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Super Andersen-Kashaev Topological Quantum Field Theory

Periodic Reporting for period 1 - SAKTQFT (Super Andersen-Kashaev Topological Quantum Field Theory)

Okres sprawozdawczy: 2020-11-01 do 2022-10-31

The interaction between mathematics and theoretical physics led to the creation of a new interdisciplinary research topic called mathematical physics where mathematical rigour and physical intuition merge in a natural way. Topological Quantum Field Theory (TQFT) is an excellent example of this fruitful interplay. A topological quantum field theory (TQFT) is a model of quantum field theory which computes topological invariants. Although TQFTs were invented by physicists, they are also of mathematical interest because of their relation to quantum topology. Particularly interesting examples of 3d TQFTs arise from Chern-Simons (CS) theory with non-compact gauge groups. A connected component of the phase space of PSL(2,R) CS theory is identified with Teichmüller space, and its quantum theory corresponds to a specific class of unitary mapping class group representations is infinite-dimensional Hilbert spaces, in contrast to more standard TQFTs. This TQFT was previously formulated in the mathematics literature by Andersen and Kashaev. 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. In this project, we had some results in 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. We then compute the perturbative expansions of the partition functions, to obtain perturbative invariants of spin 3-manifolds. The mentioned super generalization originated from the deformation theory of super Riemann surfaces which initially was motivated by superstring perturbation therefore in the second part, we focused on 2d-TQFT aspect. In particular, we focused on the mathematical techniques related to that so-called Topological Recursion(TR). We studied the spectral curve of the Super TR, so-called super spectral curve and the super geometry behind this curve. We focused on the OSP(1|2) super gauge group, which was the main interest in our original proposal.
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.
I initiate various initiatives for women mathematicians to promote their inclusion and advancement. These efforts aim to increase representation, visibility, and career opportunities for women in mathematics, leading to broader societal impacts and socio-economic benefits.
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