Abstract

We study the relation between discrete gauge symmetries in F-theory compactifications and torsion homology on the associated Calabi-Yau manifold. Focusing on the simplest example of a $$ {\mathbb{Z}}_2 $$ symmetry, we show that there are two physically distinct ways that such a discrete gauge symmetry can arise. First, compactifications of M-Theory on Calabi-Yau threefolds which support a genus-one fibration with a bi-section are known to be dual to six-dimensional F-theory vacua with a $$ {\mathbb{Z}}_2 $$ gauge symmetry. We show that the resulting five-dimensional theories do not have a $$ {\mathbb{Z}}_2 $$ symmetry but that the latter emerges only in the F-theory decompactification limit. Accordingly the genus-one fibred Calabi-Yau manifolds do not exhibit torsion in homology. Associated to the bi-section fibration is a Jacobian fibration which does support a section. Compactifying on these related but distinct varieties does lead to a $$ {\mathbb{Z}}_2 $$ symmetry in five dimensions and, accordingly, we find explicitly an associated torsion cycle. We identify the expected particle and membrane system of the discrete symmetry in terms of wrapped M2 and M5 branes and present a field-theory description of the physics for both cases in terms of circle reductions of six-dimensional theories. Our results and methods generalise straightforwardly to larger discrete symmetries and to four-dimensional compactifications.

Highlights

  • JHEP06(2015)029 precisely the discrete symmetry observed in the effective action

  • We study the relation between discrete gauge symmetries in F-theory compactifications and torsion homology on the associated Calabi-Yau manifold

  • In this paper we have studied the relation between discrete gauge symmetry and torsional homology in F/M-theory

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Summary

Z2 symmetry from genus-one fibrations

The intersection number with S − U is the charge of M2-branes wrapping the various fibre components with respect to the six-dimensional massless U(1) gauge symmetry. The Higgs field of charge ±2 with respect to the six-dimensional U(1) symmetry is associated with an M2-brane wrapping either of the fibre components over the locus CI. Let us consider M-theory compactified on PQ to five spacetime dimensions In five dimensions, both independent sections S and U give rise to a bona fide U(1) gauge symmetry from expanding C3.3 The intersection numbers in table 1 compute the charges of M2-branes wrapping the various fibre components over CI and CII. There is no notion of chirality associated with this Z2, and this is well in agreement with the property of the flux of not inducing any U(1)S+U chirality already in M-theory

Torsion from the Weierstraß fibration
Conclusions
A Blowing up the matter locus
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