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Higher Algebraic Structures and Quantization

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arxiv hep-th/9212115 v2 pith:QXZD4AYQ submitted 1992-12-18 hep-th math.QA

Daniel S. Freed This is my paper · ORCID
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keywords finiteintegralpaththeorydimensionaltopologicalactionclassical
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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.

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Cited by 3 Pith papers

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    Authors introduce a TFT-based framework for finite topological symmetries in QFT, including gauging, condensation defects, and duality defects, with an appendix on finite homotopy theories.

  3. The many faces of higher Hilbert spaces

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    Introduces G-Hermitian 2-vector spaces via fixed points of an O(2)-action on 2Vect and criteria for positive pairings to generalize the Hermitian-to-Hilbert passage, with an outline for inductive higher-dimensional versions.

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