# Type II String Theory on Calabi-Yau Manifolds with Torsion and Non-Abelian Discrete Gauge Symmetries

## TL;DR

This paper presents an explicit example of Type IIB string theory compactified on a Calabi-Yau threefold with torsion, leading to a four-dimensional theory with a non-Abelian discrete gauge symmetry, advancing understanding of string compactifications with torsion.

## Contribution

It provides the first explicit construction of a Calabi-Yau compactification with torsion that yields a non-Abelian discrete gauge symmetry in four dimensions.

## Key findings

- Cohomology contains torsion classes in various degrees.
- Cup product of second cohomology torsion elements is non-trivial.
- Specifies a non-Abelian, Heisenberg-type discrete symmetry group.

## 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 Z_2 x 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 four-dimensional theory.