Intersection spaces, perverse sheaves and type IIB string theory

TL;DR

This paper explores the mathematical structures of intersection spaces and perverse sheaves in relation to type IIB string theory, providing new insights into the cohomology of singular hypersurfaces and their physical implications.

Contribution

It establishes that the cohomology of intersection spaces corresponds to the hypercohomology of a perverse sheaf, the intersection space complex, on hypersurfaces with isolated singularities, linking geometry and physics.

Findings

Cohomology of intersection spaces equals hypercohomology of a perverse sheaf.

Intersection space complex underlies a mixed Hodge module with canonical structures.

Global Poincaré duality is induced by Verdier self-duality for the perverse sheaf.

Abstract

The method of intersection spaces associates rational Poincar\'e complexes to singular stratified spaces. For a conifold transition, the resulting cohomology theory yields the correct count of all present massless 3-branes in type IIB string theory, while intersection cohomology yields the correct count of massless 2-branes in type IIA theory. For complex projective hypersurfaces with an isolated singularity, we show that the cohomology of intersection spaces is the hypercohomology of a perverse sheaf, the intersection space complex, on the hypersurface. Moreover, the intersection space complex underlies a mixed Hodge module, so its hypercohomology groups carry canonical mixed Hodge structures. For a large class of singularities, e.g., weighted homogeneous ones, global Poincar\'e duality is induced by a more refined Verdier self-duality isomorphism for this perverse sheaf. For such singularities, we prove furthermore that the pushforward of the constant sheaf of a nearby smooth deformation under the specialization map to the singular space splits off the intersection space complex as a direct summand. The complementary summand is the contribution of the singularity. Thus, we obtain for such hypersurfaces a mirror statement of the Beilinson-Bernstein-Deligne decomposition of the pushforward of the constant sheaf under an algebraic resolution map into the intersection sheaf plus contributions from the singularities.