Higher-dimensional algebra and topological quantum field theory

Abstract

The study of topological quantum field theories increasingly relies upon concepts from higherdimensional 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.

TQFT’s and gerbes

“…The constructions and methods of proof here have a distinct state-sum flavour, and there are intriguing suggestions of a graded integration theory, involving objects of all dimensions up to the dimension being integrated. This could be useful for understanding broader dimensional ladders for TQFT [4], which normally involves just the top dimension and one dimension lower, or TQFT with corners. It would also be interesting to see whether there are links between these constructions and the state-sum models for quantum gravity proposed by Barrett and Crane [6] and Mikovic [17].…”

“…Acknowledgements Preliminary ideas on the relation between TQFT's and gerbes were presented at the XXth Workshop on Geometric Methods in Physics, Bialowieża, July [1][2][3][4][5][6][7] 2001, in the special session on Mathematical Physics at the National Meeting of the Portuguese Mathematical Society, Coimbra, Approaches to non-abelian gerbes with connection have been studied by various authors [7,2,3]. We could have obtained solutions of the higher-rank 1-dimensional embedded TQFT equations, mentioned after Definition 3.2, from nonabelian bundles with connection, via path-ordered exponentials.…”

Towards a Categorical Foundation of Mathematics

“…The difficulty is in giving explicitly the so-called "coherence conditions" the canonical arrows should satisfy. In [BD1] we read: "It is clear that new ideas are needed to do so without a combinatorial explosion, since already the explicit definition of a tricategory [in [GPS]] takes six pages, and that of a triequivalence 13 pages! ".…”

Relational Quantum Mechanics

“…There is an important literature on possible ways the category notions can be applied to physics; specifically to quantising space-time [6], attaching a formal language to a physical system [7], studying topological quantum field theories [8,9], exploring quantum issues and quantum information theory [10].…”