Topological sigma models. (English) Zbl 0674.58047

Summary: A variant of the usual supersymmetric nonlinear sigma model is described, governing maps from a Riemann surface \(\Sigma\) to an arbitrary almost complex manifold M. It possesses a fermionic BRST-like symmetry, conserved for arbitrary \(\Sigma\), and obeying \(Q^ 2=0\). In a suitable version, the quantum ground states are the \(1+1\) dimensional Floer groups. The correlation functions of the BRST-invariant operators are invariants (depending only on the homotopy type of the almost complex structure of M) similar to those that have entered in recent work of Gromov on symplectic geometry. The model can be coupled to dynamical gravitational or gauge fields while preserving the fermionic symmetry; some observations by Atiyah suggest that the latter coupling may be related to the Jones polynomial of knot theory. From the point of view of string theory, the main novelty of this type of sigma model is that the graviton vertex operator is a BRST commutator. Thus, models of this type may correspond to a realization at the level of string theory of an unbroken phase of quantum gravity.


58Z05 Applications of global analysis to the sciences
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
58C50 Analysis on supermanifolds or graded manifolds
81T60 Supersymmetric field theories in quantum mechanics
58J70 Invariance and symmetry properties for PDEs on manifolds
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 of Smooth Four-Manifolds, Oxford preprint
[2] Floer, A.: An instanton invariant for three manifolds. Courant Institute preprint (1987); Morse theory for fixed points of symplectic diffeomorphisms. Bull. Am. Math. Soc.16 279 (1987) · Zbl 0617.53042
[3] Atiyah, M. F.: New invariants of three and four dimensional manifolds. To appear in the proceedings of the Symposium on the Mathematical Heritage of Hermann Weyl (Chapel Hill, May, 1987), ed. R. Wells et. al.
[4] Witten, E.: Topological quantum field theory. Commun. Math. Phys. (in press) · Zbl 0656.53078
[5] Gromov, M.: Pseudo holomorphic curves in symplectic manifolds. Invent. Math.82 307 (1985); and in the Proceedings of the International Congress of Mathematicians, Berkeley (August, 1986), p. 81 · Zbl 0592.53025
[6] Jones, V.: A polynomial invariant for knots via von neumann algebras. Bull. Am. Math. Soc.82 103 (1985) · Zbl 0564.57006
[7] Witten, W.: Superconducting strings. Nucl. Phys.B249 557 (1985)
[8] Gross, D. J., Harvey, J. A., Martinec, E., Rohm, R.: Heterotic string theory. Nucl. Phys.B256 253 (1985)
[9] Witten, E.: Cosmic superstrings. Phys. Lett.153B 243 (1985)
[10] Witten, E.: Topological gravity. IAS preprint (February 1988)
[11] Horowitz, G. T., Lykken, J., Rohm, R., Strominger, A.: Phys. Rev. Lett.57 162 (1978)
[12] Strominger, A.: Lectures on closed string field theory. To appear in the proceedings of the ICTP spring workshop, 1987
[13] Gross, D. J.: High energy symmetries of string theory, Princeton preprint 1988
[14] Friedan, D., Martinec, E., Shenker, S.: Nucl. Phys.B271 93 (1986)
[15] Peskin, M.: Introduction to string and superstring theory. SLAC-PUB-4251 (1987)
[16] Atiyah, M. F.: Concluding remarks at the Schloss Ringberg meeting (March, 1987)
[17] Hooft, G. ?t: Computation of the quantum effects due to a four dimensional pseudoparticle. Phys. Rev.D14 3432 (1976)
[18] Witten, E.: Topological gravity: IAS preprint, February, 1988
[19] Atiyah, M. F., Bott, R.: The moment map and equivariant cohomology. Topology23 1 (1984) · Zbl 0521.58025
[20] Mathai, V., Quillen, D.: Superconnections, thom classes, and equivariant differential forms: Topology25A 85 (1986) · Zbl 0592.55015
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.