Stringy Hirzebruch classes of Weierstrass fibrations

Abstract

A Weierstrass fibration is an elliptic fibration Y → B whose total space Y may be given by a global Weierstrass equation in a 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 → B we then derive a generating function χ str y (Y ; t), whose degree-d coefficient encodes the stringy χ ygenus of Y → 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. Contents 16 7. Stringy Hirzebruch-Riemann-Roch for SU(2) fibrations 20 8. Summary of results 21 References 23 * supported by NSF grant 1618657.

Orientifold limits of singular $F$-theory vacua

“…For each singular F -theory compactification ϕ : X → B we consider, crepant resolutions τ : X → X were constructed in [11], which were then used to compute explicit formulas for ϕ * (τ * c( X)) (which may also be recovered by taking the limit as y → −1 of the stringy Hirzebruch class formulas derived in [14]). And since τ * c( X) = c str (X), we have ϕ * (τ * c( X)) = ϕ * c str (X), which will enable us to compute the LHS of the universal tadpole relation (4.3) assciated with each limit.…”