The Teichmüller theory has an interesting generalization originating from the deformation theory of super Riemann surfaces which initially was motivated by super string perturbation theory. The goal of this project was to consider the gauge groups which is replaced by the super group OSP(1|2) and study the super generalized version of TQFT. Part of the results of listed Obj O1 and O2 are published in Adv. Theor. Math. Phys., with the title of “Towards Super Teichmüller Spin TQFT” in collaboration with M.K. Pawelkiewicz and Masahito Yamazaki. In this paper we initiated the study of “supersymmetrization” of the Teichmüller TQFT, which we call the super Teichmüller spin TQFT. We obtained concrete expressions for the partition functions of the super Teichmüller spin TQFT for a class of spin 3-manifold geometries, by taking advantage of the recent results on the quantization of the super Teichmüller theory. We then computed the perturbative expansions of the partition functions, to obtain perturbative invariants of spin 3-manifolds.
Having the advantages of being in the host institutes including QM center, SDU Denmark, and the University of Geneva, which both institutes are part of the ERC grants number ERC-2018-SyG no. 810573, I had the great chance to become familiar with the modern new topic that was not directly mentioned in my original proposal, the so-called topological recursion (TR). TR is a very nice and fast-progressive topic in different branches of mathematics. The beautiful possible relationship between TR and the ultimate goal of the proposed project was a nice observation that I had during the time of this project. One of the big applications of the topological recursion procedure is the computation of correlators of Cohomological Field Theories (CohFT). A CohFT can be thought of as a family of 2d-TQFT parameterised by points t = (t1, ... tn) leaving in a manifold M. The genus 0 part of the CohFT is equivalent to equipping M with a Frobenius structure making it a Frobenius manifold. In the second part, we focused on the super generalization of theses chains of relations. Before studying the superCoh one needs to study the geometry behind the spectral curve. As the first step, we started by studying spectral curve of the Super TR, the so-called super spectral curve as initial data for super TR construction in collaboration with Nicolas Orantin, Kento Osuga and Reinier Kramer. The result of this project will be announced in the near future.
Moreover, during this time I invited my collaborators and visited them in their host institutes. I could present my work in several international conferences which established new collaborations and gave me the chance to continue my interaction with the international community. In addition, during this period I had the chance to obtain several experiences in organizing events and teaching which are valuable experiences and will help to get closer to my ultimate career goal as an academic tenure. Furthermore, I actively participated in initiatives supporting women mathematicians, broadening my network and initiating fresh collaborations.
