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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

MSC:
53C80 Applications of global differential geometry to the sciences
81T60 Supersymmetric field theories in quantum mechanics
81T08 Constructive quantum field theory
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