Communications in Number Theory and Physics

Volume 14 (2020)

Number 3

Stringy Hirzebruch classes of Weierstrass fibrations

Pages: 453 – 485

DOI: https://dx.doi.org/10.4310/CNTP.2020.v14.n3.a1

Authors

James Fullwood (School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, China)

Mark van Hoeij (Department of Mathematics, Florida State University, Tallahassee, Fl., U.S.A.)

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 $\chi^{\textrm{str}}_{y} (Y; t)$, 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.

Received 30 October 2018

Accepted 30 January 2020

Published 13 July 2020