The twisted geometry of 6d F-theory vacua with discrete gauge symmetries

We study the fate of discrete gauge groups and discrete charges of gravitational theories under twisted circle compactification. We then apply our results to six-dimensional F-theory vacua with discrete gauge symmetries and relate them to the geometry of the genus one fibered Calabi-Yau threefolds that underlie the dual M-theory compactifications. This leads us to introduce a class of geometries, which we call almost generic elliptic/genus one fibered Calabi-Yau threefolds, and to make detailed conjectures about their properties. A second twisted circle compactification relates these M-theory vacua to Type IIA compactifications with flat but topologically non-trivial B-fields along the internal geometry. The A-model topological string partition function on such configurations is intimately tied to the twisted-twined elliptic genera of the six-dimensional non-critical strings of the associated F-theory vacuum. The modular properties of the twisted-twined elliptic genera imply new twisted derived equivalences. We thus recover and significantly extend earlier results from both the physical and the mathematical literature. An important outcome of our study is that if the discrete gauge symmetry is not cyclic, then no smooth genus one fibration exists that represents the associated axio-dilaton profile.