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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.

MSC:
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
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