The Classification of Two-Dimensional Extended Topological Field Theories (1112.1000v2)

Abstract: 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.

• The paper presents a generators and relations framework that characterizes both oriented and unoriented bordism bicategories.
• It applies advanced jet transversality and higher Morse theory to address geometric singularities in the classification.
• It demonstrates that oriented TFTs correspond to separable symmetric Frobenius algebras, linking topology with algebraic structures.

An Expert Assessment of "The Classification of Two-Dimensional Extended Topological Field Theories"

The paper "The Classification of Two-Dimensional Extended Topological Field Theories" by Christopher John Schommer-Pries provides a comprehensive treatment of the classification of 2-dimensional extended topological field theories (TFTs) using the formalism of symmetric monoidal bicategories. The work is structured to interrelate differential topology, algebra, and higher category theory, offering a detailed exposition that is critical for both understanding specific low-dimensional TFTs and exploring broader implications in mathematical physics and higher category theory.

Overview and Structure

The paper addresses the classification problem of 2-dimensional extended TFTs by establishing a generators and relations presentation for the unoriented and oriented bordism bicategories as symmetric monoidal bicategories. The implications extend to classifying these theories in terms of foundational algebraic structures. The paper is meticulously assembled into three primary components, each underpinning a crucial aspect of the classification process: geometric/topological considerations, algebraic/categorical structure, and the synthesis in the setting of topological field theories. These components are methodically developed through four main chapters, with a heavy reliance on jet transversality and higher Morse theory concepts anchored in classical differential topology.

Main Contributions and Numerical Insights

• Generators and Relations: A central result of the paper is the explicit characterization of the bordism bicategory using a finite set of generators and relations, which is reminiscent of classical structures but adapted for the higher categorical context. In the unoriented case, the bicategory is presented with specific 2D Morse generators, cusp generators, and associated relations reflective of the complex structure of surfaces and their topological features.
• Geometric Singularity Theory: The passage through geometric singularity theory is handled through an advanced discussion on jet spaces and their transversality properties. The strategic application of the Thom-Boardman stratification and its refinement facilitates the classification of singularities, which is pivotal for understanding the construction of the bordism bicategory.
• Symmetric Monoidal Bicategories: The notion of symmetric monoidal bicategories is extended by establishing a coherence theorem that verifies the equivalence of these categories with their stricter variants. Such a result ensures that the generated bordism bicategory can be effectively managed and understood within the proposed algebraic framework.
• Implications for Algebraic Structures: When directing the bordism bicategory into the symmetric monoidal bicategory of algebras, bimodules, and morphisms over a fixed ring, Schommer-Pries demonstrates that in the oriented case, these extended TFTs correspond to separable symmetric Frobenius algebras. This result aligns with classical algebraic insights, reinforcing the connection between algebra and topology.

Theoretical and Practical Implications

The implications of this work are both grounded in the specifics of 2-dimensional extended TFTs and expansive into the potential for greater dimensional generalizations. The theoretical framework provides foundational insight into how higher algebraic structures correspond to topological entities, enriching the dialogue between category theory and topology. Practically, this work affects areas such as manifold invariants, where the inability of traditional unitary TQFTs to distinguish between certain manifolds is an open problem. Additionally, the connections to modular tensor categories and their role in categorifying TQFTs via Reshetikhin-Turaev constructions highlight the relevance of these findings in quantum topological models and their algebraic underpinnings.

Future Directions and Speculations

This work's meticulous development of methods that leverage traditional differential topology and algebraic category theory lays a robust groundwork for further exploration in similar classification challenges across different dimensions. The detailed singularity analysis and bicategorical strategies outlined here suggest potential for solving classification problems in higher-dimensional TFTs. Additionally, considering the Hopkins-Lurie cobordism hypothesis and its interplay with these theories suggests fertile ground for continued exploration of the exact nature of symmetry and locality in extended quantum field theories.

In summary, Schommer-Pries' work is an in-depth and technical treatise on classifying 2D extended TFTs, characterized by an adept intertwining of topology, algebra, and higher category theory. Its findings provide insights and methodologies that are essential for researchers investigating both the specificities of lower-dimensional theories and, more broadly, the abstract structures that govern field theories and their algebraic counterparts.