On A Stringy Singular Cohomology

TL;DR

The paper proposes a new cohomology theory inspired by string theory on singular varieties, which exhibits symmetry under the mirror map and could describe string theory zero modes geometrically.

Contribution

It introduces a novel stringy singular cohomology that is symmetric under mirror symmetry, differing from existing theories like intersection and L^2-cohomology.

Findings

The new cohomology has good behavior similar to intersection cohomology.

It is symmetric with respect to the mirror map.

Potentially describes string theory zero modes geometrically.

Abstract

String theory has already motivated, suggested, and sometimes well-nigh proved a number of interesting and sometimes unexpected mathematical results, such as mirror symmetry. A careful examination of the behavior of string propagation on (mildly) singular varieties similarly suggests a new type of (co)homology theory. It has the `good behavior' of the well established intersection (co)homology and $L^2$-cohomology, but is markedly different in some aspects. For one, unlike the intersection (co)homology and the $L^2$-cohomology (or any other known thus far), this new cohomology is symmetric with respect to the mirror map. Among the available choices, this makes it into a prime candidate for describing the string theory zero modes in geometrical terms.