Mathematical structures of cohomological field theories
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Submission date: 08. Mar. 2022 (revised version: January 2023)
Keywords and phrases: Supergeometry, Jet bundle theory, Topological quantum field theory
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A mathematical framework of cohomological field theories (CohFTs) is formulated in the language of bigraded manifolds. Algebraic properties of operators in CohFTs are studied. Methods of constructing CohFTs, with or without gauge symmetries, are discussed. In particular, a generalization of the Mathai-Quillen formalism is given. Examples such as topological quantum mechanics, topological sigma model, topological M-theory, and topological Yang-Mills theory can be obtained uniformly using this new formalism.