Communications in Number Theory and Physics

Volume 14 (2020)

Number 3

Stringy Hirzebruch classes of Weierstrass fibrations

Pages: 453 – 485



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.)


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.

Mark van Hoeij was supported by NSF grant 1618657.

Received 30 October 2018

Accepted 30 January 2020

Published 13 July 2020