Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
2007-10-12
J. High Energy Phys. 2009, no. 3, 032, 29 pp
Physics
High Energy Physics
High Energy Physics - Theory
26 pages, 1 figure, references added, typos corrected
Scientific paper
10.1088/1126-6708/2009/03/032
In light of Sen's weak coupling limit of F-theory as a type IIB orientifold, the compatibility of the tadpole conditions leads to a non-trivial identity relating the Euler characteristics of an elliptically fibered Calabi-Yau fourfold and of certain related surfaces. We present the physical argument leading to the identity, and a mathematical derivation of a Chern class identity which confirms it, after taking into account singularities of the relevant loci. This identity of Chern classes holds in arbitrary dimension, and for varieties that are not necessarily Calabi-Yau. Singularities are essential in both the physics and the mathematics arguments: the tadpole relation may be interpreted as an identity involving stringy invariants of a singular hypersurface, and corrections for the presence of pinch-points. The mathematical discussion is streamlined by the use of Chern-Schwartz-MacPherson classes of singular varieties. We also show how the main identity may be obtained by applying `Verdier specialization' to suitable constructible functions.
Aluffi Paolo
Esole Mboyo
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