Stringy Hirzebruch classes of Weierstrass fibrations

TL;DR

This paper computes stringy Hirzebruch classes for singular Weierstrass fibrations in F-theory, deriving generating functions for stringy $ ext{chi}_y$-genera based on base invariants.

Contribution

It introduces a method to calculate stringy Hirzebruch classes of singular Weierstrass fibrations and derives a generating function for stringy $ ext{chi}_y$-genera.

Findings

Derived a formula for characteristic classes of blowups along complete intersections.

Computed stringy Hirzebruch classes for Weierstrass fibrations.

Established a generating function for stringy $ ext{chi}_y$-genera.

Abstract

A Weierstrass fibration is an elliptic fibration $Y\to B$ whose total space $Y$ may be given by a global Weierstrass equation in a $\mathbb{P}^2$-bundle over $B$. In this note, we compute stringy Hirzebruch classes of singular Weierstrass fibrations associated with constructing non-Abelian gauge theories in $F$-theory. For each Weierstrass fibration $Y\to B$ we then derive a generating function $\chi^{\text{str}}_y(Y;t)$, whose degree-$d$ coefficient encodes the stringy $\chi_y$-genus of $Y\to B$ over an unspecified base of dimension $d$, solely in terms of invariants of the base. To facilitate our computations, we prove a formula for general characteristic classes of blowups along (possibly singular) complete intersections.