On A Stringy Singular Cohomology

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.