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Higher-dimensional analogues of Donaldson-Witten theory. (English) Zbl 0938.81035

Summary: We present a Donaldson-Witten-type field theory in eight dimensions on manifolds with Spin(7) holonomy. We prove that the stress tensor is BRST exact for metric variations preserving the holonomy and we give the invariants for this class of variations. In six and seven dimensions we propose similar theories on Calabi-Yau threefolds and manifolds of \(G_2\) holonomy, respectively. We point out that these theories arise by considering supersymmetric Yang-Mills theory defined on such manifolds. The theories are invariant under metric variations preserving the holonomy structure without the need for twisting. This statement is a higher-dimensional analogue of the fact that Donaldson-Witten field theory on hyper-Kähler 4-manifolds is topological without twisting. Higher-dimensional analogues of Floer cohomology are briefly outlined. All of these theories arise naturally within the context of string theory.

MSC:

81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
58D29 Moduli problems for topological structures
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
57R57 Applications of global analysis to structures on manifolds
81T13 Yang-Mills and other gauge theories in quantum field theory
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