Higher-dimensional Algebra and Topological Quantum Field Theory

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.

36 pages, LaTeX; this version includes all 36 figures

References in corpus (7)

• Chern-Simons Theory with Finite Gauge Group
• Higher Algebraic Structures and Quantization
• Four dimensional topological quantum field theory, Hopf categories, and the canonical bases
• Lattice Topological Field Theory in Two Dimensions
• Structure of Topological Lattice Field Theories in Three Dimensions
• Classification and construction of unitary topological field theories in two dimensions
• Topological Aspects of Spin and Statistics in Nonlinear Sigma Models

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