Higher algebraic structures and quantization

• DOI: 10.1007/bf02102643
• Publication Date: 2005-09-13T01:25:35Z
• Authors: Daniel S. Freed
• Keywords: High Energy Physics - Theory; 18D05; 58D30; FOS: Physical sciences; 58D29; 01 natural sciences; 81T70; 58F06; High Energy Physics - Theory (hep-th); 57N15; Mathematics - Quantum Algebra; FOS: Mathematics; Quantum Algebra (math.QA); 0101 mathematics

62 pages + 16 figures (revised version). In this revision we make some small corrections and clarifications<br/>We derive (quasi-)quantum groups in 2+1 dimensional topological field theory directly from the classical action and the path integral. Detailed computations are carried out for the Chern-Simons theory with finite gauge group. The principles behind our computations are presumably more general. We extend the classical action in a d+1 dimensional topological theory to manifolds of dimension less than d+1. We then ``construct'' a generalized path integral which in d+1 dimensions reduces to the standard one and in d dimensions reproduces the quantum Hilbert space. In a 2+1 dimensional topological theory the path integral over the circle is the category of representations of a quasi-quantum group. In this paper we only consider finite theories, in which the generalized path integral reduces to a finite sum. New ideas are needed to extend beyond the finite theories treated here.<br/>