On Discrete Symmetries and Torsion Homology in F-Theory

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 discrete torsion. 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 discrete torsion. 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.

Figures (4)

• Figure 1: The fibre structure over the singlet curves $C_I$ and $C_{II}$ taken from Mayrhofer:2014haa with blue denoting the section $S$ and green the section $U$.
• Figure 2: Figure showing the boundaries induced after the conifold transition in the Weierstraß hypersurface in $\mathbb P_{231}$. The divisor $S$ is denoted in blue and $U$ is denoted in green. After the transition $U$ does not develop a boundary and therefore is associated to the five-dimensional $U(1)_U$ symmetry. On the other hand $S$ develops two boundaries of the same orientation. The sum over all the points $B^i_{I}$ for each one of the two boundaries illustrated gives the torsional 3-cycle associated to the $\mathbb{Z}_2$ symmetry.
• Figure 3: Figure showing the 3-chains stretching between a point in the set of points $C_I$ and two points in the set $C_{II}$ in the resolved space. The boundary of the chain is therefore $2 B_{II} - B_{I}$. After the deformation the boundary $B_{I}$ is lost leaving a chain with a boundary $2 B_{II}$ and thereby identifying $B_{II}$ as the torsional 2-cycle.
• Figure 4: Figure showing the boundaries induced after the conifold transition in the quartic hypersurface in $\mathbb P_{112}$. The divisors $S$, denoted in blue, and $U$, denoted in green, both develop a single boundary from each of the $A_I^i$ of opposite orientation. The two boundaries are then glued together to form the divisor $S+U$ corresponding to the remnant five-dimensional $U(1)$ symmetry.