Singular cohomology from supersymmetric field theories

We show that Sullivan's model of rational differential forms on a simplicial set X may be interpreted as a (kind of) 0 vertical bar 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 vertical bar 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. (c) 2021 Elsevier Inc. All rights reserved.

Automated summary

In this paper, Sullivan's model of rational differential forms on a simplicial set X is interpreted as a kind of 0|1 dimensional supersymmetric quantum field theory, where concordance classes of such theories represent the rational cohomology of X. The notion of superalgebraic cartesian sets is introduced as a concept of space blending simplicial sets and supermanifolds valid over an arbitrary base ring. The paper explores various variations of field theory and discusses their cohomological interpretations, ultimately describing a variant that allows recovery of the cohomology H*(X; S) over any commutative ring S.