Dualizable tensor categories

Abstract: We investigate the relationship between the algebra of tensor categories and the topology of framed 3-manifolds. On the one hand, tensor categories with certain algebraic properties determine topological invariants. We prove that fusion categories of nonzero global dimension are 3-dualizable, and therefore provide 3-dimensional 3-framed local field theories. We also show that all finite tensor categories are 2-dualizable, and yield categorified 2-dimensional 3-framed local field theories. On the other hand, topological properties of 3-framed manifolds determine algebraic equations among functors of tensor categories. We show that the 1-dimensional loop bordism, which exhibits a single full rotation, acts as the double dual autofunctor of a tensor category. We prove that the 2-dimensional belt-trick bordism, which unravels a double rotation, operates on any finite tensor category, and therefore supplies a trivialization of the quadruple dual. This approach produces a quadruple-dual theorem for suitably dualizable objects in any symmetric monoidal 3-category. There is furthermore a correspondence between algebraic structures on tensor categories and homotopy fixed point structures, which in turn provide structured field theories; we describe the expected connection between pivotal tensor categories and combed fixed point structures, and between spherical tensor categories and oriented fixed point structures.

• The paper establishes that fusion categories with nonzero global dimension are 3-dualizable, enabling 3D framed local field theories.
• It demonstrates that all finite tensor categories are 2-dualizable and underscores separability as a key criterion for full dualizability.
• The authors link algebraic tensor structures to geometric constructs like loop and belt bordisms, providing insights into topological invariants.

Dualizable Tensor Categories

The paper "Dualizable Tensor Categories" rigorously explores the connection between the algebra of tensor categories and the topology of framed 3-manifolds. The authors present a detailed investigation into the topological significance of tensor categories with certain algebraic properties, demonstrating their capacity to determine topological invariants. They investigate the dualizability of tensor categories and reveal their role in enabling structured field theories.

Algebraic and Topological Interfaces

Tensor categories encapsulate complex algebraic structures, and the paper is pivotal in demonstrating how these categories contribute to topological field theories (TFTs). Notably, the authors assert that fusion categories of nonzero global dimension are 3-dualizable, thus facilitating 3-dimensional, 3-framed local field theories. Furthermore, they establish that all finite tensor categories are 2-dualizable, providing categorified 2-dimensional 3-framed local field theories.

Dualizability and Field Theory

The concept of dualizability within the 3-category of tensor categories is intricately tied to structured TFTs. The authors illuminate the notion of dualizability by showcasing tensor categories as exemplars of such structures. Specifically, a significant assertion in the paper posits that fully dualizable finite tensor categories must be separable. The paper substantiates the separability criterion through a nuanced discussion on the connection between separability and semi-simplicity in different fields, offering a path to constructing fully dualizable 3-categories.

The Loop and Belt Bordisms

One of the enlightening aspects is the association of algebraic properties with specific geometric constructs, such as the loop bordism and the belt bordism. The authors demonstrate that the 1-dimensional loop bordism acts as the double dual autofunctor of a tensor category and detail how the belt-trick bordism facilitates a trivialization hence supporting the quadruple-dual theorem for objects within symmetric monoidal 3-categories.

Handling Separability and Exactness

Building upon Etingof and Ostrik’s theory, the paper consolidates the link between exact module categories and separability conditions. This extends to a significant theorem regarding tensor products: the tensor product of exact module categories preserves exactness. Such results are instrumental in guaranteeing that appropriate dualizable objects in tensor categories can indeed construct a consistent system for local field theories.

Implications and Prospects

Practical implications of these results manifest in the classification of certain topological invariants and theoretical insights into the intersection of algebraic and topological properties. Moreover, this work opens avenues for expanded exploration into homotopy fixed point structures associated with pivotal and spherical categories, and their subsequent descent to structured field theories.

Future Engagements

Speculating on future trajectories, this paper sets a robust foundation for exploring higher-dimensional algebraic structures and their topological counterparts. The elucidation of conditions under which tensor categories are fully dualizable beckons further exploration into broader classes of algebraic objects and their applications in comprehensive topological contexts.

Overall, "Dualizable Tensor Categories" stands as a significant contribution to the landscape of mathematics and theoretical physics, delineating rich connections between algebraic tensor structures and the topology of manifolds with applicable frameworks in local field theories.