Advances in Theoretical and Mathematical Physics

Intersection spaces, perverse sheaves and type IIB string theory

Markus Banagl, Nero Budur, and Laurenţu Maxim

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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.

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

First available in Project Euclid: 27 October 2014

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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.

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