1,599 research outputs found
Iterated spans and classical topological field theories
We construct higher categories of iterated spans, possibly equipped with
extra structure in the form of "local systems", and classify their fully
dualizable objects. By the Cobordism Hypothesis, these give rise to framed
topological quantum field theories, which are the framed versions of the
"classical" TQFTs considered in the quantization programme of
Freed-Hopkins-Lurie-Teleman.
Using this machinery, we also construct an infinity-category of Lagrangian
correspondences between symplectic derived algebraic stacks and show that all
its objects are fully dualizable.Comment: Accepted version, plus corrections to Remarks 10.5 and 10.7. 64 page
Higher-dimensional Algebra and Topological Quantum Field Theory
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.Comment: 36 pages, LaTeX; this version includes all 36 figure
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