## Advances in Theoretical and Mathematical Physics

- Adv. Theor. Math. Phys.
- Volume 18, Number 2 (2014), 363-399.

### Intersection spaces, perverse sheaves and type IIB string theory

Markus Banagl, Nero Budur, and Laurenţu Maxim

#### Abstract

The method of intersection spaces associates rational Poincaré 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 selfduality 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.

#### Article information

**Source**

Adv. Theor. Math. Phys., Volume 18, Number 2 (2014), 363-399.

**Dates**

First available in Project Euclid: 27 October 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.atmp/1414414838

**Mathematical Reviews number (MathSciNet)**

MR3273317

**Zentralblatt MATH identifier**

06386218

#### Citation

Banagl, Markus; Budur, Nero; Maxim, Laurenţu. Intersection spaces, perverse sheaves and type IIB string theory. Adv. Theor. Math. Phys. 18 (2014), no. 2, 363--399. https://projecteuclid.org/euclid.atmp/1414414838