On stringy invariants of GUT vacua

We investigate aspects of certain stringy-invariants of singular elliptic fibrations which arise in engineering Grand Unified Theories in F-theory. In particular, we exploit the small resolutions of the total space of these fibrations provided recently in the physics literature to compute "stringy characteristic classes", and find that numerical invariants obtained by integrating such characteristic classes are predetermined by the topology of the base of the elliptic fibration. Moreover, we derive a simple (dimension independent) formula for pushing forward powers of the exceptional divisor of a blowup, which one may use to reduce any integral (in the sense of Chow cohomology) on a small resolution of a singular elliptic fibration to an integral on the base. We conclude with a remark on the cohomology of small resolutions of GUT vacua, where we conjecture that certain simple formulas for their Hodge numbers may be given solely in terms of the first Chern class and Hodge numbers of the base.