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Group Actions on Bicategories and Topological Quantum Field Theories

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Group Actions on Bicategories and Topological Quantum Field Theories Dissertation zur Erlangung des Doktorgrades der Fakultät für Mathematik, Informatik und Naturwissenschaften der Universität Hamburg vorgelegt am Fachbereich Mathematik von Jan Hesse Hamburg 2017 Als Dissertation angenommen vom Fachbereich Mathematik der Universität Hamburg. Auf Grund der Gutachten von Prof. Dr. Christoph Schweigert und Prof. Dr. Ingo Runkel. Datum der Disputation: 05.07.2017. Vorsitzende der Prüfungskommission: Prof. Dr. Sören Christensen. Summary Bicategories play an important rôle in the classification of fully-extended two-dimensional topological field theories. In order to describe topological field theories with more geometric structure, one needs more structure on the algebraic side, which is given by homotopy fixed points of a certain group action on a bicategory. In the first chapter of this thesis, we develop the mathematical theory of group actions on bicategories. By categorifying the notion of a group action on a set, we arrive at a suitable definition of an action of a topological group on a bicategory. Given such an action, we provide an explicit definition of the bicategory of homotopy fixed points. This allows us to explicitly compute the bicategory of homotopy fixed points of certain group actions. Two fundamental examples show that even homotopy fixed points of trivial group actions give rise to additional structure: we show that a certain bigroupoid of semisimple symmetric Frobenius algebras is equivalent to the bicategory of homotopy fixed points of the trivial SO(2)-action on the core of fully-dualizable objects of the bicategory of algebras, bimodules and intertwiners. Furthermore, we show that homotopy fixed points of the trivial SO(2)-action on the bicategory of finite, linear categories are given by Calabi-Yau categories. The next chapter deals with an additional equivariant structure on a functor between bicategories equipped with a group action. We show that such an equivariant functor induces a functor on homotopy fixed points. As an application, we consider the 2-functor which sends a finite-dimensional, semisimple algebra to its category of representations. This functor has got a natural SO(2)-equivariant structure, and thus induces a functor on homotopy fixed points. We show that this induced functor is pseudo-naturally isomorphic to an equivalence between Frobenius algebras and Calabi-Yau categories which we have constructed previously. In the last two chapters, we classify fully-extended, 2-dimensional oriented topological field theories. We begin by constructing a non-trivial SO(2)-action on the framed bordism bicategory. The cobordism hypothesis for framed manifolds allows us to transport this action to the core of fully-dualizable objects of the target bicategory. We show that this action is given by the Serre automorphism and compute the bicategory of homotopy fixed points of this action. Finally, we identify this bigroupoid of homotopy fixed points with the bicategory of fully-extended oriented topological quantum field theory with values in an arbitrary symmetric monoidal bicategory. This proves the cobordism hypothesis for two-dimensional oriented cobordisms. i Zusammenfassung In der Klassifizierung von vollständig erweiterten zweidimensionalen topologischen Feld- theorien spielen Bikategorien eine wichtige Rolle. Um topologische Feldtheorien mit zu- sätzlicher geometrischer Struktur zu beschreiben, benötigt man zusätzliche algebraische Struktur, die durch Homotopiefixpunkte einer Gruppenwirkung auf einer bestimmten Bikategorie gegeben ist. Im ersten Kapitel entwickeln wir einen mathematischen Formalismus zur Beschreibung von Gruppenwirkungen auf Bikategorien. Gegeben die Wirkung einer topologischen Gruppe auf einer Bikategorie, konstruieren wir explizit eine Bikategorie von Homoto- piefixpunkten dieser Wirkung. Dieser Formalismus ermöglicht uns, Homotopiefixpunkte von bestimmten Gruppenwirkungen explizit zu berechnen. Zwei fundamentale Beispiele zeigen nun, dass sogar Homotopiefixpunkte von trivialen Gruppenwirkungen zusätzliche Struktur sind: so ist die Bikategorie von endlichdimensionalen, halbeinfachen Frobeniu- salgebren äquivalent zu der Bikategorie von Homotopiefixpunkten der trivialen SO(2)- Wirkung auf der Bikategorie von vollständig dualisierbaren Algebren und Bimoduln. Weiterhin zeigen wir, dass Homotopiefixpunkte der trivialen SO(2)-Wirkung auf der Bi- kategorie von endlichen, linearen Kategorien äquivalent zur Bikategorie von Calabi-Yau Kategorien sind. Im nächsten Kapitel beschäftigen wir uns mit einer zusätzlichen äquivarianten Struk- tur auf einem Funktor zwischen Bikategorien mit einer Gruppenwirkung. Wir zeigen, dass solch ein äquivarianter Funktor zwischen zwei Bikategorien einen Funktor auf Ho- motopiefixpunkten induziert. Als Anwendung betrachten wir den 2-Funktor, der einer halbeinfachen Algebra ihre Darstellungskategorie zuweist. Dieser 2-Funktor hat eine natürliche SO(2)-äquivariante Struktur, und induziert daher einen Funktor auf Homo- topiefixpunkten. Sodann identifizieren wir diesen induzierten Funktor mit einer bereits zuvor konstruierten Äquivalenz zwischen Frobeniusalgebren und Calabi-Yau Kategorien. In den letzten beiden Kapiteln wenden wir uns der Klassifizierung von zweidimensio- nalen, vollständig erweiterten, orientierten topologischen Quantenfeldtheorien zu: wir konstruieren zunächst eine nicht-triviale SO(2)-Wirkung auf einem Skelett der Bika- tegorie von gerahmten Bordismen. Die Kobordismushypothese für gerahmte Mannig- faltigkeiten erlaubt uns, diese Wirkung auf den maximalen Untergruppoiden von voll- ständig dualisierbaren Objekten der Zielkategorie zu transportieren. Wir zeigen, dass diese SO(2)-Wirkung durch den Serre Automorphismus gegeben ist, und berechnen die Bikategorie von Homotopiefixpunkten. Schlussendlich identifizieren wir diese Bikategorie von Homotopiefixpunkten mit der Bikategorie von vollständig erweiterten, orientierten, zweidimensionalen topologischen Feldtheorien mit Werten in einer symmetrisch monoida- len Bikategorie. Dies beweist die Kobordismushypothese für zweidimensionale orientierte Kobordismen. iii Contents Introduction vii 1. Preliminaries 1 1.1. Symmetric monoidal bicategories . .1 1.2. Fully-dualizable objects . .5 1.3. Group actions on bicategories . .6 1.4. The cobordism hypothesis . 10 2. Frobenius algebras and homotopy fixed points of group actions on bicategories 15 2.1. Frobenius algebras and Morita contexts . 16 2.2. Group actions on bicategories and their homotopy fixed points . 22 2.3. Induced functors on homotopy fixed points . 34 2.4. Computing homotopy fixed points . 37 2.4.1. Symmetric Frobenius algebras as homotopy fixed points . 42 3. An equivalence between Frobenius algebras and Calabi-Yau categories 47 3.1. Calabi-Yau categories . 48 3.1.1. Calabi-Yau categories as homotopy fixed points . 52 3.2. Constructing an equivalence between Frobenius algebras and Calabi-Yau categories . 53 3.2.1. Separable algebras and projective modules . 53 3.2.2. A Calabi-Yau structure on the representation category of a Frobe- nius algebra . 57 3.2.3. Constructing the 2-functor Repfg on 1-morphisms . 63 3.2.4. Constructing the 2-functor Repfg on 2-morphisms . 66 3.3. Proving the equivalence . 67 3.3.1. The functor of representations as equivarinatization . 70 4. The Serre automorphism as a homotopy action 73 4.1. Fully-dualizable objects and the Serre automorphism . 73 4.1.1. The Serre automorphism . 75 4.1.2. The Serre automorphism in 2-vector spaces . 78 4.1.3. The Serre automorphism in the Morita bicategory . 80 4.1.4. Monoidality of the Serre automorphism . 81 4.2. Monoidal homotopy actions . 84 4.2.1. Invertible Field Theories . 87 v Contents 4.3. The 2-dimensional framed bordism bicategory . 90 4.3.1. An action on the framed bordism bicategory . 94 4.3.2. Induced action on functor categories . 97 4.3.3. Induced action on the core of fully-dualizable objects . 99 5. Calabi-Yau objects and the cobordism hypothesis for oriented manifolds 103 6. Outlook 111 6.1. The homotopy hypothesis . 112 6.2. Homotopy orbits . 113 A. Weak endofunctors as a monoidal bicategory 115 B. Finite linear categories as categories of modules 123 B.1. Abelian categories as Vect-modules . 123 B.2. Finite linear categories as categories of modules . 124 Bibliography 129 vi Introduction This thesis investigates structures in 2-dimensional topological quantum field theories and is based at the interface of algebra and topology. Giving a precise definition of quantum field theory is a notoriously difficult problem in mathematical physics. Historically, quantum field theory grew out of attempts to reconcile relativistic field theory, like classical electrodynamics, with quantum mechanics. Nowadays, there are essentially two main approaches for formalising quantum field theories. One approach is given by the framework of algebraic quantum field theory: here, axiomatic systems for quantum field theory were first developed by Wightman in [SW64] and then further abstracted by Haag and Kastler in [HK64]. More recent work of Brunetti, Fredenhagen and Verch in [BFV03] extends this approach to locally covariant field theories. These axiomatic systems formalise the assignment of an algebra of observables to certain patches of spacetime. In this thesis, we use the second main approach for formalising quantum field theory, which is given by functorial quantum field theory. This approach formalises the assign- ment of a state space to patches of physical space and tries to axiomatize the output of the Feynman path integral.
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