The Classification of Two-Dimensional Extended Topological Field Theories

By Christopher J. Schommer-Pries. 2011

218 citations recorded by the source corpus.

We provide a complete generators and relations presentation of the 2-dimensional extended unoriented and oriented bordism bicategories as symmetric monoidal bicategories. Thereby we classify these types of 2-dimensional extended topological field theories with arbitrary target bicategory. As an immediate corollary we obtain a concrete classification when the target is the symmetric monoidal bicategory of algebras, bimodules, and intertwiners over a fixed commutative ground ring. In the oriented case, such an extended topological field theory is equivalent to specifying a (non-commutative) separable symmetric Frobenius algebra. The text is divided into three chapters. The first develops a variant of higher Morse theory and uses it to obtain a combinatorial description of surfaces suitable for the higher categorical language used later. The second chapter is an extensive treatment of the theory of symmetric monoidal bicategories. We introduce several stricter variants on the notion of symmetric monoidal bicategory, and give a very general treatment of the notion of presentation by generators and relations. Finally we provide a host of strictification and cohernece results for symmetric monoidal bicategories. The final chapter focuses on extended tqfts. We give a precise treatment of the extended bordism bicategory equipped with additional structure (such as framings or orientations). We apply the results of the previous two chapters to obtain a simple presentation of both the oriented and unoriented bordism bicategories, and describe the general method to obtain such classifications for other choices of structure. We examine the consequences of our classification when the target is the bicategory of algebras, bimodules, and maps, over a fixed commutative ground ring.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9a2ff10621588e4ed35d23147a3f1bd619433a06

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Citations in the corpus

Local subsystems in gauge theory and gravity (2016, background): “Here we point out that such a gluing construction has effectively been carried out for a large class of rigorously-constructed diffeomorphisminvariant quantum field theories: topological quantum field theories (TQFTs).”
Compact closed bicategories (2013, unknown): “Schommer-Pries [47] gave the correct notion of a monoidal transformation between monoidal functors between monoidal bicategories.”
Braided Picard groups and graded extensions of braided tensor categories (2020, methodology): “We include a detailed description of the Eilneberg-Mac Lane cohomology [19] in low degrees and the notions of braided and symmetric monoidal 2-categories and 2-functors between them [12, 27, 32].”
The higher Morita category of n–algebras (2014, background): “A complete definition has been given by Schommer-Pries [38] in the case n = 2, but for larger n it seems that an appropriate notion of symmetric monoidal n-category has not even been defined .”