Abstract
We provide the first explicit example of Type IIB string theory compactification on a globally defined Calabi-Yau threefold with torsion which results in a four-dimensional effective theory with a non-Abelian discrete gauge symmetry. Our example is based on a particular Calabi-Yau manifold, the quotient of a product of three elliptic curves by a fixed point free action of {mathbb{Z}}_2times {mathbb{Z}}_2 . Its cohomology contains torsion classes in various degrees. The main technical novelty is in determining the multiplicative structure of the (torsion part of) the cohomology ring, and in particular showing that the cup product of second cohomology torsion elements goes non-trivially to the fourth cohomology. This specifies a non-Abelian, Heisenberg-type discrete symmetry group of the cfour-dimensional theory.
Highlights
Compactifications with discrete gauge symmetries Zn was initiated in [1] and followed-up in [2,3,4,5,6,7,8,9]
We provide the first explicit example of Type IIB string theory compactification on a globally defined Calabi-Yau threefold with torsion which results in a fourdimensional effective theory with a non-Abelian discrete gauge symmetry
The main technical novelty is in determining the multiplicative structure of the the cohomology ring, and in particular showing that the cup product of second cohomology torsion elements goes non-trivially to the fourth cohomology
Summary
We review the construction of non-Abelian discrete symmetries arising in Type IIB compactifications on a Calabi-Yau manifold X6. This cup-product integrates to ρ2 ∧ γ1 = M α3, where M ∈ Z and k M = k M. this cup-product integrates to ρ2 ∧ γ1 = M α3, where M ∈ Z and k M = k M These torsion subgroups give a priori rise to three non-commuting discrete cyclic groups in the effective four-dimensional Type IIB action. This can be seen from the following. The important fact to note is that these generators T1, T2, T3 do not commute, provided that there is a non-trivial cup-product (2.6) These discrete gauge symmetries of the effective four-dimensional action lead to the following discrete symmetry operations on a four-dimensional state ψ(x), with charges (q1, q2, q3) under (Zk, Zk, Zk ).
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