TY - JOUR
T1 - Singular cohomology from supersymmetric field theories
AU - Schommer-Pries, Christopher
AU - Stapleton, Nathaniel
N1 - Publisher Copyright:
© 2021 Elsevier Inc.
PY - 2021/10/29
Y1 - 2021/10/29
N2 - 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.
AB - 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.
KW - Quantum field theory
KW - Singular cohomology
KW - Supersymmetric
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U2 - 10.1016/j.aim.2021.107944
DO - 10.1016/j.aim.2021.107944
M3 - Article
AN - SCOPUS:85111849887
SN - 0001-8708
VL - 390
JO - Advances in Mathematics
JF - Advances in Mathematics
M1 - 107944
ER -