Abstract
Starting from a Lie group G whose Lie algebra is equipped with an invariant nondegenerate symmetric bilinear form, we show that four-dimensional BF theory with cosmological term gives rise to a TQFT satisfying a generalization of Atiyah's axioms to manifolds equipped with principal G-bundle. The case G=GL(4,ℝ) is especially interesting because every 4-manifold is then naturally equipped with a principal G-bundle, namely its frame bundle. In this case, the partition function of a compact oriented 4-manifold is the exponential of its signature, and the resulting TQFT is isomorphic to that constructed by Crane and Yetter using a state sum model, or by Broda using a surgery presentation of four-manifolds.
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References
AshtekarA.: Lectures on Non-perturbative Canonical Quantum Gravity, World Scientific, Singapore, 1991.
AtiyahM.: Topological quantum field theories, Inst. Hautes Études Sci. Publ. Math. 68 (1988), 175–186.
Baez, J.: Knots and quantum gravity: progress and prospects, to appear in the Proc. Seventh Marcel Grossman Meeting on General Relativity, Preprint available as gr-qc/9410018.
Barrett, J. and Westbury, B.: Spherical categories, Preprint available as hep-th/9310164.
Barrett, J. and Westbury, B.: The equality of 3-manifold invariants, Preprint available as hep-th/9406019.
Blanchet, C., Habegger, N., Masbaum, G., and Vogel, P.: Topological quantum field theories derived from the Kauffman bracket, Preprint.
BlauM. and ThompsonG.: Topological gauge theories of antisymmetric tensor fields, Ann. Phys. 205 (1991), 130–172.
BrodaB.: Surgical invariants of 4-manifolds, in D.Yetter (ed.), Proc. Conference on Quantum Topology, World Scientific, Singapore, 1994.
Cattaneo, A., Cotta-Ramusino, P., and Martellini, M.: Three-dimensional BF theories and the Alexander-Conway invariant of knots, Preprint available as hep-th/9506002.
Cattaneo, A., Cotta-Ramusino, P., Fröhlich, J., and Martellini, M.: Topological BF theories in three and four dimensions. To appear in J. Math. Phys., Preprint available as hep-th/9505027.
Chari, V. and Pressley, A.: A Guide to Quantum Groups, Cambridge Univ. Press, 1994.
Cotta-RamusinoP. and MartelliniM.: BF theories and two-knots, in: J.Baez. (ed), Knots and Quantum Gravity, Oxford Univ. Press, Oxford, 1994.
CraneL., KauffmanL., and YetterD.: Evaluating the Crane-Yetter invariant, in: L.Kauffman and R.Baadhio, (eds), Quantum Topology, World Scientific, Singapore, 1993.
Crane, L., Kauffman, L., and Yetter, D.: State-sum invariants of four-manifolds, I, Kansas State Univ. preprint available as hep-th/9409167.
CraneL. and YetterD.: A categorical construction of 4d topological quantum field theories, in: L.Kauffman and R.Baadhio, (eds), Quantum Topology, World Scientific, Singapore, 1993.
Crane, L. and Yetter, D.: On algebraic structures implicit in topological quantum field theories, Preprint available as hep-th/9412025.
HorowitzG.: Exactly soluble diffeomorphism-invariant theories, Comm. Math. Phys. 125 (1989), 417–437.
LawrenceR.: Triangulation, categories and extended topological field theories, in: R.Baadhio and L.Kauffman, (eds), Quantum Topology, World Scientific, Singapore, 1993, pp. 191–208.
ReshetikhinN. and TuraevV.: Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127 (1990), 1–26.
Roberts, J.: Skein theory and Turaev-Viro invariants, Cambridge Univ. preprint, 1993.
Roberts, J.: Refined state-sum invariants of three- and four-manifolds, Cambridge Univ. preprint, 1994.
Roberts, J.: Personal communication.
Sawin, S.: Links, quantum groups and TQFTs. To appear in Bull. Amer. Math. Soc., Preprint available as q-alg/9506002.
Turaev, V.: Topology of shadows. Preprint.
TuraevV. and ViroO.: State-sum invariants of three-manifolds and quantum 6j-symbols, Topology 31 (1992), 865–902.
Walker, K.: On Witten's three-manifold invariants. Preprint.
WittenE.: (2+1)-dimensional gravity as an exactly soluble system, Nuclear Phys. B311 (1988), 46–78.
WittenE.: Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989), 351–399.
YetterD.: State-sum invariants of three-manifolds associated to Artinian semisimple tortile categories, Topology Appl. 58 (1994), 47–80.
Yetter, D.: Homologically twisted invariants related to (2+1)- and (3+1)-dimensional state-sum topological quantum field theories, Preprint available as hep-th/9311082.
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Baez, J.C. Four-dimensional BF theory as a topological quantum field theory. Lett Math Phys 38, 129–143 (1996). https://doi.org/10.1007/BF00398315
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DOI: https://doi.org/10.1007/BF00398315