zbMATH — the first resource for mathematics

Topological quantum field theory. (English) Zbl 0656.53078
A twisted version of four dimensional supersymmetric gauge theory is formulated. The starting point is the connection between the Floer and Donaldson theories which has led to the conjecture that the “Morse theory” interpretation of Floer homology must be an approximation to a relativistic quantum field theory. It is shown that the Donaldson polynomial invariants of four manifolds and the Floer groups of three manifolds appear naturally.
The Floer theory is generalized to the relativistic case and then the formula for the supersymmetry current and the energy-momentum tensor are obtained. The most important result of the paper is the assertion that the stress tensor is a “BRST” commutator \(T_{\alpha \beta}=\{Q,\lambda_{\alpha \beta}\},\) where Q is a linear transformation of the space of all functionals of the field variables, and \(\lambda_{\alpha \beta}^ a \)tensor field characteristic for the model.
Finally, the possible physical interpretation of the model is presented. It is pointed out that the model is in a sense a generally covariant quantum theory in which general covariance is unbroken, there are no gravitons, and the only excitations are topological.
Reviewer: G.Zet

53C80 Applications of global differential geometry to the sciences
81T60 Supersymmetric field theories in quantum mechanics
81T08 Constructive quantum field theory
Full Text: DOI
[1] Donaldson, S.: An application of gauge theory to the topology of four manifolds. J. Differ. Geom.18, 269 (1983); The orientation of Yang-Mills moduli spaces and 4-manifold topology. J. Differ. Geom.26, 397 (1987); Polynomial invariants for smooth four-manifolds. Oxford preprint
[2] Freed, D., Uhlenbeck, K.: Instantons and four manifolds. Berlin, Heidelberg, New York: Springer 1984 · Zbl 0559.57001
[3] Belavin, A., Polyakov, A., Schwartz, A., Tyupkin, Y.: Phys. Lett. B59, 85 (1975)
[4] Taubes, C.: Self-dual Yang-Mills connections on non-self-dual 4-manifolds. J. Differ. Geom.17, 139 (1982) · Zbl 0484.53026
[5] Uhlenbeck, K.: Connections withL p bounds on curvature. Commun. Math. Phys.83, 31 (1982). Removable singularities in Yang-Mills fields. Commun. Math. Phys.83, 11 (1982) · Zbl 0499.58019 · doi:10.1007/BF01947069
[6] Floer, A.: An instanton invariant for three manifolds. Courant Institute preprint (1987); Morse theory for fixed points of symplectic diffeomorphisms. Bull. AMS16, 279 (1987) · Zbl 0617.53042
[7] Atiyah, M.F.: New invariants of three and four dimensional manifolds. In: The Symposium on the Mathematical Heritage of Hermann Weyl, Wells, R. et al. (eds.). (Univ. of North Carolina, May, 1987)
[8] Braam, P.J.: Floer homology groups for homology three spheres. University of Utrecht Mathematics preprint 484, November, 1987 · Zbl 0856.57010
[9] Witten, E.: Supersymmetry and morse theory. J. Differ. Geom.17, 661 (1982) · Zbl 0499.53056
[10] ’t Hooft, G.: Computation of the quantum effects due to a four dimensional pseudoparticle. Phys. Rev. D14, 3432 (1976)
[11] Jackiw, R., Rebbi, C.: Phys. Rev. Lett.37, 172 (1976) · doi:10.1103/PhysRevLett.37.172
[12] Callan, C.G., Dashen, R., Gross, D.J.: Phys. Lett.63 B, 334 (1976)
[13] Atiyah, M.F., Hitchin, N., Singer, I.: Self-duality in Riemannian geometry. Proc. Roy. Soc. London A362, 425 (1978) · Zbl 0389.53011
[14] Affleck, I., Dine, M., Seiberg, N.: Dynamical supersymmetry breaking in supersymmetric QCD. Nucl. Phys. B241, 493 (1984); Dynamical supersymmetry breaking in four dimensions and its phenomenological implications. Nucl. Phys. B256, 557 (1985) · doi:10.1016/0550-3213(84)90058-0
[15] Seiberg, N.: IAS preprint (to appear)
[16] Novikov, V.A., Shifman, M.A., Vainshtein, A.I., Zakharov, V.I.: Nucl. Phys. B229, 407 (1983) · doi:10.1016/0550-3213(83)90340-1
[17] Amati, D., Konishi, K., Meurice, Y., Rossi, G.C., Veneziano, G.: Non-perturbative aspects in supersymmetric gauge theories. Physics Reports (to appear)
[18] Friedan, D., Martinec, E., Shenker, S.: Nucl. Phys. B271, 93 (1986)
[19] Peskin, M.: Introduction to string and superstring theory. SLAC-PUB-4251 (1987)
[20] Green, M.B., Schwarz, J.H., Witten, E.: Superstring theory. Cambridge: Cambridge University Press 1987 · Zbl 0619.53002
[21] Witten, E.: Global anomalies in string theory. In: Symposium on anomalies, geometry, and topology. White, A., Bardeen, W. (eds.), especially pp. 90-95. Singapore: World Scientific 1985
[22] Becchi, C., Rouet, A., Stora, R.: The abelian Higgs-Kibble model, unitarity of theS-operator. Phys. Lett.69 B, 309 (1974); Renormalization of gauge theories. Ann. Phys.98, 287 (1976)
[23] Tyupin, I.V.: Gauge invariance in field theory and in statistical physics in the operator formalism. Lebedev preprint FIAN No. 39 (1975), unpublished
[24] Kugo, T., Ojima, I.: Manifestly covariant canonical formulation of Yang-Mills theories. Phys. Lett.73 B, 459 (1978); Local covariant operator formalism of non-abelian gauge theories and quark confinement problem. Supp. Prog. Theor. Phys.66, 1 (1979) · Zbl 1098.81591
[25] Polchinski, J.: Scale and conformal invariance in quantum field theory. Univ. of Texas preprint UTTG-22-87
[26] D’Adda, A., DiVecchia, P.: Supersymmetry and instantons. Phys. Lett.73 B, 162 (1978)
[27] Witten, E.: AnSU(2) anomaly. Phys. Lett.117 B, 432 (1982)
[28] Segal, G.: Oxford preprint (to appear)
[29] Horowitz, G.T., Lykken, J., Rohm, R., Strominger, A.: Phys. Rev. Lett.57, 283 (1986) · doi:10.1103/PhysRevLett.57.283
[30] Witten, E.: Topological gravity. IAS preprint, February, 1988
[31] Witten, E.: Topological sigma models. Commun. Math. Phys. (to appear) · Zbl 0674.58047
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.