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Topological Lagrangians and cohomology. (English) Zbl 0721.58056
E. Witten [Commun. Math. Phys. 117, No.3, 353-386 (1988; Zbl 0656.53078)] has interpreted the Donaldson’s invariants of 4-manifolds by means of a topological Lagrangian. His work has subsequently been widely re-interpreted in the physics literature. The present authors describe an alternative approach: they seek to explain Witten’s theory in terms of an infinite-dimensional version of the Gauss-Bonnet theorem (and its generalizations). They derive the explicit form of Witten’s Lagrangian as a direct consequence of standard formulas in differential geometry. In the second part of the paper, they use the same methods to treat Lagrangians proposed by Witten for the Casson invariant of homology 3- spheres.

MSC:
58J90 Applications of PDEs on manifolds
58A10 Differential forms in global analysis
81T13 Yang-Mills and other gauge theories in quantum field theory
53C80 Applications of global differential geometry to the sciences
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