Five-dimensional Yang-Mills-Einstein supergravity on orbifolds: Parity assignments

We discuss the options for parity assignments in (on-shell) $\mathcal{N}=2$ five-dimensional Yang-Mills-Einstein supergravity theories (YMESGTs) coupled to tensor and hypermultiplets on the orbifold spacetime ${\mathcal{M}}_{4}\ifmmode\times\else\texttimes\fi{}{S}^{1}/{\mathbb{Z}}_{2}$. Along the lines of orbifold-grand unified theories (GUTs), we allow for general breaking of the five-dimensional gauge group at the orbifold fixed points. We then extend the discussion to the case where the orbifold is ${S}^{1}/({\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2})$. We do not presume the existence of fields with support only at fixed points. As in the familiar case of (rigid) super-Yang-Mills theories on such orbifolds, only bulk hypermultiplets can lead to chiral multiplets in complex representations of the gauge group on the boundaries. Massless chiral multiplets coming from bulk vector or tensor multiplets can potentially be used as Higgs supermultiplets, though a ``doublet-triplet'' splitting via parity assignments is not available for the tensor sector. We also find parity assignments for objects other than fields that appear in the Lagrangian, which will partially determine the structure of interactions of the boundary theories. Assigning odd parities to the scalar sector of vector/tensor multiplets requires the four-dimensional boundary moduli spaces to lie on the boundary of the classical K\"ahler cone, which corresponds to collapsed Calabi-Yau 2-cycles at the orbifold fixed points in a compactification of 11-dimensional supergravity. There is an ambiguity in how to affect odd parity for the field-independent ${C}_{IJK}$ tensor of the 5D theory, which may admit a classical interpretation as Calabi-Yau 4-cycles collapsing to either 2- or 0-cycles.