Higher-dimensional Algebra and Topological Quantum Field Theory

Details Higher-dimensional Algebra and Topological Quantum Field Theory Higher-dimensional Algebra and Topological Quantum Field Theory

1995-03-05

arxiv.org/abs/q-alg/9503002v2

J.Math.Phys. 36 (1995) 6073-6105

Physics

High Energy Physics

High Energy Physics - Theory

36 pages, LaTeX; this version includes all 36 figures

Scientific paper

10.1063/1.531236

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.

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