Construction of two-dimensional topological field theories with non-invertible symmetries

Solitons in Two–dimensional Topological Field Theories

We consider a class of N = 2 supersymmetric non–unitary theories in two– dimensional Minkowski spacetime which admit classical solitonic solutions. We show how these models can be twisted into a topological sector whose energy–momentum tensor is a BRST commutator. There is an infinite number of degrees of freedom associated to the zero modes of the solitons. As explicit realizations of such mod...

State sum construction of two-dimensional open-closed Topological Quantum Field Theories

We present a state sum construction of two-dimensional extended Topological Quantum Field Theories (TQFTs), so-called open-closed TQFTs, which generalizes the state sum of Fukuma–Hosono–Kawai from triangulations of conventional two-dimensional cobordisms to those of open-closed cobordisms, i.e. smooth compact oriented 2-manifolds with corners that have a particular global structure. This constr...

Two-dimensional Topological Quantum Field Theories and Frobenius Algebras

We characterize Frobenius algebras A as algebras having a comultiplication which is a map of A-modules. This characterization allows a simple demonstration of the compatibility of Frobenius algebra structure with direct sums. We then classify the indecomposable Frobenius algebras as being either \annihilator algebras" | algebras whose socle is a principal ideal | or eld extensions. The relation...

Non-commutative Extensions of Two-dimensional Topological Field Theories and Hurwitz Numbers for Real Algebraic Curves

Contents 1. Introduction 1 2. Structure algebra 4 2.1. Definition of a structure algebra 4 2.2. Semisimple structure algebras 7 2.3. Structure algebra of a finite group 11 3. Cuts of stratified surfaces 15 3.1. Stratified surfaces 15 3.2. Cut systems 17 3.3. Basic and simple surfaces 21 3.4. Neighboring complete cut systems 25 4. Two-dimensional topological field theory 29 4.1. Definition of Kl...

On 2-dimensional Topological Field Theories