Abstract
In this work we prove a bound for the torsion in Mordell-Weil groups of smooth elliptically fibered Calabi-Yau 3- and 4-folds. In particular, we show that the set of torsion groups which can occur on a smooth elliptic Calabi-Yau n-fold is contained in the set of subgroups which appear on a rational elliptic surface if n ≥ 3 and is slightly larger for n = 2. The key idea in our proof is showing that any elliptic fibration with sufficiently large torsion1 has singularities in codimension 2 which do not admit a crepant resolution. We prove this by explicitly constructing and studying maps to a modular curve whose existence is predicted by a universal property. We use the geometry of modular curves to explain the minimal singularities that appear on an elliptic fibration with prescribed torsion, and to determine the degree of the fundamental line bundle (hence the Kodaira dimension) of the universal elliptic surface which we show to be consistent with explicit Weierstrass models. The constraints from the modular curves are used to bound the fundamental group of any gauge group G in a supergravity theory obtained from F-theory. We comment on the isolated 8-dimensional theories, obtained from extremal K3’s, that are able to circumvent lower dimensional bounds. These theories neither have a heterotic dual, nor can they be compactified to lower dimensional minimal SUGRA theories. We also comment on the maximal, discrete gauged symmetries obtained from certain Calabi-Yau threefold quotients.
Highlights
Calabi-Yau manifolds play a central role in the description of lower dimensional field and supergravity (SUGRA) theories through their use as compactification backgrounds of String/M/F-theory
We use the geometry of modular curves to explain the minimal singularities that appear on an elliptic fibration with prescribed torsion, and to determine the degree of the fundamental line bundle of the universal elliptic surface which we show to be consistent with explicit Weierstrass models
In this work we proved that the Mordell-Weil torsion group T of elliptically fibered CalabiYau n-folds, with n > 2, is no larger than T = Z6 in the cyclic case and T = Z3 × Z3 or
Summary
Calabi-Yau manifolds play a central role in the description of lower dimensional field and supergravity (SUGRA) theories through their use as compactification backgrounds of String/M/F-theory. We want to prove that the list of MW torsion models appearing in Aspinwall and Morrison [54] contains every group that can be realized on a smooth, Calabi-Yau n-fold with n = 3, 4 This geometric result allows us to put sharp swampland constraints on the fundamental group of gauge groups in SUGRA theories constructed from [p, q]-7 branes in F-theory. Torsion sections in the Mordell-Weil group can be used as a building block to construct the covering geometry of a specific class of smooth, non- connected Calabi-Yau quotient torsors [22] Their associated F-theory physics admits discrete symmetries coupled to superconformal matter and gravity [18, 19, 21] of the same order as the torsion factor of the covering geometry.
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