---
address:
- |
    Department of Mathematics\
    University of Texas\
    1 University Station C1200\
    Austin, TX 78712-0257
- |
    Department of Mathematics\
    Harvard University\
    1 Oxford Street\
    Cambridge, MA 02138
- |
    Department of mathematics\
    Massachusetts Institute of Technology\
    77 Massachusetts Avenue\
    Cambridge, MA 02139-4307
- |
    Department of Mathematics\
    University of California\
    970 Evans Hall \#3840\
    Berkeley, CA 94720-3840
author:
- 'Daniel S. Freed'
- 'Michael J. Hopkins'
- Jacob Lurie
- Constantin Teleman
date: 'June 19, 2009'
title: Topological Quantum Field Theories from Compact Lie Groups
---

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Let $G$ be a compact Lie group and $BG$ a classifying space for $G$. Then a class in $H^{n+1}(BG;{\mathbb Z})$ leads to an $n$-dimensional topological quantum field theory (TQFT), at least for $n=1,2,3$. The theory for $n=1$ is trivial, but we include it for completeness. The theory for $n=2$ has some infinities if $G$ is not a finite group; it is a topological limit of 2-dimensional Yang-Mills theory. The most direct analog for $n=3$ is an $L^2$ version of the topological quantum field theory based on the classical Chern-Simons invariant, which is only partially defined. The TQFT constructed by 