Characteristic numbers of crepant resolutions of Weierstrass models

Abstract

We compute characteristic numbers of crepant resolutions of Weierstrass models corresponding to elliptically fibered fourfolds $Y$ dual in F-theory to a gauge theory with gauge group $G$. In contrast to the case of fivefolds, Chern and Pontryagin numbers of fourfolds are invariant under crepant birational maps. It follows that Chern and Pontryagin numbers are independent on a choice of a crepant resolution. We present the results for the Euler characteristic, the holomorphic genera, the Todd-genus, the $L$-genus, the $\hat{A}$-genus, and the curvature invariant $X_8$ that appears in M-theory. We also show that certain characteristic classes are independent on the choice of the Kodaria fiber characterizing the group $G$. That is the case of ${\int }_Y{c}_1^2{c}_2$, the arithmetic genus, and the $\hat{A}$-genus. Thus, it is enough to know ${\int }_Y{c}_2^2$ and the Euler characteristic $\chi (Y)$ to determine all the Chern numbers of an elliptically fibered fourfold. We consider the cases of $G=$ SU($n$) for ($n=2,3,4,5,6,7$), USp($4$), Spin($7$), Spin($8$), Spin($10$), G${}_2$, F${}_4$, E${}_6$, E${}_7$, or E${}_8$.

34 pages, 9 Tables

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• Les Houches Lectures on Constructing String Vacua
• Mordell-Weil Torsion and the Global Structure of Gauge Groups in F-theory
• A New Model for Elliptic Fibrations with a Rank One Mordell-Weil Group: I. Singular Fibers and Semi-Stable Degenerations
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