Dimensional reduction in supersymmetric field theories

A class of quantum field theories invariant with respect to the action of an odd vector field Q on a source supermanifold $Σ$ is considered. We suppose that Q satisfies the conditions under which an integral of any Q-invariant function over $Σ$ localizes to the zero locus of Q. The Q-invariant sector of a field theory from the class above is shown then to be equivalent to the quantum field theory defined on zero locus of the vector field Q.

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