Singular cohomology from supersymmetric field theories

Abstract

We show that Sullivan's model of rational differential forms on a simplicial set X may be interpreted as a (kind of) 0|1-dimensional supersymmetric quantum field theory over X, and, as a consequence, concordance classes of such theories represent the rational cohomology of X.We introduce the notion of superalgebraic cartesian sets, a concept of space which should roughly be thought of as a blend of simplicial sets and supermanifolds, but valid over an arbitrary base ring. Every simplicial set gives rise to a superalgebraic cartesian set and so we can formulate the notion of 0|1-dimensional supersymmetric quantum field theory over X, entirely within the language of such spaces. We explore several variations in the kind of field theory and discuss their cohomological interpretations.Finally, utilizing a theorem of Cartan-Miller, we describe a variant of our theory which is valid over any commutative ring S and allows one to recover the S-cohomology H⁎(X;S) additively and with multiples of the cup product structure.