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.
Highlights
Given an equivalence between two k-manifolds, the field theory provides an equivalence between the two seemingly distinct algebraic operations corresponding to the manifolds. We apply this approach with the aforementioned partial 3-dimensional 3-framed local topological field theory FC associated to any finite tensor category C. (Here “3-framed” means that the manifolds are equipped with trivializations of the stabilizations of their tangent bundles up to dimension 3.) The simplest nontrivial 3-framed manifold is the following 1-dimensional interval, called “the loop bordism”: (Comparing a normal framing of this immersion with the blackboard framing of the paper provides the bordism with a 2-framing, and a 3-framing by stabilization.) We prove that the field theory invariant of this bordism is the C-C bimodule obtained by twisting the left action on the identity bimodule by the right double dual functor
We say that a finite semisimple tensor category C over a perfect field is separable if the identity C–C-bimodule category C can be expressed as the category of modules for a separable algebra object within the tensor category C ⊠ Cmp. (Here, Cmp is the category C with the opposite monoidal structure, and − ⊠ − denotes the Deligne tensor product, that is the linear category corepresenting bilinear functors.) This separability condition is satisfied by any finite semisimple tensor category over a field of characteristic zero, and by any fusion category of nonzero global dimension over any algebraically closed field
The belt bordism trivializing the square of the loop bordism, depicted above, can itself be decomposed into two pieces, each of which can be identified as a certain adjoint or witness to an adjunction in the bordism category. (This decomposition is depicted in Figure 3.) The Radford property of a finite tensor category ensures that the associated field theory takes values on these pieces, and provides an algebraic trivialization of the bimodule associated to the quadruple dual
Summary
Quantum field theories associate to a manifold, thought of as the underlying physical space of a system, a vector space of quantum field states on that manifold. Atiyah and Segal abstracted this situation into the formal notion of a topological (quantum) field theory: an n-dimensional topological field theory is a symmetric monoidal functor from the category of (n − 1)-dimensional manifolds and their bordisms to the category of vector spaces [Ati88b, Seg04]. Not every object of the target n-category is an allowable value for this algebraic invariant of a point; rather, there is a restrictive algebraic condition called full dualizability (or n-dualizability) that ensures an object of an n-category extends to a consistent system of invariants providing a local field theory This is the content of the cobordism hypothesis, a classification result conjectured by Baez–Dolan [BD95] and proven by Hopkins–Lurie [Lur09a]: an n-dimensional local framed topological field theory is determined by its value on a point, and any fully dualizable object of a symmetric monoidal n-category provides a local framed topological field theory whose point-value is that object
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