## Abstract

A variant of the usual supersymmetric nonlinear sigma model is described, governing maps from a Riemann surface*Σ* to an arbitrary almost complex manifold*M*. It possesses a fermionic BRST-like symmetry, conserved for arbitrary*Σ*, 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.

## Keywords

Vertex Operator Sigma Model Gauge Field Homotopy Type Symplectic Geometry## Preview

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## References

- 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 preprintGoogle Scholar - 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)Google Scholar - 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.Google Scholar
- 4.Witten, E.: Topological quantum field theory. Commun. Math. Phys. (in press)Google Scholar
- 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. 81Google Scholar - 6.Jones, V.: A polynomial invariant for knots via von neumann algebras. Bull. Am. Math. Soc.
**82**103 (1985)Google Scholar - 7.Witten, W.: Superconducting strings. Nucl. Phys.
**B249**557 (1985)Google Scholar - 8.Gross, D. J., Harvey, J. A., Martinec, E., Rohm, R.: Heterotic string theory. Nucl. Phys.
**B256**253 (1985)Google Scholar - 9.Witten, E.: Cosmic superstrings. Phys. Lett.
**153B**243 (1985)Google Scholar - 10.Witten, E.: Topological gravity. IAS preprint (February 1988)Google Scholar
- 11.Horowitz, G. T., Lykken, J., Rohm, R., Strominger, A.: Phys. Rev. Lett.
**57**162 (1978)Google Scholar - 12.Strominger, A.: Lectures on closed string field theory. To appear in the proceedings of the ICTP spring workshop, 1987Google Scholar
- 13.Gross, D. J.: High energy symmetries of string theory, Princeton preprint 1988Google Scholar
- 14.Friedan, D., Martinec, E., Shenker, S.: Nucl. Phys.
**B271**93 (1986)Google Scholar - 15.Peskin, M.: Introduction to string and superstring theory. SLAC-PUB-4251 (1987)Google Scholar
- 16.Atiyah, M. F.: Concluding remarks at the Schloss Ringberg meeting (March, 1987)Google Scholar
- 17.Hooft, G. ‘t: Computation of the quantum effects due to a four dimensional pseudoparticle. Phys. Rev.
**D14**3432 (1976)Google Scholar - 18.Witten, E.: Topological gravity: IAS preprint, February, 1988Google Scholar
- 19.Atiyah, M. F., Bott, R.: The moment map and equivariant cohomology. Topology
**23**1 (1984)Google Scholar - 20.Mathai, V., Quillen, D.: Superconnections, thom classes, and equivariant differential forms: Topology
**25A**85 (1986)Google Scholar