Chern class identities from tadpole matching in type IIB and F-theory
Paolo Aluffi, Mboyo Esole
TL;DR
The paper analyzes the consistency of D3-tadpole cancellation between type IIB orientifolds and F-theory in Sen's weak coupling limit, showing that singularities on the D7-brane locus D necessitate corrections to naive Euler-characteristic matching. It casts the problem in a rigorous mathematical framework using Chern-Schwartz-MacPherson classes and Verdier specialization, proving a general Chern-class identity 2 φ_* c(Y) = π_* c(ar D) + 4 c(O) − ρ_* c(S) and its top-degree consequence 2 χ(Y) = χ(ar D) + 4 χ(O) − χ(S). The work provides a bridge between physics intuition and singularity-sensitive invariants, revealing that the natural quantities to compare are stringy-like corrections to Euler characteristics, such as χ(ar D) corrected by the pinch locus S. It further extends the classical Sethi–Vafa–Witten relation to arbitrary dimension and non-Calabi–Yau settings, and speculates on a generalized family of invariants c^{(m)}(D) with a limiting case c^{(∞)}(D) that recovers the tadpole identity, hinting at a deeper, universally applicable mathematical structure for singular brane geometries.
Abstract
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.