita context. We also found that the category $2\C ob_{2o\da c}$ of two coloured open surfaces is the free monoidal category on a bimodule btween two Cardy pairs. We moved then to the category $2\C ob_{d}$ of surfaces with defect of codimension 1. We gave a dicribtion of Cardy correspondences in this category and ended up with a conjecture stating that the category $2\C ob_{d}$ is free on a Cardy correspondence in a separable case, i.e. all Frobenius algebras are separable. We gave an example of a 2 dimensional topological field theory which we call a set-theoretic 2 dimensional topological field theory. The target of this functor is the category of correspondences $\C orr$. We showed that any Frobenius monoid in $\C orr$ is a multi-fusion ring.
Alexei Davydov, Professor (Advisor)
Nancy Sandler (Committee Member)
Marcel Bischoff, Professor (Committee Member)
Vladimir Uspenskiy, Professor (Committee Member)
98 p.

Recommended Citations

Citations

  • Zailai, M. (2022). Topological Field Theories with Defects [Doctoral dissertation, Ohio University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1646737048910495

    APA Style (7th edition)

  • Zailai, Mohmmad. Topological Field Theories with Defects . 2022. Ohio University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1646737048910495.

    MLA Style (8th edition)

  • Zailai, Mohmmad. "Topological Field Theories with Defects ." Doctoral dissertation, Ohio University, 2022. http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1646737048910495

    Chicago Manual of Style (17th edition)