Chern class identities from tadpole matching in type IIB and F-theory

TL;DR

This paper establishes a Chern class identity derived from tadpole matching conditions in type IIB and F-theory, linking physical consistency with mathematical invariants of singular algebraic varieties.

Contribution

It provides a mathematical derivation of a Chern class identity from physical tadpole conditions, including singularities, using advanced algebraic geometry tools.

Findings

Derived a Chern class identity confirming tadpole relations

Connected stringy invariants with algebraic geometry methods

Extended the identity to arbitrary dimensions and non-Calabi-Yau varieties

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.