Higher-dimensional Algebra and Topological Quantum Field Theory

TL;DR

The paper argues that topological quantum field theories should be understood as representations of higher-dimensional algebra, specifically $n$-categories, and lays out a program to model unitary extended $n$-dimensional TQFTs as weak $n$-functors into $n$-Hilbert spaces. It introduces the suspension operation on $n$-categories and formulates the Stabilization Hypothesis, connecting higher-categorical structures to stable homotopy theory. Central conjectures—the Tangle Hypothesis and Extended TQFT Hypotheses—link framed tangles and duality-rich $n$-categories to concrete realizations of TQFTs via $n$-Hilb as a linear-algebraic target. The work also discusses deformation quantization, quantum doubles, and higher-dimensional centers as pathways to quantization in this unified higher-categorical setting, aiming to provide a coherent framework for extended TQFTs across dimensions.

Abstract

The study of topological quantum field theories increasingly relies upon concepts from higher-dimensional algebra such as n-categories and n-vector spaces. We review progress towards a definition of n-category suited for this purpose, and outline a program in which n-dimensional TQFTs are to be described as n-category representations. First we describe a "suspension" operation on n-categories, and hypothesize that the k-fold suspension of a weak n-category stabilizes for k >= n+2. We give evidence for this hypothesis and describe its relation to stable homotopy theory. We then propose a description of n-dimensional unitary extended TQFTs as weak n-functors from the "free stable weak n-category with duals on one object" to the n-category of "n-Hilbert spaces". We conclude by describing n-categorical generalizations of deformation quantization and the quantum double construction.