Abstract
We argue that M-theory compactified on an arbitrary genus-one fibration, that is, an elliptic fibration which need not have a section, always has an F-theory limit when the area of the genus-one fiber approaches zero. Such genus-one fibrations can be easily constructed as toric hypersurfaces, and various $SU(5)\times U(1)^n$ and $E_6$ models are presented as examples. To each genus-one fibration one can associate a $\tau$-function on the base as well as an $SL(2,\mathbb{Z})$ representation which together define the IIB axio-dilaton and 7-brane content of the theory. The set of genus-one fibrations with the same $\tau$-function and $SL(2,\mathbb{Z})$ representation, known as the Tate-Shafarevich group, supplies an important degree of freedom in the corresponding F-theory model which has not been studied carefully until now. Six-dimensional anomaly cancellation as well as Witten's zero-mode count on wrapped branes both imply corrections to the usual F-theory dictionary for some of these models. In particular, neutral hypermultiplets which are localized at codimension-two fibers can arise. (All previous known examples of localized hypermultiplets were charged under the gauge group of the theory.) Finally, in the absence of a section some novel monodromies of Kodaira fibers are allowed which lead to new breaking patterns of non-Abelian gauge groups.
Highlights
Six-dimensional anomaly cancellation as well as Witten’s zero-mode count on wrapped branes both imply corrections to the usual F-theory dictionary for some of these models
We argue that M-theory compactified on an arbitrary genus-one fibration, that is, an elliptic fibration which need not have a section, always has an F-theory limit when the area of the genus-one fiber approaches zero
The basic F-theory construction comes along with F-theory/M-theory duality: if Ftheory is further compactified on a circle, the resulting theory should be dual to M-theory compactified on a Calabi-Yau variety Y which is fibered over B by curves of genus one, with the curves becoming singular over ∆, and with the function b → τ (b) corresponding to the ratio of periods of the holomorphic 1-form on the fiber over b
Summary
The argument presented above is clearly naıve: the fiber complex structure τ varies holomorphically, so if it is non-constant it must have zeros and poles where the ansatz eq (4.1) cannot be valid. That choice has no physical significance: possible Dehn twists on the F-theory elliptic curve just correspond to the changing S-duality frame of the IIB axion-dilaton. This obviously does not preserve the zero-section (i.e., the locus of points serving as “0” in the group structure on each fiber), so the ensuing fibration will, in general, only be a genus-one fibration. At first sight, allowing translations seems to be very boring: τ does not change if we translate along the torus, so no physical quantity appears to know about it This argument really only tells us that no field knows about the translations locally, which is tautologically true, as the geometry has local sections. Global monodromies can and will depend on this additional freedom, and in section 7.5 we will see an explicit example
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