Traube, Matthias (2021): Algebraic constructions related to quantum field theories. Dissertation, LMU München: Faculty of Physics
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### Abstract

Quantum field theory is the main technical tool in understanding modern theoretical high energy physics. After nearly a century of quantum field theory its mathematics remains mysterious, though. A firm grip on the mathematics of quantum field theory seems ever more desirable since the upcoming of string theory with its rich and challenging mathematical structure. Among the best understood quantum field theories are two dimensional rational quantum field theories. In this work we contribute to a better mathematical understanding of such theories by providing a mathematically rigorous but intuitive description in terms of string-net models. Heuristically string-net models give a Feynman diagram framework for rational conformal field theories on all genus $g$ surfaces. We prove a uniqueness and existence result for open-closed rational conformal field theories with fixed boundary condition making extensive use of the category theory underlying string-nets. Secondly, we give a construction of consistent correlators in rational conformal field theories with arbitrary topological defects and symmetry preserving boundary conditions using string-nets. As a proof of principle we compute torus and annulus partition functions in the string-net framework, thereby reproducing established results. Compared to earlier categorical approaches the use of string-nets almost completely avoids three dimensional considerations, rendering the use of categorical tools very intuitive. The second part of the thesis deals with homotopy algebras and their appearance in quantum field theories. Roughly speaking every consistent classical field theory having some gauge freedom produces a strong homotopy Lie algebra through the Batalin Vilkovisky formalism. Hence by studying strong homotopy Lie algebras (or $L_\infty$ algebras) one can learn something about field theories. The first result presented in that direction is a theorem closing every skewsymmetric bilinear bracket on a vector space into a finite term $L_\infty$ algebra. This is a generalization of the $L_\infty$ structure of the Courant algebroid. The second result is a theorem relating quasi-isomorphisms of $L_\infty$ algebras to Seiberg-Witten maps, linking the mathematics of homotopy algebras closer to physical notions.