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, theÂ-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 ∫ Y c 2 1 c 2 , the arithmetic genus, and theÂ-genus. Thus, it is enough to know ∫ Y c 2 2 and the Euler characteristic χ(Y ) to determine all the Chern numbers of an elliptically fibered fourfold. We consider the cases of G = SU(n) for (n =

We reformulate entanglement wedge reconstruction in the language of operator-algebra quantum error correction with infinite-dimensional physical and code Hilbert spaces. Von Neumann algebras are used to characterize observables in a boundary subregion and its entanglement wedge. Assuming that the infinite-dimensional von Neumann algebras associated with an entanglement wedge and its complement may both be reconstructed in their corresponding boundary subregions, we prove that the relative entropies measured with respect to the bulk and boundary observables are equal. We also prove the converse: when the relative entropies measured in an entanglement wedge and its complement equal the relative entropies measured in their respective boundary subregions, entanglement wedge reconstruction is possible. Along the way, we show that the bulk and boundary modular operators act on the code subspace in the same way. For holographic theories with a well-defined entanglement wedge, this result provides a well-defined notion of holographic relative entropy.

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We study elliptic fibrations that geometrically engineer an SU(2)×G 2 gauge theory realized by a Weierstrass model for the collision III+I * ns 0 . We find all the distinct crepant resolutions of such a model and the flops connecting them. We compute the generating function for the Euler characteristic of the SU(2)×G 2 -model. In the case of a Calabi-Yau threefold, we consider the compactification of M-theory and F-theory on an SU(2)×G 2 -model to a five and six-dimensional supergravity with eight supercharges. By matching each crepant resolution with each Coulomb chamber of the fivedimensional theory, we determine the number of multiplets and compute the prepotential in each Coulomb chamber. In particular, we discuss counting number of hypermultiplets in presence of singularities. We discuss in detail the cancellation of anomalies of the six-dimensional theory.