zbMATH — the first resource for mathematics

Topological quantum field theories. (English) Zbl 0692.53053
The starting point to topological quantum field theory was given by E. Witten [J. Differ. Geom. 17, 661-692 (1982; Zbl 0499.53056)] where he explained the geometric meaning of super-symmetry, pointing out that for super-symmetric quantum mechanics the Hamiltonian is just the Hodge- Laplacian. He then outlined the corresponding ideas for super-symmetric quantum field theories viewed as differential geometry of certain infinite dimensional manifolds, including the associated analysis and topology. This shows that there may be interesting topological aspects of quantum field theory and that these should be important for physics. On the other hand one can use these quantum field theories as a conceptual tool to suggest new mathematical results. Indeed this reversed process led to spectacular progress in the understanding of geometry in low dimensions.
The author starts with presenting a set of axioms for topological quantum field theories following G. B. Segal [The definition of conformal field theory, Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165-171 (1988; Zbl 0657.53060)], then lists examples of theories (known to exist) satisfying such axioms. These include in \(d=1\) the Floer/Gromov theory and holomorphic conformal field theories (Segal); in \(d=2\) the Jones/Witten theory, Casson theory, Johnson theory and Thursten theory and in \(d=3\) the Floer/Donaldson theory.
Reviewer: R.Schmid

53C80 Applications of global differential geometry to the sciences
57N99 Topological manifolds
81T99 Quantum field theory; related classical field theories
Full Text: DOI Numdam EuDML
[1] M. F. Atiyah, New invariants of three and four dimensional manifolds, inThe Mathematical Heritage of Herman Weyl, Proc. Symp. Pure Math.,48, American Math. Soc. (1988), 285–299.
[2] S. K. Donaldson, Polynomial invariants for smooth four-manifolds, to appear inTopology.
[3] A. Floer, Morse theory for fixed points of symplectic diffeomorphisms,Bull. A.M.S.,16 (1987), 279–281. · Zbl 0617.53042 · doi:10.1090/S0273-0979-1987-15517-0
[4] A. Floer,An instanton invariant for three manifolds, Courant Institute preprint, to appear. · Zbl 0684.53027
[5] M. Gromov, Pseudo-holomorphic curves in symplectic manifolds,Invent. Math.,82 (1985), 307–347. · Zbl 0592.53025 · doi:10.1007/BF01388806
[6] N. J. Hitchin, The self-duality equations on a Riemann surface,Proc. London Math. Soc. (3),55 (1987), 59–126. · Zbl 0634.53045 · doi:10.1112/plms/s3-55.1.59
[7] D. Johnson,A geometric form of Casson’s invariant and its connection with Reidemeister torsion, unpublished lecture notes.
[8] V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials,Ann. of Math.,126 (1987), 335–388. · Zbl 0631.57005 · doi:10.2307/1971403
[9] A. Pressley andG. B. Segal,Loop Groups, Oxford University Press (1988). · Zbl 0638.22009
[10] G. B. Segal,The definition of conformal field theory (to appear). · Zbl 0657.53060
[11] E. Witten, Super-symmetry and Morse theory,J. Diff. Geom.,17 (4) (1982), 661–692. · Zbl 0499.53056
[12] E. Witten, Quantum field theory and the Jones polynomial,Comm. Math. Phys. (to appear). · Zbl 0726.57010
[13] E. Witten, Topological quantum field theory,Comm. Math. Phys.,117 (1988), 353–386. · Zbl 0656.53078 · doi:10.1007/BF01223371
[14] E. Witten, Topological sigma models,Comm. Math. Phys.,118 (1988), 411–449. · Zbl 0674.58047 · doi:10.1007/BF01466725
[15] E. Witten, 2 + 1 dimensional gravity as an exactly soluble system,Nucl. Phys. B,311 (1988/89), 46–78. · Zbl 1258.83032 · doi:10.1016/0550-3213(88)90143-5
[16] E. Witten, Topology changing amplitudes in 2 + 1 dimensional gravity,Nucl. Phys. B (to appear).
[17] E. Witten, Elliptic genera and quantum field theory,Comm. Math. Phys.,109 (1987), 525–536. · Zbl 0625.57008 · doi:10.1007/BF01208956
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.