More precisely, while the tadpole bound in the limit of a large number of complex-structure moduli goes like 1/4 of the number of moduli, we conjecture that the amount of charge induced by fluxes stabilizing all moduli grows faster than this, and is therefore larger than the allowed amount. We conjecture that one cannot stabilize all complex-structure moduli in F-theory at a generic point in moduli space (away from singularities) by fluxes that satisfy the bound imposed by the tadpole cancelation condition. We examine the mechanism of moduli stabilization by fluxes in the limit of a large number of moduli. This allows us to discuss moduli stabilization explicitly and establish the relevant scaling constraints for the tadpole. Furthermore, we find that for Calabi-Yau four-folds all but one representation can be identified with representations occurring on two-folds. We show that the number of stabilized moduli scales with the number of sl(2)-representations supported by fluxes, and that each representation fixes a single modulus. In particular, we use the fact that in each asymptotic regime an orthogonal sl(2)-block structure emerges that allows us to group fluxes into sl(2)-representations and decouple complex structure directions. Our approach relies on the use of asymptotic Hodge theory. Restricting to the asymptotic regions of the complex structure moduli space, we give the first conceptual argument that explains this linear scaling setting and clarifies why it sets in only for a large number of stabilized moduli. More precisely, it states that the stabilization of a large number of moduli requires a flux background with a tadpole that scales linearly in the number of stabilized fields. The tadpole conjecture suggests that the complete stabilization of complex structure deformations in Type IIB and F-theory flux compactifications is severely obstructed by the tadpole bound on the fluxes. M-theory analysis identifies consistent flux configurations in four-dimensionalį-theory compactifications and flat directions in the deformation space of Global fourfold compactifications with flux. For special geometries and backgroundĬonfigurations, the local transitions extend to extremal transitions between A local analysis of the flux superpotential shows that the potential hasįlat directions for a subset of these fluxes and the topologically different Including new solutions in which the flux is neither of horizontal nor vertical Which solve the local quantization condition on G for a given geometry, We identify a set of canonical, minimal flux quanta Transition is generically a genus g curve of conifold singularities, whichĮngineers a 3d gauge theory with four supercharges, near the intersection ofĬoulomb and Higgs branches. We consider topology changing transitions for M-theory compactifications onĬalabi-Yau fourfolds with background G-flux.
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