@phdthesis{5b6ada0a40424d82b6848f7334c960ac,
title = "State-sum construction of two-dimensional functorial field theories",
abstract = "In this thesis we study two classes of 2-dimensional functorial field theories and give a state-sum construction of these theories. In the first part of this thesis we look at topological field theories on r-spin surfaces. We define a combinatorial model of r-spin surfaces, which is suitable for for the state-sum construction. The latter takes a Frobenius algebra A, whose window element is invertible and whose Nakayama automorphism N satisfies N\textasciicircum{}r = id, as an input and produces an r-spin topological field theory Z\_A . For r even we give an example of such a state-sum topological field theory with values in super vector spaces, where A = Cl is the Clifford algebra with one odd generator and we show that Z\_Cl computes the Arf invariant of r-spin surfaces. As an application of the combinatorial model and this r-spin topological field theory we compute mapping class group orbits of r-spin structures extending results of Randal-Williams and Geiges, Gonzalo. In the second part of the thesis we consider area-dependent quantum field theories. An important feature of these theories that, contrary to topological field theories, they allow for infinite-dimensional state-spaces. We classify these theories in terms of regularised Frobenius algebras and give a state-sum construction of them, for which the input data is now a strongly separable regularised Frobenius algebra. We then extend the state-sum construction to include defect lines, which we label with bimodules over strongly separable regularised Frobenius algebras. We show that the fusion of defect lines corresponds to the tensor product of bimodules over regularised algebras. The main example of area-dependent quantum field theories is 2-dimensional Yang-Mills theory with a compact semisimple Lie group G with Wilson lines as defects, which we treat in great detail. We finally introduce other defect lines by twisting by outer automorphisms of G.",
author = "Lorant Szegedy",
year = "2018",
language = "English",
school = "University of Hamburg",
}