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On finiteness of type IIB compactifications: magnetized branes on elliptic Calabi-Yau threefolds. (English) Zbl 1333.81321
Summary: The string landscape satisfies interesting finiteness properties imposed by supersymmetry and string-theoretical consistency conditions. We study \(\mathcal N=1\) supersymmetric compactifications of Type IIB string theory on smooth elliptically fibered Calabi-Yau threefolds at large volume with magnetized D9-branes and D5-branes. We prove that supersymmetry and tadpole cancellation conditions imply that there is a finite number of such configurations. In particular, we derive an explicitly computable bound on the number of magnetic flux quanta, as well as the number of D5-branes, which is independent of the continuous moduli of the setup. The proof applies if a number of easy to check geometric conditions of the twofold base are met. We show that these geometric conditions are satisfied for the almost Fano twofold bases given by each toric variety associated to a reflexive two-dimensional polytope as well as by the generic del Pezzo surfaces \(dP_n\) with \(n=0,\dots, 8\). Physically, this finiteness proof shows that there exist a finite collection of four-dimensional gauge groups and chiral matter spectra in the 4D supergravity theories realized by these compactifications. As a by-product we explicitly construct all generators of the Kähler cones of \(dP_n\) and work out their relation to representation theory.

MSC:
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
14J81 Relationships with physics
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
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