Cohomological Field Theories and First-Order Nonlinear PDEs

We introduce a formalism for constructing cohomological field theories (CohFTs), inspired by Witten’s paradigm of “symmetries, fields, and equations”. We apply this formalism to various first-order nonlinear PDEs and show that the resulting CohFTs agree with those previously proposed by physicists. In particular, applying it to the generalized Seiberg–Witten equations provides a unified perspective on the supersymmetric action functionals of the Donaldson–Witten, Seiberg–Witten, and Kapustin–Witten theories.